Some Data for C. Breuil About the Local Representation Attached to a Newform of Medium Weight

David Savitt  William A. Stein

April 7, 2002

1 Introduction

This document contains data about ρ_{f}|D_{ℓ}, where f is a mod-ℓ newform whose weight k satisfies ℓ+1 < k ≤ 2ℓ. The data is organized as follows. Section ℓ, for ℓ = 2, 3, 5, 7, 11, 13, 17 contains examples in characteristioc ℓ. Each section is divided up into subsections corresponding to levels N. In each subsection, there is a bullet corresponding to each mod-ℓ newform f whose weight k satisfies ℓ + 1 < k ≤ 2ℓ. If f is ordinary or the representation attached to f is reducible, then we do no further analysis. If f is supersingular, we search for a cuspidal newform g of weight k′, where 2 ≤ k′ ≤ ℓ+1, such that ρ_{f} is a twist of ρ_{g} by a power of the mod-ℓ cyclotomic character. In fact, we really only check that trace((ρ_{f} ⊗ χ^{i})(Frob_{p})) = trace(ρ_{g}(Frob_{p})) for all primes p ≠ ℓ that are less than [SL_{2}(Z) : Γ_{0}(N)] · (ℓ/6) + 10. If no such g exists, then (I think!) this implies that ρ_{f} is reducible (or old?), so we do not consider it further. When such a g exists, we apply results of Deligne and Fontaine to read off the semisimplification of ρ_{g}|D_{ℓ} from the q-expansion and weight of g. By twisting, we obtain the semisimplification of ρ_{f}|D_{ℓ}.

WARNING: This document is very long.

1.1 Answers to some questions of Breuil about the data

Many thanks. I [Breuil] also have 2 questions:

1. Assuming f is a new form on Γ_{0}(N) (thus eigenvector for all T_{p} and U_{p}), I guess so is g? The program computes the eigenforms f and g as follows. First it lists the newforms (for all T_{p} and U_{p}) in S_{2}(Γ_{0}(N); C). Then it computes all of the reductions of these newforms to mod-ℓ eigenforms. Thus both f and g are attached to new maximal ideals of residue characteristic ℓ of the full Hecke algebra associated to S_{2}(Γ_{0}(N); C).

2. Is the lift of f you give a classical eigenform or a genuine ℓ-adic one? (I want to be sure that when the slope of f is 1, (a_{ℓ}/ℓ) mod ℓ computed on the ℓ-adic lift you give is equal to (a_{ℓ}/ℓ) mod ℓ computed on a “classical” lift. It seems to be so in several cases I have checked). I do not completely understand the question. However, I can tell you how the lift f_{0} of f is computed by my program. The program computed f by finding the newforms in S_{2}(Γ_{0}(N); C), so it knows a newform f̃ ∈ S_{2}(Γ_{0}(N); \overline{Q}) that has f as one of its reductions. To find f_{0}, the program first computes each embeddings of f̃into \overline{Q}_{ℓ}[[q]]. It then compares the mod-ℓ reduction of each of these ℓ-adic modular forms with f, and lets f_{0} be the first form it finds with mod-ℓ reduction equal to f.

1.2 Technical remarks

This data was created using the MAGMA program that can be found at the end of this document. It was necessary to write a number of routines to convert the output of MAGMA into something that tex understands. This got really tedious, so I haven’t aimed for perfection. For example, sometimes you’ll see things like O(q^{2}05), which should really be O(q^{205})! I hope this doesn’t cause undue confusion.

I have no reason to suspect that this data is incorrect in any way. However, it relies on several layers of very complicated computer programs, so be very careful!!

Contents

1 Introduction 1 1.1 Answers to some questions of Breuil about the data . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Technical remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 2 ≤ N ≤ 50, ℓ = 2  2

2.1 N = 3, ℓ = 2 (no forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 N = 5, ℓ = 2 (1 form) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.3 N = 7, ℓ = 2 (1 form) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.4 N = 9, ℓ = 2 (1 form) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.5 N = 11, ℓ = 2 (1 form) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.6 N = 13, ℓ = 2 (3 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.7 N = 15, ℓ = 2 (2 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.8 N = 17, ℓ = 2 (3 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.9 N = 19, ℓ = 2 (3 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.10 N = 21, ℓ = 2 (4 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.11 N = 23, ℓ = 2 (5 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.12 N = 25, ℓ = 2 (3 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.13 N = 27, ℓ = 2 (3 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.14 N = 29, ℓ = 2 (4 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.15 N = 31, ℓ = 2 (5 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.16 N = 33, ℓ = 2 (6 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.17 N = 35, ℓ = 2 (4 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.18 N = 37, ℓ = 2 (5 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.19 N = 39, ℓ = 2 (4 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.20 N = 41, ℓ = 2 (4 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.21 N = 43, ℓ = 2 (4 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.22 N = 45, ℓ = 2 (5 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.23 N = 47, ℓ = 2 (5 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.24 N = 49, ℓ = 2 (6 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 2 ≤ N ≤ 70, ℓ = 3  9

3.1 N = 2, ℓ = 3 (no forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 N = 4, ℓ = 3 (1 form) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 N = 5, ℓ = 3 (1 form) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.4 N = 7, ℓ = 3 (2 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.5 N = 8, ℓ = 3 (1 form) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.6 N = 10, ℓ = 3 (3 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.7 N = 11, ℓ = 3 (3 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.8 N = 13, ℓ = 3 (4 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.9 N = 14, ℓ = 3 (2 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.10 N = 16, ℓ = 3 (2 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.11 N = 17, ℓ = 3 (4 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.12 N = 19, ℓ = 3 (4 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.13 N = 20, ℓ = 3 (1 form) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.14 N = 22, ℓ = 3 (5 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.15 N = 23, ℓ = 3 (6 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.16 N = 25, ℓ = 3 (7 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.17 N = 26, ℓ = 3 (4 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.18 N = 28, ℓ = 3 (2 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.19 N = 29, ℓ = 3 (7 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.20 N = 31, ℓ = 3 (9 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.21 N = 32, ℓ = 3 (4 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.22 N = 34, ℓ = 3 (6 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.23 N = 35, ℓ = 3 (8 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.24 N = 37, ℓ = 3 (6 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.25 N = 38, ℓ = 3 (6 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.26 N = 40, ℓ = 3 (4 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.27 N = 41, ℓ = 3 (10 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.28 N = 43, ℓ = 3 (10 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.29 N = 44, ℓ = 3 (3 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.30 N = 46, ℓ = 3 (8 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.31 N = 47, ℓ = 3 (10 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.32 N = 49, ℓ = 3 (12 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.33 N = 50, ℓ = 3 (7 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.34 N = 52, ℓ = 3 (4 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.35 N = 53, ℓ = 3 (9 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.36 N = 55, ℓ = 3 (12 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.37 N = 56, ℓ = 3 (6 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.38 N = 58, ℓ = 3 (11 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.39 N = 59, ℓ = 3 (13 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.40 N = 61, ℓ = 3 (11 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.41 N = 62, ℓ = 3 (9 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.42 N = 64, ℓ = 3 (8 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.43 N = 65, ℓ = 3 (12 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.44 N = 67, ℓ = 3 (12 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.45 N = 68, ℓ = 3 (5 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.46 N = 70, ℓ = 3 (9 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5 2 ≤ N ≤ 22, ℓ = 5  35

5.1 N = 2, ℓ = 5 (2 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.2 N = 3, ℓ = 5 (3 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.3 N = 4, ℓ = 5 (1 form) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.4 N = 6, ℓ = 5 (2 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.5 N = 7, ℓ = 5 (6 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.6 N = 8, ℓ = 5 (4 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.7 N = 9, ℓ = 5 (5 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.8 N = 11, ℓ = 5 (11 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.9 N = 12, ℓ = 5 (3 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.10 N = 13, ℓ = 5 (16 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.11 N = 14, ℓ = 5 (7 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.12 N = 16, ℓ = 5 (7 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.13 N = 17, ℓ = 5 (20 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.14 N = 18, ℓ = 5 (6 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.15 N = 19, ℓ = 5 (22 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.16 N = 21, ℓ = 5 (13 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.17 N = 22, ℓ = 5 (12 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

7 2 ≤ N ≤ 16, ℓ = 7  51

7.1 N = 2, ℓ = 7 (3 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 7.2 N = 3, ℓ = 7 (6 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 7.3 N = 4, ℓ = 7 (3 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 7.4 N = 5, ℓ = 7 (11 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 7.5 N = 6, ℓ = 7 (5 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 7.6 N = 8, ℓ = 7 (8 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 7.7 N = 9, ℓ = 7 (11 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 7.8 N = 10, ℓ = 7 (11 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 7.9 N = 11, ℓ = 7 (23 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 7.10 N = 12, ℓ = 7 (5 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 7.11 N = 13, ℓ = 7 (33 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 7.12 N = 15, ℓ = 7 (21 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 7.13 N = 16, ℓ = 7 (15 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

11 2 ≤ N ≤ 16, ℓ = 11  72

11.1 N = 2, ℓ = 11 (8 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 11.2 N = 3, ℓ = 11 (13 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 11.3 N = 4, ℓ = 11 (6 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 11.4 N = 5, ℓ = 11 (29 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 11.5 N = 6, ℓ = 11 (13 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 11.6 N = 7, ℓ = 11 (43 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 11.7 N = 8, ℓ = 11 (19 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 11.8 N = 9, ℓ = 11 (28 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 11.9 N = 10, ℓ = 11 (24 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 11.10N = 12, ℓ = 11 (14 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 11.11N = 13, ℓ = 11 (83 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 11.12N = 14, ℓ = 11 (38 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 11.13N = 15, ℓ = 11 (55 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 11.14N = 16, ℓ = 11 (37 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

13 2 ≤ N ≤ 10, ℓ = 13  168

13.1 N = 2, ℓ = 13 (10 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 13.2 N = 3, ℓ = 13 (20 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 13.3 N = 4, ℓ = 13 (10 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 13.4 N = 5, ℓ = 13 (37 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 13.5 N = 6, ℓ = 13 (20 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 13.6 N = 7, ℓ = 13 (56 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 13.7 N = 8, ℓ = 13 (30 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 13.8 N = 9, ℓ = 13 (42 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 13.9 N = 10, ℓ = 13 (40 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

17 2 ≤ N ≤ 6, ℓ = 17  227

17.1 N = 2, ℓ = 17 (18 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 17.2 N = 3, ℓ = 17 (34 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 17.3 N = 4, ℓ = 17 (17 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 17.4 N = 5, ℓ = 17 (67 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 17.5 N = 6, ℓ = 17 (34 forms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

18 Source code 271

2 2 ≤ N ≤ 50, ℓ = 2

2.1 N = 3, ℓ = 2 (no forms)

There are no forms to analyze.

2.2 N = 5, ℓ = 2 (1 form)

- —————————————————————————————————- (Uninteresting cases below.)

1. Consider f = q + ⋯ ∈ S4(Γ0(5); \overline{F}2). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

2.3 N = 7, ℓ = 2 (1 form)

- —————————————————————————————————- (Uninteresting cases below.)

1. Consider f = q + q2 + ⋯ ∈ S4(Γ0(7); \overline{F}2). Since f is ordinary, we do not continue our analysis.

2.4 N = 9, ℓ = 2 (1 form)

- —————————————————————————————————- (Uninteresting cases below.)

1. Consider f = q + ⋯ ∈ S4(Γ0(9); \overline{F}2). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

2.5 N = 11, ℓ = 2 (1 form)

1. Consider f = q + q^{3} + ⋯ ∈ S_{4}(Γ_{0}(11); \overline{F}_{2}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{1}. The form f_{0} = q + aq^{2} + −2^{2} ∗ a + 3q^{3} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −2x + −2. The slope of f_{0} is 1/2.

—————————————————————————————————- (Uninteresting cases below.)

2.6 N = 13, ℓ = 2 (3 forms)

- —————————————————————————————————- (Uninteresting cases below.)

1. Consider f = q + q2 + q3 + ⋯ ∈ S4(Γ0(13); \overline{F}2). Since f is ordinary, we do not continue our analysis.

2. Consider f = q + ⋯ ∈ S4(Γ0(13); \overline{F}2). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

3. Consider f = q + q2 + q3 + ⋯ ∈ S4(Γ0(13); \overline{F}2). Since f is ordinary, we do not continue our analysis.

2.7 N = 15, ℓ = 2 (2 forms)

- —————————————————————————————————- (Uninteresting cases below.)

1. Consider f = q + q2 + q3 + ⋯ ∈ S4(Γ0(15); \overline{F}2). Since f is ordinary, we do not continue our analysis.

2. Consider f = q + q2 + q3 + ⋯ ∈ S4(Γ0(15); \overline{F}2). Since f is ordinary, we do not continue our analysis.

2.8 N = 17, ℓ = 2 (3 forms)

1. Consider f = q + ⋯ ∈ S_{4}(Γ_{0}(17); \overline{F}_{2}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + −(21839 ∗ 2^{−}2 + O(2^{1}8)) ∗ a − 60961 ∗ 2 + O(2^{1}8)q^{3} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −2729 ∗ 2^{3} + O(2^{2}0)x + −15241 ∗ 2^{5} + O(2^{2}0). The slope of f_{0} is 5/2. —————————————————————————————————- (Uninteresting cases below.)

2. Consider f = q + q2 + ⋯ ∈ S4(Γ0(17); \overline{F}2). Since f is ordinary, we do not continue our analysis.

3. Consider f = q + q2 + ⋯ ∈ S4(Γ0(17); \overline{F}2). Since f is ordinary, we do not continue our analysis.

2.9 N = 19, ℓ = 2 (3 forms)

1. Consider f = q + ⋯ ∈ S_{4}(Γ_{0}(19); \overline{F}_{2}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{1}. The form f_{0} = q+aq^{2}+(38281∗2^{2}+O(2^{2}0))∗a−54941∗2+O(2^{2}0)q^{3}+⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} +28711∗2^{4} +O(2^{2}0)x+−164833∗2+O(2^{2}0). The slope of f_{0} is 1/2. —————————————————————————————————- (Uninteresting cases below.)

2. Consider f = q + q2 + q3 + ⋯ ∈ S4(Γ0(19); \overline{F}2). Since f is ordinary, we do not continue our analysis.

3. Consider f = q + q2 + q3 + ⋯ ∈ S4(Γ0(19); \overline{F}2). Since f is ordinary, we do not continue our analysis.

2.10 N = 21, ℓ = 2 (4 forms)

1. Consider f = q + q^{3} + ⋯ ∈ S_{4}(Γ_{0}(21); \overline{F}_{2}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 2^{2}q^{2} + −3q^{3} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 2.

2. Consider f = q + q^{3} + ⋯ ∈ S_{4}(Γ_{0}(21); \overline{F}_{2}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −1355 ∗ 2^{2} + O(2^{2}0)q^{2} + 3q^{3} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 2. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + q2 + q3 + ⋯ ∈ S4(Γ0(21); \overline{F}2). Since f is ordinary, we do not continue our analysis.

4. Consider f = q + q2 + q3 + ⋯ ∈ S4(Γ0(21); \overline{F}2). Since f is ordinary, we do not continue our analysis.

2.11 N = 23, ℓ = 2 (5 forms)

1. Consider f = q + q^{3} + ⋯ ∈ S_{4}(Γ_{0}(23); \overline{F}_{2}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + zq^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2})ω^{0} & 0 \\ 0 & u(z)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −2q^{2} + −5q^{3} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + q^{3} + ⋯ ∈ S_{4}(Γ_{0}(23); \overline{F}_{2}). Also, z satisfies the equation x^{2} + x+ = 0. We have ρ_{f} ⊗χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q +zq^{2} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2})ω^{0} & 0 \\ 0 & u(z)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −91733 ∗ 2 + O(2^{2}0)q^{2} + −283929 + O(2^{2}0)q^{3} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + q2 + ⋯ ∈ S4(Γ0(23); \overline{F}2). Also, z satisfies the equation x2 + x+ = 0. Since f is ordinary, we do not continue our analysis.

4. Consider f = q + zq2 + q3 + ⋯ ∈ S4(Γ0(23); \overline{F}2). Also, z satisfies the equation x2 + x+ = 0. Since f is ordinary, we do not continue our analysis.

5. Consider f = q + z2q2 + q3 + ⋯ ∈ S4(Γ0(23); \overline{F}2). Also, z satisfies the equation x2 + x+ = 0. Since f is ordinary, we do not continue our analysis.

2.12 N = 25, ℓ = 2 (3 forms)

- —————————————————————————————————- (Uninteresting cases below.)

1. Consider f = q + q2 + q3 + ⋯ ∈ S4(Γ0(25); \overline{F}2). Since f is ordinary, we do not continue our analysis.

2. Consider f = q + q2 + q3 + ⋯ ∈ S4(Γ0(25); \overline{F}2). Since f is ordinary, we do not continue our analysis.

3. Consider f = q + ⋯ ∈ S4(Γ0(25); \overline{F}2). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

2.13 N = 27, ℓ = 2 (3 forms)

1. Consider f = q + ⋯ ∈ S_{4}(Γ_{0}(27); \overline{F}_{2}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{1}. The form f_{0} = q + aq^{2} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −9 ∗ 2. The slope of f_{0} is 1/2. —————————————————————————————————- (Uninteresting cases below.)

2. Consider f = q + q2 + ⋯ ∈ S4(Γ0(27); \overline{F}2). Since f is ordinary, we do not continue our analysis.

3. Consider f = q + q2 + ⋯ ∈ S4(Γ0(27); \overline{F}2). Since f is ordinary, we do not continue our analysis.

2.14 N = 29, ℓ = 2 (4 forms)

1. Consider f = q + q^{3} + ⋯ ∈ S_{4}(Γ_{0}(29); \overline{F}_{2}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + −(112943 ∗ 2^{−}2 + O(2^{1}7)) ∗ a + 30177 + O(2^{1}7)q^{3} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 7625 ∗ 2^{6} + O(2^{2}0)x + 5849 ∗ 2^{6} + O(2^{2}0). The slope of f_{0} is 3. —————————————————————————————————- (Uninteresting cases below.)

2. Consider f = q + q2 + q3 + ⋯ ∈ S4(Γ0(29); \overline{F}2). Since f is ordinary, we do not continue our analysis.

3. Consider f = q + q2 + q3 + ⋯ ∈ S4(Γ0(29); \overline{F}2). Also, z satisfies the equation x2 + x+ = 0. Since f is ordinary, we do not continue our analysis.

4. Consider f = q + ⋯ ∈ S4(Γ0(29); \overline{F}2). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

2.15 N = 31, ℓ = 2 (5 forms)

1. Consider f = q + ⋯ ∈ S_{4}(Γ_{0}(31); \overline{F}_{2}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + zq^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2})ω^{0} & 0 \\ 0 & u(z)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −210503 ∗ 2 + O(2^{2}0)q^{2} + 421003 ∗ 2 + O(2^{2}1)q^{3} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + ⋯ ∈ S_{4}(Γ_{0}(31); \overline{F}_{2}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + zq^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2})ω^{0} & 0 \\ 0 & u(z)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −58155 ∗ 2 + O(2^{2}0)q^{2} + 11495 ∗ 2^{2} + O(2^{1}7)q^{3} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + q2 + ⋯ ∈ S4(Γ0(31); \overline{F}2). Since f is ordinary, we do not continue our analysis.

4. Consider f = q + zq2 + ⋯ ∈ S4(Γ0(31); \overline{F}2). Also, z satisfies the equation x2 + x+ = 0. Since f is ordinary, we do not continue our analysis.

5. Consider f = q + q2 + ⋯ ∈ S4(Γ0(31); \overline{F}2). Since f is ordinary, we do not continue our analysis.

2.16 N = 33, ℓ = 2 (6 forms)

1. Consider f = q + q^{3} + ⋯ ∈ S_{4}(Γ_{0}(33); \overline{F}_{2}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −33595 ∗ 2^{3} + O(2^{2}0)q^{2} + −3q^{3} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 3.

2. Consider f = q + q^{3} + ⋯ ∈ S_{4}(Γ_{0}(33); \overline{F}_{2}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 10631 ∗ 2^{3} + O(2^{2}0)q^{2} + 3q^{3} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 3. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + q2 + q3 + ⋯ ∈ S4(Γ0(33); \overline{F}2). Since f is ordinary, we do not continue our analysis.

4. Consider f = q + q2 + q3 + ⋯ ∈ S4(Γ0(33); \overline{F}2). Since f is ordinary, we do not continue our analysis.

5. Consider f = q + q2 + q3 + ⋯ ∈ S4(Γ0(33); \overline{F}2). Since f is ordinary, we do not continue our analysis.

6. Consider f = q + q2 + q3 + ⋯ ∈ S4(Γ0(33); \overline{F}2). Since f is ordinary, we do not continue our analysis.

2.17 N = 35, ℓ = 2 (4 forms)

1. Consider f = q + q^{3} + ⋯ ∈ S_{4}(Γ_{0}(35); \overline{F}_{2}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{1}. The form f_{0} = q + aq^{2} + −2^{2} ∗ a + 17q^{3} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −2^{3}x + 7 ∗ 2. The slope of f_{0} is 1/2.

2. Consider f = q +q^{3} +⋯ ∈ S_{4}(Γ_{0}(35); \overline{F}_{2}). We have ρ_{f} ⊗χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{1}. The form f_{0} = q+aq^{2}+−(107937∗2^{2}+O(2^{2}0))∗a+178075+O(2^{2}0)q^{3}+⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} +−107937∗2^{2} +O(2^{2}0)x+89031∗2+O(2^{2}0). The slope of f_{0} is 1/2. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + q2 + ⋯ ∈ S4(Γ0(35); \overline{F}2). Since f is ordinary, we do not continue our analysis.

4. Consider f = q + q2 + ⋯ ∈ S4(Γ0(35); \overline{F}2). Since f is ordinary, we do not continue our analysis.

1. Consider f = q + q^{3} + ⋯ ∈ S_{4}(Γ_{0}(37); \overline{F}_{2}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{1}. The form f_{0} = q+aq^{2}+(2503∗2^{3}+O(2^{1}7))∗a+57997+O(2^{1}7)q^{3}+⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −38327 ∗ 2^{2} + O(2^{2}0)x + 1619 ∗ 2 + O(2^{2}0). The slope of f_{0} is 1/2.

2. Consider f = q + q^{3} + ⋯ ∈ S_{4}(Γ_{0}(37); \overline{F}_{2}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{1}. The form f_{0} = q+aq^{2}+−(6451∗2^{2}+O(2^{1}7))∗a−42319+O(2^{1}8)q^{3}+⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} +−239867∗2+O(2^{2}0)x+82871∗2+O(2^{2}0). The slope of f_{0} is 1/2. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + q2 + zq3 + ⋯ ∈ S4(Γ0(37); \overline{F}2). Also, z satisfies the equation x2 + x+ = 0. Since f is ordinary, we do not continue our analysis.

4. Consider f = q + ⋯ ∈ S4(Γ0(37); \overline{F}2). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

5. Consider f = q + q2 + zq3 + ⋯ ∈ S4(Γ0(37); \overline{F}2). Also, z satisfies the equation x2 + x+ = 0. Since f is ordinary, we do not continue our analysis.

2.19 N = 39, ℓ = 2 (4 forms)

1. Consider f = q + q^{3} + ⋯ ∈ S_{4}(Γ_{0}(39); \overline{F}_{2}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −3q^{3} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is Infinity.

2. Consider f = q + q^{3} + ⋯ ∈ S_{4}(Γ_{0}(39); \overline{F}_{2}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+5405∗2^{3}+O(2^{2}0)q^{2}+ 3q^{3} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 3. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + q2 + q3 + ⋯ ∈ S4(Γ0(39); \overline{F}2). Since f is ordinary, we do not continue our analysis.

4. Consider f = q + q2 + q3 + ⋯ ∈ S4(Γ0(39); \overline{F}2). Since f is ordinary, we do not continue our analysis.

2.20 N = 41, ℓ = 2 (4 forms)

1. Consider f = q + ⋯ ∈ S_{4}(Γ_{0}(41); \overline{F}_{2}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + (31327 ∗ 2^{−}6 + O(2^{1}4)) ∗ a^{3} + (16127 ∗ 2^{−}4 + O(2^{1}4)) ∗ a^{2} + (6699 ∗ 2^{−}2 + O(2^{1}4)) ∗ a + 23 ∗ 2^{5} + O(2^{1}4)q^{3} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{4} + 7961 ∗ 2^{4} + O(2^{2}0)x^{3} + −5623 ∗ 2^{6} + O(2^{2}0)x^{2} + 475 ∗ 2^{7} + O(2^{2}0)x + 145 ∗ 2^{9} + O(2^{2}0). The slope of f_{0} is 9/4. —————————————————————————————————- (Uninteresting cases below.)

2. Consider f = q + q2 + ⋯ ∈ S4(Γ0(41); \overline{F}2). Since f is ordinary, we do not continue our analysis.

3. Consider f = q + q2 + ⋯ ∈ S4(Γ0(41); \overline{F}2). Since f is ordinary, we do not continue our analysis.

4. Consider f = q + q2 + ⋯ ∈ S4(Γ0(41); \overline{F}2). Since f is ordinary, we do not continue our analysis.

2.21 N = 43, ℓ = 2 (4 forms)

1. Consider f = q + ⋯ ∈ S_{4}(Γ_{0}(43); \overline{F}_{2}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{1}. The form f_{0} = q+aq^{2}+−(2533∗2+O(2^{1}7))∗a+17∗2+O(2^{1}7)q^{3}+⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} +216203∗2+O(2^{2}0)x+−87679∗2+O(2^{2}0). The slope of f_{0} is 1/2.

2. Consider f = q+⋯ ∈ S_{4}(Γ_{0}(43); \overline{F}_{2}). Also, z satisfies the equation x^{2}+x+ = 0. We have ρ_{f} ⊗χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{1}. The form f_{0} = q + aq^{2} + −(89971 ∗ 2^{−}1 + O(2^{1}7)) ∗ a^{3} + (1121 ∗ 2^{2} + O(2^{1}7)) ∗ a^{2} − (5067 ∗ 2^{2} + O(2^{1}7)) ∗ a + 10999 ∗ 2^{2} + O(2^{1}7)q^{3} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{4} + −29505 ∗ 2^{3} + O(2^{2}0)x^{3} + −199705 ∗ 2 + O(2^{2}0)x^{2} + −52211 ∗ 2^{3} + O(2^{2}0)x + 37431 ∗ 2^{2} + O(2^{2}0). The slope of f_{0} is 1/2. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + q2 + zq3 + ⋯ ∈ S4(Γ0(43); \overline{F}2). Also, z satisfies the equation x2 + x+ = 0. Since f is ordinary, we do not continue our analysis.

4. Consider f = q + q2 + z2q3 + ⋯ ∈ S4(Γ0(43); \overline{F}2). Also, z satisfies the equation x2 + x+ = 0. Since f is ordinary, we do not continue our analysis.

2.22 N = 45, ℓ = 2 (5 forms)

1. Consider f = q + ⋯ ∈ S_{4}(Γ_{0}(45); \overline{F}_{2}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 2^{2}q^{2} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 2. —————————————————————————————————- (Uninteresting cases below.)

2. Consider f = q + q2 + ⋯ ∈ S4(Γ0(45); \overline{F}2). Since f is ordinary, we do not continue our analysis.

3. Consider f = q + q2 + ⋯ ∈ S4(Γ0(45); \overline{F}2). Since f is ordinary, we do not continue our analysis.

4. Consider f = q + q2 + ⋯ ∈ S4(Γ0(45); \overline{F}2). Since f is ordinary, we do not continue our analysis.

5. Consider f = q + q2 + ⋯ ∈ S4(Γ0(45); \overline{F}2). Since f is ordinary, we do not continue our analysis.

2.23 N = 47, ℓ = 2 (5 forms)

1. Consider f = q + a^{2}q^{3} + ⋯ ∈ S_{4}(Γ_{0}(47); \overline{F}_{2}). Also, z satisfies the equation x^{2} + x+ = 0. We have ρ_{f} ⊗χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q+z^{6}q^{2}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{9})ω^{0} & 0 \\ 0 & u(z^{6})ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + −(342759 ∗ 2^{−}1 + O(2^{1}9)) ∗ a − 218379 + O(2^{1}9)q^{3} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −171377 ∗ 2 + O(2^{2}0)x + −109189 ∗ 2^{2} + O(2^{2}0). The slope of f_{0} is 1.

2. Consider f = q + a^{2}q^{3} + ⋯ ∈ S_{4}(Γ_{0}(47); \overline{F}_{2}). Also, z satisfies the equation x^{2} + x+ = 0. We have ρ_{f} ⊗χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q+z^{6}q^{2}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{9})ω^{0} & 0 \\ 0 & u(z^{6})ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +aq^{2} +(29517∗2^{−}1+ O(2^{1}6)) ∗ a + 21745 + O(2^{1}6)q^{3} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −46513 ∗ 2 + O(2^{2}0)x + 111751 ∗ 2^{2} + O(2^{2}0). The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + q2 + ⋯ ∈ S4(Γ0(47); \overline{F}2). Since f is ordinary, we do not continue our analysis.

4. Consider f = q + z3q2 + z5q3 + ⋯ ∈ S4(Γ0(47); \overline{F}2). Also, z satisfies the equation x4 + x+ = 0. Since f is ordinary, we do not continue our analysis.

5. Consider f = q + q2 + ⋯ ∈ S4(Γ0(47); \overline{F}2). Since f is ordinary, we do not continue our analysis.

2.24 N = 49, ℓ = 2 (6 forms)

1. Consider f = q + ⋯ ∈ S_{4}(Γ_{0}(49); \overline{F}_{2}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + O(2^{2}0) ∗ a + 19983 ∗ 2^{4} + O(2^{2}0)q^{2} + −(1 + O(2^{2}0)) ∗ a + 19983 ∗ 2^{4} + O(2^{2}0)q^{3} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 12785 ∗ 2^{5} + O(2^{2}0)x + 17615 ∗ 2^{3} + O(2^{2}0). The slope of f_{0} is 4. —————————————————————————————————- (Uninteresting cases below.)

2. Consider f = q + q2 + ⋯ ∈ S4(Γ0(49); \overline{F}2). Since f is ordinary, we do not continue our analysis.

3. Consider f = q + q3 + ⋯ ∈ S4(Γ0(49); \overline{F}2). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

4. Consider f = q + q3 + ⋯ ∈ S4(Γ0(49); \overline{F}2). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

5. Consider f = q + q2 + ⋯ ∈ S4(Γ0(49); \overline{F}2). Since f is ordinary, we do not continue our analysis.

6. Consider f = q + q2 + ⋯ ∈ S4(Γ0(49); \overline{F}2). Since f is ordinary, we do not continue our analysis.

3 2 ≤ N ≤ 70, ℓ = 3

3.1 N = 2, ℓ = 3 (no forms)

There are no forms to analyze.

3.2 N = 4, ℓ = 3 (1 form)

- —————————————————————————————————- (Uninteresting cases below.)

1. Consider f = q + ⋯ ∈ S6(Γ0(4); \overline{F}3). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

3.3 N = 5, ℓ = 3 (1 form)

- —————————————————————————————————- (Uninteresting cases below.)

1. Consider f = q + 2q2 + 2q3 + 2q4 + q5 + ⋯ ∈ S6(Γ0(5); \overline{F}3). Since f is ordinary, we do not continue our analysis.

3.4 N = 7, ℓ = 3 (2 forms)

- —————————————————————————————————- (Uninteresting cases below.)

1. Consider f = q + 2q2 + q3 + 2q4 + q5 + ⋯ ∈ S6(Γ0(7); \overline{F}3). Since f is ordinary, we do not continue our analysis.

2. Consider f = q + q4 + ⋯ ∈ S6(Γ0(7); \overline{F}3). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

3.5 N = 8, ℓ = 3 (1 form)

- —————————————————————————————————- (Uninteresting cases below.)

1. Consider f = q + 2q3 + q5 + ⋯ ∈ S6(Γ0(8); \overline{F}3). Since f is ordinary, we do not continue our analysis.

3.6 N = 10, ℓ = 3 (3 forms)

1. Consider f = q + q^{2} + q^{4} + 2q^{5} + ⋯ ∈ S_{6}(Γ_{0}(10); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-4 newform g = q + 2q^{2} + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 4q^{2} + 2 ∗ 3q^{3} + 16q^{4} + −25q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 2q^{2} + q^{4} + q^{5} + ⋯ ∈ S_{6}(Γ_{0}(10); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-4 newform g = q + 2q^{2} + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −4q^{2} + 8 ∗ 3q^{3} + 16q^{4} + 25q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + 2q2 + q3 + q4 + 2q5 + ⋯ ∈ S6(Γ0(10); \overline{F}3). Since f is ordinary, we do not continue our analysis.

3.7 N = 11, ℓ = 3 (3 forms)

1. Consider f = q + 2q^{2} + 2q^{4} + 2q^{5} + ⋯ ∈ S_{6}(Γ_{0}(11); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + 2q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{0} & 0 \\ 0 & u(2)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −4q^{2} + −5 ∗ 3q^{3} + −16q^{4} + −19q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + q^{2} + 2q^{4} + q^{5} + ⋯ ∈ S_{6}(Γ_{0}(11); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + 2q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(2)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 768257473 + O(3^{2}0)q^{2} + 96924388 ∗ 3 + O(3^{1}9)q^{3} + 1033139585 + O(3^{2}0)q^{4} + −1697335037 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + q2 + 2q3 + 2q4 + q5 + ⋯ ∈ S6(Γ0(11); \overline{F}3). Since f is ordinary, we do not continue our analysis.

3.8 N = 13, ℓ = 3 (4 forms)

- —————————————————————————————————- (Uninteresting cases below.)

1. Consider f = q + zq2 + q3 + z7q4 + z2q5 + ⋯ ∈ S6(Γ0(13); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

2. Consider f = q + z3q2 + q3 + z5q4 + z6q5 + ⋯ ∈ S6(Γ0(13); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

3. Consider f = q + q4 + ⋯ ∈ S6(Γ0(13); \overline{F}3). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

4. Consider f = q + q2 + 2q3 + 2q4 + 2q5 + ⋯ ∈ S6(Γ0(13); \overline{F}3). Since f is ordinary, we do not continue our analysis.

3.9 N = 14, ℓ = 3 (2 forms)

- —————————————————————————————————- (Uninteresting cases below.)

1. Consider f = q + q2 + 2q3 + q4 + q5 + ⋯ ∈ S6(Γ0(14); \overline{F}3). Since f is ordinary, we do not continue our analysis.

2. Consider f = q + 2q2 + q3 + q4 + ⋯ ∈ S6(Γ0(14); \overline{F}3). Since f is ordinary, we do not continue our analysis.

3.10 N = 16, ℓ = 3 (2 forms)

- —————————————————————————————————- (Uninteresting cases below.)

1. Consider f = q + ⋯ ∈ S6(Γ0(16); \overline{F}3). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

2. Consider f = q + q3 + q5 + ⋯ ∈ S6(Γ0(16); \overline{F}3). Since f is ordinary, we do not continue our analysis.

3.11 N = 17, ℓ = 3 (4 forms)

1. Consider f = q + q^{2} + 2q^{4} + 2q^{5} + ⋯ ∈ S_{6}(Γ_{0}(17); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + 2q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{5}. The form f_{0} = q + q^{2} + −2 ∗ 3^{2}q^{3} + −31q^{4} + −16q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 2.

2. Consider f = q + 2q^{2} + 2q^{4} + q^{5} + ⋯ ∈ S_{6}(Γ_{0}(17); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + 2q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{1}. The form f_{0} = q +aq^{2} +(828968360∗3^{−}1+O(3^{1}9))∗a+397823729∗3^{−}1+O(3^{1}9)q^{3} + −(1686638999 + O(3^{2}0)) ∗ a + 172039336 ∗ 3 + O(3^{2}0)q^{4} + (1630537157 + O(3^{2}0)) ∗ a − 76122263 ∗ 3^{2} + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 1686638999 + O(3^{2}0)x + −516118040 + O(3^{2}0). The slope of f_{0} is 1/2. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + q3 + q4 + ⋯ ∈ S6(Γ0(17); \overline{F}3). Since f is ordinary, we do not continue our analysis.

4. Consider f = q + z5q2 + 2q3 + z7q4 + z2q5 + ⋯ ∈ S6(Γ0(17); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

3.12 N = 19, ℓ = 3 (4 forms)

1. Consider f = q + q^{4} + ⋯ ∈ S_{6}(Γ_{0}(19); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + 3/22 ∗ a^{3} − 5 ∗ 3/22∗a^{2}−35∗3/11∗a+181∗3/11q^{3}+a^{2}−32q^{4}+−23/22∗a^{3}+3^{3}/22∗a^{2}+893/11∗a−56∗3^{3}/11q^{5}+⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{4} + −3^{2}x^{3} + −8 ∗ 3^{2}x^{2} + 86 ∗ 3^{2}x + −380 ∗ 3. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

2. Consider f = q + q2 + 2q3 + 2q4 + ⋯ ∈ S6(Γ0(19); \overline{F}3). Since f is ordinary, we do not continue our analysis.

3. Consider f = q + q3 + q4 + ⋯ ∈ S6(Γ0(19); \overline{F}3). Since f is ordinary, we do not continue our analysis.

4. Consider f = q + q2 + q3 + 2q4 + q5 + ⋯ ∈ S6(Γ0(19); \overline{F}3). Since f is ordinary, we do not continue our analysis.

3.13 N = 20, ℓ = 3 (1 form)

- —————————————————————————————————- (Uninteresting cases below.)

1. Consider f = q + q3 + 2q5 + ⋯ ∈ S6(Γ0(20); \overline{F}3). Since f is ordinary, we do not continue our analysis.

3.14 N = 22, ℓ = 3 (5 forms)

1. Consider f = q + 2q^{2} + q^{4} + ⋯ ∈ S_{6}(Γ_{0}(22); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-4 newform g = q + 2q^{2} + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −4q^{2} + −7 ∗ 3q^{3} + 16q^{4} + 3^{4}q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q +q^{2} +q^{4} +⋯ ∈ S_{6}(Γ_{0}(22); \overline{F}_{3}). We have ρ_{f} ⊗χ^{1} ∼ ρ_{g}, where g is the weight-4 newform g = q + 2q^{2} + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 4q^{2} + −372933190 ∗ 3 + O(3^{2}0)q^{3} + 16q^{4} + −51095218 ∗ 3^{3} + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + 2q2 + q3 + q4 + ⋯ ∈ S6(Γ0(22); \overline{F}3). Since f is ordinary, we do not continue our analysis.

4. Consider f = q + q2 + q3 + q4 + 2q5 + ⋯ ∈ S6(Γ0(22); \overline{F}3). Since f is ordinary, we do not continue our analysis.

5. Consider f = q + q2 + 2q3 + q4 + 2q5 + ⋯ ∈ S6(Γ0(22); \overline{F}3). Since f is ordinary, we do not continue our analysis.

3.15 N = 23, ℓ = 3 (6 forms)

1. Consider f = q +zq^{2} +z^{7}q^{4} +z^{5}q^{5} +⋯ ∈ S_{6}(Γ_{0}(23); \overline{F}_{3}). Also, z satisfies the equation x^{2} +2x+2 = 0. We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + z^{5}q^{2} + z^{6}q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2})ω^{0} & 0 \\ 0 & u(z^{6})ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + −(14041298 ∗ 3 + O(3^{2}0)) ∗ a + 301865704 ∗ 3 + O(3^{2}0)q^{3} + (168495562 + O(3^{2}0)) ∗ a − 135604087 + O(3^{2}0)q^{4} + (84247781 + O(3^{2}0)) ∗ a + 558530045 ∗ 3 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −168495562 + O(3^{2}0)x + 135604055 + O(3^{2}0). The slope of f_{0} is 1.

2. Consider f = q+z^{3}q^{2} +z^{5}q^{4} +z^{7}q^{5} +⋯ ∈ S_{6}(Γ_{0}(23); \overline{F}_{3}). Also, z satisfies the equation x^{2} +2x+2 = 0. We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + z^{7}q^{2} + z^{2}q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{6})ω^{0} & 0 \\ 0 & u(z^{2})ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + −(14041298 ∗ 3 + O(3^{2}0)) ∗ a + 301865704 ∗ 3 + O(3^{2}0)q^{3} + (168495562 + O(3^{2}0)) ∗ a − 135604087 + O(3^{2}0)q^{4} + (84247781 + O(3^{2}0)) ∗ a + 558530045 ∗ 3 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −168495562 + O(3^{2}0)x + 135604055 + O(3^{2}0). The slope of f_{0} is 1.

3. Consider f = q +z^{5}q^{2} +z^{7}q^{4} +zq^{5} +⋯ ∈ S_{6}(Γ_{0}(23); \overline{F}_{3}). Also, z satisfies the equation x^{2} +2x+2 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + z^{5}q^{2} + z^{6}q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2})ω^{1} & 0 \\ 0 & u(z^{6})ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + (47741734 ∗ 3 + O(3^{1}9)) ∗ a − 108721528 ∗ 3 + O(3^{1}9)q^{3} + −(1041843502 + O(3^{2}0)) ∗ a + 534064556 + O(3^{2}0)q^{4} + (835880963 + O(3^{2}0)) ∗ a − 162616748 ∗ 3 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} +1041843502+O(3^{2}0)x+−534064588+O(3^{2}0). The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

4. Consider f = q + q2 + q3 + 2q4 + ⋯ ∈ S6(Γ0(23); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

5. Consider f = q + z2q2 + 2q3 + z3q5 + ⋯ ∈ S6(Γ0(23); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

6. Consider f = q + z5q2 + z6q3 + z7q4 + zq5 + ⋯ ∈ S6(Γ0(23); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

3.16 N = 25, ℓ = 3 (7 forms)

- —————————————————————————————————- (Uninteresting cases below.)

1. Consider f = q + q2 + q3 + 2q4 + ⋯ ∈ S6(Γ0(25); \overline{F}3). Since f is ordinary, we do not continue our analysis.

2. Consider f = q + z6q2 + ⋯ ∈ S6(Γ0(25); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

3. Consider f = q + z2q2 + ⋯ ∈ S6(Γ0(25); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

4. Consider f = q + q4 + ⋯ ∈ S6(Γ0(25); \overline{F}3). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

5. Consider f = q + 2q2 + 2q3 + 2q4 + ⋯ ∈ S6(Γ0(25); \overline{F}3). Since f is ordinary, we do not continue our analysis.

6. Consider f = q + q4 + ⋯ ∈ S6(Γ0(25); \overline{F}3). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

7. Consider f = q + q2 + q3 + 2q4 + ⋯ ∈ S6(Γ0(25); \overline{F}3). Since f is ordinary, we do not continue our analysis.

3.17 N = 26, ℓ = 3 (4 forms)

1. Consider f = q + 2q^{2} + q^{4} + q^{5} + ⋯ ∈ S_{6}(Γ_{0}(26); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{5}. The form f_{0} = q + −4q^{2} + 16q^{4} + −14q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is Infinity.

2. Consider f = q + q^{2} + q^{4} + 2q^{5} + ⋯ ∈ S_{6}(Γ_{0}(26); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{1}. The form f_{0} = q + 4q^{2} + aq^{3} + 16q^{4} + −3 ∗ a + 50q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −3^{2}x + −64 ∗ 3. The slope of f_{0} is 1/2. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + 2q2 + 2q3 + q4 + 2q5 + ⋯ ∈ S6(Γ0(26); \overline{F}3). Since f is ordinary, we do not continue our analysis.

4. Consider f = q + 2q2 + q3 + q4 + ⋯ ∈ S6(Γ0(26); \overline{F}3). Since f is ordinary, we do not continue our analysis.

3.18 N = 28, ℓ = 3 (2 forms)

- —————————————————————————————————- (Uninteresting cases below.)

1. Consider f = q + q3 + ⋯ ∈ S6(Γ0(28); \overline{F}3). Since f is ordinary, we do not continue our analysis.

2. Consider f = q + 2q3 + q5 + ⋯ ∈ S6(Γ0(28); \overline{F}3). Since f is ordinary, we do not continue our analysis.

3.19 N = 29, ℓ = 3 (7 forms)

1. Consider f = q + c^{5}q^{2} + c^{7}q^{4} + q^{5} + ⋯ ∈ S_{6}(Γ_{0}(29); \overline{F}_{3}). Also, z satisfies the equation x^{2} + 2x + 2 = 0. We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + cq^{2} + c^{5}q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(c^{3})ω^{0} & 0 \\ 0 & u(c^{5})ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + (485908895 ∗ 3 + O(3^{2}0)) ∗ a + 47013614 ∗ 3^{3} + O(3^{2}0)q^{3} + −(343712983 + O(3^{2}0)) ∗ a + 205336370 + O(3^{2}0)q^{4} + −(177711077∗3^{2} +O(3^{2}0))∗a−1158569672+O(3^{2}0)q^{5} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 343712983 + O(3^{2}0)x + −205336402 + O(3^{2}0). The slope of f_{0} is 1.

2. Consider f = q + c^{7}q^{2} + c^{5}q^{4} + q^{5} + ⋯ ∈ S_{6}(Γ_{0}(29); \overline{F}_{3}). Also, z satisfies the equation x^{2} + 2x + 2 = 0. We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + c^{3}q^{2} + c^{7}q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(c)ω^{0} & 0 \\ 0 & u(c^{7})ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + (485908895 ∗ 3 + O(3^{2}0)) ∗ a + 47013614 ∗ 3^{3} + O(3^{2}0)q^{3} + −(343712983 + O(3^{2}0)) ∗ a + 205336370 + O(3^{2}0)q^{4} + −(177711077∗3^{2} +O(3^{2}0))∗a−1158569672+O(3^{2}0)q^{5} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 343712983 + O(3^{2}0)x + −205336402 + O(3^{2}0). The slope of f_{0} is 1.

3. Consider f = q + cq^{2} + c^{7}q^{4} + 2q^{5} + ⋯ ∈ S_{6}(Γ_{0}(29); \overline{F}_{3}). Also, z satisfies the equation x^{2} + 2x + 2 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + cq^{2} + c^{5}q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(c^{3})ω^{1} & 0 \\ 0 & u(c^{5})ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + −(35132759 ∗ 3 + O(3^{1}9)) ∗ a − 104509003 ∗ 3 + O(3^{1}9)q^{3} + −(568806134 + O(3^{2}0)) ∗ a − 842362495 + O(3^{2}0)q^{4} + (56814136 ∗ 3^{3} + O(3^{2}0)) ∗ a − 636425824 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 568806134 + O(3^{2}0)x + 842362463 + O(3^{2}0). The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

4. Consider f = q + cq2 + q3 + c7q4 + c7q5 + ⋯ ∈ S6(Γ0(29); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

5. Consider f = q + c3q2 + q3 + c5q4 + c5q5 + ⋯ ∈ S6(Γ0(29); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

6. Consider f = q + a7q2 + 2q3 + a16q4 + a22q5 + ⋯ ∈ S6(Γ0(29); \overline{F}3). Also, z satisfies the equation x3 + 2x+ = 0. Since f is ordinary, we do not continue our analysis.

7. Consider f = q + cq2 + c5q3 + c7q4 + 2q5 + ⋯ ∈ S6(Γ0(29); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

1. Consider f = q + zq^{2} + z^{7}q^{4} + q^{5} + ⋯ ∈ S_{6}(Γ_{0}(31); \overline{F}_{3}). Also, z satisfies the equation x^{2} + 2x + 2 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + zq^{2} + zq^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{7})ω^{1} & 0 \\ 0 & u(z)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + −(59399854 ∗ 3 + O(3^{1}8)) ∗ a − 60392801 ∗ 3 + O(3^{1}8)q^{3} + −(26993603 + O(3^{2}0)) ∗ a + 507311864 + O(3^{2}0)q^{4} + (60958945 ∗ 3 + O(3^{1}9)) ∗ a − 437126999 + O(3^{1}9)q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 26993603 + O(3^{2}0)x + −507311896 + O(3^{2}0). The slope of f_{0} is 1.

2. Consider f = q +z^{5}q^{2} +z^{7}q^{4} +2q^{5} +⋯ ∈ S_{6}(Γ_{0}(31); \overline{F}_{3}). Also, z satisfies the equation x^{2} +2x+2 = 0. We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + zq^{2} + zq^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{7})ω^{0} & 0 \\ 0 & u(z)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + −(61242059 ∗ 3 + O(3^{1}8)) ∗ a + 14785886 ∗ 3^{2} + O(3^{1}8)q^{3} + −(839661361 + O(3^{2}0)) ∗ a + 332501318 + O(3^{2}0)q^{4} + −(150252932 ∗ 3 + O(3^{1}9)) ∗ a + 95195480 + O(3^{1}9)q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 839661361 + O(3^{2}0)x + −332501350 + O(3^{2}0). The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + z546q2 + 2q3 + z455q5 + ⋯ ∈ S6(Γ0(31); \overline{F}3). Also, z satisfies the equation x6 + 2x4 + x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

4. Consider f = q + z28q2 + q3 + z588q4 + z532q5 + ⋯ ∈ S6(Γ0(31); \overline{F}3). Also, z satisfies the equation x6 + 2x4 + x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

5. Consider f = q + z84q2 + q3 + z308q4 + z140q5 + ⋯ ∈ S6(Γ0(31); \overline{F}3). Also, z satisfies the equation x6 + 2x4 + x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

6. Consider f = q + z182q2 + 2q3 + z637q5 + ⋯ ∈ S6(Γ0(31); \overline{F}3). Also, z satisfies the equation x6 + 2x4 + x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

7. Consider f = q + z252q2 + q3 + z196q4 + z420q5 + ⋯ ∈ S6(Γ0(31); \overline{F}3). Also, z satisfies the equation x6 + 2x4 + x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

8. Consider f = q + zq2 + zq3 + z7q4 + q5 + ⋯ ∈ S6(Γ0(31); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

9. Consider f = q + q4 + ⋯ ∈ S6(Γ0(31); \overline{F}3). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

3.21 N = 32, ℓ = 3 (4 forms)

1. Consider f = q + 2q^{5} + ⋯ ∈ S_{6}(Γ_{0}(32); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{5}. The form f_{0} = q + −82q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is Infinity.

2. Consider f = q + q^{5} + ⋯ ∈ S_{6}(Γ_{0}(32); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{1}. The form f_{0} = q + aq^{3} + 46q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −256 ∗ 3. The slope of f_{0} is 1/2. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + 2q3 + 2q5 + ⋯ ∈ S6(Γ0(32); \overline{F}3). Since f is ordinary, we do not continue our analysis.

4. Consider f = q + q3 + 2q5 + ⋯ ∈ S6(Γ0(32); \overline{F}3). Since f is ordinary, we do not continue our analysis.

1. Consider f = q + 2q^{2} + q^{4} + ⋯ ∈ S_{6}(Γ_{0}(34); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −4q^{2} + aq^{3} + 16q^{4} + 4 ∗ 3^{−}1 ∗ a − 14q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 2 ∗ 3x + −68 ∗ 3^{2}. The slope of f_{0} is 1.

2. Consider f = q +q^{2} +q^{4} +⋯ ∈ S_{6}(Γ_{0}(34); \overline{F}_{3}). We have ρ_{f} ⊗χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 4q^{2} + aq^{3} + 16q^{4} + −(1309201658 ∗ 3^{−}1 + O(3^{1}9)) ∗ a + 385290877 + O(3^{1}9)q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −142520176 ∗ 3 + O(3^{2}0)x + 132689539 ∗ 3^{2} + O(3^{2}0). The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + q2 + q3 + q4 + q5 + ⋯ ∈ S6(Γ0(34); \overline{F}3). Since f is ordinary, we do not continue our analysis.

4. Consider f = q + 2q2 + 2q3 + q4 + 2q5 + ⋯ ∈ S6(Γ0(34); \overline{F}3). Since f is ordinary, we do not continue our analysis.

5. Consider f = q + 2q2 + q3 + q4 + ⋯ ∈ S6(Γ0(34); \overline{F}3). Since f is ordinary, we do not continue our analysis.

6. Consider f = q + q2 + q3 + q4 + ⋯ ∈ S6(Γ0(34); \overline{F}3). Since f is ordinary, we do not continue our analysis.

3.23 N = 35, ℓ = 3 (8 forms)

1. Consider f = q + zq^{2} + z^{7}q^{4} + 2q^{5} + ⋯ ∈ S_{6}(Γ_{0}(35); \overline{F}_{3}). Also, z satisfies the equation x^{2} + 2x + 2 = 0. We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + z^{5}q^{2} + z^{7}q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z)ω^{0} & 0 \\ 0 & u(z^{7})ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + −3 ∗ a + 3q^{3} + a − 16q^{4} + −25q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −1x + −16. The slope of f_{0} is 1.

2. Consider f = q +z^{3}q^{2} +z^{5}q^{4} +2q^{5} +⋯ ∈ S_{6}(Γ_{0}(35); \overline{F}_{3}). Also, z satisfies the equation x^{2} +2x+2 = 0. We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + z^{7}q^{2} + z^{5}q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{3})ω^{0} & 0 \\ 0 & u(z^{5})ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + −3 ∗ a + 3q^{3} + a − 16q^{4} + −25q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −1x + −16. The slope of f_{0} is 1.

3. Consider f = q + z^{5}q^{2} + z^{7}q^{4} + q^{5} + ⋯ ∈ S_{6}(Γ_{0}(35); \overline{F}_{3}). Also, z satisfies the equation x^{2} + 2x + 2 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + z^{5}q^{2} + z^{7}q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z)ω^{1} & 0 \\ 0 & u(z^{7})ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(145944434∗3+ O(3^{1}9))∗a+10184609∗3^{3} +O(3^{1}9)q^{3} +−(76109761+O(3^{2}0))∗a−1069901098+O(3^{2}0)q^{4} +25q^{5} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 76109761 + O(3^{2}0)x + 1069901066 + O(3^{2}0). The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

4. Consider f = q + q2 + q3 + 2q4 + q5 + ⋯ ∈ S6(Γ0(35); \overline{F}3). Since f is ordinary, we do not continue our analysis.

5. Consider f = q + q3 + q4 + 2q5 + ⋯ ∈ S6(Γ0(35); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

6. Consider f = q + z6q2 + 2q3 + 2q5 + ⋯ ∈ S6(Γ0(35); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

7. Consider f = q + z2q2 + 2q3 + 2q5 + ⋯ ∈ S6(Γ0(35); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

8. Consider f = q + z5q2 + z7q3 + z7q4 + q5 + ⋯ ∈ S6(Γ0(35); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

3.24 N = 37, ℓ = 3 (6 forms)

1. Consider f = q + 2q^{2} + 2q^{4} + 2q^{5} + ⋯ ∈ S_{6}(Γ_{0}(37); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{5}. The form f_{0} = q + −261495886 + O(3^{2}0)q^{2} + −24617477 ∗ 3^{2} + O(3^{1}9)q^{3} + −1072438657 + O(3^{2}0)q^{4} + −411596752 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 2.

2. Consider f = q + q^{2} + 2q^{4} + q^{5} + ⋯ ∈ S_{6}(Γ_{0}(37); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{1}. The form f_{0} = q + aq^{2} + −(1699627921 ∗ 3^{−}1 + O(3^{1}9)) ∗ a + 1522894744 ∗ 3^{−}1 + O(3^{1}9)q^{3} + −(84265174 + O(3^{2}0)) ∗ a + 257270885 ∗ 3 + O(3^{2}0)q^{4} + −(1551044939 + O(3^{2}0)) ∗ a − 529209481 ∗ 3 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 84265174 + O(3^{2}0)x + −771812687 + O(3^{2}0). The slope of f_{0} is 1/2.

3. Consider f = q + q^{4} + ⋯ ∈ S_{6}(Γ_{0}(37); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +aq^{2} +(5386375∗3^{2} + O(3^{1}9))∗a^{3} +(162643129∗3+O(3^{1}9))∗a^{2} −(15843080∗3+O(3^{1}9))∗a−172566433∗3+O(3^{1}9)q^{3} +a^{2} − 32q^{4}+−(844412614+O(3^{2}0))∗a^{3}−(150025085∗3^{2}+O(3^{2}0))∗a^{2}−(44174068+O(3^{2}0))∗a+14467880∗3^{5}+ O(3^{2}1)q^{5}+⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{4}+7178639∗3^{3}+O(3^{2}0)x^{3}+−227042∗3+O(3^{2}0)x^{2}+171993760∗3^{2}+O(3^{2}0)x+−9566960∗3+O(3^{2}0). The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

4. Consider f = q + z13q2 + q3 + z72q4 + z54q5 + ⋯ ∈ S6(Γ0(37); \overline{F}3). Also, z satisfies the equation x4 + 2x3 + 2 = 0. Since f is ordinary, we do not continue our analysis.

5. Consider f = q + q3 + q4 + ⋯ ∈ S6(Γ0(37); \overline{F}3). Since f is ordinary, we do not continue our analysis.

6. Consider f = q + z14q2 + 2q3 + z21q4 + z8q5 + ⋯ ∈ S6(Γ0(37); \overline{F}3). Also, z satisfies the equation x3 + 2x+ = 0. Since f is ordinary, we do not continue our analysis.

1. Consider f = q + 2q^{2} + q^{4} + q^{5} + ⋯ ∈ S_{6}(Γ_{0}(38); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + 2q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{0} & 0 \\ 0 & u(2)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −4q^{2} + −2 ∗ 3q^{3} + 16q^{4} + 31q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + q^{2} + q^{4} + 2q^{5} + ⋯ ∈ S_{6}(Γ_{0}(38); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + 2q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(2)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 4q^{2} + −9661139 ∗ 3 + O(3^{2}0)q^{3} + 16q^{4} + 1167061640 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + q2 + q3 + q4 + ⋯ ∈ S6(Γ0(38); \overline{F}3). Since f is ordinary, we do not continue our analysis.

4. Consider f = q + 2q2 + 2q3 + q4 + ⋯ ∈ S6(Γ0(38); \overline{F}3). Since f is ordinary, we do not continue our analysis.

5. Consider f = q + 2q2 + q3 + q4 + ⋯ ∈ S6(Γ0(38); \overline{F}3). Since f is ordinary, we do not continue our analysis.

6. Consider f = q + q2 + 2q3 + q4 + 2q5 + ⋯ ∈ S6(Γ0(38); \overline{F}3). Since f is ordinary, we do not continue our analysis.

3.26 N = 40, ℓ = 3 (4 forms)

1. Consider f = q + 2q^{5} + ⋯ ∈ S_{6}(Γ_{0}(40); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{5}. The form f_{0} = q + −2 ∗ 3^{2}q^{3} + −25q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 2.

2. Consider f = q + q^{5} + ⋯ ∈ S_{6}(Γ_{0}(40); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{1}. The form f_{0} = q + aq^{3} + 25q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 4 ∗ 3x + −160 ∗ 3. The slope of f_{0} is 1/2. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + q3 + 2q5 + ⋯ ∈ S6(Γ0(40); \overline{F}3). Since f is ordinary, we do not continue our analysis.

4. Consider f = q + q3 + q5 + ⋯ ∈ S6(Γ0(40); \overline{F}3). Since f is ordinary, we do not continue our analysis.

3.27 N = 41, ℓ = 3 (10 forms)

1. Consider f = q + z^{4}20q^{2} + z^{5}04q^{4} + z^{5}88q^{5} + ⋯ ∈ S_{6}(Γ_{0}(41); \overline{F}_{3}). Also, z satisfies the equation x^{6} + 2x^{4}+x^{2}+2x+2 = 0. We have ρ_{f} ⊗χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q+z^{2}q^{2}+z^{1}4q^{3}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}2)ω^{0} & 0 \\ 0 & u(z^{1}4)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + (361506644 ∗ 3 + O(3^{2}0)) ∗ a^{2} + (123336743 ∗ 3 + O(3^{2}0)) ∗ a − 142810786 ∗ 3^{2} + O(3^{2}0)q^{3} + a^{2} − 32q^{4} + −(4062758 ∗ 3^{3} + O(3^{2}0)) ∗ a^{2} + (1416069860 + O(3^{2}0)) ∗ a + 171085594 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + −1732513015 + O(3^{2}0)x^{2} + −369378572 + O(3^{2}0)x + −261992342 + O(3^{2}0). The slope of f_{0} is 1.

2. Consider f = q+z^{5}32q^{2}+z^{5}6q^{4}+z^{3}08q^{5}+⋯ ∈ S_{6}(Γ_{0}(41); \overline{F}_{3}). Also, z satisfies the equation x^{6}+2x^{4}+ x^{2} + 2x + 2 = 0. We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + z^{6}q^{2} + z^{1}6q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}0)ω^{0} & 0 \\ 0 & u(z^{1}6)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + (361506644 ∗ 3 + O(3^{2}0)) ∗ a^{2} + (123336743 ∗ 3 + O(3^{2}0)) ∗ a − 142810786 ∗ 3^{2} + O(3^{2}0)q^{3} + a^{2} − 32q^{4} + −(4062758 ∗ 3^{3} + O(3^{2}0)) ∗ a^{2} + (1416069860 + O(3^{2}0)) ∗ a + 171085594 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + −1732513015 + O(3^{2}0)x^{2} + −369378572 + O(3^{2}0)x + −261992342 + O(3^{2}0). The slope of f_{0} is 1.

3. Consider f = q + z^{1}40q^{2} + z^{1}68q^{4} + z^{1}96q^{5} + ⋯ ∈ S_{6}(Γ_{0}(41); \overline{F}_{3}). Also, z satisfies the equation x^{6} + 2x^{4}+x^{2}+2x+2 = 0. We have ρ_{f} ⊗χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q+z^{1}8q^{2}+z^{2}2q^{3}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{4})ω^{0} & 0 \\ 0 & u(z^{2}2)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + (361506644 ∗ 3 + O(3^{2}0)) ∗ a^{2} + (123336743 ∗ 3 + O(3^{2}0)) ∗ a − 142810786 ∗ 3^{2} + O(3^{2}0)q^{3} + a^{2} − 32q^{4} + −(4062758 ∗ 3^{3} + O(3^{2}0)) ∗ a^{2} + (1416069860 + O(3^{2}0)) ∗ a + 171085594 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + −1732513015 + O(3^{2}0)x^{2} + −369378572 + O(3^{2}0)x + −261992342 + O(3^{2}0). The slope of f_{0} is 1.

4. Consider f = q +z^{2}q^{2} +z^{1}8q^{4} +z^{8}q^{5} +⋯ ∈ S_{6}(Γ_{0}(41); \overline{F}_{3}). Also, z satisfies the equation x^{3} +2x+ = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + z^{2}q^{2} + z^{1}4q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}2)ω^{1} & 0 \\ 0 & u(z^{1}4)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + −(62479930 ∗ 3^{2} + O(3^{1}9)) ∗ a^{2} − (176107054 ∗ 3 + O(3^{1}9)) ∗ a + 131184580 ∗ 3 + O(3^{1}9)q^{3} + a^{2} − 32q^{4} + −(144889612 ∗ 3^{2} + O(3^{2}0)) ∗ a^{2} − (775411618 + O(3^{2}0)) ∗ a − 524825446 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 802283635 + O(3^{2}0)x^{2} + −1403525993 + O(3^{2}0)x + 963205139 + O(3^{2}0). The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

5. Consider f = q + 2q3 + q4 + q5 + ⋯ ∈ S6(Γ0(41); \overline{F}3). Also, z satisfies the equation x6 + 2x4 + x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

6. Consider f = q + z546q2 + q3 + z91q5 + ⋯ ∈ S6(Γ0(41); \overline{F}3). Also, z satisfies the equation x6 + 2x4 + x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

7. Consider f = q + z182q2 + q3 + z273q5 + ⋯ ∈ S6(Γ0(41); \overline{F}3). Also, z satisfies the equation x6 + 2x4 + x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

8. Consider f = q + z2q2 + z14q3 + z18q4 + z8q5 + ⋯ ∈ S6(Γ0(41); \overline{F}3). Also, z satisfies the equation x3 + 2x+ = 0. Since f is ordinary, we do not continue our analysis.

9. Consider f = q + z8q2 + 2q3 + z22q4 + z5q5 + ⋯ ∈ S6(Γ0(41); \overline{F}3). Also, z satisfies the equation x3 + 2x+ = 0. Since f is ordinary, we do not continue our analysis.

10. Consider f = q + q2 + q3 + 2q4 + 2q5 + ⋯ ∈ S6(Γ0(41); \overline{F}3). Since f is ordinary, we do not continue our analysis.

3.28 N = 43, ℓ = 3 (10 forms)

1. Consider f = q + 2q^{2} + 2q^{4} + q^{5} + ⋯ ∈ S_{6}(Γ_{0}(43); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −1151662513 + O(3^{2}0)q^{2} + −66064810 ∗ 3 + O(3^{1}9)q^{3} + −556198801 + O(3^{2}0)q^{4} + −1501437224 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + q^{2} + 2q^{4} + 2q^{5} + ⋯ ∈ S_{6}(Γ_{0}(43); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −1385818214 + O(3^{2}0)q^{2} + −109835833 ∗ 3 + O(3^{1}9)q^{3} + −1686754798 + O(3^{2}0)q^{4} + 147075986 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q + z^{2}q^{2} + z^{5}q^{5} + ⋯ ∈ S_{6}(Γ_{0}(43); \overline{F}_{3}). Also, z satisfies the equation x^{2} + 2x + 2 = 0. We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + z^{6}q^{2} + z^{2}q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{6})ω^{0} & 0 \\ 0 & u(z^{2})ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + (36548588 ∗ 3 + O(3^{1}8)) ∗ a − 7430932 ∗ 3^{2} + O(3^{1}8)q^{3} + (21441881 ∗ 3^{3} + O(3^{2}0)) ∗ a + 30681451 ∗ 3 + O(3^{2}0)q^{4} + −(41934607 + O(3^{1}7)) ∗ a − 63117281 + O(3^{1}7)q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2}+−21441881∗3^{3}+O(3^{2}0)x+−92044385+O(3^{2}0). The slope of f_{0} is 1.

4. Consider f = q + z^{2}q^{2} + z^{3}q^{5} + ⋯ ∈ S_{6}(Γ_{0}(43); \overline{F}_{3}). Also, z satisfies the equation x^{2} + 2x + 2 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + z^{2}q^{2} + z^{6}q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2})ω^{1} & 0 \\ 0 & u(z^{6})ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + (1140656 ∗ 3 + O(3^{1}8)) ∗ a + 12331391 ∗ 3^{2} + O(3^{1}8)q^{3} + (50344022 ∗ 3^{2} + O(3^{2}0)) ∗ a − 451817726 ∗ 3 + O(3^{2}0)q^{4} + −(41413471 + O(3^{1}7)) ∗ a + 18344459 + O(3^{1}7)q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2}+−50344022∗3^{2}+O(3^{2}0)x+1355453146+O(3^{2}0). The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

5. Consider f = q + 2q3 + q4 + q5 + ⋯ ∈ S6(Γ0(43); \overline{F}3). Since f is ordinary, we do not continue our analysis.

6. Consider f = q + z5q2 + q3 + z6q4 + z7q5 + ⋯ ∈ S6(Γ0(43); \overline{F}3). Also, z satisfies the equation x3 + 2x+ = 0. Since f is ordinary, we do not continue our analysis.

7. Consider f = q + q2 + q3 + 2q4 + 2q5 + ⋯ ∈ S6(Γ0(43); \overline{F}3). Since f is ordinary, we do not continue our analysis.

8. Consider f = q + z2q2 + z6q3 + z3q5 + ⋯ ∈ S6(Γ0(43); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

9. Consider f = q + z5q2 + 2q3 + z7q4 + z7q5 + ⋯ ∈ S6(Γ0(43); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

10. Consider f = q + q4 + ⋯ ∈ S6(Γ0(43); \overline{F}3). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

3.29 N = 44, ℓ = 3 (3 forms)

- —————————————————————————————————- (Uninteresting cases below.)

1. Consider f = q + q3 + 2q5 + ⋯ ∈ S6(Γ0(44); \overline{F}3). Since f is ordinary, we do not continue our analysis.

2. Consider f = q + 2q3 + 2q5 + ⋯ ∈ S6(Γ0(44); \overline{F}3). Since f is ordinary, we do not continue our analysis.

3. Consider f = q + q3 + ⋯ ∈ S6(Γ0(44); \overline{F}3). Since f is ordinary, we do not continue our analysis.

1. Consider f = q + 2q^{2} + q^{4} + ⋯ ∈ S_{6}(Γ_{0}(46); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-4 newform g = q + 2q^{2} + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −4q^{2} + 294421781 ∗ 3 + O(3^{2}0)q^{3} + 16q^{4} + 19362385 ∗ 3 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 2q^{2} + q^{4} + q^{5} + ⋯ ∈ S_{6}(Γ_{0}(46); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + 2q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{1}. The form f_{0} = q + −4q^{2} + aq^{3} + 16q^{4} + (49875845 ∗ 3 + O(3^{1}9)) ∗ a + 446738977 + O(3^{1}9)q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 32713531 ∗ 3^{3} + O(3^{2}0)x + −34110940 ∗ 3 + O(3^{2}0). The slope of f_{0} is 1/2.

3. Consider f = q + q^{2} + q^{4} + 2q^{5} + ⋯ ∈ S_{6}(Γ_{0}(46); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + 2q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{5}. The form f_{0} = q+4q^{2} +−11817551∗3^{4} +O(3^{2}0)q^{3} +16q^{4} +393171758+O(3^{1}9)q^{5} +⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 4.

4. Consider f = q +q^{2} +q^{4} +⋯ ∈ S_{6}(Γ_{0}(46); \overline{F}_{3}). We have ρ_{f} ⊗χ^{1} ∼ ρ_{g}, where g is the weight-4 newform g = q + 2q^{2} + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 4q^{2} + 127609667 ∗ 3 + O(3^{2}0)q^{3} + 16q^{4} + 142014293 ∗ 3 + O(3^{1}9)q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

5. Consider f = q + 2q2 + 2q3 + q4 + q5 + ⋯ ∈ S6(Γ0(46); \overline{F}3). Since f is ordinary, we do not continue our analysis.

6. Consider f = q + 2q2 + q3 + q4 + ⋯ ∈ S6(Γ0(46); \overline{F}3). Since f is ordinary, we do not continue our analysis.

7. Consider f = q + q2 + q3 + q4 + zq5 + ⋯ ∈ S6(Γ0(46); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

8. Consider f = q + q2 + 2q3 + q4 + 2q5 + ⋯ ∈ S6(Γ0(46); \overline{F}3). Since f is ordinary, we do not continue our analysis.

3.31 N = 47, ℓ = 3 (10 forms)

1. Consider f = q + q^{2} + 2q^{4} + ⋯ ∈ S_{6}(Γ_{0}(47); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + 2q^{2} + 2q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{0} & 0 \\ 0 & u(2)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 1653526378 + O(3^{2}0)q^{2} + −130264028 ∗ 3 + O(3^{2}0)q^{3} + −1693489162 + O(3^{2}0)q^{4} + 402185927 ∗ 3 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q+e^{1}7q^{2} +e^{1}5q^{4} +e^{7}q^{5} +⋯ ∈ S_{6}(Γ_{0}(47); \overline{F}_{3}). Also, z satisfies the equation x^{3} +2x+ = 0. We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + e^{4}q^{2} + e^{2}5q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(e)ω^{0} & 0 \\ 0 & u(e^{2}5)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + (509900633 ∗ 3 + O(3^{2}0)) ∗ a^{2} − (118738282 ∗ 3^{2} + O(3^{2}0)) ∗ a + 236689189 ∗ 3 + O(3^{2}0)q^{3} + a^{2} − 32q^{4} + −(164358022 ∗ 3^{2} + O(3^{2}0)) ∗ a^{2} − (879986902 + O(3^{2}0)) ∗ a − 748983769 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 46652102 + O(3^{2}0)x^{2} + −32551807 ∗ 3 + O(3^{2}0)x + 419601076 + O(3^{2}0). The slope of f_{0} is 1.

3. Consider f = q + 2q^{2} + 2q^{4} + ⋯ ∈ S_{6}(Γ_{0}(47); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + 2q^{2} + 2q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(2)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 314431748 + O(3^{2}0)q^{2} + 1138795 ∗ 3 + O(3^{1}5)q^{3} + −1231616200 + O(3^{2}0)q^{4} + 1393093 ∗ 3 + O(3^{1}5)q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

4. Consider f = q+e^{4}q^{2} +e^{1}5q^{4} +e^{2}0q^{5} +⋯ ∈ S_{6}(Γ_{0}(47); \overline{F}_{3}). Also, z satisfies the equation x^{3} +2x+ = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + e^{4}q^{2} + e^{2}5q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(e)ω^{1} & 0 \\ 0 & u(e^{2}5)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + −(681929 ∗ 3 + O(3^{1}5)) ∗ a^{2} − (1580135 ∗ 3 + O(3^{1}5)) ∗ a − 2299130 ∗ 3 + O(3^{1}5)q^{3} + a^{2} − 32q^{4} + (248440 ∗ 3^{3} + O(3^{1}5))∗a^{2} +(2601347+O(3^{1}5))∗a−3114497+O(3^{1}5)q^{5} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + −1494047123 + O(3^{2}0)x^{2} + −462015928 ∗ 3 + O(3^{2}0)x + 1621688861 + O(3^{2}0). The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

5. Consider f = q + 2q3 + q4 + 2q5 + ⋯ ∈ S6(Γ0(47); \overline{F}3). Since f is ordinary, we do not continue our analysis.

6. Consider f = q + 2q2 + q3 + 2q4 + 2q5 + ⋯ ∈ S6(Γ0(47); \overline{F}3). Since f is ordinary, we do not continue our analysis.

7. Consider f = q + b5q2 + 2q3 + b7q4 + b3q5 + ⋯ ∈ S6(Γ0(47); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

8. Consider f = q + q2 + q3 + 2q4 + ⋯ ∈ S6(Γ0(47); \overline{F}3). Since f is ordinary, we do not continue our analysis.

9. Consider f = q + e4q2 + e25q3 + e15q4 + e20q5 + ⋯ ∈ S6(Γ0(47); \overline{F}3). Also, z satisfies the equation x3 + 2x+ = 0. Since f is ordinary, we do not continue our analysis.

10. Consider f = q + 2q2 + 2q3 + 2q4 + ⋯ ∈ S6(Γ0(47); \overline{F}3). Since f is ordinary, we do not continue our analysis.

3.32 N = 49, ℓ = 3 (12 forms)

1. Consider f = q + 2q^{2} + 2q^{4} + ⋯ ∈ S_{6}(Γ_{0}(49); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{5}. The form f_{0} = q + 11q^{2} + 89q^{4} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is Infinity.

2. Consider f = q + q^{2} + 2q^{4} + ⋯ ∈ S_{6}(Γ_{0}(49); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{1}. The form f_{0} = q + −2q^{2} + −1/2 ∗ a − 1q^{3} + −28q^{4} + −3/2 ∗ a − 3q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 4x + −2492. The slope of f_{0} is 1/2. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + 2q2 + 2q3 + 2q4 + 2q5 + ⋯ ∈ S6(Γ0(49); \overline{F}3). Since f is ordinary, we do not continue our analysis.

4. Consider f = q + q4 + ⋯ ∈ S6(Γ0(49); \overline{F}3). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

5. Consider f = q + 2q2 + 2q3 + 2q4 + 2q5 + ⋯ ∈ S6(Γ0(49); \overline{F}3). Since f is ordinary, we do not continue our analysis.

6. Consider f = q + q4 + ⋯ ∈ S6(Γ0(49); \overline{F}3). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

7. Consider f = q + 2q2 + q3 + 2q4 + q5 + ⋯ ∈ S6(Γ0(49); \overline{F}3). Since f is ordinary, we do not continue our analysis.

8. Consider f = q + q4 + ⋯ ∈ S6(Γ0(49); \overline{F}3). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

9. Consider f = q + z3q2 + 2q3 + z5q4 + z7q5 + ⋯ ∈ S6(Γ0(49); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

10. Consider f = q + z3q2 + q3 + z5q4 + z3q5 + ⋯ ∈ S6(Γ0(49); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

11. Consider f = q + zq2 + 2q3 + z7q4 + z5q5 + ⋯ ∈ S6(Γ0(49); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

12. Consider f = q + zq2 + q3 + z7q4 + zq5 + ⋯ ∈ S6(Γ0(49); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

3.33 N = 50, ℓ = 3 (7 forms)

1. Consider f = q + 2q^{2} + q^{4} + ⋯ ∈ S_{6}(Γ_{0}(50); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + 2q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{0} & 0 \\ 0 & u(2)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −4q^{2} + −2 ∗ 3q^{3} + 16q^{4} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q +q^{2} +q^{4} +⋯ ∈ S_{6}(Γ_{0}(50); \overline{F}_{3}). We have ρ_{f} ⊗χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + 2q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{0} & 0 \\ 0 & u(2)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 4q^{2} + −8 ∗ 3q^{3} + 16q^{4} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + q2 + 2q3 + q4 + ⋯ ∈ S6(Γ0(50); \overline{F}3). Since f is ordinary, we do not continue our analysis.

4. Consider f = q + 2q2 + q3 + q4 + ⋯ ∈ S6(Γ0(50); \overline{F}3). Since f is ordinary, we do not continue our analysis.

5. Consider f = q + 2q2 + 2q3 + q4 + ⋯ ∈ S6(Γ0(50); \overline{F}3). Since f is ordinary, we do not continue our analysis.

6. Consider f = q + q2 + q3 + q4 + ⋯ ∈ S6(Γ0(50); \overline{F}3). Since f is ordinary, we do not continue our analysis.

7. Consider f = q + q2 + 2q3 + q4 + ⋯ ∈ S6(Γ0(50); \overline{F}3). Since f is ordinary, we do not continue our analysis.

3.34 N = 52, ℓ = 3 (4 forms)

1. Consider f = q + q^{5} + ⋯ ∈ S_{6}(Γ_{0}(52); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{5}. The form f_{0} = q + 26369093 ∗ 3^{2} + O(3^{2}0)q^{3} + −454783301 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 2.

2. Consider f = q + 2q^{5} + ⋯ ∈ S_{6}(Γ_{0}(52); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{1}. The form f_{0} = q +aq^{3} +−(47205994∗3+O(3^{1}9))∗a−258276565+O(3^{1}9)q^{5} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 79107275 ∗ 3 + O(3^{2}0)x + −129148660 ∗ 3 + O(3^{2}0). The slope of f_{0} is 1/2. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + q3 + ⋯ ∈ S6(Γ0(52); \overline{F}3). Since f is ordinary, we do not continue our analysis.

4. Consider f = q + 2q3 + 2q5 + ⋯ ∈ S6(Γ0(52); \overline{F}3). Since f is ordinary, we do not continue our analysis.

1. Consider f = q+e^{1}7q^{2}+e^{1}5q^{4}+e^{2}1q^{5}+⋯ ∈ S_{6}(Γ_{0}(53); \overline{F}_{3}). Also, z satisfies the equation x^{3}+2x+ = 0. We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + e^{4}q^{2} + e^{9}q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(e^{1}7)ω^{0} & 0 \\ 0 & u(e^{9})ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + (5331601 ∗ 3 + O(3^{1}7)) ∗ a^{2} + (8579489 ∗ 3 + O(3^{1}7)) ∗ a − 5093699 ∗ 3^{2} + O(3^{1}7)q^{3} + a^{2} − 32q^{4} + −(22609735 + O(3^{1}8)) ∗ a^{2} − (1849567 ∗ 3^{4} + O(3^{1}8)) ∗ a + 12349451 ∗ 3 + O(3^{1}8)q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + −1111115428 + O(3^{2}0)x^{2} + −32142394 ∗ 3^{3} + O(3^{2}0)x + 1484898871 + O(3^{2}0). The slope of f_{0} is 1.

2. Consider f = q+2q^{2}+2q^{4}+⋯ ∈ S_{6}(Γ_{0}(53); \overline{F}_{3}). We have ρ_{f} ⊗χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q+2q^{2}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{1}. The form f_{0} = q + aq^{2} + (671653523 ∗ 3^{−}3 + O(3^{1}7)) ∗ a + 626262563 ∗ 3^{−}3 + O(3^{1}7)q^{3} + −(394255091 + O(3^{2}0))∗a+446556856∗3+O(3^{2}0)q^{4}+(1060874695∗3^{−}2+O(3^{1}8))∗a−321686045∗3^{−}2+O(3^{1}8)q^{5}+⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 394255091 + O(3^{2}0)x + −1339670600 + O(3^{2}0). The slope of f_{0} is 1/2.

3. Consider f = q + q^{2} + 2q^{4} + ⋯ ∈ S_{6}(Γ_{0}(53); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + 2q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{5}. The form f_{0} = q + 701788618 + O(3^{2}0)q^{2} + 12813718 ∗ 3^{2} + O(3^{1}8)q^{3} + −198717412 + O(3^{2}0)q^{4} + −12944290 ∗ 3^{2} + O(3^{1}8)q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 2.

4. Consider f = q +e^{4}q^{2} +e^{1}5q^{4} +e^{8}q^{5} +⋯ ∈ S_{6}(Γ_{0}(53); \overline{F}_{3}). Also, z satisfies the equation x^{3} +2x+ = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + e^{4}q^{2} + e^{9}q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(e^{1}7)ω^{1} & 0 \\ 0 & u(e^{9})ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + −(18288626 ∗ 3^{2} + O(3^{1}8)) ∗ a^{2} − (56043461 ∗ 3 + O(3^{1}8)) ∗ a + 284704 ∗ 3^{5} + O(3^{1}8)q^{3} + a^{2} − 32q^{4} + (192596398 + O(3^{1}8)) ∗ a^{2} + (18249620 ∗ 3^{2} + O(3^{1}8)) ∗ a + 30613450 ∗ 3 + O(3^{1}8)q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + −1067853935 + O(3^{2}0)x^{2} + 50270663 ∗ 3^{2} + O(3^{2}0)x + 433715999 + O(3^{2}0). The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

5. Consider f = q + q3 + q4 + ⋯ ∈ S6(Γ0(53); \overline{F}3). Since f is ordinary, we do not continue our analysis.

6. Consider f = q + eq2 + q3 + e21q4 + e6q5 + ⋯ ∈ S6(Γ0(53); \overline{F}3). Also, z satisfies the equation x3 + 2x+ = 0. Since f is ordinary, we do not continue our analysis.

7. Consider f = q + q2 + 2q3 + 2q4 + 2q5 + ⋯ ∈ S6(Γ0(53); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

8. Consider f = q + e8q2 + 2q3 + e22q4 + e7q5 + ⋯ ∈ S6(Γ0(53); \overline{F}3). Also, z satisfies the equation x3 + 2x+ = 0. Since f is ordinary, we do not continue our analysis.

9. Consider f = q + e4q2 + e9q3 + e15q4 + e8q5 + ⋯ ∈ S6(Γ0(53); \overline{F}3). Also, z satisfies the equation x3 + 2x+ = 0. Since f is ordinary, we do not continue our analysis.

3.36 N = 55, ℓ = 3 (12 forms)

1. Consider f = q + zq^{2} + z^{7}q^{4} + q^{5} + ⋯ ∈ S_{6}(Γ_{0}(55); \overline{F}_{3}). Also, z satisfies the equation x^{2} + 2x + 2 = 0. We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + z^{5}q^{2} + z^{6}q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2})ω^{0} & 0 \\ 0 & u(z^{6})ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +aq^{2} +−(149963012∗ 3+O(3^{1}9))∗a−86865929∗3+O(3^{1}9)q^{3} +(101575171+O(3^{2}0))∗a+1064982128+O(3^{2}0)q^{4} +25q^{5} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −101575171 + O(3^{2}0)x + −1064982160 + O(3^{2}0). The slope of f_{0} is 1.

2. Consider f = q +q^{4} +q^{5} +⋯ ∈ S_{6}(Γ_{0}(55); \overline{F}_{3}). We have ρ_{f} ⊗χ^{1} ∼ ρ_{g}, where g is the weight-4 newform g = q + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −389782084 ∗ 3 + O(3^{2}0)q^{2} + −98318002 ∗ 3 + O(3^{1}9)q^{3} + 1917043798 + O(3^{2}1)q^{4} + 25q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q + q^{2} + 2q^{4} + q^{5} + ⋯ ∈ S_{6}(Γ_{0}(55); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{1}. The form f_{0} = q+aq^{2}+−(913249939∗3^{−}1+O(3^{1}9))∗a+678332182∗3^{−}1+O(3^{1}9)q^{3}+ (1067771090+O(3^{2}0))∗a+520777885∗3+O(3^{2}0)q^{4}+25q^{5}+⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2}+−1067771090+O(3^{2}0)x+−1562333687+O(3^{2}0). The slope of f_{0} is 1/2.

4. Consider f = q + q^{4} + 2q^{5} + ⋯ ∈ S_{6}(Γ_{0}(55); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-4 newform g = q + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −172888007 ∗ 3 + O(3^{2}0)q^{2} + 423607919 ∗ 3 + O(3^{2}0)q^{3} + 981427216 + O(3^{2}1)q^{4} + −25q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

5. Consider f = q +z^{5}q^{2} +z^{7}q^{4} +2q^{5} +⋯ ∈ S_{6}(Γ_{0}(55); \overline{F}_{3}). Also, z satisfies the equation x^{2} +2x+2 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + z^{5}q^{2} + z^{6}q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2})ω^{1} & 0 \\ 0 & u(z^{6})ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +aq^{2} +(11697022∗3+ O(3^{1}9))∗a−178316809∗3+O(3^{1}9)q^{3} +(263260781+O(3^{2}0))∗a+163222646+O(3^{2}0)q^{4} +−25q^{5} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −263260781 + O(3^{2}0)x + −163222678 + O(3^{2}0). The slope of f_{0} is 1.

6. Consider f = q + 2q^{2} + 2q^{4} + 2q^{5} + ⋯ ∈ S_{6}(Γ_{0}(55); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{5}. The form f_{0} = q + −1452883384 + O(3^{2}0)q^{2} + 54271223 ∗ 3^{2} + O(3^{1}9)q^{3} + 655483784 + O(3^{2}0)q^{4} + −25q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 2. —————————————————————————————————- (Uninteresting cases below.)

7. Consider f = q + z18q2 + q3 + z6q4 + q5 + ⋯ ∈ S6(Γ0(55); \overline{F}3). Also, z satisfies the equation x3 + 2x+ = 0. Since f is ordinary, we do not continue our analysis.

8. Consider f = q + z2q2 + q3 + z18q4 + q5 + ⋯ ∈ S6(Γ0(55); \overline{F}3). Also, z satisfies the equation x3 + 2x+ = 0. Since f is ordinary, we do not continue our analysis.

9. Consider f = q + z6q2 + q3 + z2q4 + q5 + ⋯ ∈ S6(Γ0(55); \overline{F}3). Also, z satisfies the equation x3 + 2x+ = 0. Since f is ordinary, we do not continue our analysis.

10. Consider f = q + q3 + q4 + 2q5 + ⋯ ∈ S6(Γ0(55); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

11. Consider f = q + zq2 + 2q3 + z7q4 + 2q5 + ⋯ ∈ S6(Γ0(55); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

12. Consider f = q + z5q2 + z6q3 + z7q4 + 2q5 + ⋯ ∈ S6(Γ0(55); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

1. Consider f = q + q^{5} + ⋯ ∈ S_{6}(Γ_{0}(56); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + 2q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{0} & 0 \\ 0 & u(2)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −2 ∗ 3q^{3} + 4q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 2q^{5} + ⋯ ∈ S_{6}(Γ_{0}(56); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + 2q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(2)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 10 ∗ 3q^{3} + 32q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q + 2q^{5} + ⋯ ∈ S_{6}(Γ_{0}(56); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{1}. The form f_{0} = q + aq^{3} + 3 ∗ a + 50q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 2 ∗ 3x + −112 ∗ 3. The slope of f_{0} is 1/2.

4. Consider f = q + q^{5} + ⋯ ∈ S_{6}(Γ_{0}(56); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{5}. The form f_{0} = q + 33868382 ∗ 3^{2} + O(3^{2}0)q^{3} + −1524077204 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 2. —————————————————————————————————- (Uninteresting cases below.)

5. Consider f = q + q3 + 2q5 + ⋯ ∈ S6(Γ0(56); \overline{F}3). Since f is ordinary, we do not continue our analysis.

6. Consider f = q + 2q3 + 2q5 + ⋯ ∈ S6(Γ0(56); \overline{F}3). Since f is ordinary, we do not continue our analysis.

3.38 N = 58, ℓ = 3 (11 forms)

1. Consider f = q +q^{2} +q^{4} +⋯ ∈ S_{6}(Γ_{0}(58); \overline{F}_{3}). We have ρ_{f} ⊗χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + 2q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{5}. The form f_{0} = q + 4q^{2} + −53470127 ∗ 3^{3} + O(3^{2}0)q^{3} + 16q^{4} + 163265525 ∗ 3 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 3.

2. Consider f = q + 2q^{2} + q^{4} + ⋯ ∈ S_{6}(Γ_{0}(58); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-4 newform g = q + 2q^{2} + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−4q^{2} +−347715772∗ 3 + O(3^{2}0)q^{3} + 16q^{4} + −95531146 ∗ 3 + O(3^{1}9)q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q + 2q^{2} + q^{4} + 2q^{5} + ⋯ ∈ S_{6}(Γ_{0}(58); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + 2q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{0} & 0 \\ 0 & u(2)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−4q^{2} +−445443440∗ 3 + O(3^{2}0)q^{3} + 16q^{4} + 558702869 + O(3^{1}9)q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

4. Consider f = q +q^{2} +q^{4} +⋯ ∈ S_{6}(Γ_{0}(58); \overline{F}_{3}). We have ρ_{f} ⊗χ^{1} ∼ ρ_{g}, where g is the weight-4 newform g = q + 2q^{2} + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 4q^{2} + 325654298 ∗ 3 + O(3^{2}0)q^{3} + 16q^{4} + −49606112 ∗ 3 + O(3^{1}9)q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

5. Consider f = q + q^{2} + q^{4} + q^{5} + ⋯ ∈ S_{6}(Γ_{0}(58); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + 2q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(2)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 4q^{2} + 252497056 ∗ 3 + O(3^{2}0)q^{3} + 16q^{4} + 203645182 + O(3^{1}9)q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

6. Consider f = q+2q^{2}+q^{4}+⋯ ∈ S_{6}(Γ_{0}(58); \overline{F}_{3}). We have ρ_{f} ⊗χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + 2q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{1}. The form f_{0} = q +−4q^{2} +aq^{3} +16q^{4} +(185620919∗3+O(3^{1}9))∗a+155168791∗3+O(3^{1}9)q^{5} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −3085555 ∗ 3^{5} + O(3^{2}0)x + 577826741 ∗ 3 + O(3^{2}0). The slope of f_{0} is 1/2. —————————————————————————————————- (Uninteresting cases below.)

7. Consider f = q + q2 + q3 + q4 + 2q5 + ⋯ ∈ S6(Γ0(58); \overline{F}3). Since f is ordinary, we do not continue our analysis.

8. Consider f = q + 2q2 + 2q3 + q4 + ⋯ ∈ S6(Γ0(58); \overline{F}3). Since f is ordinary, we do not continue our analysis.

9. Consider f = q + q2 + 2q3 + q4 + q5 + ⋯ ∈ S6(Γ0(58); \overline{F}3). Since f is ordinary, we do not continue our analysis.

10. Consider f = q + 2q2 + q3 + q4 + ⋯ ∈ S6(Γ0(58); \overline{F}3). Since f is ordinary, we do not continue our analysis.

11. Consider f = q + 2q2 + q3 + q4 + 2q5 + ⋯ ∈ S6(Γ0(58); \overline{F}3). Since f is ordinary, we do not continue our analysis.

3.39 N = 59, ℓ = 3 (13 forms)

1. Consider f = q + z^{4}6848q^{2} + z^{5}5876q^{4} + z^{3}904q^{5} + ⋯ ∈ S_{6}(Γ_{0}(59); \overline{F}_{3}). Also, z satisfies the equation x^{10} + 2x^{6} + 2x^{5} + 2x^{4} + x + 2 = 0. We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + c^{7}1q^{2} + c^{4}8q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(c^{1}94)ω^{0} & 0 \\ 0 & u(c^{4}8)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +aq^{2} +−(514302740∗ 3 + O(3^{2}0)) ∗ a^{4} − (47575549 ∗ 3^{3} + O(3^{2}0)) ∗ a^{3} + (43334773 ∗ 3 + O(3^{2}0)) ∗ a^{2} + (576300484 ∗ 3 + O(3^{2}0)) ∗ a + 125419504 ∗ 3^{2} + O(3^{2}0)q^{3} + a^{2} − 32q^{4} + (168545018 ∗ 3^{2} + O(3^{2}0)) ∗ a^{4} + (75663416 + O(3^{2}0)) ∗ a^{3} + (47506472 + O(3^{2}0)) ∗ a^{2} − (344288051 ∗ 3 + O(3^{2}0)) ∗ a − 926330182 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{5} + −229419539 ∗ 3 + O(3^{2}0)x^{4} + 580411007 ∗ 3 + O(3^{2}0)x^{3} + 612744433 + O(3^{2}0)x^{2} + −726172448 + O(3^{2}0)x + −622416646 + O(3^{2}0). The slope of f_{0} is 1.

2. Consider f = q + z^{8}296q^{2} + z^{3}0500q^{4} + z^{3}5136q^{5} + ⋯ ∈ S_{6}(Γ_{0}(59); \overline{F}_{3}). Also, z satisfies the equation x^{10} + 2x^{6} + 2x^{5} + 2x^{4} + x + 2 = 0. We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + c^{1}55q^{2} + c^{1}90q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(c^{5}2)ω^{0} & 0 \\ 0 & u(c^{1}90)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +aq^{2} +−(514302740∗ 3 + O(3^{2}0)) ∗ a^{4} − (47575549 ∗ 3^{3} + O(3^{2}0)) ∗ a^{3} + (43334773 ∗ 3 + O(3^{2}0)) ∗ a^{2} + (576300484 ∗ 3 + O(3^{2}0)) ∗ a + 125419504 ∗ 3^{2} + O(3^{2}0)q^{3} + a^{2} − 32q^{4} + (168545018 ∗ 3^{2} + O(3^{2}0)) ∗ a^{4} + (75663416 + O(3^{2}0)) ∗ a^{3} + (47506472 + O(3^{2}0)) ∗ a^{2} − (344288051 ∗ 3 + O(3^{2}0)) ∗ a − 926330182 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{5} + −229419539 ∗ 3 + O(3^{2}0)x^{4} + 580411007 ∗ 3 + O(3^{2}0)x^{3} + 612744433 + O(3^{2}0)x^{2} + −726172448 + O(3^{2}0)x + −622416646 + O(3^{2}0). The slope of f_{0} is 1.

3. Consider f = q + z^{1}5616q^{2} + z^{3}8308q^{4} + z^{2}0984q^{5} + ⋯ ∈ S_{6}(Γ_{0}(59); \overline{F}_{3}). Also, z satisfies the equation x^{10} + 2x^{6} + 2x^{5} + 2x^{4} + x + 2 = 0. We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + c^{1}85q^{2} + c^{1}6q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(c^{2}26)ω^{0} & 0 \\ 0 & u(c^{1}6)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +aq^{2} +−(514302740∗ 3 + O(3^{2}0)) ∗ a^{4} − (47575549 ∗ 3^{3} + O(3^{2}0)) ∗ a^{3} + (43334773 ∗ 3 + O(3^{2}0)) ∗ a^{2} + (576300484 ∗ 3 + O(3^{2}0)) ∗ a + 125419504 ∗ 3^{2} + O(3^{2}0)q^{3} + a^{2} − 32q^{4} + (168545018 ∗ 3^{2} + O(3^{2}0)) ∗ a^{4} + (75663416 + O(3^{2}0)) ∗ a^{3} + (47506472 + O(3^{2}0)) ∗ a^{2} − (344288051 ∗ 3 + O(3^{2}0)) ∗ a − 926330182 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{5} + −229419539 ∗ 3 + O(3^{2}0)x^{4} + 580411007 ∗ 3 + O(3^{2}0)x^{3} + 612744433 + O(3^{2}0)x^{2} + −726172448 + O(3^{2}0)x + −622416646 + O(3^{2}0). The slope of f_{0} is 1.

4. Consider f = q + z^{2}2448q^{2} + z^{4}9532q^{4} + z^{1}1712q^{5} + ⋯ ∈ S_{6}(Γ_{0}(59); \overline{F}_{3}). Also, z satisfies the equation x^{10} + 2x^{6} + 2x^{5} + 2x^{4} + x + 2 = 0. We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + c^{2}13q^{2} + c^{1}44q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(c^{9}8)ω^{0} & 0 \\ 0 & u(c^{1}44)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +aq^{2} +−(514302740∗ 3 + O(3^{2}0)) ∗ a^{4} − (47575549 ∗ 3^{3} + O(3^{2}0)) ∗ a^{3} + (43334773 ∗ 3 + O(3^{2}0)) ∗ a^{2} + (576300484 ∗ 3 + O(3^{2}0)) ∗ a + 125419504 ∗ 3^{2} + O(3^{2}0)q^{3} + a^{2} − 32q^{4} + (168545018 ∗ 3^{2} + O(3^{2}0)) ∗ a^{4} + (75663416 + O(3^{2}0)) ∗ a^{3} + (47506472 + O(3^{2}0)) ∗ a^{2} − (344288051 ∗ 3 + O(3^{2}0)) ∗ a − 926330182 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{5} + −229419539 ∗ 3 + O(3^{2}0)x^{4} + 580411007 ∗ 3 + O(3^{2}0)x^{3} + 612744433 + O(3^{2}0)x^{2} + −726172448 + O(3^{2}0)x + −622416646 + O(3^{2}0). The slope of f_{0} is 1.

5. Consider f = q + z^{2}4888q^{2} + z^{3}2452q^{4} + z^{4}6360q^{5} + ⋯ ∈ S_{6}(Γ_{0}(59); \overline{F}_{3}). Also, z satisfies the equation x^{10} + 2x^{6} + 2x^{5} + 2x^{4} + x + 2 = 0. We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + c^{2}23q^{2} + c^{8}6q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(c^{1}56)ω^{0} & 0 \\ 0 & u(c^{8}6)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +aq^{2} +−(514302740∗ 3 + O(3^{2}0)) ∗ a^{4} − (47575549 ∗ 3^{3} + O(3^{2}0)) ∗ a^{3} + (43334773 ∗ 3 + O(3^{2}0)) ∗ a^{2} + (576300484 ∗ 3 + O(3^{2}0)) ∗ a + 125419504 ∗ 3^{2} + O(3^{2}0)q^{3} + a^{2} − 32q^{4} + (168545018 ∗ 3^{2} + O(3^{2}0)) ∗ a^{4} + (75663416 + O(3^{2}0)) ∗ a^{3} + (47506472 + O(3^{2}0)) ∗ a^{2} − (344288051 ∗ 3 + O(3^{2}0)) ∗ a − 926330182 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{5} + −229419539 ∗ 3 + O(3^{2}0)x^{4} + 580411007 ∗ 3 + O(3^{2}0)x^{3} + 612744433 + O(3^{2}0)x^{2} + −726172448 + O(3^{2}0)x + −622416646 + O(3^{2}0). The slope of f_{0} is 1.

6. Consider f = q+c^{7}1q^{2}+c^{2}29q^{4}+c^{1}37q^{5}+⋯ ∈ S_{6}(Γ_{0}(59); \overline{F}_{3}). Also, z satisfies the equation x^{5}+2x+ = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + c^{7}1q^{2} + c^{4}8q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(c^{1}94)ω^{1} & 0 \\ 0 & u(c^{4}8)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + (166918577 ∗ 3+O(3^{1}9))∗a^{4} −(107477639∗3+O(3^{1}9))∗a^{3} −(27977363∗3^{2} +O(3^{1}9))∗a^{2} +(111122560∗3+O(3^{1}9))∗ a + 175854971 ∗ 3 + O(3^{1}9)q^{3} + a^{2} − 32q^{4} + −(13334003 ∗ 3^{2} + O(3^{2}0)) ∗ a^{4} + (506287667 + O(3^{2}0)) ∗ a^{3} + (226975510+O(3^{2}0))∗a^{2}−(237537488∗3+O(3^{2}0))∗a−789117095+O(3^{2}0)q^{5}+⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{5}+324487241∗3+O(3^{2}0)x^{4}+ 116829959 ∗ 3 + O(3^{2}0)x^{3} + 363803114 + O(3^{2}0)x^{2} + −683914757 + O(3^{2}0)x + 744462514 + O(3^{2}0). The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

7. Consider f = q + z36905q2 + 2q3 + z51667q4 + z44286q5 + ⋯ ∈ S6(Γ0(59); \overline{F}3). Also, z satisfies the equation x10 + 2x6 + 2x5 + 2x4 + x + 2 = 0. Since f is ordinary, we do not continue our analysis.

8. Consider f = q + z51667q2 + 2q3 + z36905q4 + z14762q5 + ⋯ ∈ S6(Γ0(59); \overline{F}3). Also, z satisfies the equation x10 + 2x6 + 2x5 + 2x4 + x + 2 = 0. Since f is ordinary, we do not continue our analysis.

9. Consider f = q + z7381q2 + q3 + z51667q4 + q5 + ⋯ ∈ S6(Γ0(59); \overline{F}3). Also, z satisfies the equation x10 + 2x6 + 2x5 + 2x4 + x + 2 = 0. Since f is ordinary, we do not continue our analysis.

10. Consider f = q + z22143q2 + q3 + z36905q4 + q5 + ⋯ ∈ S6(Γ0(59); \overline{F}3). Also, z satisfies the equation x10 + 2x6 + 2x5 + 2x4 + x + 2 = 0. Since f is ordinary, we do not continue our analysis.

11. Consider f = q + a2q2 + q3 + ⋯ ∈ S6(Γ0(59); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

12. Consider f = q + c71q2 + c48q3 + c229q4 + c137q5 + ⋯ ∈ S6(Γ0(59); \overline{F}3). Also, z satisfies the equation x5 + 2x+ = 0. Since f is ordinary, we do not continue our analysis.

13. Consider f = q + d14q2 + 2q3 + d21q4 + d19q5 + ⋯ ∈ S6(Γ0(59); \overline{F}3). Also, z satisfies the equation x3 + 2x+ = 0. Since f is ordinary, we do not continue our analysis.

3.40 N = 61, ℓ = 3 (11 forms)

1. Consider f = q + q^{2} + 2q^{4} + ⋯ ∈ S_{6}(Γ_{0}(61); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + 2q^{2} + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 1599162835 + O(3^{2}0)q^{2} + 90647756 ∗ 3 + O(3^{1}9)q^{3} + −1407332329 + O(3^{2}0)q^{4} + −41465042 ∗ 3^{3} + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 2q^{2} + 2q^{4} + ⋯ ∈ S_{6}(Γ_{0}(61); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + 2q^{2} + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −808135042 + O(3^{2}0)q^{2} + 163128800 ∗ 3 + O(3^{1}9)q^{3} + −166480027 + O(3^{2}0)q^{4} + −37734250 ∗ 3^{2} + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q+d^{1}7q^{2}+d^{1}5q^{4}+d^{1}0q^{5}+⋯ ∈ S_{6}(Γ_{0}(61); \overline{F}_{3}). Also, z satisfies the equation x^{3}+2x+ = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + d^{1}7q^{2} + d^{2}1q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(d^{5})ω^{1} & 0 \\ 0 & u(d^{2}1)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + −(2194184 ∗ 3^{4} + O(3^{1}8)) ∗ a^{2} + (35077376 ∗ 3 + O(3^{1}8)) ∗ a − 28512515 ∗ 3 + O(3^{1}8)q^{3} + a^{2} − 32q^{4} + −(72789638 + O(3^{1}8)) ∗ a^{2} − (110602664 + O(3^{1}8)) ∗ a + 151534504 + O(3^{1}8)q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + −779482108 + O(3^{2}0)x^{2} + 12823322 ∗ 3^{4} + O(3^{2}0)x + 1719292888 + O(3^{2}0). The slope of f_{0} is 1.

4. Consider f = q+d^{4}q^{2}+d^{1}5q^{4}+d^{2}3q^{5}+⋯ ∈ S_{6}(Γ_{0}(61); \overline{F}_{3}). Also, z satisfies the equation x^{3}+2x+ = 0. We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + d^{1}7q^{2} + d^{2}1q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(d^{5})ω^{0} & 0 \\ 0 & u(d^{2}1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + −(50470253 ∗ 3 + O(3^{1}8)) ∗ a^{2} + (7236025 ∗ 3^{2} + O(3^{1}8)) ∗ a + 8308406 ∗ 3^{2} + O(3^{1}8)q^{3} + a^{2} − 32q^{4} + −(114074392 + O(3^{1}8)) ∗ a^{2} − (124065863 + O(3^{1}8)) ∗ a − 129268900 + O(3^{1}8)q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3}+−1183229405+O(3^{2}0)x^{2}+−65561014∗ 3^{2} + O(3^{2}0)x + −1053207253 + O(3^{2}0). The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

5. Consider f = q + cq2 + 2q3 + c7q4 + 2q5 + ⋯ ∈ S6(Γ0(61); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

6. Consider f = q + c2q2 + q3 + c7q5 + ⋯ ∈ S6(Γ0(61); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

7. Consider f = q + d5q2 + q3 + d6q4 + d16q5 + ⋯ ∈ S6(Γ0(61); \overline{F}3). Also, z satisfies the equation x3 + 2x+ = 0. Since f is ordinary, we do not continue our analysis.

8. Consider f = q + 2q2 + q3 + 2q4 + ⋯ ∈ S6(Γ0(61); \overline{F}3). Since f is ordinary, we do not continue our analysis.

9. Consider f = q + d17q2 + 2q3 + d15q4 + 2q5 + ⋯ ∈ S6(Γ0(61); \overline{F}3). Also, z satisfies the equation x3 + 2x+ = 0. Since f is ordinary, we do not continue our analysis.

10. Consider f = q + d17q2 + d21q3 + d15q4 + d10q5 + ⋯ ∈ S6(Γ0(61); \overline{F}3). Also, z satisfies the equation x3 + 2x+ = 0. Since f is ordinary, we do not continue our analysis.

11. Consider f = q + q4 + ⋯ ∈ S6(Γ0(61); \overline{F}3). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

3.41 N = 62, ℓ = 3 (9 forms)

1. Consider f = q + 2q^{2} + q^{4} + 2q^{5} + ⋯ ∈ S_{6}(Γ_{0}(62); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{5}. The form f_{0} = q+−4q^{2}+−78220336∗3^{2}+O(3^{2}0)q^{3}+16q^{4}+1407966041+O(3^{2}0)q^{5}+⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 2.

2. Consider f = q +q^{2} +q^{4} +⋯ ∈ S_{6}(Γ_{0}(62); \overline{F}_{3}). We have ρ_{f} ⊗χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + 2q^{2} + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 4q^{2} + 325496924 ∗ 3 + O(3^{2}0)q^{3} + 16q^{4} + 511267598 ∗ 3 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q + 2q^{2} + q^{4} + ⋯ ∈ S_{6}(Γ_{0}(62); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + 2q^{2} + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−4q^{2} +−510084745∗ 3 + O(3^{2}0)q^{3} + 16q^{4} + −152279534 ∗ 3 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

4. Consider f = q + q^{2} + q^{4} + q^{5} + ⋯ ∈ S_{6}(Γ_{0}(62); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{1}. The form f_{0} = q + 4q^{2} + aq^{3} + 16q^{4} + (157788533 ∗ 3 + O(3^{1}9)) ∗ a − 417681821 + O(3^{1}9)q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −207667840 ∗ 3 + O(3^{2}0)x + 502161137 ∗ 3 + O(3^{2}0). The slope of f_{0} is 1/2. —————————————————————————————————- (Uninteresting cases below.)

5. Consider f = q + 2q2 + q3 + q4 + ⋯ ∈ S6(Γ0(62); \overline{F}3). Since f is ordinary, we do not continue our analysis.

6. Consider f = q + q2 + q3 + q4 + q5 + ⋯ ∈ S6(Γ0(62); \overline{F}3). Since f is ordinary, we do not continue our analysis.

7. Consider f = q + 2q2 + q3 + q4 + ⋯ ∈ S6(Γ0(62); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

8. Consider f = q + 2q2 + 2q3 + q4 + 2q5 + ⋯ ∈ S6(Γ0(62); \overline{F}3). Since f is ordinary, we do not continue our analysis.

9. Consider f = q + q2 + 2q3 + q4 + z6q5 + ⋯ ∈ S6(Γ0(62); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

3.42 N = 64, ℓ = 3 (8 forms)

1. Consider f = q + q^{5} + ⋯ ∈ S_{6}(Γ_{0}(64); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{5}. The form f_{0} = q + 82q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is Infinity.

2. Consider f = q + 2q^{5} + ⋯ ∈ S_{6}(Γ_{0}(64); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{1}. The form f_{0} = q + aq^{3} + −46q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −256 ∗ 3. The slope of f_{0} is 1/2. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + 2q3 + q5 + ⋯ ∈ S6(Γ0(64); \overline{F}3). Since f is ordinary, we do not continue our analysis.

4. Consider f = q + q3 + q5 + ⋯ ∈ S6(Γ0(64); \overline{F}3). Since f is ordinary, we do not continue our analysis.

5. Consider f = q + ⋯ ∈ S6(Γ0(64); \overline{F}3). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

6. Consider f = q + ⋯ ∈ S6(Γ0(64); \overline{F}3). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

7. Consider f = q + 2q3 + 2q5 + ⋯ ∈ S6(Γ0(64); \overline{F}3). Since f is ordinary, we do not continue our analysis.

8. Consider f = q + q3 + 2q5 + ⋯ ∈ S6(Γ0(64); \overline{F}3). Since f is ordinary, we do not continue our analysis.

3.43 N = 65, ℓ = 3 (12 forms)

1. Consider f = q + 2q^{2} + 2q^{4} + 2q^{5} + ⋯ ∈ S_{6}(Γ_{0}(65); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + 2q^{2} + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 5q^{2} + 2 ∗ 3q^{3} + −7q^{4} + −25q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q +q^{4} +q^{5} +⋯ ∈ S_{6}(Γ_{0}(65); \overline{F}_{3}). We have ρ_{f} ⊗χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 13758946 ∗ 3^{2} + O(3^{2}0)q^{2} + −144843037 ∗ 3 + O(3^{2}0)q^{3} + −13388777987 + O(3^{2}2)q^{4} + 25q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q +a^{5}q^{2} +a^{7}q^{4} +2q^{5} +⋯ ∈ S_{6}(Γ_{0}(65); \overline{F}_{3}). Also, z satisfies the equation x^{2} +2x+2 = 0. We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + aq^{2} + a^{2}q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(a^{6})ω^{0} & 0 \\ 0 & u(a^{2})ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(166345072∗3+ O(3^{2}0))∗a+375315836∗3+O(3^{2}0)q^{3} +(238194770+O(3^{2}0))∗a−738647491+O(3^{2}0)q^{4} +−25q^{5} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −238194770 + O(3^{2}0)x + 738647459 + O(3^{2}0). The slope of f_{0} is 1.

4. Consider f = q + q^{4} + 2q^{5} + ⋯ ∈ S_{6}(Γ_{0}(65); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 58971128 ∗ 3^{3} + O(3^{2}0)q^{2} + 510193673 ∗ 3 + O(3^{2}1)q^{3} + 39783429013 + O(3^{2}3)q^{4} + −25q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

5. Consider f = q + q^{2} + 2q^{4} + q^{5} + ⋯ ∈ S_{6}(Γ_{0}(65); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + 2q^{2} + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 177024910 + O(3^{2}0)q^{2} + −34429429 ∗ 3 + O(3^{1}9)q^{3} + −1235135131 + O(3^{2}0)q^{4} + 25q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

6. Consider f = q + aq^{2} + a^{7}q^{4} + q^{5} + ⋯ ∈ S_{6}(Γ_{0}(65); \overline{F}_{3}). Also, z satisfies the equation x^{2} + 2x + 2 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + aq^{2} + a^{2}q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(a^{6})ω^{1} & 0 \\ 0 & u(a^{2})ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + (180951337 ∗ 3+O(3^{1}9))∗a−23569679∗3+O(3^{1}9)q^{3} +(347870677+O(3^{2}0))∗a+32218088+O(3^{2}0)q^{4} +25q^{5} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −347870677 + O(3^{2}0)x + −32218120 + O(3^{2}0). The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

7. Consider f = q + 2q2 + q3 + 2q4 + q5 + ⋯ ∈ S6(Γ0(65); \overline{F}3). Since f is ordinary, we do not continue our analysis.

8. Consider f = q + 2q2 + q3 + 2q4 + 2q5 + ⋯ ∈ S6(Γ0(65); \overline{F}3). Since f is ordinary, we do not continue our analysis.

9. Consider f = q + q3 + q4 + 2q5 + ⋯ ∈ S6(Γ0(65); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

10. Consider f = q + cq2 + 2q3 + c21q4 + 2q5 + ⋯ ∈ S6(Γ0(65); \overline{F}3). Also, z satisfies the equation x3 + 2x+ = 0. Since f is ordinary, we do not continue our analysis.

11. Consider f = q + aq2 + a2q3 + a7q4 + q5 + ⋯ ∈ S6(Γ0(65); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

12. Consider f = q + 2q2 + 2q3 + 2q4 + q5 + ⋯ ∈ S6(Γ0(65); \overline{F}3). Since f is ordinary, we do not continue our analysis.

1. Consider f = q + b^{2}q^{2} + ⋯ ∈ S_{6}(Γ_{0}(67); \overline{F}_{3}). Also, z satisfies the equation x^{2} + 2x + 2 = 0. We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + b^{6}q^{2} + b^{2}q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(b^{6})ω^{0} & 0 \\ 0 & u(b^{2})ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +aq^{2} +(90220453∗3+ O(3^{1}9))∗a^{3} −(25678615∗3^{2} +O(3^{1}9))∗a^{2} −(21488869∗3^{2} +O(3^{1}9))∗a+6176140∗3^{3} +O(3^{1}9)q^{3} +a^{2} − 32q^{4}+−(801566359+O(3^{2}0))∗a^{3}−(96457579∗3^{2}+O(3^{2}0))∗a^{2}−(58037176+O(3^{2}0))∗a+20846537∗3^{2}+ O(3^{2}0)q^{5}+⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{4}+5459413∗3^{2}+O(3^{2}0)x^{3}+−1355760613+O(3^{2}0)x^{2}+6379591∗3^{2}+O(3^{2}0)x+−837687920+O(3^{2}0). The slope of f_{0} is 1.

2. Consider f = q + bq^{2} + b^{7}q^{4} + b^{7}q^{5} + ⋯ ∈ S_{6}(Γ_{0}(67); \overline{F}_{3}). Also, z satisfies the equation x^{2} + 2x + 2 = 0. We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + b^{5}q^{2} + b^{3}q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(b^{5})ω^{0} & 0 \\ 0 & u(b^{3})ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +aq^{2} +−(175848968∗ 3 + O(3^{1}9)) ∗ a + 167631758 ∗ 3 + O(3^{1}9)q^{3} + (84575302 + O(3^{2}0)) ∗ a + 1493788952 + O(3^{2}0)q^{4} + (1068538375 + O(3^{2}0)) ∗ a − 8516467 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} +−84575302+O(3^{2}0)x+−1493788984+O(3^{2}0). The slope of f_{0} is 1.

3. Consider f = q +b^{5}q^{2} +b^{7}q^{4} +b^{3}q^{5} +⋯ ∈ S_{6}(Γ_{0}(67); \overline{F}_{3}). Also, z satisfies the equation x^{2} +2x+2 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + b^{5}q^{2} + b^{3}q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(b^{5})ω^{1} & 0 \\ 0 & u(b^{3})ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +aq^{2} +−(122064892∗ 3 + O(3^{1}9)) ∗ a + 62471720 ∗ 3^{2} + O(3^{1}9)q^{3} + −(261898081 + O(3^{2}0)) ∗ a − 412676854 + O(3^{2}0)q^{4} + (1392873787 + O(3^{2}0)) ∗ a − 278585045 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 261898081 + O(3^{2}0)x + 412676822 + O(3^{2}0). The slope of f_{0} is 1.

4. Consider f = q + q^{2} + 2q^{4} + q^{5} + ⋯ ∈ S_{6}(Γ_{0}(67); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + 2q^{2} + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −191169581 + O(3^{2}0)q^{2} + 6715778 ∗ 3 + O(3^{1}9)q^{3} + −1488284107 + O(3^{2}0)q^{4} + −1075677026 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

5. Consider f = q + 2q^{2} + 2q^{4} + 2q^{5} + ⋯ ∈ S_{6}(Γ_{0}(67); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + 2q^{2} + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 1337552924 + O(3^{2}0)q^{2} + −119441722 ∗ 3 + O(3^{1}9)q^{3} + 298773839 + O(3^{2}0)q^{4} + −50094526 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

6. Consider f = q + b2q2 + b6q3 + ⋯ ∈ S6(Γ0(67); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

7. Consider f = q + c2q2 + q3 + c189q4 + c223q5 + ⋯ ∈ S6(Γ0(67); \overline{F}3). Also, z satisfies the equation x5 + 2x+ = 0. Since f is ordinary, we do not continue our analysis.

8. Consider f = q + q2 + 2q3 + 2q4 + ⋯ ∈ S6(Γ0(67); \overline{F}3). Since f is ordinary, we do not continue our analysis.

9. Consider f = q + b5q2 + b3q3 + b7q4 + b3q5 + ⋯ ∈ S6(Γ0(67); \overline{F}3). Also, z satisfies the equation x2 + 2x + 2 = 0. Since f is ordinary, we do not continue our analysis.

10. Consider f = q + c14q2 + 2q3 + c21q4 + c23q5 + ⋯ ∈ S6(Γ0(67); \overline{F}3). Also, z satisfies the equation x3 + 2x+ = 0. Since f is ordinary, we do not continue our analysis.

11. Consider f = q + 2q2 + q3 + 2q4 + 2q5 + ⋯ ∈ S6(Γ0(67); \overline{F}3). Since f is ordinary, we do not continue our analysis.

12. Consider f = q + q4 + ⋯ ∈ S6(Γ0(67); \overline{F}3). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

3.45 N = 68, ℓ = 3 (5 forms)

1. Consider f = q + ⋯ ∈ S_{6}(Γ_{0}(68); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 114227840 ∗ 3 + O(3^{2}0)q^{3} + −228455686 ∗ 3 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + ⋯ ∈ S_{6}(Γ_{0}(68); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 90734957 ∗ 3 + O(3^{2}0)q^{3} + −35592890 ∗ 3 + O(3^{2}0)q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + q3 + q5 + ⋯ ∈ S6(Γ0(68); \overline{F}3). Since f is ordinary, we do not continue our analysis.

4. Consider f = q + 2q3 + 2q5 + ⋯ ∈ S6(Γ0(68); \overline{F}3). Since f is ordinary, we do not continue our analysis.

5. Consider f = q + q3 + ⋯ ∈ S6(Γ0(68); \overline{F}3). Since f is ordinary, we do not continue our analysis.

3.46 N = 70, ℓ = 3 (9 forms)

1. Consider f = q + 2q^{2} + q^{4} + 2q^{5} + ⋯ ∈ S_{6}(Γ_{0}(70); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-4 newform g = q + q^{2} + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −4q^{2} + −3q^{3} + 16q^{4} + −25q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 2q^{2} + q^{4} + q^{5} + ⋯ ∈ S_{6}(Γ_{0}(70); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{5}. The form f_{0} = q + −4q^{2} + −3^{2}q^{3} + 16q^{4} + 25q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 2.

3. Consider f = q + q^{2} + q^{4} + 2q^{5} + ⋯ ∈ S_{6}(Γ_{0}(70); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-2 newform g = q + q^{2} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{1}. The form f_{0} = q + 4q^{2} + aq^{3} + 16q^{4} + −25q^{5} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −3x + −280 ∗ 3. The slope of f_{0} is 1/2.

4. Consider f = q + q^{2} + q^{4} + q^{5} + ⋯ ∈ S_{6}(Γ_{0}(70); \overline{F}_{3}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-4 newform g = q + q^{2} + q^{3} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 4q^{2} + 22429019 ∗ 3 + O(3^{2}0)q^{3} + 16q^{4} + 25q^{5} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

5. Consider f = q + 2q2 + 2q3 + q4 + 2q5 + ⋯ ∈ S6(Γ0(70); \overline{F}3). Since f is ordinary, we do not continue our analysis.

6. Consider f = q + q2 + q3 + q4 + 2q5 + ⋯ ∈ S6(Γ0(70); \overline{F}3). Since f is ordinary, we do not continue our analysis.

7. Consider f = q + q2 + q3 + q4 + q5 + ⋯ ∈ S6(Γ0(70); \overline{F}3). Since f is ordinary, we do not continue our analysis.

8. Consider f = q + 2q2 + q3 + q4 + q5 + ⋯ ∈ S6(Γ0(70); \overline{F}3). Since f is ordinary, we do not continue our analysis.

9. Consider f = q + q2 + 2q3 + q4 + q5 + ⋯ ∈ S6(Γ0(70); \overline{F}3). Since f is ordinary, we do not continue our analysis.

5 2 ≤ N ≤ 22, ℓ = 5

5.1 N = 2, ℓ = 5 (2 forms)

- —————————————————————————————————- (Uninteresting cases below.)

1. Consider f = q + 2q2 + 2q3 + 4q4 + 4q6 + q7 + 3q8 + 2q9 + ⋯ ∈ S8(Γ0(2); \overline{F}5). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

2. Consider f = q + q2 + 4q3 + q4 + 4q6 + 3q7 + q8 + 3q9 + ⋯ ∈ S10(Γ0(2); \overline{F}5). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

5.2 N = 3, ℓ = 5 (3 forms)

- —————————————————————————————————- (Uninteresting cases below.)

1. Consider f = q + q2 + 3q3 + 3q4 + 3q6 + q7 + 4q9 + ⋯ ∈ S8(Γ0(3); \overline{F}5). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

2. Consider f = q + 3q2 + q3 + 2q4 + 3q6 + 3q7 + q9 + ⋯ ∈ S10(Γ0(3); \overline{F}5). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

3. Consider f = q + 4q2 + 4q3 + 4q4 + q5 + q6 + 3q8 + q9 + ⋯ ∈ S10(Γ0(3); \overline{F}5). Since f is ordinary, we do not continue our analysis.

5.3 N = 4, ℓ = 5 (1 form)

- —————————————————————————————————- (Uninteresting cases below.)

1. Consider f = q + 3q3 + 4q5 + 2q7 + q9 + ⋯ ∈ S10(Γ0(4); \overline{F}5). Since f is ordinary, we do not continue our analysis.

5.4 N = 6, ℓ = 5 (2 forms)

- —————————————————————————————————- (Uninteresting cases below.)

1. Consider f = q + 3q2 + 2q3 + 4q4 + q5 + q6 + 4q7 + 2q8 + 4q9 + ⋯ ∈ S8(Γ0(6); \overline{F}5). Since f is ordinary, we do not continue our analysis.

2. Consider f = q + 4q2 + q3 + q4 + 4q5 + 4q6 + q7 + 4q8 + q9 + ⋯ ∈ S10(Γ0(6); \overline{F}5). Since f is ordinary, we do not continue our analysis.

1. Consider f = q + q^{2} + 2q^{3} + 3q^{4} + 2q^{6} + 2q^{7} + 2q^{9} + ⋯ ∈ S_{8}(Γ_{0}(7); \overline{F}_{5}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 4q^{2} + 3q^{3} + 3q^{4} + q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +aq^{2} +−2∗a+44q^{3} + −3∗a+86q^{4}+−2∗5∗a+6∗5^{2}q^{5}+2∗5^{2}∗a−428q^{6}+−343q^{7}+−33∗a−642q^{8}+−188∗a+121∗5q^{9}+⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} +3x+−214. The slope of f_{0} is 1.

2. Consider f = q + 3q^{2} + 4q^{3} + 2q^{4} + 2q^{6} + q^{7} + 3q^{9} + ⋯ ∈ S_{10}(Γ_{0}(7); \overline{F}_{5}). We have ρ_{f} ⊗ χ^{3} ∼ ρ_{g}, where g is the weight-4 newform g = q + 4q^{2} + 3q^{3} + 3q^{4} + q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(38028478599507+ O(5^{2}0))∗a−41676169300512+O(5^{2}0)q^{3} +(19902944725341+O(5^{2}0))∗a+5630890182319+O(5^{2}0)q^{4} + (3506612718082 ∗ 5 + O(5^{2}0)) ∗ a + 6082877787528 ∗ 5 + O(5^{2}0)q^{5} + (299488066526 ∗ 5^{3} + O(5^{2}0)) ∗ a + 36045279120567 + O(5^{2}0)q^{6} + 2401q^{7} + (36492112669963 + O(5^{2}0)) ∗ a + 22881262179746 + O(5^{2}0)q^{8} + (23216721506416 + O(5^{2}0)) ∗ a − 1197718328164 ∗ 5 + O(5^{2}0)q^{9} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −19902944725341 + O(5^{2}0)x + −5630890182831 + O(5^{2}0). The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + 4q2 + 3q3 + 3q4 + q5 + 2q6 + 3q7 + 2q9 + ⋯ ∈ S8(Γ0(7); \overline{F}5). Since f is ordinary, we do not continue our analysis.

4. Consider f = q + z16q2 + z8q3 + z21q4 + q5 + q6 + 4q7 + z20q8 + z19q9 + ⋯ ∈ S10(Γ0(7); \overline{F}5). Also, z satisfies the equation x2 + 4x + 2 = 0. Since f is ordinary, we do not continue our analysis.

5. Consider f = q + z8q2 + z16q3 + z9q4 + q5 + q6 + 4q7 + z4q8 + z23q9 + ⋯ ∈ S10(Γ0(7); \overline{F}5). Also, z satisfies the equation x2 + 4x + 2 = 0. Since f is ordinary, we do not continue our analysis.

6. Consider f = q + q3 + 3q4 + 4q5 + q7 + 3q9 + ⋯ ∈ S10(Γ0(7); \overline{F}5). Since f is ordinary, we do not continue our analysis.

5.6 N = 8, ℓ = 5 (4 forms)

1. Consider f = q + 4q^{3} + q^{7} + 4q^{9} + ⋯ ∈ S_{8}(Γ_{0}(8); \overline{F}_{5}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + q^{3} + 3q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(3)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 44q^{3} + 86 ∗ 5q^{5} + −1224q^{7} + −251q^{9} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 3q^{3} + 3q^{7} + q^{9} + ⋯ ∈ S_{10}(Γ_{0}(8); \overline{F}_{5}). We have ρ_{f} ⊗ χ^{3} ∼ ρ_{g}, where g is the weight-4 newform g = q + q^{3} + 3q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{0} & 0 \\ 0 & u(3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 68q^{3} + 302 ∗ 5q^{5} + 10248q^{7} + −15059q^{9} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + q3 + 3q5 + 4q7 + 4q9 + ⋯ ∈ S8(Γ0(8); \overline{F}5). Since f is ordinary, we do not continue our analysis.

4. Consider f = q + q5 + q7 + 2q9 + ⋯ ∈ S10(Γ0(8); \overline{F}5). Since f is ordinary, we do not continue our analysis.

1. Consider f = q + 2q^{4} + ⋯ ∈ S_{8}(Γ_{0}(9); \overline{F}_{5}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 2q^{4} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{15}. The form f_{0} = q + aq^{2} + 232q^{4} + −16 ∗ aq^{5} + 52 ∗ 5q^{7} + 104 ∗ aq^{8} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −72 ∗ 5. The slope of f_{0} is 1/2.

2. Consider f = q + 3q^{4} + ⋯ ∈ S_{10}(Γ_{0}(9); \overline{F}_{5}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-4 newform g = q + 2q^{4} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{21}. The form f_{0} = q + −512q^{4} + −2516 ∗ 5q^{7} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is Infinity. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + 4q2 + 3q4 + q7 + ⋯ ∈ S8(Γ0(9); \overline{F}5). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

4. Consider f = q + 2q2 + 2q4 + 3q7 + ⋯ ∈ S10(Γ0(9); \overline{F}5). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

5. Consider f = q + q2 + 4q4 + 4q5 + 2q8 + ⋯ ∈ S10(Γ0(9); \overline{F}5). Since f is ordinary, we do not continue our analysis.

5.8 N = 11, ℓ = 5 (11 forms)

1. Consider f = q + q^{2} + 2q^{3} + 3q^{4} + 2q^{6} + q^{7} + 2q^{9} + ⋯ ∈ S_{8}(Γ_{0}(11); \overline{F}_{5}). We have ρ_{f} ⊗ χ^{3} ∼ ρ_{g}, where g is the weight-2 newform g = q + 3q^{2} + 4q^{3} + 2q^{4} + q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{2} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +aq^{2} +−6∗a−27q^{3} + −8∗a−84q^{4}+4∗5∗a−31∗5q^{5}+21∗a−264q^{6}+82∗a−286q^{7}+−148∗a−352q^{8}+36∗a+126q^{9}+⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} +8x+−44. The slope of f_{0} is 1.

2. Consider f = q + z^{1}9q^{2} + z^{1}1q^{3} + z^{1}0q^{4} + 2q^{6} + z^{9}q^{7} + q^{8} + z^{1}7q^{9} + ⋯ ∈ S_{8}(Γ_{0}(11); \overline{F}_{5}). Also, z satisfies the equation x^{2} + 4x + 2 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + z^{7}q^{2} + z^{2}3q^{3} + z^{1}0q^{4} + z^{2}q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}2)ω^{1} & 0 \\ 0 & u(z^{2})ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(14723740761909+ O(5^{2}0))∗a−18457990057768+O(5^{2}0)q^{3}+−(3345838299947+O(5^{2}0))∗a−9712541111131+O(5^{2}0)q^{4}+ (8567596081731 ∗ 5 + O(5^{2}0)) ∗ a + 772294121228 ∗ 5 + O(5^{2}0)q^{5} + −(28780860585289 ∗ 5 + O(5^{2}1)) ∗ a − 55798651659023 + O(5^{2}1)q^{6} + (21074961207461 + O(5^{2}0)) ∗ a − 46141526117749 + O(5^{2}0)q^{7} + (1381066229287 ∗ 5^{2} + O(5^{2}0)) ∗ a − 9284799530034 + O(5^{2}0)q^{8} + −(41877807585508 + O(5^{2}0)) ∗ a − 12433774789831 + O(5^{2}0)q^{9} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 3345838299947 + O(5^{2}0)x + 9712541111003 + O(5^{2}0). The slope of f_{0} is 1.

3. Consider f = q + z^{2}3q^{2} + z^{7}q^{3} + z^{2}q^{4} + 2q^{6} + z^{2}1q^{7} + q^{8} + z^{1}3q^{9} + ⋯ ∈ S_{8}(Γ_{0}(11); \overline{F}_{5}). Also, z satisfies the equation x^{2} + 4x + 2 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + z^{1}1q^{2} + z^{1}9q^{3} + z^{2}q^{4} + z^{1}0q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}4)ω^{1} & 0 \\ 0 & u(z^{1}0)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(14723740761909+ O(5^{2}0))∗a−18457990057768+O(5^{2}0)q^{3}+−(3345838299947+O(5^{2}0))∗a−9712541111131+O(5^{2}0)q^{4}+ (8567596081731 ∗ 5 + O(5^{2}0)) ∗ a + 772294121228 ∗ 5 + O(5^{2}0)q^{5} + −(28780860585289 ∗ 5 + O(5^{2}1)) ∗ a − 55798651659023 + O(5^{2}1)q^{6} + (21074961207461 + O(5^{2}0)) ∗ a − 46141526117749 + O(5^{2}0)q^{7} + (1381066229287 ∗ 5^{2} + O(5^{2}0)) ∗ a − 9284799530034 + O(5^{2}0)q^{8} + −(41877807585508 + O(5^{2}0)) ∗ a − 12433774789831 + O(5^{2}0)q^{9} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 3345838299947 + O(5^{2}0)x + 9712541111003 + O(5^{2}0). The slope of f_{0} is 1.

4. Consider f = q + z^{1}3q^{2} + z^{1}7q^{3} + z^{2}2q^{4} + 2q^{6} + z^{3}q^{7} + 2q^{8} + z^{5}q^{9} + ⋯ ∈ S_{10}(Γ_{0}(11); \overline{F}_{5}). Also, z satisfies the equation x^{2} + 4x + 2 = 0. We have ρ_{f} ⊗ χ^{3} ∼ ρ_{g}, where g is the weight-4 newform g = q + z^{7}q^{2} + z^{2}3q^{3} + z^{1}0q^{4} + z^{2}q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}2)ω^{0} & 0 \\ 0 & u(z^{2})ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(30954438160334+ O(5^{2}0))∗a+29996059870094+O(5^{2}0)q^{3}+(38466789639159+O(5^{2}0))∗a+46133815960931+O(5^{2}0)q^{4}+ −(500823431929 ∗ 5 + O(5^{2}0)) ∗ a − 1075850406153 ∗ 5 + O(5^{2}0)q^{5} + (1074049443443 ∗ 5^{2} + O(5^{2}0)) ∗ a − 15317408685538 + O(5^{2}0)q^{6} + −(13425618042939 + O(5^{2}0)) ∗ a + 7658704338523 + O(5^{2}0)q^{7} + 8 ∗ 5^{2} ∗ a−6688q^{8} +−(213904856443304+O(5^{2}1))∗a+87984304968211+O(5^{2}1)q^{9} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −38466789639159 + O(5^{2}0)x + −46133815961443 + O(5^{2}0). The slope of f_{0} is 1.

5. Consider f = q + z^{1}7q^{2} + z^{1}3q^{3} + z^{1}4q^{4} + 2q^{6} + z^{1}5q^{7} + 2q^{8} + zq^{9} + ⋯ ∈ S_{10}(Γ_{0}(11); \overline{F}_{5}). Also, z satisfies the equation x^{2} + 4x + 2 = 0. We have ρ_{f} ⊗ χ^{3} ∼ ρ_{g}, where g is the weight-4 newform g = q + z^{1}1q^{2} + z^{1}9q^{3} + z^{2}q^{4} + z^{1}0q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}4)ω^{0} & 0 \\ 0 & u(z^{1}0)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(30954438160334+ O(5^{2}0))∗a+29996059870094+O(5^{2}0)q^{3}+(38466789639159+O(5^{2}0))∗a+46133815960931+O(5^{2}0)q^{4}+ −(500823431929 ∗ 5 + O(5^{2}0)) ∗ a − 1075850406153 ∗ 5 + O(5^{2}0)q^{5} + (1074049443443 ∗ 5^{2} + O(5^{2}0)) ∗ a − 15317408685538 + O(5^{2}0)q^{6} + −(13425618042939 + O(5^{2}0)) ∗ a + 7658704338523 + O(5^{2}0)q^{7} + 8 ∗ 5^{2} ∗ a−6688q^{8} +−(213904856443304+O(5^{2}1))∗a+87984304968211+O(5^{2}1)q^{9} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −38466789639159 + O(5^{2}0)x + −46133815961443 + O(5^{2}0). The slope of f_{0} is 1.

6. Consider f = q +3q^{2} +4q^{3} +2q^{4} +2q^{6} +3q^{7} +3q^{9} +⋯ ∈ S_{10}(Γ_{0}(11); \overline{F}_{5}). We have ρ_{f} ⊗χ^{4} ∼ ρ_{g}, where g is the weight-2 newform g = q + 3q^{2} + 4q^{3} + 2q^{4} + q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(23313874539791+ O(5^{2}0)) ∗ a − 25574079328894 + O(5^{2}0)q^{3} + −(31889281285684 + O(5^{2}0)) ∗ a + 41588807916519 + O(5^{2}0)q^{4} + −(323394749909 ∗ 5 + O(5^{1}9)) ∗ a + 1895456261351 ∗ 5 + O(5^{1}9)q^{5} + −(38488229496563 + O(5^{2}0))∗a+29471527877396+O(5^{2}0)q^{6} +(32286121866192+O(5^{2}0))∗a+578218072002+O(5^{2}0)q^{7} + (44537475795113 + O(5^{2}0)) ∗ a − 45654127349829 + O(5^{2}0)q^{8} + (42143088724888 + O(5^{2}0)) ∗ a + 15292834551539 + O(5^{2}0)q^{9} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 31889281285684 + O(5^{2}0)x + −41588807917031 + O(5^{2}0). The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

7. Consider f = q + z7q2 + z23q3 + z10q4 + z2q5 + 2q6 + z21q7 + 4q8 + z17q9 + ⋯ ∈ S8(Γ0(11); \overline{F}5). Also, z satisfies the equation x2 + 4x + 2 = 0. Since f is ordinary, we do not continue our analysis.

8. Consider f = q + z11q2 + z19q3 + z2q4 + z10q5 + 2q6 + z9q7 + 4q8 + z13q9 + ⋯ ∈ S8(Γ0(11); \overline{F}5). Also, z satisfies the equation x2 + 4x + 2 = 0. Since f is ordinary, we do not continue our analysis.

9. Consider f = q + q2 + 4q4 + q5 + 2q8 + 2q9 + ⋯ ∈ S10(Γ0(11); \overline{F}5). Also, z satisfies the equation x2 + 4x + 2 = 0. Since f is ordinary, we do not continue our analysis.

10. Consider f = q + 3q2 + 4q3 + 2q4 + q5 + 2q6 + 3q7 + 3q9 + ⋯ ∈ S10(Γ0(11); \overline{F}5). Since f is ordinary, we do not continue our analysis.

11. Consider f = q + z2q2 + z9q3 + z7q4 + 4q5 + z11q6 + 3q7 + zq8 + ⋯ ∈ S10(Γ0(11); \overline{F}5). Also, z satisfies the equation x2 + 4x + 2 = 0. Since f is ordinary, we do not continue our analysis.

5.9 N = 12, ℓ = 5 (3 forms)

1. Consider f = q + 2q^{3} + 2q^{7} + 4q^{9} + ⋯ ∈ S_{8}(Γ_{0}(12); \overline{F}_{5}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 3q^{3} + 2q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(3)ω^{1} & 0 \\ 0 & u(2)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 27q^{3} + 54 ∗ 5q^{5} + 1112q^{7} + 729q^{9} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 4q^{3} + q^{7} + q^{9} + ⋯ ∈ S_{10}(Γ_{0}(12); \overline{F}_{5}). We have ρ_{f} ⊗ χ^{3} ∼ ρ_{g}, where g is the weight-4 newform g = q + 3q^{3} + 2q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(3)ω^{0} & 0 \\ 0 & u(2)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −81q^{3} + 198 ∗ 5q^{5} + 8576q^{7} + 6561q^{9} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + 3q3 + 2q5 + 3q7 + 4q9 + ⋯ ∈ S8(Γ0(12); \overline{F}5). Since f is ordinary, we do not continue our analysis.

5.10 N = 13, ℓ = 5 (16 forms)

1. Consider f = q + 2q^{3} + 2q^{4} + 3q^{7} + 2q^{9} + ⋯ ∈ S_{8}(Γ_{0}(13); \overline{F}_{5}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 3q^{3} + 2q^{4} + 3q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(3)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 2 ∗ 5q^{2} + −73q^{3} + −28q^{4} + −59 ∗ 5q^{5} + −146 ∗ 5q^{6} + 1373q^{7} + −312 ∗ 5q^{8} + 3142q^{9} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + z^{1}6q^{2} + z^{1}5q^{3} + z^{2}3q^{4} + z^{7}q^{6} + z^{1}6q^{7} + z^{2}3q^{8} + ⋯ ∈ S_{8}(Γ_{0}(13); \overline{F}_{5}). Also, z satisfies the equation x^{2} + 4x + 2 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + z^{4}q^{2} + z^{3}q^{3} + z^{2}3q^{4} + z^{7}q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}7)ω^{1} & 0 \\ 0 & u(z^{7})ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(11502674914997+ O(5^{2}0))∗a+25254120162651+O(5^{2}0)q^{3}+(27006403655779+O(5^{2}0))∗a−39794541661534+O(5^{2}0)q^{4}+ −(5094180181544 ∗ 5 + O(5^{2}0)) ∗ a + 1166223899184 ∗ 5 + O(5^{2}0)q^{5} + (27281430480314 + O(5^{2}0)) ∗ a + 16834204650468 + O(5^{2}0)q^{6} + (21849841762476 + O(5^{2}0)) ∗ a + 8841132770709 ∗ 5 + O(5^{2}0)q^{7} + −(31662478986696 + O(5^{2}0)) ∗ a + 18367144334726 + O(5^{2}0)q^{8} + (3403574756096 ∗ 5 + O(5^{2}0)) ∗ a + 4409996916592 ∗ 5 + O(5^{2}0)q^{9} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −27006403655779 + O(5^{2}0)x + 39794541661406 + O(5^{2}0). The slope of f_{0} is 1.

3. Consider f = q + z^{8}q^{2} + z^{3}q^{3} + z^{1}9q^{4} + z^{1}1q^{6} + z^{8}q^{7} + z^{1}9q^{8} + ⋯ ∈ S_{8}(Γ_{0}(13); \overline{F}_{5}). Also, z satisfies the equation x^{2} + 4x + 2 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + z^{2}0q^{2} + z^{1}5q^{3} + z^{1}9q^{4} + z^{1}1q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}3)ω^{1} & 0 \\ 0 & u(z^{1}1)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(11502674914997+ O(5^{2}0))∗a+25254120162651+O(5^{2}0)q^{3}+(27006403655779+O(5^{2}0))∗a−39794541661534+O(5^{2}0)q^{4}+ −(5094180181544 ∗ 5 + O(5^{2}0)) ∗ a + 1166223899184 ∗ 5 + O(5^{2}0)q^{5} + (27281430480314 + O(5^{2}0)) ∗ a + 16834204650468 + O(5^{2}0)q^{6} + (21849841762476 + O(5^{2}0)) ∗ a + 8841132770709 ∗ 5 + O(5^{2}0)q^{7} + −(31662478986696 + O(5^{2}0)) ∗ a + 18367144334726 + O(5^{2}0)q^{8} + (3403574756096 ∗ 5 + O(5^{2}0)) ∗ a + 4409996916592 ∗ 5 + O(5^{2}0)q^{9} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −27006403655779 + O(5^{2}0)x + 39794541661406 + O(5^{2}0). The slope of f_{0} is 1.

4. Consider f = q+4q^{3}+3q^{4}+4q^{7}+3q^{9}+⋯ ∈ S_{10}(Γ_{0}(13); \overline{F}_{5}). Also, z satisfies the equation x^{2}+4x+2 = 0. We have ρ_{f} ⊗ χ^{3} ∼ ρ_{g}, where g is the weight-4 newform g = q + 3q^{3} + 2q^{4} + 3q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{0} & 0 \\ 0 & u(3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 48473920096 ∗ 5^{3} + O(5^{2}0)q^{2} + 33839733665394 + O(5^{2}0)q^{3} + −3822264514203637 + O(5^{2}3)q^{4} + 9313460303297 ∗ 5 + O(5^{2}0)q^{5} + −981957704676 ∗ 5^{3} + O(5^{2}1)q^{6} + −28930472149396 + O(5^{2}0)q^{7} + 173951696696 ∗ 5^{3} + O(5^{2}0)q^{8} + −201941289422572 + O(5^{2}1)q^{9} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

5. Consider f = q + z^{2}q^{2} + z^{9}q^{3} + z^{7}q^{4} + z^{1}1q^{6} + z^{2}q^{7} + zq^{8} + ⋯ ∈ S_{10}(Γ_{0}(13); \overline{F}_{5}). Also, z satisfies the equation x^{2} + 4x + 2 = 0. We have ρ_{f} ⊗ χ^{3} ∼ ρ_{g}, where g is the weight-4 newform g = q + z^{2}0q^{2} + z^{1}5q^{3} + z^{1}9q^{4} + z^{1}1q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}3)ω^{0} & 0 \\ 0 & u(z^{1}1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(14051061280893+ O(5^{2}0))∗a+18335295278812+O(5^{2}0)q^{3} +(28458143436482+O(5^{2}0))∗a+2522525476144+O(5^{2}0)q^{4} + (163535058257 ∗ 5 + O(5^{2}0)) ∗ a − 1708064033718 ∗ 5 + O(5^{2}0)q^{5} + −(34842188435887 + O(5^{2}0)) ∗ a + 42125701036933 + O(5^{2}0)q^{6} + (4510139771486 + O(5^{2}0)) ∗ a + 1182832232168 ∗ 5^{2} + O(5^{2}0)q^{7} + −(4779729034919 + O(5^{2}0)) ∗ a + 22621052342317 + O(5^{2}0)q^{8} + −(1120777451344 ∗ 5^{2} + O(5^{2}0)) ∗ a − 4113534249059 ∗ 5 + O(5^{2}0)q^{9} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −28458143436482 + O(5^{2}0)x + −2522525476656 + O(5^{2}0). The slope of f_{0} is 1.

6. Consider f = q + z^{1}0q^{2} + z^{2}1q^{3} + z^{1}1q^{4} + z^{7}q^{6} + z^{1}0q^{7} + z^{5}q^{8} + ⋯ ∈ S_{10}(Γ_{0}(13); \overline{F}_{5}). Also, z satisfies the equation x^{2} + 4x + 2 = 0. We have ρ_{f} ⊗ χ^{3} ∼ ρ_{g}, where g is the weight-4 newform g = q + z^{4}q^{2} + z^{3}q^{3} + z^{2}3q^{4} + z^{7}q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}7)ω^{0} & 0 \\ 0 & u(z^{7})ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(14051061280893+ O(5^{2}0))∗a+18335295278812+O(5^{2}0)q^{3} +(28458143436482+O(5^{2}0))∗a+2522525476144+O(5^{2}0)q^{4} + (163535058257 ∗ 5 + O(5^{2}0)) ∗ a − 1708064033718 ∗ 5 + O(5^{2}0)q^{5} + −(34842188435887 + O(5^{2}0)) ∗ a + 42125701036933 + O(5^{2}0)q^{6} + (4510139771486 + O(5^{2}0)) ∗ a + 1182832232168 ∗ 5^{2} + O(5^{2}0)q^{7} + −(4779729034919 + O(5^{2}0)) ∗ a + 22621052342317 + O(5^{2}0)q^{8} + −(1120777451344 ∗ 5^{2} + O(5^{2}0)) ∗ a − 4113534249059 ∗ 5 + O(5^{2}0)q^{9} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −28458143436482 + O(5^{2}0)x + −2522525476656 + O(5^{2}0). The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

7. Consider f = q + z20q2 + z15q3 + z19q4 + z11q5 + z11q6 + z20q7 + z7q8 + ⋯ ∈ S8(Γ0(13); \overline{F}5). Also, z satisfies the equation x2 + 4x + 2 = 0. Since f is ordinary, we do not continue our analysis.

8. Consider f = q + z4q2 + z3q3 + z23q4 + z7q5 + z7q6 + z4q7 + z11q8 + ⋯ ∈ S8(Γ0(13); \overline{F}5). Also, z satisfies the equation x2 + 4x + 2 = 0. Since f is ordinary, we do not continue our analysis.

9. Consider f = q + 3q3 + 2q4 + 3q5 + 2q7 + 2q9 + ⋯ ∈ S8(Γ0(13); \overline{F}5). Also, z satisfies the equation x2 + 4x + 2 = 0. Since f is ordinary, we do not continue our analysis.

10. Consider f = q + q2 + 2q3 + 3q4 + 2q6 + q7 + 2q9 + ⋯ ∈ S8(Γ0(13); \overline{F}5). Also, z satisfies the equation x2 + 4x + 2 = 0. I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

11. Consider f = q + 4q2 + 3q3 + 4q4 + 4q5 + 2q6 + q7 + 3q8 + q9 + ⋯ ∈ S10(Γ0(13); \overline{F}5). Also, z satisfies the equation x2 + 4x + 2 = 0. Since f is ordinary, we do not continue our analysis.

12. Consider f = q + z14q2 + z3q3 + z7q4 + q5 + z17q6 + z8q7 + z13q8 + 4q9 + ⋯ ∈ S10(Γ0(13); \overline{F}5). Also, z satisfies the equation x2 + 4x + 2 = 0. Since f is ordinary, we do not continue our analysis.

13. Consider f = q + z22q2 + z15q3 + z11q4 + q5 + z13q6 + z16q7 + z17q8 + 4q9 + ⋯ ∈ S10(Γ0(13); \overline{F}5). Also, z satisfies the equation x2 + 4x + 2 = 0. Since f is ordinary, we do not continue our analysis.

14. Consider f = q + 3q2 + 4q3 + 2q4 + 2q6 + 3q7 + 3q9 + ⋯ ∈ S10(Γ0(13); \overline{F}5). Also, z satisfies the equation x2 + 4x + 2 = 0. I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

15. Consider f = q + z21q2 + z11q3 + q4 + 4q5 + z8q6 + 2q7 + z9q8 + z2q9 + ⋯ ∈ S10(Γ0(13); \overline{F}5). Also, z satisfies the equation x2 + 4x + 2 = 0. Since f is ordinary, we do not continue our analysis.

16. Consider f = q + z9q2 + z7q3 + q4 + 4q5 + z16q6 + 2q7 + z21q8 + z10q9 + ⋯ ∈ S10(Γ0(13); \overline{F}5). Also, z satisfies the equation x2 + 4x + 2 = 0. Since f is ordinary, we do not continue our analysis.

5.11 N = 14, ℓ = 5 (7 forms)

1. Consider f = q + 3q^{2} + 4q^{3} + 4q^{4} + 2q^{6} + 2q^{7} + 2q^{8} + 4q^{9} + ⋯ ∈ S_{8}(Γ_{0}(14); \overline{F}_{5}). We have ρ_{f} ⊗ χ^{3} ∼ ρ_{g}, where g is the weight-2 newform g = q + 4q^{2} + 3q^{3} + q^{4} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{7}. The form f_{0} = q + 8q^{2} + −66q^{3} + 64q^{4} + −16 ∗ 5^{2}q^{5} + −528q^{6} + −343q^{7} + 512q^{8} + 2169q^{9} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 2.

2. Consider f = q+3q^{2}+2q^{3}+4q^{4}+q^{6}+3q^{7}+2q^{8}+2q^{9}+⋯ ∈ S_{8}(Γ_{0}(14); \overline{F}_{5}). We have ρ_{f} ⊗χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 2q^{2} + 3q^{3} + 4q^{4} + 3q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(3)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+8q^{2}+44140511564472+ O(5^{2}0)q^{3} + 64q^{4} + −3158975503474 ∗ 5 + O(5^{2}0)q^{5} + −28345634046724 + O(5^{2}0)q^{6} + 343q^{7} + 512q^{8} + 228812860292847 + O(5^{2}1)q^{9} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q+4q^{2}+4q^{3}+q^{4}+q^{6}+4q^{7}+4q^{8}+3q^{9}+⋯ ∈ S_{10}(Γ_{0}(14); \overline{F}_{5}). We have ρ_{f} ⊗χ^{3} ∼ ρ_{g}, where g is the weight-4 newform g = q + 2q^{2} + 3q^{3} + 4q^{4} + 3q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{0} & 0 \\ 0 & u(3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −16q^{2} + −6q^{3} + 256q^{4} + 112 ∗ 5q^{5} + 96q^{6} + −2401q^{7} + −4096q^{8} + −19647q^{9} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

4. Consider f = q + 4q^{2} + 3q^{3} + q^{4} + 2q^{6} + q^{7} + 4q^{8} + q^{9} + ⋯ ∈ S_{10}(Γ_{0}(14); \overline{F}_{5}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-2 newform g = q + 4q^{2} + 3q^{3} + q^{4} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{1}. The form f_{0} = q + −16q^{2} + aq^{3} + 256q^{4} + 21 ∗ 5^{−}1 ∗ a − 6678 ∗ 5^{−}1q^{5} + −16 ∗ aq^{6} + 2401q^{7} + −4096q^{8} + −14 ∗ a + 37893q^{9} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 14x + −57576. The slope of f_{0} is 1/2. —————————————————————————————————- (Uninteresting cases below.)

5. Consider f = q + 2q2 + 3q3 + 4q4 + 3q5 + q6 + 2q7 + 3q8 + 2q9 + ⋯ ∈ S8(Γ0(14); \overline{F}5). Since f is ordinary, we do not continue our analysis.

6. Consider f = q + 3q2 + 3q3 + 4q4 + q5 + 4q6 + 3q7 + 2q8 + 2q9 + ⋯ ∈ S8(Γ0(14); \overline{F}5). Since f is ordinary, we do not continue our analysis.

7. Consider f = q + q2 + q4 + 4q5 + 4q7 + q8 + 2q9 + ⋯ ∈ S10(Γ0(14); \overline{F}5). Since f is ordinary, we do not continue our analysis.

5.12 N = 16, ℓ = 5 (7 forms)

1. Consider f = q + q^{3} + 4q^{7} + 4q^{9} + ⋯ ∈ S_{8}(Γ_{0}(16); \overline{F}_{5}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 4q^{3} + 3q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(3)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −44q^{3} + 86 ∗ 5q^{5} + 1224q^{7} + −251q^{9} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 2q^{3} + 2q^{7} + q^{9} + ⋯ ∈ S_{10}(Γ_{0}(16); \overline{F}_{5}). We have ρ_{f} ⊗ χ^{3} ∼ ρ_{g}, where g is the weight-4 newform g = q + 4q^{3} + 3q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{0} & 0 \\ 0 & u(3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −68q^{3} + 302 ∗ 5q^{5} + −10248q^{7} + −15059q^{9} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + 3q3 + 4q7 + 2q9 + ⋯ ∈ S8(Γ0(16); \overline{F}5). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

4. Consider f = q + 4q3 + 3q5 + q7 + 4q9 + ⋯ ∈ S8(Γ0(16); \overline{F}5). Since f is ordinary, we do not continue our analysis.

5. Consider f = q + q5 + 4q7 + 2q9 + ⋯ ∈ S10(Γ0(16); \overline{F}5). Since f is ordinary, we do not continue our analysis.

6. Consider f = q + q3 + 2q7 + 3q9 + ⋯ ∈ S10(Γ0(16); \overline{F}5). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

7. Consider f = q + 2q3 + 4q5 + 3q7 + q9 + ⋯ ∈ S10(Γ0(16); \overline{F}5). Since f is ordinary, we do not continue our analysis.

5.13 N = 17, ℓ = 5 (20 forms)

1. Consider f = q + 3q^{2} + 3q^{3} + q^{4} + 4q^{6} + 3q^{7} + 4q^{8} + 2q^{9} + ⋯ ∈ S_{8}(Γ_{0}(17); \overline{F}_{5}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q+2q^{2} +2q^{3} +q^{4} +q^{5} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −2q^{2} + 18q^{3} + −124q^{4} + −2 ∗ 5q^{5} + −36q^{6} + −902q^{7} + 504q^{8} + −1863q^{9} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + z^{6}9q^{2} + z^{5}8q^{3} + z^{6}3q^{4} + z^{3}q^{6} + z^{1}21q^{7} + z^{2}q^{8} + z^{1}04q^{9} + ⋯ ∈ S_{8}(Γ_{0}(17); \overline{F}_{5}). Also, z satisfies the equation x^{3} + 3x + 3 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + z^{7}q^{2} + z^{1}20q^{3} + z^{6}3q^{4} + z^{9}5q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}9)ω^{1} & 0 \\ 0 & u(z^{9}5)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(36918740115879+ O(5^{2}0))∗a^{2}+(26995823805512+O(5^{2}0))∗a+11766912184049+O(5^{2}0)q^{3}+a^{2}−128q^{4}+−(284390270399∗ 5^{2}+O(5^{2}0))∗a^{2}−(5207406561004∗5+O(5^{2}0))∗a+6598790002149∗5+O(5^{2}0)q^{5}+−(45914080300817+ O(5^{2}0)) ∗ a^{2} + (602744931637 ∗ 5^{2} + O(5^{2}0)) ∗ a + 25133829749243 + O(5^{2}0)q^{6} + (717142301933 ∗ 5^{2} + O(5^{2}0)) ∗ a^{2} − (43015855277311 + O(5^{2}0)) ∗ a − 10618306124786 + O(5^{2}0)q^{7} + −(18371543705426 + O(5^{2}0)) ∗ a^{2} + (12405291764338 + O(5^{2}0)) ∗ a + 9307391057342 + O(5^{2}0)q^{8} + −(588118752198 ∗ 5 + O(5^{2}0)) ∗ a^{2} − (42988300097032 + O(5^{2}0)) ∗ a − 47451623568776 + O(5^{2}0)q^{9} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 18371543705426 + O(5^{2}0)x^{2} + −12405291764594 + O(5^{2}0)x + −9307391057342 + O(5^{2}0). The slope of f_{0} is 1.

3. Consider f = q + 3q^{2} + q^{4} + 3q^{7} + 4q^{8} + 3q^{9} + ⋯ ∈ S_{8}(Γ_{0}(17); \overline{F}_{5}). Also, z satisfies the equation x^{3} + 3x + 3 = 0. We have ρ_{f} ⊗ χ^{3} ∼ ρ_{g}, where g is the weight-2 newform g = q + 4q^{2} + 4q^{4} + 3q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{2} & 0 \\ 0 & u(3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +−26036239370422+ O(5^{2}0)q^{2} + −697784769483 ∗ 5^{2} + O(5^{2}0)q^{3} + −19607012948294 + O(5^{2}0)q^{4} + −8579596927316 ∗ 5 + O(5^{2}0)q^{5} + 1643849681826 ∗ 5^{2} + O(5^{2}0)q^{6} + −21107172468197 + O(5^{2}0)q^{7} + −43956978422791 + O(5^{2}0)q^{8} + 27216089709688 + O(5^{2}0)q^{9} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

4. Consider f = q + z^{9}7q^{2} + z^{4}2q^{3} + z^{6}7q^{4} + z^{1}5q^{6} + z^{1}09q^{7} + z^{1}0q^{8} + z^{2}4q^{9} + ⋯ ∈ S_{8}(Γ_{0}(17); \overline{F}_{5}). Also, z satisfies the equation x^{3} + 3x + 3 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + z^{3}5q^{2} + z^{1}04q^{3} + z^{6}7q^{4} + z^{1}03q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}1)ω^{1} & 0 \\ 0 & u(z^{1}03)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(36918740115879+ O(5^{2}0))∗a^{2}+(26995823805512+O(5^{2}0))∗a+11766912184049+O(5^{2}0)q^{3}+a^{2}−128q^{4}+−(284390270399∗ 5^{2}+O(5^{2}0))∗a^{2}−(5207406561004∗5+O(5^{2}0))∗a+6598790002149∗5+O(5^{2}0)q^{5}+−(45914080300817+ O(5^{2}0)) ∗ a^{2} + (602744931637 ∗ 5^{2} + O(5^{2}0)) ∗ a + 25133829749243 + O(5^{2}0)q^{6} + (717142301933 ∗ 5^{2} + O(5^{2}0)) ∗ a^{2} − (43015855277311 + O(5^{2}0)) ∗ a − 10618306124786 + O(5^{2}0)q^{7} + −(18371543705426 + O(5^{2}0)) ∗ a^{2} + (12405291764338 + O(5^{2}0)) ∗ a + 9307391057342 + O(5^{2}0)q^{8} + −(588118752198 ∗ 5 + O(5^{2}0)) ∗ a^{2} − (42988300097032 + O(5^{2}0)) ∗ a − 47451623568776 + O(5^{2}0)q^{9} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 18371543705426 + O(5^{2}0)x^{2} + −12405291764594 + O(5^{2}0)x + −9307391057342 + O(5^{2}0). The slope of f_{0} is 1.

5. Consider f = q + z^{1}13q^{2} + z^{8}6q^{3} + z^{8}7q^{4} + z^{7}5q^{6} + z^{4}9q^{7} + z^{5}0q^{8} + z^{1}20q^{9} + ⋯ ∈ S_{8}(Γ_{0}(17); \overline{F}_{5}). Also, z satisfies the equation x^{3} + 3x + 3 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + z^{5}1q^{2} + z^{2}4q^{3} + z^{8}7q^{4} + z^{1}9q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}05)ω^{1} & 0 \\ 0 & u(z^{1}9)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(36918740115879+ O(5^{2}0))∗a^{2}+(26995823805512+O(5^{2}0))∗a+11766912184049+O(5^{2}0)q^{3}+a^{2}−128q^{4}+−(284390270399∗ 5^{2}+O(5^{2}0))∗a^{2}−(5207406561004∗5+O(5^{2}0))∗a+6598790002149∗5+O(5^{2}0)q^{5}+−(45914080300817+ O(5^{2}0)) ∗ a^{2} + (602744931637 ∗ 5^{2} + O(5^{2}0)) ∗ a + 25133829749243 + O(5^{2}0)q^{6} + (717142301933 ∗ 5^{2} + O(5^{2}0)) ∗ a^{2} − (43015855277311 + O(5^{2}0)) ∗ a − 10618306124786 + O(5^{2}0)q^{7} + −(18371543705426 + O(5^{2}0)) ∗ a^{2} + (12405291764338 + O(5^{2}0)) ∗ a + 9307391057342 + O(5^{2}0)q^{8} + −(588118752198 ∗ 5 + O(5^{2}0)) ∗ a^{2} − (42988300097032 + O(5^{2}0)) ∗ a − 47451623568776 + O(5^{2}0)q^{9} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 18371543705426 + O(5^{2}0)x^{2} + −12405291764594 + O(5^{2}0)x + −9307391057342 + O(5^{2}0). The slope of f_{0} is 1.

6. Consider f = q+4q^{2}+q^{3}+4q^{4}+4q^{6}+4q^{7}+3q^{8}+3q^{9}+⋯ ∈ S_{10}(Γ_{0}(17); \overline{F}_{5}). Also, z satisfies the equation x^{2} +4x+2 = 0. We have ρ_{f} ⊗χ^{3} ∼ ρ_{g}, where g is the weight-4 newform g = q+2q^{2} +2q^{3} +q^{4} +q^{5} +· · · .

We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +−12031215643341+ O(5^{2}0)q^{2} +−31762337230194+O(5^{2}0)q^{3} +5809705141769+O(5^{2}0)q^{4} +2707153813359∗5+O(5^{2}0)q^{5} + −26731482886846+O(5^{2}0)q^{6}+6827855005699+O(5^{2}0)q^{7}+−23928464154012+O(5^{2}0)q^{8}+−47618770367047+ O(5^{2}0)q^{9} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

7. Consider f = q + 4q^{2} + 4q^{4} + 4q^{7} + 3q^{8} + 2q^{9} + ⋯ ∈ S_{10}(Γ_{0}(17); \overline{F}_{5}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-2 newform g = q + 4q^{2} + 4q^{4} + 3q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(3)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +−24256877065986+ O(5^{2}0)q^{2} + −2340702392 ∗ 5^{3} + O(5^{1}7)q^{3} + 41474723932934 + O(5^{2}0)q^{4} + 10665633938 ∗ 5 + O(5^{1}6)q^{5} + 2590788512∗5^{3}+O(5^{1}7)q^{6}+998184332879+O(5^{1}8)q^{7}+7198157451908+O(5^{2}0)q^{8}+−1152493550933+ O(5^{1}8)q^{9} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

8. Consider f = q + z^{3}8q^{2} + z^{8}9q^{3} + zq^{4} + z^{3}q^{6} + z^{9}0q^{7} + z^{3}3q^{8} + z^{4}2q^{9} + ⋯ ∈ S_{10}(Γ_{0}(17); \overline{F}_{5}). Also, z satisfies the equation x^{3} + 3x + 3 = 0. We have ρ_{f} ⊗ χ^{3} ∼ ρ_{g}, where g is the weight-4 newform g = q + z^{7}q^{2} + z^{1}20q^{3} + z^{6}3q^{4} + z^{9}5q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}9)ω^{0} & 0 \\ 0 & u(z^{9}5)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(252585850582+ O(5^{1}7)) ∗ a^{2} + (364359119143 + O(5^{1}7)) ∗ a + 263690622298 + O(5^{1}7)q^{3} + a^{2} − 512q^{4} + (12487674137 ∗ 5 + O(5^{1}6)) ∗ a^{2} − (5855982743 ∗ 5 + O(5^{1}6)) ∗ a + 6573414564 ∗ 5 + O(5^{1}6)q^{5} + −(168574853363 + O(5^{1}7))∗a^{2} +(25886343752∗5+O(5^{1}7))∗a−372201466412+O(5^{1}7)q^{6} +(183939945599∗5+O(5^{1}8))∗ a^{2} − (1771660233631 + O(5^{1}8)) ∗ a + 1386595429422 + O(5^{1}8)q^{7} + −(7799436247558 + O(5^{2}0)) ∗ a^{2} − (27850795755433 + O(5^{2}0)) ∗ a − 44768521347816 + O(5^{2}0)q^{8} + (49342961971 ∗ 5^{2} + O(5^{1}8)) ∗ a^{2} − (1736275877071 + O(5^{1}8)) ∗ a + 448626612261 + O(5^{1}8)q^{9} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 7799436247558 + O(5^{2}0)x^{2} + 27850795754409 + O(5^{2}0)x + 44768521347816 + O(5^{2}0). The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

9. Consider f = q+z7q2 +z120q3 +z63q4 +z95q5 +z3q6 +z59q7 +z64q8 +z104q9 +⋯ ∈ S8(Γ0(17); \overline{F}5). Also, z satisfies the equation x3 +3x+3 = 0. Since f is ordinary, we do not continue our analysis.

10. Consider f = q + z35q2 + z104q3 + z67q4 + z103q5 + z15q6 + z47q7 + z72q8 + z24q9 + ⋯ ∈ S8(Γ0(17); \overline{F}5). Also, z satisfies the equation x3 + 3x + 3 = 0. Since f is ordinary, we do not continue our analysis.

11. Consider f = q + z51q2 + z24q3 + z87q4 + z19q5 + z75q6 + z111q7 + z112q8 + z120q9 + ⋯ ∈ S8(Γ0(17); \overline{F}5). Also, z satisfies the equation x3 + 3x + 3 = 0. Since f is ordinary, we do not continue our analysis.

12. Consider f = q + q2 + 2q3 + 3q4 + 2q6 + q7 + 2q9 + ⋯ ∈ S8(Γ0(17); \overline{F}5). Also, z satisfies the equation x3 + 3x + 3 = 0. I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

13. Consider f = q + 2q2 + 2q3 + q4 + q5 + 4q6 + 2q7 + q8 + 2q9 + ⋯ ∈ S8(Γ0(17); \overline{F}5). Also, z satisfies the equation x3 + 3x + 3 = 0. Since f is ordinary, we do not continue our analysis.

14. Consider f = q + z14q2 + zq3 + z7q4 + 4q5 + z15q6 + z5q7 + z13q8 + zq9 + ⋯ ∈ S10(Γ0(17); \overline{F}5). Also, z satisfies the equation x2 + 4x + 2 = 0. Since f is ordinary, we do not continue our analysis.

15. Consider f = q + z21q2 + z11q3 + q4 + q5 + z8q6 + z23q7 + z9q8 + z2q9 + ⋯ ∈ S10(Γ0(17); \overline{F}5). Also, z satisfies the equation x2 + 4x + 2 = 0. Since f is ordinary, we do not continue our analysis.

16. Consider f = q + z22q2 + z5q3 + z11q4 + 4q5 + z3q6 + zq7 + z17q8 + z5q9 + ⋯ ∈ S10(Γ0(17); \overline{F}5). Also, z satisfies the equation x2 + 4x + 2 = 0. Since f is ordinary, we do not continue our analysis.

17. Consider f = q + z9q2 + z7q3 + q4 + q5 + z16q6 + z19q7 + z21q8 + z10q9 + ⋯ ∈ S10(Γ0(17); \overline{F}5). Also, z satisfies the equation x2 + 4x + 2 = 0. Since f is ordinary, we do not continue our analysis.

18. Consider f = q + 4q2 + 4q4 + 3q5 + 4q7 + 3q8 + 2q9 + ⋯ ∈ S10(Γ0(17); \overline{F}5). Since f is ordinary, we do not continue our analysis.

19. Consider f = q + q2 + 2q3 + 4q4 + 4q5 + 2q6 + 3q7 + 2q8 + q9 + ⋯ ∈ S10(Γ0(17); \overline{F}5). Since f is ordinary, we do not continue our analysis.

20. Consider f = q + 3q2 + 4q3 + 2q4 + 2q6 + 3q7 + 3q9 + ⋯ ∈ S10(Γ0(17); \overline{F}5). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

1. Consider f = q + 3q^{2} + 4q^{4} + q^{7} + 2q^{8} + ⋯ ∈ S_{8}(Γ_{0}(18); \overline{F}_{5}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 2q^{2} + 4q^{4} + 4q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{1} & 0 \\ 0 & u(4)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 8q^{2} + 64q^{4} + 42 ∗ 5q^{5} + 1016q^{7} + 512q^{8} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 4q^{2} + q^{4} + 3q^{7} + 4q^{8} + ⋯ ∈ S_{10}(Γ_{0}(18); \overline{F}_{5}). We have ρ_{f} ⊗ χ^{3} ∼ ρ_{g}, where g is the weight-4 newform g = q + 2q^{2} + 4q^{4} + 4q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{0} & 0 \\ 0 & u(4)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −16q^{2} + 256q^{4} + −174 ∗ 5q^{5} + −952q^{7} + −4096q^{8} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + 2q2 + 4q4 + 4q5 + 4q7 + 3q8 + ⋯ ∈ S8(Γ0(18); \overline{F}5). Since f is ordinary, we do not continue our analysis.

4. Consider f = q + q2 + q4 + 4q5 + 2q7 + q8 + ⋯ ∈ S10(Γ0(18); \overline{F}5). Since f is ordinary, we do not continue our analysis.

5. Consider f = q + 4q2 + q4 + q5 + 2q7 + 4q8 + ⋯ ∈ S10(Γ0(18); \overline{F}5). Since f is ordinary, we do not continue our analysis.

6. Consider f = q + q2 + q4 + q5 + q7 + q8 + ⋯ ∈ S10(Γ0(18); \overline{F}5). Since f is ordinary, we do not continue our analysis.

5.15 N = 19, ℓ = 5 (22 forms)

1. Consider f = q + 4q^{3} + 2q^{4} + 3q^{7} + 4q^{9} + ⋯ ∈ S_{8}(Γ_{0}(19); \overline{F}_{5}). We have ρ_{f} ⊗ χ^{3} ∼ ρ_{g}, where g is the weight-2 newform g = q + 3q^{3} + 3q^{4} + 3q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{2} & 0 \\ 0 & u(3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +8701404509683∗5+ O(5^{2}0)q^{2} + 21790466780719 + O(5^{2}0)q^{3} + 1446097137097 + O(5^{2}1)q^{4} + 606323604979 ∗ 5 + O(5^{2}0)q^{5} + −2967279922923 ∗ 5 + O(5^{2}0)q^{6} + −16194637143797 + O(5^{2}0)q^{7} + −1378813791673 ∗ 5 + O(5^{2}0)q^{8} + 46146593045399 + O(5^{2}0)q^{9} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 3q^{2} + q^{4} + 4q^{7} + 4q^{8} + 3q^{9} + ⋯ ∈ S_{8}(Γ_{0}(19); \overline{F}_{5}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 2q^{2} + q^{4} + 3q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(3)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +−30170798838292+ O(5^{2}0)q^{2} + 1763344147083 ∗ 5 + O(5^{2}0)q^{3} + −1849572413489 + O(5^{2}0)q^{4} + −7174251360219 ∗ 5 + O(5^{2}0)q^{5} +2193548794639∗5+O(5^{2}0)q^{6} +36789556652224+O(5^{2}0)q^{7} +23408701900289+O(5^{2}0)q^{8} + 24998366860663 + O(5^{2}0)q^{9} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q + q^{2} + 2q^{3} + 3q^{4} + 2q^{6} + q^{7} + 2q^{9} + ⋯ ∈ S_{8}(Γ_{0}(19); \overline{F}_{5}). We have ρ_{f} ⊗ χ^{3} ∼ ρ_{g}, where g is the weight-6 newform g = q + 3q^{2} + 4q^{3} + 2q^{4} + q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{2} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +−47142929741184+ O(5^{2}0)q^{2}+45484846421867+O(5^{2}0)q^{3}+15643243299853+O(5^{2}0)q^{4}+−7960661195137∗5+O(5^{2}0)q^{5}+ 23573663101347 + O(5^{2}0)q^{6} + −24029740261109 + O(5^{2}0)q^{7} + −1619551238101 ∗ 5^{2} + O(5^{2}0)q^{8} + 40947535372877 + O(5^{2}0)q^{9} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

4. Consider f = q + z^{1}11q^{2} + z^{9}7q^{3} + z^{1}12q^{4} + z^{8}4q^{6} + z^{3}2q^{7} + z^{5}5q^{8} + z^{8}1q^{9} + ⋯ ∈ S_{8}(Γ_{0}(19); \overline{F}_{5}). Also, z satisfies the equation x^{3} + 3x + 3 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + z^{4}9q^{2} + z^{3}5q^{3} + z^{1}12q^{4} + z^{1}3q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}11)ω^{1} & 0 \\ 0 & u(z^{1}3)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(42870822084872+ O(5^{2}0))∗a^{2}−(10478265673048+O(5^{2}0))∗a+764099362189∗5^{2}+O(5^{2}0)q^{3}+a^{2}−128q^{4}+−(129547771276∗ 5^{2}+O(5^{2}0))∗a^{2}−(3199196608842∗5+O(5^{2}0))∗a−173638668846∗5^{2}+O(5^{2}0)q^{5}+−(44132226891159+ O(5^{2}0))∗a^{2}+(1754163133361+O(5^{2}0))∗a−15147680600994+O(5^{2}0)q^{6}+(30452230833798+O(5^{2}0))∗ a^{2} + (47134683199706 + O(5^{2}0)) ∗ a − 731488242534 + O(5^{2}0)q^{7} + −(16061936479463 + O(5^{2}0)) ∗ a^{2} − (42699945597218 + O(5^{2}0)) ∗ a + 5126210581248 + O(5^{2}0)q^{8} + (85707063004523 + O(5^{2}1)) ∗ a^{2} + (17795599498521 ∗ 5 + O(5^{2}1)) ∗ a − 23603788717304 + O(5^{2}1)q^{9} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 16061936479463 + O(5^{2}0)x^{2} + 42699945596962 + O(5^{2}0)x + −5126210581248 + O(5^{2}0). The slope of f_{0} is 1.

5. Consider f = q + z^{4}7q^{2} + z^{6}9q^{3} + z^{7}2q^{4} + z^{1}16q^{6} + z^{5}6q^{7} + z^{1}1q^{8} + z^{4}1q^{9} + ⋯ ∈ S_{8}(Γ_{0}(19); \overline{F}_{5}). Also, z satisfies the equation x^{3} + 3x + 3 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + z^{1}09q^{2} + z^{7}q^{3} + z^{7}2q^{4} + z^{7}7q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{4}7)ω^{1} & 0 \\ 0 & u(z^{7}7)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(42870822084872+ O(5^{2}0))∗a^{2}−(10478265673048+O(5^{2}0))∗a+764099362189∗5^{2}+O(5^{2}0)q^{3}+a^{2}−128q^{4}+−(129547771276∗ 5^{2}+O(5^{2}0))∗a^{2}−(3199196608842∗5+O(5^{2}0))∗a−173638668846∗5^{2}+O(5^{2}0)q^{5}+−(44132226891159+ O(5^{2}0))∗a^{2}+(1754163133361+O(5^{2}0))∗a−15147680600994+O(5^{2}0)q^{6}+(30452230833798+O(5^{2}0))∗ a^{2} + (47134683199706 + O(5^{2}0)) ∗ a − 731488242534 + O(5^{2}0)q^{7} + −(16061936479463 + O(5^{2}0)) ∗ a^{2} − (42699945597218 + O(5^{2}0)) ∗ a + 5126210581248 + O(5^{2}0)q^{8} + (85707063004523 + O(5^{2}1)) ∗ a^{2} + (17795599498521 ∗ 5 + O(5^{2}1)) ∗ a − 23603788717304 + O(5^{2}1)q^{9} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 16061936479463 + O(5^{2}0)x^{2} + 42699945596962 + O(5^{2}0)x + −5126210581248 + O(5^{2}0). The slope of f_{0} is 1.

6. Consider f = q + z^{5}9q^{2} + z^{1}13q^{3} + z^{6}4q^{4} + z^{4}8q^{6} + z^{3}6q^{7} + z^{2}7q^{8} + z^{3}3q^{9} + ⋯ ∈ S_{8}(Γ_{0}(19); \overline{F}_{5}). Also, z satisfies the equation x^{3} + 3x + 3 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + z^{1}21q^{2} + z^{5}1q^{3} + z^{6}4q^{4} + z^{6}5q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{5}9)ω^{1} & 0 \\ 0 & u(z^{6}5)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(42870822084872+ O(5^{2}0))∗a^{2}−(10478265673048+O(5^{2}0))∗a+764099362189∗5^{2}+O(5^{2}0)q^{3}+a^{2}−128q^{4}+−(129547771276∗ 5^{2}+O(5^{2}0))∗a^{2}−(3199196608842∗5+O(5^{2}0))∗a−173638668846∗5^{2}+O(5^{2}0)q^{5}+−(44132226891159+ O(5^{2}0))∗a^{2}+(1754163133361+O(5^{2}0))∗a−15147680600994+O(5^{2}0)q^{6}+(30452230833798+O(5^{2}0))∗ a^{2} + (47134683199706 + O(5^{2}0)) ∗ a − 731488242534 + O(5^{2}0)q^{7} + −(16061936479463 + O(5^{2}0)) ∗ a^{2} − (42699945597218 + O(5^{2}0)) ∗ a + 5126210581248 + O(5^{2}0)q^{8} + (85707063004523 + O(5^{2}1)) ∗ a^{2} + (17795599498521 ∗ 5 + O(5^{2}1)) ∗ a − 23603788717304 + O(5^{2}1)q^{9} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 16061936479463 + O(5^{2}0)x^{2} + 42699945596962 + O(5^{2}0)x + −5126210581248 + O(5^{2}0). The slope of f_{0} is 1.

7. Consider f = q + z^{1}0080q^{2} + z^{5}04q^{3} + z^{6}300q^{4} + z^{1}0584q^{6} + z^{1}26q^{7} + z^{1}0836q^{8} + z^{2}394q^{9} + ⋯ ∈ S_{10}(Γ_{0}(19); \overline{F}_{5}). Also, z satisfies the equation x^{6} + x^{4} + 4x^{3} + x^{2} + 2 = 0. We have ρ_{f} ⊗ χ^{3} ∼ ρ_{g}, where g is the weight-4 newform g = q + z^{4}9q^{2} + z^{3}5q^{3} + z^{1}12q^{4} + z^{1}3q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}11)ω^{0} & 0 \\ 0 & u(z^{1}3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(42776768664626+ O(5^{2}0))∗a^{2}+(20533639891373+O(5^{2}0))∗a−982583612877∗5^{2}+O(5^{2}0)q^{3}+a^{2}−512q^{4}+−(3627661505853∗ 5+O(5^{2}0))∗a^{2} +(2830407649658∗5+O(5^{2}0))∗a−1132183846419∗5+O(5^{2}0)q^{5} +(41570386338084+ O(5^{2}0))∗a^{2}−(25933811061843+O(5^{2}0))∗a−1502564554204+O(5^{2}0)q^{6}+−(26603483204304+O(5^{2}0))∗ a^{2} + (8112460822211 + O(5^{2}0)) ∗ a − 8429453082652 + O(5^{2}0)q^{7} + (15420721960836 + O(5^{2}0)) ∗ a^{2} + (10021068884808 + O(5^{2}0)) ∗ a + 32385995966796 + O(5^{2}0)q^{8} + (28780724881538 + O(5^{2}0)) ∗ a^{2} + (750332938969 ∗ 5 + O(5^{2}0)) ∗ a − 29427738551286 + O(5^{2}0)q^{9} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + −15420721960836 + O(5^{2}0)x^{2} + −10021068885832 + O(5^{2}0)x + −32385995966796 + O(5^{2}0). The slope of f_{0} is 1.

8. Consider f = q + z^{2}016q^{2} + z^{1}2600q^{3} + z^{1}260q^{4} + z^{1}4616q^{6} + z^{3}150q^{7} + z^{5}292q^{8} + z^{1}2978q^{9} + ⋯ ∈ S_{10}(Γ_{0}(19); \overline{F}_{5}). Also, z satisfies the equation x^{6} + x^{4} + 4x^{3} + x^{2} + 2 = 0. We have ρ_{f} ⊗ χ^{3} ∼ ρ_{g}, where g is the weight-4 newform g = q + z^{1}09q^{2} + z^{7}q^{3} + z^{7}2q^{4} + z^{7}7q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{4}7)ω^{0} & 0 \\ 0 & u(z^{7}7)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(42776768664626+ O(5^{2}0))∗a^{2}+(20533639891373+O(5^{2}0))∗a−982583612877∗5^{2}+O(5^{2}0)q^{3}+a^{2}−512q^{4}+−(3627661505853∗ 5+O(5^{2}0))∗a^{2} +(2830407649658∗5+O(5^{2}0))∗a−1132183846419∗5+O(5^{2}0)q^{5} +(41570386338084+ O(5^{2}0))∗a^{2}−(25933811061843+O(5^{2}0))∗a−1502564554204+O(5^{2}0)q^{6}+−(26603483204304+O(5^{2}0))∗ a^{2} + (8112460822211 + O(5^{2}0)) ∗ a − 8429453082652 + O(5^{2}0)q^{7} + (15420721960836 + O(5^{2}0)) ∗ a^{2} + (10021068884808 + O(5^{2}0)) ∗ a + 32385995966796 + O(5^{2}0)q^{8} + (28780724881538 + O(5^{2}0)) ∗ a^{2} + (750332938969 ∗ 5 + O(5^{2}0)) ∗ a − 29427738551286 + O(5^{2}0)q^{9} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + −15420721960836 + O(5^{2}0)x^{2} + −10021068885832 + O(5^{2}0)x + −32385995966796 + O(5^{2}0). The slope of f_{0} is 1.

9. Consider f = q + z^{3}528q^{2} + z^{2}520q^{3} + z^{2}52q^{4} + z^{6}048q^{6} + z^{6}30q^{7} + z^{7}308q^{8} + z^{1}1970q^{9} + ⋯ ∈ S_{10}(Γ_{0}(19); \overline{F}_{5}). Also, z satisfies the equation x^{6} + x^{4} + 4x^{3} + x^{2} + 2 = 0. We have ρ_{f} ⊗ χ^{3} ∼ ρ_{g}, where g is the weight-4 newform g = q + z^{1}21q^{2} + z^{5}1q^{3} + z^{6}4q^{4} + z^{6}5q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{5}9)ω^{0} & 0 \\ 0 & u(z^{6}5)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(42776768664626+ O(5^{2}0))∗a^{2}+(20533639891373+O(5^{2}0))∗a−982583612877∗5^{2}+O(5^{2}0)q^{3}+a^{2}−512q^{4}+−(3627661505853∗ 5+O(5^{2}0))∗a^{2} +(2830407649658∗5+O(5^{2}0))∗a−1132183846419∗5+O(5^{2}0)q^{5} +(41570386338084+ O(5^{2}0))∗a^{2}−(25933811061843+O(5^{2}0))∗a−1502564554204+O(5^{2}0)q^{6}+−(26603483204304+O(5^{2}0))∗ a^{2} + (8112460822211 + O(5^{2}0)) ∗ a − 8429453082652 + O(5^{2}0)q^{7} + (15420721960836 + O(5^{2}0)) ∗ a^{2} + (10021068884808 + O(5^{2}0)) ∗ a + 32385995966796 + O(5^{2}0)q^{8} + (28780724881538 + O(5^{2}0)) ∗ a^{2} + (750332938969 ∗ 5 + O(5^{2}0)) ∗ a − 29427738551286 + O(5^{2}0)q^{9} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + −15420721960836 + O(5^{2}0)x^{2} + −10021068885832 + O(5^{2}0)x + −32385995966796 + O(5^{2}0). The slope of f_{0} is 1.

10. Consider f = q + 3q^{3} + 3q^{4} + 4q^{7} + q^{9} + ⋯ ∈ S_{10}(Γ_{0}(19); \overline{F}_{5}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-2 newform g = q + 3q^{3} + 3q^{4} + 3q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(3)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 5576470313142 ∗ 5 + O(5^{2}0)q^{2} + 44783519563928 + O(5^{2}0)q^{3} + −15230936962037 + O(5^{2}1)q^{4} + 405632409603 ∗ 5 + O(5^{1}9)q^{5}+1743580354276∗5+O(5^{2}0)q^{6}+−41370428642036+O(5^{2}0)q^{7}+4226288012292∗5+O(5^{2}0)q^{8}+ 22338317832001 + O(5^{2}0)q^{9} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

11. Consider f = q + 4q^{2} + 4q^{4} + 2q^{7} + 3q^{8} + 2q^{9} + ⋯ ∈ S_{10}(Γ_{0}(19); \overline{F}_{5}). We have ρ_{f} ⊗ χ^{3} ∼ ρ_{g}, where g is the weight-4 newform g = q + 2q^{2} + q^{4} + 3q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{0} & 0 \\ 0 & u(3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 47507616025214 + O(5^{2}0)q^{2}+1393955669912∗5+O(5^{1}9)q^{3}+38285381401534+O(5^{2}0)q^{4}+1274870049527∗5+O(5^{1}9)q^{5}+ −1826165588832∗5+O(5^{1}9)q^{6}+−3248556141933+O(5^{1}9)q^{7}+7983068462458+O(5^{2}0)q^{8}+−9076979076083+ O(5^{1}9)q^{9} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

12. Consider f = q +3q^{2} +4q^{3} +2q^{4} +2q^{6} +3q^{7} +3q^{9} +⋯ ∈ S_{10}(Γ_{0}(19); \overline{F}_{5}). We have ρ_{f} ⊗χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 3q^{2} + 4q^{3} + 2q^{4} + q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +−10776133410992+ O(5^{2}0)q^{2}+−6416144924401+O(5^{1}9)q^{3}+32317509079802+O(5^{2}0)q^{4}+−1250964593171∗5+O(5^{1}9)q^{5}+ −8986182740458+O(5^{1}9)q^{6}+−2916495708102+O(5^{1}9)q^{7}+−1808542111136∗5+O(5^{2}0)q^{8}+−1254744810882+ O(5^{1}9)q^{9} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

13. Consider f = q + 2q2 + q4 + 3q5 + q7 + q8 + 3q9 + ⋯ ∈ S8(Γ0(19); \overline{F}5). Since f is ordinary, we do not continue our analysis.

14. Consider f = q + z109q2 + z7q3 + z72q4 + z77q5 + z116q6 + z118q7 + z73q8 + z41q9 + ⋯ ∈ S8(Γ0(19); \overline{F}5). Also, z satisfies the equation x3 + 3x + 3 = 0. Since f is ordinary, we do not continue our analysis.

15. Consider f = q + z121q2 + z51q3 + z64q4 + z65q5 + z48q6 + z98q7 + z89q8 + z33q9 + ⋯ ∈ S8(Γ0(19); \overline{F}5). Also, z satisfies the equation x3 + 3x + 3 = 0. Since f is ordinary, we do not continue our analysis.

16. Consider f = q + z49q2 + z35q3 + z112q4 + z13q5 + z84q6 + z94q7 + z117q8 + z81q9 + ⋯ ∈ S8(Γ0(19); \overline{F}5). Also, z satisfies the equation x3 + 3x + 3 = 0. Since f is ordinary, we do not continue our analysis.

17. Consider f = q + 3q2 + 4q3 + 2q4 + q5 + 2q6 + 3q7 + 3q9 + ⋯ ∈ S10(Γ0(19); \overline{F}5). Also, z satisfies the equation x6 + x4 + 4x3 + x2 + 2 = 0. Since f is ordinary, we do not continue our analysis.

18. Consider f = q + z12369q2 + z14322q3 + z8463q4 + q5 + z11067q6 + z4557q7 + z14322q8 + z5859q9 + ⋯ ∈ S10(Γ0(19); \overline{F}5). Also, z satisfies the equation x6 + x4 + 4x3 + x2 + 2 = 0. Since f is ordinary, we do not continue our analysis.

19. Consider f = q + z14973q2 + z9114q3 + z11067q4 + q5 + z8463q6 + z7161q7 + z9114q8 + z13671q9 + ⋯ ∈ S10(Γ0(19); \overline{F}5). Also, z satisfies the equation x6 + x4 + 4x3 + x2 + 2 = 0. Since f is ordinary, we do not continue our analysis.

20. Consider f = q + 3q3 + 3q4 + 3q5 + 4q7 + q9 + ⋯ ∈ S10(Γ0(19); \overline{F}5). Since f is ordinary, we do not continue our analysis.

21. Consider f = q + 4q2 + 4q3 + 4q4 + 4q5 + q6 + 3q7 + 3q8 + 3q9 + ⋯ ∈ S10(Γ0(19); \overline{F}5). Since f is ordinary, we do not continue our analysis.

22. Consider f = q + z24q2 + z26q3 + z120q4 + 4q5 + z50q6 + z122q7 + z67q8 + z53q9 + ⋯ ∈ S10(Γ0(19); \overline{F}5). Also, z satisfies the equation x3 + 3x + 3 = 0. Since f is ordinary, we do not continue our analysis.

5.16 N = 21, ℓ = 5 (13 forms)

1. Consider f = q + 3q^{2} + 3q^{3} + q^{4} + 4q^{6} + 3q^{7} + 4q^{8} + 4q^{9} + ⋯ ∈ S_{8}(Γ_{0}(21); \overline{F}_{5}). We have ρ_{f} ⊗ χ^{3} ∼ ρ_{g}, where g is the weight-2 newform g = q+4q^{2}+q^{3}+4q^{4}+3q^{5}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{2} & 0 \\ 0 & u(3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + −27q^{3} + −9 ∗ a + 118q^{4} + −4 ∗ 5 ∗ a − 54 ∗ 5q^{5} + −27 ∗ aq^{6} + 343q^{7} + 71 ∗ a − 2214q^{8} + 729q^{9} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 9x + −246. The slope of f_{0} is 1.

2. Consider f = q + z^{1}9q^{2} + 2q^{3} + z^{1}0q^{4} + zq^{6} + 3q^{7} + q^{8} + 4q^{9} + ⋯ ∈ S_{8}(Γ_{0}(21); \overline{F}_{5}). Also, z satisfies the equation x^{2} + 4x + 2 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + z^{7}q^{2} + 3q^{3} + z^{1}0q^{4} + zq^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}3)ω^{1} & 0 \\ 0 & u(z)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + 27q^{3} + −(6160244277797 + O(5^{2}0)) ∗ a − 1300495433396 + O(5^{2}0)q^{4} + (2464097711122 ∗ 5 + O(5^{2}0)) ∗ a + 520198173384 ∗ 5 + O(5^{2}0)q^{5} + 27 ∗ aq^{6} + 343q^{7} + (3696146566687 ∗ 5 + O(5^{2}0)) ∗ a + 3901486300596 + O(5^{2}0)q^{8} +729q^{9} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 6160244277797 + O(5^{2}0)x + 1300495433268 + O(5^{2}0). The slope of f_{0} is 1.

3. Consider f = q + z^{2}3q^{2} + 2q^{3} + z^{2}q^{4} + z^{5}q^{6} + 3q^{7} + q^{8} + 4q^{9} + ⋯ ∈ S_{8}(Γ_{0}(21); \overline{F}_{5}). Also, z satisfies the equation x^{2} + 4x + 2 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + z^{1}1q^{2} + 3q^{3} + z^{2}q^{4} + z^{5}q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}9)ω^{1} & 0 \\ 0 & u(z^{5})ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + 27q^{3} + −(6160244277797 + O(5^{2}0)) ∗ a − 1300495433396 + O(5^{2}0)q^{4} + (2464097711122 ∗ 5 + O(5^{2}0)) ∗ a + 520198173384 ∗ 5 + O(5^{2}0)q^{5} + 27 ∗ aq^{6} + 343q^{7} + (3696146566687 ∗ 5 + O(5^{2}0)) ∗ a + 3901486300596 + O(5^{2}0)q^{8} +729q^{9} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 6160244277797 + O(5^{2}0)x + 1300495433268 + O(5^{2}0). The slope of f_{0} is 1.

4. Consider f = q + z^{1}3q^{2} + 4q^{3} + z^{2}2q^{4} + zq^{6} + 4q^{7} + 2q^{8} + q^{9} + ⋯ ∈ S_{10}(Γ_{0}(21); \overline{F}_{5}). Also, z satisfies the equation x^{2} + 4x + 2 = 0. We have ρ_{f} ⊗ χ^{3} ∼ ρ_{g}, where g is the weight-4 newform g = q + z^{7}q^{2} + 3q^{3} + z^{1}0q^{4} + zq^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}3)ω^{0} & 0 \\ 0 & u(z)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + −81q^{3} + 9 ∗ a + 56q^{4} + −14 ∗ 5 ∗ a + 36 ∗ 5^{2}q^{5} + −81 ∗ aq^{6} + −2401q^{7} + −3 ∗ 5^{3} ∗ a + 5112q^{8} + 6561q^{9} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −9x + −568. The slope of f_{0} is 1.

5. Consider f = q + z^{1}7q^{2} + 4q^{3} + z^{1}4q^{4} + z^{5}q^{6} + 4q^{7} + 2q^{8} + q^{9} + ⋯ ∈ S_{10}(Γ_{0}(21); \overline{F}_{5}). Also, z satisfies the equation x^{2} + 4x + 2 = 0. We have ρ_{f} ⊗ χ^{3} ∼ ρ_{g}, where g is the weight-4 newform g = q + z^{1}1q^{2} + 3q^{3} + z^{2}q^{4} + z^{5}q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}9)ω^{0} & 0 \\ 0 & u(z^{5})ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + −81q^{3} + 9 ∗ a + 56q^{4} + −14 ∗ 5 ∗ a + 36 ∗ 5^{2}q^{5} + −81 ∗ aq^{6} + −2401q^{7} + −3 ∗ 5^{3} ∗ a + 5112q^{8} + 6561q^{9} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −9x + −568. The slope of f_{0} is 1.

6. Consider f = q + 4q^{2} + q^{3} + 4q^{4} + 4q^{6} + 4q^{7} + 3q^{8} + q^{9} + ⋯ ∈ S_{10}(Γ_{0}(21); \overline{F}_{5}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-2 newform g = q+4q^{2}+q^{3}+4q^{4}+3q^{5}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(3)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + 81q^{3} + (6667289243573 + O(5^{2}0)) ∗ a − 26029455799893 + O(5^{2}0)q^{4} + −(533383139484 ∗ 5 + O(5^{1}9)) ∗ a − 1732340801421 ∗ 5 + O(5^{1}9)q^{5} + 81 ∗ aq^{6} + −2401q^{7} + (8692671474674 + O(5^{2}0)) ∗ a − 43086801159563 + O(5^{2}0)q^{8} + 6561q^{9} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −6667289243573 + O(5^{2}0)x + 26029455799381 + O(5^{2}0). The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

7. Consider f = q + 2q2 + 2q3 + q4 + 2q5 + 4q6 + 2q7 + q8 + 4q9 + ⋯ ∈ S8(Γ0(21); \overline{F}5). Since f is ordinary, we do not continue our analysis.

8. Consider f = q + z7q2 + 3q3 + z10q4 + zq5 + zq6 + 2q7 + 4q8 + 4q9 + ⋯ ∈ S8(Γ0(21); \overline{F}5). Also, z satisfies the equation x2 + 4x + 2 = 0. Since f is ordinary, we do not continue our analysis.

9. Consider f = q + z11q2 + 3q3 + z2q4 + z5q5 + z5q6 + 2q7 + 4q8 + 4q9 + ⋯ ∈ S8(Γ0(21); \overline{F}5). Also, z satisfies the equation x2 + 4x + 2 = 0. Since f is ordinary, we do not continue our analysis.

10. Consider f = q + 4q2 + 2q3 + 3q4 + q5 + 3q6 + 3q7 + 4q9 + ⋯ ∈ S8(Γ0(21); \overline{F}5). Also, z satisfies the equation x2 + 4x + 2 = 0. Since f is ordinary, we do not continue our analysis.

11. Consider f = q + q2 + q3 + 4q4 + q5 + q6 + q7 + 2q8 + q9 + ⋯ ∈ S10(Γ0(21); \overline{F}5). Since f is ordinary, we do not continue our analysis.

12. Consider f = q + 4q3 + 3q4 + 4q5 + q7 + q9 + ⋯ ∈ S10(Γ0(21); \overline{F}5). Since f is ordinary, we do not continue our analysis.

13. Consider f = q + 4q2 + q3 + 4q4 + 3q5 + 4q6 + 4q7 + 3q8 + q9 + ⋯ ∈ S10(Γ0(21); \overline{F}5). Since f is ordinary, we do not continue our analysis.

5.17 N = 22, ℓ = 5 (12 forms)

1. Consider f = q+2q^{2}+q^{3}+4q^{4}+2q^{6}+3q^{7}+3q^{8}+4q^{9}+⋯ ∈ S_{8}(Γ_{0}(22); \overline{F}_{5}). We have ρ_{f} ⊗χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 3q^{2} + 4q^{3} + 4q^{4} + 4q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{1} & 0 \\ 0 & u(4)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −8q^{2} + 91q^{3} + 64q^{4} + 37 ∗ 5q^{5} + −728q^{6} + −722q^{7} + −512q^{8} + 6094q^{9} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 3q^{2} + 4q^{3} + 4q^{4} + 2q^{6} + 2q^{8} + 4q^{9} + ⋯ ∈ S_{8}(Γ_{0}(22); \overline{F}_{5}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 2q^{2} + q^{3} + 4q^{4} + 2q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(3)ω^{1} & 0 \\ 0 & u(2)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+8q^{2}+−7360483388691+ O(5^{2}0)q^{3} + 64q^{4} + 1472096677769 ∗ 5 + O(5^{2}0)q^{5} + 36483564531097 + O(5^{2}0)q^{6} + 1535867160357 ∗ 5 + O(5^{2}0)q^{7} + 512q^{8} + −21443745339956 + O(5^{2}0)q^{9} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q +q^{2} +2q^{3} +q^{4} +2q^{6} +4q^{7} +q^{8} +q^{9} +⋯ ∈ S_{10}(Γ_{0}(22); \overline{F}_{5}). We have ρ_{f} ⊗χ^{3} ∼ ρ_{g}, where g is the weight-4 newform g = q + 3q^{2} + 4q^{3} + 4q^{4} + 4q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{0} & 0 \\ 0 & u(4)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 16q^{2} + 137q^{3} + 256q^{4} + −119 ∗ 5q^{5} + 2192q^{6} + 11354q^{7} + 4096q^{8} + −914q^{9} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

4. Consider f = q + 4q^{2} + 3q^{3} + q^{4} + 2q^{6} + 4q^{8} + q^{9} + ⋯ ∈ S_{10}(Γ_{0}(22); \overline{F}_{5}). We have ρ_{f} ⊗ χ^{3} ∼ ρ_{g}, where g is the weight-4 newform g = q + 2q^{2} + q^{3} + 4q^{4} + 2q^{5} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(3)ω^{0} & 0 \\ 0 & u(2)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −16q^{2} + 28098984650238 + O(5^{2}0)q^{3} + 256q^{4} + −9367194068197 ∗ 5 + O(5^{2}0)q^{5} + 27253403799317 + O(5^{2}0)q^{6} + 3064871497298 ∗ 5^{2} + O(5^{2}1)q^{7} + −4096q^{8} + −17874087813039 + O(5^{2}0)q^{9} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

5. Consider f = q + 2q2 + q3 + 4q4 + 2q5 + 2q6 + 3q8 + 4q9 + ⋯ ∈ S8(Γ0(22); \overline{F}5). Since f is ordinary, we do not continue our analysis.

6. Consider f = q + 3q2 + 4q3 + 4q4 + 4q5 + 2q6 + 2q7 + 2q8 + 4q9 + ⋯ ∈ S8(Γ0(22); \overline{F}5). Since f is ordinary, we do not continue our analysis.

7. Consider f = q + 3q2 + 3q3 + 4q4 + q5 + 4q6 + 4q7 + 2q8 + 2q9 + ⋯ ∈ S8(Γ0(22); \overline{F}5). Since f is ordinary, we do not continue our analysis.

8. Consider f = q + q2 + 4q3 + q4 + q5 + 4q6 + 3q7 + q8 + 3q9 + ⋯ ∈ S10(Γ0(22); \overline{F}5). Since f is ordinary, we do not continue our analysis.

9. Consider f = q + q2 + q3 + q4 + 4q5 + q6 + 4q7 + q8 + 3q9 + ⋯ ∈ S10(Γ0(22); \overline{F}5). Since f is ordinary, we do not continue our analysis.

10. Consider f = q + 4q2 + q3 + q4 + 4q5 + 4q6 + 4q8 + 3q9 + ⋯ ∈ S10(Γ0(22); \overline{F}5). Since f is ordinary, we do not continue our analysis.

11. Consider f = q + 4q2 + z13q3 + q4 + q5 + zq6 + zq7 + 4q8 + zq9 + ⋯ ∈ S10(Γ0(22); \overline{F}5). Also, z satisfies the equation x2 + 4x + 2 = 0. Since f is ordinary, we do not continue our analysis.

12. Consider f = q + 4q2 + z17q3 + q4 + q5 + z5q6 + z5q7 + 4q8 + z5q9 + ⋯ ∈ S10(Γ0(22); \overline{F}5). Also, z satisfies the equation x2 + 4x + 2 = 0. Since f is ordinary, we do not continue our analysis.

7 2 ≤ N ≤ 16, ℓ = 7

7.1 N = 2, ℓ = 7 (3 forms)

- —————————————————————————————————- (Uninteresting cases below.)

1. Consider f = q + 2q2 + 5q3 + 4q4 + 2q5 + 3q6 + q8 + 5q9 + 4q10 + 6q11 + 6q12 + ⋯ ∈ S10(Γ0(2); \overline{F}7). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

2. Consider f = q + q2 + 4q3 + q4 + 6q5 + 4q6 + q8 + 6q9 + 6q10 + 5q11 + 4q12 + ⋯ ∈ S14(Γ0(2); \overline{F}7). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

3. Consider f = q + 6q2 + 5q3 + q4 + 2q6 + q7 + 6q8 + q9 + 5q12 + 3q13 + ⋯ ∈ S14(Γ0(2); \overline{F}7). Since f is ordinary, we do not continue our analysis.

7.2 N = 3, ℓ = 7 (6 forms)

1. Consider f = q + 4q^{2} + 4q^{3} + q^{4} + 3q^{5} + 2q^{6} + 2q^{9} + 5q^{10} + 6q^{11} + 4q^{12} + q^{13} + ⋯ ∈ S_{10}(Γ_{0}(3); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + q^{2} + 2q^{3} + 4q^{4} + 6q^{5} + 2q^{6} + 2q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{1} & 0 \\ 0 & u(2)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+18q^{2}+81q^{3}+−188q^{4}+ −1530q^{5}+1458q^{6}+1304∗7q^{7}+−1800∗7q^{8}+6561q^{9}+−27540q^{10}+21132q^{11}+−15228q^{12}+31214q^{13}+⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 2q^{2} + 6q^{3} + 2q^{4} + 2q^{5} + 5q^{6} + q^{9} + 4q^{10} + 5q^{11} + 5q^{12} + 6q^{13} + ⋯ ∈ S_{14}(Γ_{0}(3); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-6 newform g = q + q^{2} + 2q^{3} + 4q^{4} + 6q^{5} + 2q^{6} + 2q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{0} & 0 \\ 0 & u(2)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −12q^{2} + −729q^{3} + −8048q^{4} + −30210q^{5} + 8748q^{6} + 33584 ∗ 7q^{7} + 27840 ∗ 7q^{8} + 531441q^{9} + 362520q^{10} + −11182908q^{11} + 5866992q^{12} + 8049614q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + 6q2 + 3q3 + 2q5 + 4q6 + q8 + 2q9 + 5q10 + 6q11 + ⋯ ∈ S10(Γ0(3); \overline{F}7). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

4. Consider f = q + q2 + 2q3 + 4q4 + 6q5 + 2q6 + 2q7 + 4q9 + 6q10 + 3q11 + q12 + q13 + ⋯ ∈ S12(Γ0(3); \overline{F}7). Since f is ordinary, we do not continue our analysis.

5. Consider f = q + 6q2 + q3 + 6q4 + 5q5 + 6q6 + 6q7 + 3q8 + q9 + 2q10 + 4q11 + 6q12 + 5q13 + ⋯ ∈ S14(Γ0(3); \overline{F}7). Since f is ordinary, we do not continue our analysis.

6. Consider f = q + 3q2 + q3 + 6q5 + 3q6 + q8 + q9 + 4q10 + 5q11 + ⋯ ∈ S14(Γ0(3); \overline{F}7). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

7.3 N = 4, ℓ = 7 (3 forms)

1. Consider f = q + 4q^{3} + 6q^{5} + 3q^{9} + 2q^{11} + 2q^{13} + ⋯ ∈ S_{10}(Γ_{0}(4); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 2q^{3} + 5q^{5} + 3q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{1} & 0 \\ 0 & u(3)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 228q^{3} + −666q^{5} + −904 ∗ 7q^{7} + 32301q^{9} + −30420q^{11} + −32338q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 6q^{3} + 4q^{5} + 5q^{9} + 4q^{11} + 5q^{13} + ⋯ ∈ S_{14}(Γ_{0}(4); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-6 newform g = q + 2q^{3} + 5q^{5} + 3q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{0} & 0 \\ 0 & u(3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 468q^{3} + 56214q^{5} + 47576 ∗ 7q^{7} + −1375299q^{9} + −6397380q^{11} + 15199742q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + 2q3 + 5q5 + 3q7 + 6q9 + q11 + 2q13 + ⋯ ∈ S12(Γ0(4); \overline{F}7). Since f is ordinary, we do not continue our analysis.

7.4 N = 5, ℓ = 7 (11 forms)

1. Consider f = q + q^{2} + 6q^{3} + 2q^{5} + 6q^{6} + 6q^{8} + 2q^{9} + 2q^{10} + 5q^{11} + 6q^{13} + ⋯ ∈ S_{10}(Γ_{0}(5); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 2q^{2} + 3q^{3} + 4q^{5} + 6q^{6} + 3q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{1} & 0 \\ 0 & u(3)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−36436499247640568+ O(7^{2}0)q^{2}+−6919267802330745+O(7^{2}0)q^{3}+−4942334144521979∗7+O(7^{2}0)q^{4}+625q^{5}+−74197117920125425+ O(7^{2}1)q^{6}+−3175894024413538∗7+O(7^{2}0)q^{7}+−31798764822919893+O(7^{2}0)q^{8}+36212496239049776+ O(7^{2}0)q^{9} + −32016134955934715 + O(7^{2}0)q^{10} + 30338575810159160 + O(7^{2}0)q^{11} + 4204185432103806 ∗ 7 + O(7^{2}0)q^{12} + 30318677183494069 + O(7^{2}0)q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 6q^{2} + 6q^{3} + 4q^{4} + 3q^{5} + q^{6} + 3q^{9} + 4q^{10} + 2q^{11} + 3q^{12} + 3q^{13} + ⋯ ∈ S_{12}(Γ_{0}(5); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-4 newform g = q + 3q^{2} + 2q^{3} + q^{4} + 2q^{5} + 6q^{6} + 6q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{4} & 0 \\ 0 & u(6)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 34q^{2} + −792q^{3} + −892q^{4} + 3125q^{5} + −26928q^{6} + −2508 ∗ 7q^{7} + −2040 ∗ 7^{2}q^{8} + 450117q^{9} + 106250q^{10} + −468788q^{11} + 706464q^{12} + −374042q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q + 6q^{2} + q^{3} + 4q^{4} + 4q^{5} + 6q^{6} + 3q^{9} + 3q^{10} + 2q^{11} + 4q^{12} + 4q^{13} + ⋯ ∈ S_{12}(Γ_{0}(5); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 3q^{2} + 2q^{3} + q^{4} + 2q^{5} + 6q^{6} + 6q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{1} & 0 \\ 0 & u(6)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+22817967818630025+ O(7^{2}0)q^{2}+15306139969210742+O(7^{2}0)q^{3}+22394241413074794+O(7^{2}0)q^{4}+−3125q^{5}+1486159987734441774+ O(7^{2}2)q^{6}+3764145885473580∗7+O(7^{2}0)q^{7}+−157953003842027∗7+O(7^{2}0)q^{8}+−16075608726693821+ O(7^{2}0)q^{9} + 28136636846300769 + O(7^{2}0)q^{10} + 39017467144789429 + O(7^{2}0)q^{11} + 16983759292187958 + O(7^{2}0)q^{12} + −18127525411569724 + O(7^{2}0)q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

4. Consider f = q + 4q^{2} + 2q^{3} + 6q^{5} + q^{6} + 6q^{8} + q^{9} + 3q^{10} + 3q^{11} + q^{13} + ⋯ ∈ S_{14}(Γ_{0}(5); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-6 newform g = q + 2q^{2} + 3q^{3} + 4q^{5} + 6q^{6} + 3q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{0} & 0 \\ 0 & u(3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+33691825955258356+ O(7^{2}0)q^{2}+−5340579975040357+O(7^{2}0)q^{3}+358999544859954∗7^{2}+O(7^{2}0)q^{4}+−15625q^{5}+28126849851088745+ O(7^{2}0)q^{6}+−2375535549637855∗7+O(7^{2}0)q^{7}+−6328666560350800+O(7^{2}0)q^{8}+−16454533056251135+ O(7^{2}0)q^{9} + 34592480732170098 + O(7^{2}0)q^{10} + 23623575968533075 + O(7^{2}0)q^{11} + −307350181470413 ∗ 7^{2} + O(7^{2}0)q^{12} + −1444793968139479 + O(7^{2}0)q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

5. Consider f = q + 6q2 + 5q3 + 5q5 + 2q6 + q8 + 5q9 + 2q10 + 6q11 + ⋯ ∈ S10(Γ0(5); \overline{F}7). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

6. Consider f = q + 3q2 + 2q3 + q4 + 2q5 + 6q6 + 6q7 + 5q9 + 6q10 + 4q11 + 2q12 + 4q13 + ⋯ ∈ S10(Γ0(5); \overline{F}7). Since f is ordinary, we do not continue our analysis.

7. Consider f = q + 2q2 + 3q3 + 4q5 + 6q6 + 3q7 + 6q8 + 4q9 + q10 + 6q11 + 6q13 + ⋯ ∈ S12(Γ0(5); \overline{F}7). Since f is ordinary, we do not continue our analysis.

8. Consider f = q + q3 + 5q4 + 6q5 + q7 + 5q9 + 4q11 + 5q12 + 5q13 + ⋯ ∈ S14(Γ0(5); \overline{F}7). Since f is ordinary, we do not continue our analysis.

9. Consider f = q + z25q2 + z31q3 + z38q4 + q5 + 3q6 + 6q7 + z26q8 + z35q9 + z25q10 + z7q11 + z21q12 + z26q13 + ⋯ ∈ S14(Γ0(5); \overline{F}7). Also, z satisfies the equation x2 + 6x + 3 = 0. Since f is ordinary, we do not continue our analysis.

10. Consider f = q + z31q2 + z25q3 + z26q4 + q5 + 3q6 + 6q7 + z38q8 + z5q9 + z31q10 + zq11 + z3q12 + z38q13 + ⋯ ∈ S14(Γ0(5); \overline{F}7). Also, z satisfies the equation x2 + 6x + 3 = 0. Since f is ordinary, we do not continue our analysis.

11. Consider f = q + 3q2 + 4q3 + q5 + 5q6 + q8 + 6q9 + 3q10 + 5q11 + ⋯ ∈ S14(Γ0(5); \overline{F}7). Also, z satisfies the equation x2 + 6x + 3 = 0. I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

7.5 N = 6, ℓ = 7 (5 forms)

1. Consider f = q +3q^{2} +2q^{3} +2q^{4} +5q^{5} +6q^{6} +6q^{8} +4q^{9} +q^{10} +6q^{11} +4q^{12} +3q^{13} +⋯ ∈ S_{12}(Γ_{0}(6); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 5q^{2} + 4q^{3} + 4q^{4} + 6q^{5} + 6q^{6} + 5q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(3)ω^{1} & 0 \\ 0 & u(5)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −32q^{2} + −243q^{3} + 1024q^{4}+5766q^{5}+7776q^{6}+10352∗7q^{7}+−32768q^{8}+59049q^{9}+−184512q^{10}+−408948q^{11}+−248832q^{12}+ 1367558q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q +3q^{2} +5q^{3} +2q^{4} +2q^{5} +q^{6} +6q^{8} +4q^{9} +6q^{10} +6q^{11} +3q^{12} +4q^{13} +⋯ ∈ S_{12}(Γ_{0}(6); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-4 newform g = q + 5q^{2} + 4q^{3} + 4q^{4} + 6q^{5} + 6q^{6} + 5q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(3)ω^{4} & 0 \\ 0 & u(5)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −32q^{2} + 243q^{3} + 1024q^{4}+−11730q^{5}+−7776q^{6}+−7144∗7q^{7}+−32768q^{8}+59049q^{9}+375360q^{10}+−531420q^{11}+248832q^{12}+ 1332566q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + 5q2 + 4q3 + 4q4 + 6q5 + 6q6 + 5q7 + 6q8 + 2q9 + 2q10 + 5q11 + 2q12 + 3q13 + ⋯ ∈ S10(Γ0(6); \overline{F}7). Since f is ordinary, we do not continue our analysis.

4. Consider f = q + 4q2 + 5q3 + 2q4 + 4q5 + 6q6 + q7 + q8 + 4q9 + 2q10 + 3q11 + 3q12 + ⋯ ∈ S12(Γ0(6); \overline{F}7). Since f is ordinary, we do not continue our analysis.

5. Consider f = q + q2 + 6q3 + q4 + 5q5 + 6q6 + 6q7 + q8 + q9 + 5q10 + 3q11 + 6q12 + 6q13 + ⋯ ∈ S14(Γ0(6); \overline{F}7). Since f is ordinary, we do not continue our analysis.

7.6 N = 8, ℓ = 7 (8 forms)

1. Consider f = q + 5q^{3} + 5q^{5} + 5q^{9} + 3q^{11} + 2q^{13} + ⋯ ∈ S_{10}(Γ_{0}(8); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 6q^{3} + 3q^{5} + 4q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(4)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 68q^{3} + 1510q^{5} + 1464 ∗ 7q^{7} + −15059q^{9} + 3916q^{11} + −176594q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 5q^{3} + 3q^{5} + 6q^{9} + 6q^{11} + q^{13} + ⋯ ∈ S_{12}(Γ_{0}(8); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 3q^{3} + 5q^{5} + 3q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{1} & 0 \\ 0 & u(3)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−26125321683010496+ O(7^{2}0)q^{3} +−5665204994317782+O(7^{2}0)q^{5} +2832602497167362∗7+O(7^{2}1)q^{7} +212619578001532778+ O(7^{2}1)q^{9} +231380275661284448+O(7^{2}1)q^{11} +−31202403233731204+O(7^{2}0)q^{13} +⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q + 2q^{3} + 4q^{5} + 6q^{9} + 6q^{11} + 6q^{13} + ⋯ ∈ S_{12}(Γ_{0}(8); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-4 newform g = q + 3q^{3} + 5q^{5} + 3q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{4} & 0 \\ 0 & u(3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+26125321683010552+ O(7^{2}0)q^{3}+5665204994325650+O(7^{2}0)q^{5}+−2832602497154354∗7+O(7^{2}1)q^{7}+−212619578000992576+ O(7^{2}1)q^{9} + −231380275661125368 + O(7^{2}1)q^{11} + 31202403234781680 + O(7^{2}0)q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

4. Consider f = q + 4q^{3} + q^{5} + 6q^{9} + 6q^{11} + 5q^{13} + ⋯ ∈ S_{14}(Γ_{0}(8); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-6 newform g = q + 6q^{3} + 3q^{5} + 4q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{0} & 0 \\ 0 & u(4)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+19813083833991809+ O(7^{2}0)q^{3}+−1619792884930289+O(7^{2}0)q^{5}+−1418566396767041∗7+O(7^{2}0)q^{7}+−37912683339532212+ O(7^{2}0)q^{9} + −11831635427128049 + O(7^{2}0)q^{11} + −25421190565315715 + O(7^{2}0)q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

5. Consider f = q + 3q3 + 5q5 + 3q7 + 3q9 + 5q11 + q13 + ⋯ ∈ S10(Γ0(8); \overline{F}7). Since f is ordinary, we do not continue our analysis.

6. Consider f = q + 6q3 + 3q5 + 4q7 + 3q9 + 5q11 + 2q13 + ⋯ ∈ S12(Γ0(8); \overline{F}7). Since f is ordinary, we do not continue our analysis.

7. Consider f = q + 2q3 + 3q5 + q7 + q9 + ⋯ ∈ S14(Γ0(8); \overline{F}7). Since f is ordinary, we do not continue our analysis.

8. Consider f = q + 2q5 + 6q7 + 4q9 + 3q11 + 2q13 + ⋯ ∈ S14(Γ0(8); \overline{F}7). Since f is ordinary, we do not continue our analysis.

7.7 N = 9, ℓ = 7 (11 forms)

1. Consider f = q + 3q^{2} + q^{4} + 4q^{5} + 5q^{10} + q^{11} + q^{13} + ⋯ ∈ S_{10}(Γ_{0}(9); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 6q^{2} + 4q^{4} + q^{5} + 2q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{1} & 0 \\ 0 & u(2)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −18q^{2} + −188q^{4} + 1530q^{5} + 1304 ∗ 7q^{7} + 1800 ∗ 7q^{8} + −27540q^{10} + −21132q^{11} + 31214q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 3q^{4} + ⋯ ∈ S_{12}(Γ_{0}(9); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-4 newform g = q + 6q^{4} + 6q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{2} & 0 \\ 0 & u(6)ω^{5} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + 472q^{4} + 32 ∗ 7 ∗ aq^{5} + 8300 ∗ 7q^{7} + −1576 ∗ aq^{8} + 11520 ∗ 7^{2}q^{10} + −3200 ∗ aq^{11} + 108950 ∗ 7q^{13} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −360 ∗ 7. The slope of f_{0} is 1.

3. Consider f = q+5q^{2} +2q^{4} +5q^{5} +4q^{10} +2q^{11} +6q^{13} +⋯ ∈ S_{14}(Γ_{0}(9); \overline{F}_{7}). We have ρ_{f} ⊗χ^{5} ∼ ρ_{g}, where g is the weight-6 newform g = q + 6q^{2} + 4q^{4} + q^{5} + 2q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{0} & 0 \\ 0 & u(2)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 12q^{2} + −8048q^{4} + 30210q^{5} +33584∗7q^{7} +−27840∗7q^{8} +362520q^{10} +11182908q^{11} +8049614q^{13} +⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

4. Consider f = q + 6q4 + 6q7 + ⋯ ∈ S10(Γ0(9); \overline{F}7). Since f is ordinary, we do not continue our analysis.

5. Consider f = q + q2 + 5q5 + 6q8 + 5q10 + q11 + ⋯ ∈ S10(Γ0(9); \overline{F}7). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

6. Consider f = q + 3q2 + 5q4 + 3q8 + 6q11 + ⋯ ∈ S12(Γ0(9); \overline{F}7). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

7. Consider f = q + 6q2 + 4q4 + q5 + 2q7 + 6q10 + 4q11 + q13 + ⋯ ∈ S12(Γ0(9); \overline{F}7). Since f is ordinary, we do not continue our analysis.

8. Consider f = q + z28q2 + q4 + z20q5 + q7 + z4q8 + q10 + z44q11 + 2q13 + ⋯ ∈ S14(Γ0(9); \overline{F}7). Also, z satisfies the equation x2 + 6x + 3 = 0. Since f is ordinary, we do not continue our analysis.

9. Consider f = q + z4q2 + q4 + z44q5 + q7 + z28q8 + q10 + z20q11 + 2q13 + ⋯ ∈ S14(Γ0(9); \overline{F}7). Also, z satisfies the equation x2 + 6x + 3 = 0. Since f is ordinary, we do not continue our analysis.

10. Consider f = q + 4q2 + q5 + 6q8 + 4q10 + 2q11 + ⋯ ∈ S14(Γ0(9); \overline{F}7). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

11. Consider f = q + q2 + 6q4 + 2q5 + 6q7 + 4q8 + 2q10 + 3q11 + 5q13 + ⋯ ∈ S14(Γ0(9); \overline{F}7). Since f is ordinary, we do not continue our analysis.

1. Consider f = q+5q^{2}+4q^{3}+4q^{4}+5q^{5}+6q^{6}+6q^{8}+3q^{9}+4q^{10}+4q^{11}+2q^{12}+3q^{13}+⋯ ∈ S_{10}(Γ_{0}(10); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 3q^{2} + 2q^{3} + 2q^{4} + 3q^{5} + 6q^{6} + 6q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{1} & 0 \\ 0 & u(6)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−16q^{2}+46q^{3}+256q^{4}+ −625q^{5}+−736q^{6}+−1474∗7q^{7}+−4096q^{8}+−17567q^{9}+10000q^{10}+−5568q^{11}+11776q^{12}+45986q^{13}+⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q+5q^{2}+6q^{3}+4q^{4}+2q^{5}+2q^{6}+6q^{8}+2q^{9}+3q^{10}+5q^{11}+3q^{12}+6q^{13}+⋯ ∈ S_{10}(Γ_{0}(10); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 3q^{2} + 3q^{3} + 2q^{4} + 4q^{5} + 2q^{6} + 3q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{1} & 0 \\ 0 & u(3)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −16q^{2} + −204q^{3} + 256q^{4} + 625q^{5} + 3264q^{6} + 776 ∗ 7q^{7} + −4096q^{8} + 21933q^{9} + −10000q^{10} + 73932q^{11} + −52224q^{12} + −114514q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q +4q^{2} +4q^{3} +2q^{4} +4q^{5} +2q^{6} +q^{8} +4q^{9} +2q^{10} +6q^{11} +q^{12} +2q^{13} +⋯ ∈ S_{12}(Γ_{0}(10); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-4 newform g = q + 2q^{2} + 6q^{3} + 4q^{4} + 5q^{5} + 5q^{6} + 3q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{4} & 0 \\ 0 & u(3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 32q^{2} + −318q^{3} + 1024q^{4} + −3125q^{5} + −10176q^{6} + −10102 ∗ 7q^{7} + 32768q^{8} + −76023q^{9} + −100000q^{10} + 238272q^{11} + −325632q^{12} + −2097478q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

4. Consider f = q +4q^{2} +3q^{3} +2q^{4} +3q^{5} +5q^{6} +q^{8} +4q^{9} +5q^{10} +6q^{11} +6q^{12} +5q^{13} +⋯ ∈ S_{12}(Γ_{0}(10); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 2q^{2} + 6q^{3} + 4q^{4} + 5q^{5} + 5q^{6} + 3q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{1} & 0 \\ 0 & u(3)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+32q^{2}+−16199033841643874+ O(7^{2}0)q^{3} + 1024q^{4} + 3125q^{5} + −39615485146931962 + O(7^{2}0)q^{6} + −756085249000834 ∗ 7 + O(7^{2}0)q^{7} + 32768q^{8}+30232314253304776+O(7^{2}0)q^{9}+100000q^{10}+−33439468469052759+O(7^{2}0)q^{11}+8980736059969232+ O(7^{2}0)q^{12} + 17721419729062005 + O(7^{2}0)q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

5. Consider f = q + 6q^{2} + 2q^{3} + q^{4} + 6q^{5} + 5q^{6} + 6q^{8} + q^{9} + q^{10} + 3q^{11} + 2q^{12} + q^{13} + ⋯ ∈ S_{14}(Γ_{0}(10); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-6 newform g = q + 3q^{2} + 3q^{3} + 2q^{4} + 4q^{5} + 2q^{6} + 3q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{0} & 0 \\ 0 & u(3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −64q^{2} + −26q^{3} + 4096q^{4} + −15625q^{5} + 1664q^{6} + 76934 ∗ 7q^{7} + −262144q^{8} + −1593647q^{9} + 1000000q^{10} + −3994848q^{11} + −106496q^{12} + −23834446q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

6. Consider f = q + 6q^{2} + 6q^{3} + q^{4} + q^{5} + q^{6} + 6q^{8} + 5q^{9} + 6q^{10} + q^{11} + 6q^{12} + 4q^{13} + ⋯ ∈ S_{14}(Γ_{0}(10); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-6 newform g = q + 3q^{2} + 2q^{3} + 2q^{4} + 3q^{5} + 6q^{6} + 6q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{0} & 0 \\ 0 & u(6)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −64q^{2} + 1224q^{3} + 4096q^{4} + 15625q^{5} + −78336q^{6} + −9316 ∗ 7q^{7} + −262144q^{8} + −96147q^{9} + −1000000q^{10} + 7427652q^{11} + 5013504q^{12} + 32243054q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

7. Consider f = q + 2q2 + 6q3 + 4q4 + 5q5 + 5q6 + 3q7 + q8 + 2q9 + 3q10 + 5q11 + 3q12 + 5q13 + ⋯ ∈ S10(Γ0(10); \overline{F}7). Since f is ordinary, we do not continue our analysis.

8. Consider f = q + 3q2 + 2q3 + 2q4 + 3q5 + 6q6 + 6q7 + 6q8 + 6q9 + 2q10 + 2q11 + 4q12 + 3q13 + ⋯ ∈ S12(Γ0(10); \overline{F}7). Since f is ordinary, we do not continue our analysis.

9. Consider f = q + 3q2 + 3q3 + 2q4 + 4q5 + 2q6 + 3q7 + 6q8 + 4q9 + 5q10 + 6q11 + 6q12 + 6q13 + ⋯ ∈ S12(Γ0(10); \overline{F}7). Since f is ordinary, we do not continue our analysis.

10. Consider f = q + 4q2 + 6q3 + 2q4 + 3q5 + 3q6 + q7 + q8 + 3q9 + 5q10 + 3q11 + 5q12 + ⋯ ∈ S12(Γ0(10); \overline{F}7). Since f is ordinary, we do not continue our analysis.

11. Consider f = q + q2 + q4 + 6q5 + 6q7 + q8 + 4q9 + 6q10 + 4q11 + q13 + ⋯ ∈ S14(Γ0(10); \overline{F}7). Since f is ordinary, we do not continue our analysis.

7.9 N = 11, ℓ = 7 (23 forms)

1. Consider f = q + 5q^{2} + 5q^{3} + 3q^{4} + q^{5} + 4q^{6} + 3q^{8} + 5q^{9} + 5q^{10} + 3q^{11} + q^{12} + ⋯ ∈ S_{10}(Γ_{0}(11); \overline{F}_{7}). Also, z satisfies the equation x^{2} + 6x + 3 = 0. We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 3q^{2} + 6q^{3} + 5q^{4} + 2q^{5} + 4q^{6} + 3q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{1} & 0 \\ 0 & u(3)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−30871450180857876+ O(7^{2}0)q^{2} +4773729341824567+O(7^{2}0)q^{3} +−36314553836254201+O(7^{2}0)q^{4} +−24637304674161458+ O(7^{2}0)q^{5}+−22136466302621933+O(7^{2}0)q^{6}+−5365660500392004∗7+O(7^{2}0)q^{7}+−30285531255457811+ O(7^{2}0)q^{8}+−2251524041307151+O(7^{2}0)q^{9}+35992958622474859+O(7^{2}0)q^{10}+−14641q^{11}+235045208610571934+ O(7^{2}1)q^{12} + 390369396082832 ∗ 7 + O(7^{2}0)q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 3q^{2} + 4q^{3} + q^{4} + 5q^{5} + 5q^{6} + 3q^{9} + q^{10} + 4q^{11} + 4q^{12} + 3q^{13} + ⋯ ∈ S_{10}(Γ_{0}(11); \overline{F}_{7}). Also, z satisfies the equation x^{3} + 6x^{2} + 4 = 0. We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-2 newform g = q + 5q^{2} + 6q^{3} + 2q^{4} + q^{5} + 2q^{6} + 5q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(3)ω^{2} & 0 \\ 0 & u(5)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+24777401154309464+ O(7^{2}0)q^{2} + 9068164705809584 + O(7^{2}0)q^{3} + 5636087969631820 + O(7^{2}0)q^{4} + −11708894028337977 + O(7^{2}0)q^{5}+13799444035639533+O(7^{2}0)q^{6}+−38139629513736171∗7+O(7^{2}1)q^{7}+−2086150468467182∗ 7 + O(7^{2}0)q^{8} + −17767431542269455 + O(7^{2}0)q^{9} + −22385172834451748 + O(7^{2}0)q^{10} + 14641q^{11} + −15003190329497842 + O(7^{2}0)q^{12} + 22403546794310594 + O(7^{2}0)q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q + z^{7}2q^{2} + z^{9}q^{3} + z^{1}05q^{4} + z^{3}0q^{5} + z^{8}1q^{6} + z^{5}8q^{8} + z^{2}66q^{9} + z^{1}02q^{10} + 4q^{11} + 2q^{12} + z^{5}4q^{13} + ⋯ ∈ S_{10}(Γ_{0}(11); \overline{F}_{7}). Also, z satisfies the equation x^{3} + 6x^{2} + 4 = 0. We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{1}86q^{2} + z^{2}37q^{3} + z^{3}33q^{4} + z^{1}44q^{5} + z^{8}1q^{6} + z^{7}6q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}66)ω^{1} & 0 \\ 0 & u(z^{7}6)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(27837678167433989+ O(7^{2}0)) ∗ a^{2} + (4061054072657776 + O(7^{2}0)) ∗ a + 35042109046646096 + O(7^{2}0)q^{3} + a^{2} − 512q^{4} + (38034084380105790+O(7^{2}0))∗a^{2}−(5149784015446046∗7+O(7^{2}0))∗a+41755180074486∗7+O(7^{2}0)q^{5}+ −(39119502853433172+O(7^{2}0))∗a^{2}+(4540937842883394+O(7^{2}0))∗a+7981711151246624+O(7^{2}0)q^{6}+ −(8277926740717399∗7+O(7^{2}1))∗a^{2} +(33564610154706479∗7+O(7^{2}1))∗a−39032964622270567∗7+ O(7^{2}1)q^{7}+−(51222887969695∗7+O(7^{2}0))∗a^{2}−(24130096886840683+O(7^{2}0))∗a+37908942984174977+ O(7^{2}0)q^{8}+(18245030635494335+O(7^{2}0))∗a^{2}−(24253935591846439+O(7^{2}0))∗a+28618913636236755+ O(7^{2}0)q^{9}+(30118047187886713∗7+O(7^{2}1))∗a^{2}+(12374366438588216+O(7^{2}1))∗a−159431918881716372+ O(7^{2}1)q^{10} + 14641q^{11} + −(5619986227906564 ∗ 7 + O(7^{2}0)) ∗ a^{2} + (418007817858361 ∗ 7 + O(7^{2}0)) ∗ a − 38235950949598526+O(7^{2}0)q^{12} +(12287174735587712+O(7^{2}0))∗a^{2} +(26605374927772883+O(7^{2}0))∗ a − 36462484578826797 + O(7^{2}0)q^{13} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 51222887969695 ∗ 7 + O(7^{2}0)x^{2} + 24130096886839659 + O(7^{2}0)x + −37908942984174977 + O(7^{2}0). The slope of f_{0} is 1.

4. Consider f = q + z^{1}08q^{2} + z^{9}9q^{3} + z^{1}5q^{4} + z^{1}02q^{5} + z^{2}07q^{6} + z^{1}06q^{8} + z^{3}8q^{9} + z^{2}10q^{10} + 4q^{11} + 2q^{12} + z^{2}52q^{13} + ⋯ ∈ S_{10}(Γ_{0}(11); \overline{F}_{7}). Also, z satisfies the equation x^{3} + 6x^{2} + 4 = 0. We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{2}22q^{2} + z^{3}27q^{3} + z^{2}43q^{4} + z^{2}16q^{5} + z^{2}07q^{6} + z^{3}04q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{3}8)ω^{1} & 0 \\ 0 & u(z^{3}04)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(27837678167433989+ O(7^{2}0)) ∗ a^{2} + (4061054072657776 + O(7^{2}0)) ∗ a + 35042109046646096 + O(7^{2}0)q^{3} + a^{2} − 512q^{4} + (38034084380105790+O(7^{2}0))∗a^{2}−(5149784015446046∗7+O(7^{2}0))∗a+41755180074486∗7+O(7^{2}0)q^{5}+ −(39119502853433172+O(7^{2}0))∗a^{2}+(4540937842883394+O(7^{2}0))∗a+7981711151246624+O(7^{2}0)q^{6}+ −(8277926740717399∗7+O(7^{2}1))∗a^{2} +(33564610154706479∗7+O(7^{2}1))∗a−39032964622270567∗7+ O(7^{2}1)q^{7}+−(51222887969695∗7+O(7^{2}0))∗a^{2}−(24130096886840683+O(7^{2}0))∗a+37908942984174977+ O(7^{2}0)q^{8}+(18245030635494335+O(7^{2}0))∗a^{2}−(24253935591846439+O(7^{2}0))∗a+28618913636236755+ O(7^{2}0)q^{9}+(30118047187886713∗7+O(7^{2}1))∗a^{2}+(12374366438588216+O(7^{2}1))∗a−159431918881716372+ O(7^{2}1)q^{10} + 14641q^{11} + −(5619986227906564 ∗ 7 + O(7^{2}0)) ∗ a^{2} + (418007817858361 ∗ 7 + O(7^{2}0)) ∗ a − 38235950949598526+O(7^{2}0)q^{12} +(12287174735587712+O(7^{2}0))∗a^{2} +(26605374927772883+O(7^{2}0))∗ a − 36462484578826797 + O(7^{2}0)q^{13} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 51222887969695 ∗ 7 + O(7^{2}0)x^{2} + 24130096886839659 + O(7^{2}0)x + −37908942984174977 + O(7^{2}0). The slope of f_{0} is 1.

5. Consider f = q + z^{1}62q^{2} + z^{6}3q^{3} + z^{5}1q^{4} + z^{2}10q^{5} + z^{2}25q^{6} + z^{6}4q^{8} + z^{1}52q^{9} + z^{3}0q^{10} + 4q^{11} + 2q^{12} + z^{3}6q^{13} + ⋯ ∈ S_{10}(Γ_{0}(11); \overline{F}_{7}). Also, z satisfies the equation x^{3} + 6x^{2} + 4 = 0. We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{2}76q^{2} + z^{2}91q^{3} + z^{2}79q^{4} + z^{3}24q^{5} + z^{2}25q^{6} + z^{1}90q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}52)ω^{1} & 0 \\ 0 & u(z^{1}90)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(27837678167433989+ O(7^{2}0)) ∗ a^{2} + (4061054072657776 + O(7^{2}0)) ∗ a + 35042109046646096 + O(7^{2}0)q^{3} + a^{2} − 512q^{4} + (38034084380105790+O(7^{2}0))∗a^{2}−(5149784015446046∗7+O(7^{2}0))∗a+41755180074486∗7+O(7^{2}0)q^{5}+ −(39119502853433172+O(7^{2}0))∗a^{2}+(4540937842883394+O(7^{2}0))∗a+7981711151246624+O(7^{2}0)q^{6}+ −(8277926740717399∗7+O(7^{2}1))∗a^{2} +(33564610154706479∗7+O(7^{2}1))∗a−39032964622270567∗7+ O(7^{2}1)q^{7}+−(51222887969695∗7+O(7^{2}0))∗a^{2}−(24130096886840683+O(7^{2}0))∗a+37908942984174977+ O(7^{2}0)q^{8}+(18245030635494335+O(7^{2}0))∗a^{2}−(24253935591846439+O(7^{2}0))∗a+28618913636236755+ O(7^{2}0)q^{9}+(30118047187886713∗7+O(7^{2}1))∗a^{2}+(12374366438588216+O(7^{2}1))∗a−159431918881716372+ O(7^{2}1)q^{10} + 14641q^{11} + −(5619986227906564 ∗ 7 + O(7^{2}0)) ∗ a^{2} + (418007817858361 ∗ 7 + O(7^{2}0)) ∗ a − 38235950949598526+O(7^{2}0)q^{12} +(12287174735587712+O(7^{2}0))∗a^{2} +(26605374927772883+O(7^{2}0))∗ a − 36462484578826797 + O(7^{2}0)q^{13} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 51222887969695 ∗ 7 + O(7^{2}0)x^{2} + 24130096886839659 + O(7^{2}0)x + −37908942984174977 + O(7^{2}0). The slope of f_{0} is 1.

6. Consider f = q+z^{3}3q^{2} +z^{1}0q^{3} +z^{6}q^{4} +z^{3}3q^{5} +z^{4}3q^{6} +z^{2}q^{8} +z^{3}3q^{9} +z^{1}8q^{10} +5q^{11} +2q^{12} +z^{6}q^{13} +⋯ ∈ S_{12}(Γ_{0}(11); \overline{F}_{7}). Also, z satisfies the equation x^{2} + 6x + 3 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + z^{1}7q^{2} + z^{2}6q^{3} + z^{2}2q^{4} + z^{1}7q^{5} + z^{4}3q^{6} + z^{2}5q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}3)ω^{1} & 0 \\ 0 & u(z^{2}5)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(1526861999260261+ O(7^{1}9))∗a−1594789776487798+O(7^{1}9)q^{3}+−(25739792713100455+O(7^{2}0))∗a−17835578556245635+ O(7^{2}0)q^{4} + (3427501219660162 + O(7^{1}9)) ∗ a − 282490867335 ∗ 7^{5} + O(7^{1}9)q^{5} + −(1930629739857657 + O(7^{1}9))∗a+2692481377313055+O(7^{1}9)q^{6}+(1254710459798158∗7+O(7^{2}0))∗a−2409975892731410∗7+ O(7^{2}0)q^{7}+(13645395825474968+O(7^{2}0))∗a−26382865636246640+O(7^{2}0)q^{8}+−(26049323056597189+ O(7^{2}0)) ∗ a + 3159716978932714 ∗ 7 + O(7^{2}0)q^{9} + (1565797003174233 + O(7^{1}9)) ∗ a − 72946278900823 + O(7^{1}9)q^{10} + −161051q^{11} + −(28812312361307 ∗ 7 + O(7^{1}9)) ∗ a + 1112425761109889 + O(7^{1}9)q^{12} + −(5290249945585664 + O(7^{1}9)) ∗ a + 191659696259903 + O(7^{1}9)q^{13} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2}+25739792713100455+O(7^{2}0)x+ 17835578556243587 + O(7^{2}0). The slope of f_{0} is 1.

7. Consider f = q+z^{3}3q^{2} +z^{3}4q^{3} +z^{6}q^{4} +z^{9}q^{5} +z^{1}9q^{6} +z^{2}q^{8} +z^{3}3q^{9} +z^{4}2q^{10} +5q^{11} +5q^{12} +z^{3}0q^{13} +⋯ ∈ S_{12}(Γ_{0}(11); \overline{F}_{7}). Also, z satisfies the equation x^{2} + 6x + 3 = 0. We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-4 newform g = q + z^{1}7q^{2} + z^{2}6q^{3} + z^{2}2q^{4} + z^{1}7q^{5} + z^{4}3q^{6} + z^{2}5q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}3)ω^{4} & 0 \\ 0 & u(z^{2}5)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(2403633796070289+ O(7^{1}9)) ∗ a + 2979174601081107 + O(7^{1}9)q^{3} + (21662035446559841 + O(7^{2}0)) ∗ a − 5148414536771278 + O(7^{2}0)q^{4} +−(872951253068868+O(7^{1}9))∗a+332432666028755∗7+O(7^{1}9)q^{5} +(4906377734381290+ O(7^{1}9))∗a−3806331877795355+O(7^{1}9)q^{6}+(793451168953439∗7+O(7^{2}0))∗a−208804192747914∗7^{2}+ O(7^{2}0)q^{7} + (12379833008748851 + O(7^{2}0)) ∗ a + 2876882332411434 + O(7^{2}0)q^{8} + (11185735187663562 + O(7^{2}0))∗a−504684607040532∗7+O(7^{2}0)q^{9}+−(4586788428634899+O(7^{1}9))∗a+5129554202989789+ O(7^{1}9)q^{10} + −161051q^{11} + (123098947583271 ∗ 7 + O(7^{1}9)) ∗ a − 5547712083207163 + O(7^{1}9)q^{12} + (5156693005075650 + O(7^{1}9)) ∗ a + 4604942271178595 + O(7^{1}9)q^{13} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −21662035446559841 + O(7^{2}0)x + 5148414536769230 + O(7^{2}0). The slope of f_{0} is 1.

8. Consider f = q + 6q^{2} + 4q^{3} + 6q^{4} + 3q^{5} + 3q^{6} + 3q^{8} + 6q^{9} + 4q^{10} + 6q^{11} + 3q^{12} + ⋯ ∈ S_{14}(Γ_{0}(11); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-6 newform g = q + 3q^{2} + 6q^{3} + 5q^{4} + 2q^{5} + 4q^{6} + 3q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{0} & 0 \\ 0 & u(3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−11525808211370020+ O(7^{2}0)q^{2} + 2916290527251742 + O(7^{1}9)q^{3} + 1427163846693458 + O(7^{2}0)q^{4} + 1676976543073898 + O(7^{1}9)q^{5} + −9644281299473818 + O(7^{2}0)q^{6} + 626026221346467 ∗ 7 + O(7^{1}9)q^{7} + 32074643719028767 + O(7^{2}0)q^{8}+2501352335802795+O(7^{1}9)q^{9}+−3579614233088400+O(7^{2}0)q^{10}+−1771561q^{11}+4901558653528491+ O(7^{1}9)q^{12} + 651732676710568 ∗ 7^{2} + O(7^{2}0)q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

9. Consider f = q + z^{4}8q^{2} + z^{6}q^{3} + z^{1}65q^{4} + z^{2}67q^{5} + z^{5}4q^{6} + z^{6}4q^{8} + z^{3}8q^{9} + z^{3}15q^{10} + q^{11} + 6q^{12} + z^{2}07q^{13} + ⋯ ∈ S_{14}(Γ_{0}(11); \overline{F}_{7}). Also, z satisfies the equation x^{3} + 6x^{2} + 4 = 0. We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{2}76q^{2} + z^{2}91q^{3} + z^{2}79q^{4} + z^{3}24q^{5} + z^{2}25q^{6} + z^{1}90q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}52)ω^{0} & 0 \\ 0 & u(z^{1}90)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(23564898988406929+ O(7^{2}0)) ∗ a^{2} + (23319969183186628 + O(7^{2}0)) ∗ a + 446287777056913 + O(7^{2}0)q^{3} + a^{2} − 8192q^{4} + (37408620879864044 + O(7^{2}0)) ∗ a^{2} − (4633362809955459 ∗ 7 + O(7^{2}0)) ∗ a − 790954768982913 ∗ 7 + O(7^{2}0)q^{5}+(26073395790775999+O(7^{2}0))∗a^{2}−(32504652521287725+O(7^{2}0))∗a−28311397716515978+ O(7^{2}0)q^{6}+(433180026532684∗7+O(7^{1}9))∗a^{2}+(591007389784562∗7+O(7^{1}9))∗a+33937562169769∗7+ O(7^{1}9)q^{7}+−(4905342319407886∗7+O(7^{2}0))∗a^{2}+(15150576161993628+O(7^{2}0))∗a−37594680433097990+ O(7^{2}0)q^{8}+(15484323206606431+O(7^{2}0))∗a^{2}−(9777854276633943+O(7^{2}0))∗a−32595562337427093+ O(7^{2}0)q^{9}+(78962326303016∗7^{3}+O(7^{2}0))∗a^{2}−(8844806773811312+O(7^{2}0))∗a+10601817371394682+ O(7^{2}0)q^{10} + 1771561q^{11} + (2876993263679547 ∗ 7 + O(7^{2}0)) ∗ a^{2} + (2917317402699678 ∗ 7 + O(7^{2}0)) ∗ a − 39858171996673458 + O(7^{2}0)q^{12} + −(29364245624675578 + O(7^{2}0)) ∗ a^{2} + (25076818954911898 + O(7^{2}0))∗a−4414131619729502+O(7^{2}0)q^{13} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} +4905342319407886∗7+O(7^{2}0)x^{2} +−15150576162010012+ O(7^{2}0)x + 37594680433097990 + O(7^{2}0). The slope of f_{0} is 1.

10. Consider f = q + 5q^{2} + 6q^{3} + 2q^{4} + q^{5} + 2q^{6} + 5q^{9} + 5q^{10} + q^{11} + 5q^{12} + 4q^{13} + ⋯ ∈ S_{14}(Γ_{0}(11); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-2 newform g = q + 5q^{2} + 6q^{3} + 2q^{4} + q^{5} + 2q^{6} + 5q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(3)ω^{1} & 0 \\ 0 & u(5)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+33517001184949483+ O(7^{2}0)q^{2} + 33231850863240343 + O(7^{2}0)q^{3} + −32058486536294461 + O(7^{2}0)q^{4} + 20294115003009107 + O(7^{2}0)q^{5} + 37470174623551378 + O(7^{2}0)q^{6} + 671106251284312 ∗ 7 + O(7^{1}9)q^{7} + 3896255426396357 ∗ 7 + O(7^{2}0)q^{8} + 11661003998567964 + O(7^{2}0)q^{9} + −7262792801023510 + O(7^{2}0)q^{10} + 1771561q^{11} + −2198220083637951 + O(7^{2}0)q^{12} + 3555824576434372 + O(7^{2}0)q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

11. Consider f = q + z17q2 + z26q3 + z22q4 + z17q5 + z43q6 + z25q7 + z2q8 + z17q9 + z34q10 + 3q11 + q12 + z6q13 + ⋯ ∈ S10(Γ0(11); \overline{F}7). Also, z satisfies the equation x2 + 6x + 3 = 0. Since f is ordinary, we do not continue our analysis.

12. Consider f = q + z23q2 + z38q3 + z10q4 + z23q5 + z13q6 + z31q7 + z14q8 + z23q9 + z46q10 + 3q11 + q12 + z42q13 + ⋯ ∈ S10(Γ0(11); \overline{F}7). Also, z satisfies the equation x2 + 6x + 3 = 0. Since f is ordinary, we do not continue our analysis.

13. Consider f = q + 6q2 + 5q3 + 2q5 + 2q6 + q8 + 5q9 + 5q10 + 4q11 + ⋯ ∈ S10(Γ0(11); \overline{F}7). Also, z satisfies the equation x3 + 6x2 + 4 = 0. I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

14. Consider f = q + z186q2 + z237q3 + z333q4 + z144q5 + z81q6 + z76q7 + z58q8 + z38q9 + z330q10 + 2q11 + 4q12 + z54q13 + ⋯ ∈ S12(Γ0(11); \overline{F}7). Also, z satisfies the equation x3 + 6x2 + 4 = 0. Since f is ordinary, we do not continue our analysis.

15. Consider f = q +z222q2 +z327q3 +z243q4 +z216q5 +z207q6 +z304q7 +z106q8 +z152q9 +z96q10 +2q11 +4q12 +z252q13 +⋯ ∈ S12(Γ0(11); \overline{F}7). Also, z satisfies the equation x3 + 6x2 + 4 = 0. Since f is ordinary, we do not continue our analysis.

16. Consider f = q + z276q2 + z291q3 + z279q4 + z324q5 + z225q6 + z190q7 + z64q8 + z266q9 + z258q10 + 2q11 + 4q12 + z36q13 + ⋯ ∈ S12(Γ0(11); \overline{F}7). Also, z satisfies the equation x3 + 6x2 + 4 = 0. Since f is ordinary, we do not continue our analysis.

17. Consider f = q + 3q2 + 6q3 + 5q4 + 2q5 + 4q6 + 3q7 + 3q8 + 3q9 + 6q10 + 5q11 + 2q12 + ⋯ ∈ S12(Γ0(11); \overline{F}7). Since f is ordinary, we do not continue our analysis.

18. Consider f = q + 4q3 + 5q4 + 6q5 + 6q7 + 6q9 + 6q11 + 6q12 + 3q13 + ⋯ ∈ S14(Γ0(11); \overline{F}7). Since f is ordinary, we do not continue our analysis.

19. Consider f = q + q3 + 5q4 + 3q5 + q7 + 5q9 + 6q11 + 5q12 + 3q13 + ⋯ ∈ S14(Γ0(11); \overline{F}7). Since f is ordinary, we do not continue our analysis.

20. Consider f = q + z44q2 + z43q3 + 3q4 + 5q5 + z39q6 + q7 + z44q8 + z31q9 + z36q10 + 6q11 + z3q12 + z13q13 + ⋯ ∈ S14(Γ0(11); \overline{F}7). Also, z satisfies the equation x2 + 6x + 3 = 0. Since f is ordinary, we do not continue our analysis.

21. Consider f = q + 5q2 + 6q3 + 2q4 + q5 + 2q6 + 5q7 + 5q9 + 5q10 + q11 + 5q12 + 4q13 + ⋯ ∈ S14(Γ0(11); \overline{F}7). Since f is ordinary, we do not continue our analysis.

22. Consider f = q + 3q2 + 4q3 + 6q5 + 5q6 + q8 + 6q9 + 4q10 + q11 + ⋯ ∈ S14(Γ0(11); \overline{F}7). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

23. Consider f = q + q2 + 2q3 + 6q4 + 5q5 + 2q6 + 6q7 + 4q8 + q9 + 5q10 + q11 + 5q12 + 4q13 + ⋯ ∈ S14(Γ0(11); \overline{F}7). Since f is ordinary, we do not continue our analysis.

1. Consider f = q + 5q^{3} + 6q^{5} + 4q^{9} + 4q^{11} + 4q^{13} + ⋯ ∈ S_{12}(Γ_{0}(12); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 3q^{3} + 3q^{5} + q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 243q^{3} + 2862q^{5} + 1304 ∗ 7q^{7} + 59049q^{9} + 668196q^{11} + 2052950q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 2q^{3} + q^{5} + 4q^{9} + 4q^{11} + 3q^{13} + ⋯ ∈ S_{12}(Γ_{0}(12); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-4 newform g = q + 3q^{3} + 3q^{5} + q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{4} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −243q^{3} + 9990q^{5} + −12304 ∗ 7q^{7} + 59049q^{9} + −806004q^{11} + −960250q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

3. Consider f = q + 3q3 + 3q5 + q7 + 2q9 + q11 + 4q13 + ⋯ ∈ S10(Γ0(12); \overline{F}7). Since f is ordinary, we do not continue our analysis.

4. Consider f = q + 6q3 + 4q5 + 6q7 + q9 + 2q11 + q13 + ⋯ ∈ S14(Γ0(12); \overline{F}7). Since f is ordinary, we do not continue our analysis.

5. Consider f = q + q3 + q7 + q9 + q11 + 2q13 + ⋯ ∈ S14(Γ0(12); \overline{F}7). Since f is ordinary, we do not continue our analysis.

7.11 N = 13, ℓ = 7 (33 forms)

1. Consider f = q + 6q^{2} + 5q^{3} + 2q^{5} + 2q^{6} + q^{8} + 5q^{9} + 5q^{10} + 6q^{11} + 6q^{13} + ⋯ ∈ S_{10}(Γ_{0}(13); \overline{F}_{7}). Also, z satisfies the equation x^{2} + 6x + 3 = 0. We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-8 newform g = q + 3q^{2} + 4q^{3} + 6q^{5} + 5q^{6} + q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{2} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+18238764059747678+ O(7^{2}0)q^{2} +−4719551943454812+O(7^{2}0)q^{3} +3745269092044937∗7+O(7^{2}0)q^{4} +14364434234263297+ O(7^{2}0)q^{5} + 756366197430006 + O(7^{2}0)q^{6} + −3904975213256372 ∗ 7 + O(7^{2}0)q^{7} + −8090606606283109 + O(7^{2}0)q^{8}+−22748373580684873+O(7^{2}0)q^{9}+22643564907970256+O(7^{2}0)q^{10}+−33291813972541027+ O(7^{2}0)q^{11} + 414186205162166 ∗ 7 + O(7^{2}0)q^{12} + −28561q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q +z^{3}4q^{2} +z^{4}q^{3} +z^{3}7q^{4} +z^{1}2q^{5} +z^{3}8q^{6} +zq^{8} +4q^{9} +z^{4}6q^{10} +2q^{11} +z^{4}1q^{12} +6q^{13} +⋯ ∈ S_{10}(Γ_{0}(13); \overline{F}_{7}). Also, z satisfies the equation x^{2} + 6x + 3 = 0. We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{2}q^{2} + z^{3}6q^{3} + z^{2}1q^{4} + z^{2}8q^{5} + z^{3}8q^{6} + 3q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{1} & 0 \\ 0 & u(3)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(26407275770547997+ O(7^{2}0))∗a−21497043120728541+O(7^{2}0)q^{3}+(10926801044427675+O(7^{2}0))∗a+37583490849978439+ O(7^{2}0)q^{4}+−(2409569079113520+O(7^{2}0))∗a−33628975283469285+O(7^{2}0)q^{5}+(11644729577740079+ O(7^{2}0))∗a−21494208035613427+O(7^{2}0)q^{6}+−(701708991362082∗7^{2}+O(7^{2}0))∗a−925439688618299∗ 7+O(7^{2}0)q^{7} +(6835984795072813+O(7^{2}0))∗a+10278889398300043+O(7^{2}0)q^{8} +(4307782437780713∗ 7+O(7^{2}0))∗a+38038822815563854+O(7^{2}0)q^{9}+(14919318776696172+O(7^{2}0))∗a−17637363433105905+ O(7^{2}0)q^{10}+(3578384245395751∗7+O(7^{2}0))∗a−5147909980767930+O(7^{2}0)q^{11}+(19513696373731640+ O(7^{2}0)) ∗ a + 29422974934658103 + O(7^{2}0)q^{12} + −28561q^{13} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −10926801044427675 + O(7^{2}0)x + −37583490849978951 + O(7^{2}0). The slope of f_{0} is 1.

3. Consider f = q+z^{4}6q^{2} +z^{2}8q^{3} +z^{1}9q^{4} +z^{3}6q^{5} +z^{2}6q^{6} +z^{7}q^{8} +4q^{9} +z^{3}4q^{10} +2q^{11} +z^{4}7q^{12} +6q^{13} +⋯ ∈ S_{10}(Γ_{0}(13); \overline{F}_{7}). Also, z satisfies the equation x^{2} + 6x + 3 = 0. We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{1}4q^{2} + z^{1}2q^{3} + z^{3}q^{4} + z^{4}q^{5} + z^{2}6q^{6} + 3q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{1} & 0 \\ 0 & u(3)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(26407275770547997+ O(7^{2}0))∗a−21497043120728541+O(7^{2}0)q^{3}+(10926801044427675+O(7^{2}0))∗a+37583490849978439+ O(7^{2}0)q^{4}+−(2409569079113520+O(7^{2}0))∗a−33628975283469285+O(7^{2}0)q^{5}+(11644729577740079+ O(7^{2}0))∗a−21494208035613427+O(7^{2}0)q^{6}+−(701708991362082∗7^{2}+O(7^{2}0))∗a−925439688618299∗ 7+O(7^{2}0)q^{7} +(6835984795072813+O(7^{2}0))∗a+10278889398300043+O(7^{2}0)q^{8} +(4307782437780713∗ 7+O(7^{2}0))∗a+38038822815563854+O(7^{2}0)q^{9}+(14919318776696172+O(7^{2}0))∗a−17637363433105905+ O(7^{2}0)q^{10}+(3578384245395751∗7+O(7^{2}0))∗a−5147909980767930+O(7^{2}0)q^{11}+(19513696373731640+ O(7^{2}0)) ∗ a + 29422974934658103 + O(7^{2}0)q^{12} + −28561q^{13} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −10926801044427675 + O(7^{2}0)x + −37583490849978951 + O(7^{2}0). The slope of f_{0} is 1.

4. Consider f = q + z^{6}5360q^{2} + z^{4}4032q^{3} + z^{6}8112q^{4} + z^{4}1968q^{5} + z^{1}09392q^{6} + z^{1}10768q^{8} + z^{1}03888q^{9} + z^{1}07328q^{10} + z^{6}7424q^{11} + z^{1}12144q^{12} + q^{13} + ⋯ ∈ S_{10}(Γ_{0}(13); \overline{F}_{7}). Also, z satisfies the equation x^{6} + x^{4} + 5x^{3} + 4x^{2} + 6x + 3 = 0. We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{3}04q^{2} + z^{1}4q^{3} + z^{8}4q^{4} + z^{2}36q^{5} + z^{3}18q^{6} + z^{7}1q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}71)ω^{1} & 0 \\ 0 & u(z^{7}1)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(1077691251316388+ O(7^{2}0)) ∗ a^{2} + (31241474456528662 + O(7^{2}0)) ∗ a + 38749208152044498 + O(7^{2}0)q^{3} + a^{2} − 512q^{4} + −(12135981513572536 + O(7^{2}0)) ∗ a^{2} + (9620833608511015 + O(7^{2}0)) ∗ a − 3771160044357978 ∗ 7 + O(7^{2}0)q^{5} +(8392678051344892+O(7^{2}0))∗a^{2} −(1485726033938591+O(7^{2}0))∗a−11253048580188871+ O(7^{2}0)q^{6}+(1450989311459988∗7+O(7^{2}0))∗a^{2}+(2836750518682972∗7+O(7^{2}0))∗a+1611869847230560∗ 7+O(7^{2}0)q^{7}+−(374868796274107∗7^{2}+O(7^{2}0))∗a^{2}−(38355287372638980+O(7^{2}0))∗a+6313684964894703+ O(7^{2}0)q^{8}+(22040599890190837+O(7^{2}0))∗a^{2}+(31303957070841656+O(7^{2}0))∗a+30865658895888175+ O(7^{2}0)q^{9}+(26813774858682205+O(7^{2}0))∗a^{2}−(55356017523619∗7^{3}+O(7^{2}0))∗a+15682896395583688+ O(7^{2}0)q^{10}+−(696043347116974+O(7^{2}0))∗a^{2}+(3854098865074884+O(7^{2}0))∗a+30324208109128348+ O(7^{2}0)q^{11}+−(36542067066020378+O(7^{2}0))∗a^{2}+(24849727955094926+O(7^{2}0))∗a−32971616367268097+ O(7^{2}0)q^{12} + 28561q^{13} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 374868796274107 ∗ 7^{2} + O(7^{2}0)x^{2} + 5479326767519708 ∗ 7 + O(7^{2}0)x + −6313684964894703 + O(7^{2}0). The slope of f_{0} is 1.

5. Consider f = q + z^{1}04576q^{2} + z^{7}2928q^{3} + z^{6}192q^{4} + z^{5}8480q^{5} + z^{5}9856q^{6} + z^{6}9488q^{8} + z^{2}1328q^{9} + z^{4}5408q^{10} + z^{1}376q^{11} + z^{7}9120q^{12} + q^{13} + ⋯ ∈ S_{10}(Γ_{0}(13); \overline{F}_{7}). Also, z satisfies the equation x^{6} + x^{4} + 5x^{3} + 4x^{2} + 6x + 3 = 0. We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{7}6q^{2} + z^{9}8q^{3} + z^{2}46q^{4} + z^{2}84q^{5} + z^{1}74q^{6} + z^{1}55q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}87)ω^{1} & 0 \\ 0 & u(z^{1}55)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(1077691251316388+ O(7^{2}0)) ∗ a^{2} + (31241474456528662 + O(7^{2}0)) ∗ a + 38749208152044498 + O(7^{2}0)q^{3} + a^{2} − 512q^{4} + −(12135981513572536 + O(7^{2}0)) ∗ a^{2} + (9620833608511015 + O(7^{2}0)) ∗ a − 3771160044357978 ∗ 7 + O(7^{2}0)q^{5} +(8392678051344892+O(7^{2}0))∗a^{2} −(1485726033938591+O(7^{2}0))∗a−11253048580188871+ O(7^{2}0)q^{6}+(1450989311459988∗7+O(7^{2}0))∗a^{2}+(2836750518682972∗7+O(7^{2}0))∗a+1611869847230560∗ 7+O(7^{2}0)q^{7}+−(374868796274107∗7^{2}+O(7^{2}0))∗a^{2}−(38355287372638980+O(7^{2}0))∗a+6313684964894703+ O(7^{2}0)q^{8}+(22040599890190837+O(7^{2}0))∗a^{2}+(31303957070841656+O(7^{2}0))∗a+30865658895888175+ O(7^{2}0)q^{9}+(26813774858682205+O(7^{2}0))∗a^{2}−(55356017523619∗7^{3}+O(7^{2}0))∗a+15682896395583688+ O(7^{2}0)q^{10}+−(696043347116974+O(7^{2}0))∗a^{2}+(3854098865074884+O(7^{2}0))∗a+30324208109128348+ O(7^{2}0)q^{11}+−(36542067066020378+O(7^{2}0))∗a^{2}+(24849727955094926+O(7^{2}0))∗a−32971616367268097+ O(7^{2}0)q^{12} + 28561q^{13} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 374868796274107 ∗ 7^{2} + O(7^{2}0)x^{2} + 5479326767519708 ∗ 7 + O(7^{2}0)x + −6313684964894703 + O(7^{2}0). The slope of f_{0} is 1.

6. Consider f = q + z^{2}6144q^{2} + z^{3}9904q^{3} + z^{4}3344q^{4} + z^{5}6416q^{5} + z^{6}6048q^{6} + z^{1}5824q^{8} + z^{3}1648q^{9} + z^{8}2560q^{10} + z^{9}632q^{11} + z^{8}3248q^{12} + q^{13} + ⋯ ∈ S_{10}(Γ_{0}(13); \overline{F}_{7}). Also, z satisfies the equation x^{6} + x^{4} + 5x^{3} + 4x^{2} + 6x + 3 = 0. We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{1}90q^{2} + z^{2}q^{3} + z^{1}2q^{4} + z^{2}78q^{5} + z^{1}92q^{6} + z^{5}9q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}83)ω^{1} & 0 \\ 0 & u(z^{5}9)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(1077691251316388+ O(7^{2}0)) ∗ a^{2} + (31241474456528662 + O(7^{2}0)) ∗ a + 38749208152044498 + O(7^{2}0)q^{3} + a^{2} − 512q^{4} + −(12135981513572536 + O(7^{2}0)) ∗ a^{2} + (9620833608511015 + O(7^{2}0)) ∗ a − 3771160044357978 ∗ 7 + O(7^{2}0)q^{5} +(8392678051344892+O(7^{2}0))∗a^{2} −(1485726033938591+O(7^{2}0))∗a−11253048580188871+ O(7^{2}0)q^{6}+(1450989311459988∗7+O(7^{2}0))∗a^{2}+(2836750518682972∗7+O(7^{2}0))∗a+1611869847230560∗ 7+O(7^{2}0)q^{7}+−(374868796274107∗7^{2}+O(7^{2}0))∗a^{2}−(38355287372638980+O(7^{2}0))∗a+6313684964894703+ O(7^{2}0)q^{8}+(22040599890190837+O(7^{2}0))∗a^{2}+(31303957070841656+O(7^{2}0))∗a+30865658895888175+ O(7^{2}0)q^{9}+(26813774858682205+O(7^{2}0))∗a^{2}−(55356017523619∗7^{3}+O(7^{2}0))∗a+15682896395583688+ O(7^{2}0)q^{10}+−(696043347116974+O(7^{2}0))∗a^{2}+(3854098865074884+O(7^{2}0))∗a+30324208109128348+ O(7^{2}0)q^{11}+−(36542067066020378+O(7^{2}0))∗a^{2}+(24849727955094926+O(7^{2}0))∗a−32971616367268097+ O(7^{2}0)q^{12} + 28561q^{13} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 374868796274107 ∗ 7^{2} + O(7^{2}0)x^{2} + 5479326767519708 ∗ 7 + O(7^{2}0)x + −6313684964894703 + O(7^{2}0). The slope of f_{0} is 1.

7. Consider f = q +4q^{2} +5q^{4} +4q^{8} +2q^{9} +q^{11} +6q^{13} +⋯ ∈ S_{12}(Γ_{0}(13); \overline{F}_{7}). Also, z satisfies the equation x^{2} + 6x + 3 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 2q^{2} + 3q^{4} + q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−36355254830082583+ O(7^{2}0)q^{2} + 3196627519955022 ∗ 7 + O(7^{2}0)q^{3} + 29346106439894291 + O(7^{2}0)q^{4} + 298588357548281 ∗ 7 + O(7^{2}0)q^{5}+1708032065327724∗7+O(7^{2}0)q^{6}+−4284339829459093∗7+O(7^{2}0)q^{7}+−33300029093768685+ O(7^{2}0)q^{8}+−1035727297615050+O(7^{2}0)q^{9}+−2705438795536675∗7+O(7^{2}0)q^{10}+−28621322355903654+ O(7^{2}0)q^{11} + 2023042756413492 ∗ 7 + O(7^{2}0)q^{12} + 371293q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

8. Consider f = q+z^{1}7q^{2}+z^{4}5q^{3}+z^{2}0q^{4}+z^{3}q^{5}+z^{1}4q^{6}+z^{3}9q^{8}+z^{2}8q^{9}+z^{2}0q^{10}+z^{2}6q^{11}+z^{1}7q^{12}+6q^{13}+⋯ ∈ S_{12}(Γ_{0}(13); \overline{F}_{7}). Also, z satisfies the equation x^{2} + 6x + 3 = 0. We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-4 newform g = q + zq^{2} + z^{3}7q^{3} + z^{3}6q^{4} + z^{1}1q^{5} + z^{3}8q^{6} + z^{3}7q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}1)ω^{4} & 0 \\ 0 & u(z^{3}7)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(28021382744190732+ O(7^{2}0))∗a+29075221695248335+O(7^{2}0)q^{3}+(37062850254546356+O(7^{2}0))∗a−25336027171839197+ O(7^{2}0)q^{4} + (9137506709279255 + O(7^{2}0)) ∗ a + 21130329446418833 + O(7^{2}0)q^{5} + (27050151177395587 + O(7^{2}0))∗a−8588893950849347+O(7^{2}0)q^{6}+−(679107457631557∗7+O(7^{2}0))∗a−1532777315004389∗7+ O(7^{2}0)q^{7}+−(3196942254541653+O(7^{2}0))∗a+3685562706407208+O(7^{2}0)q^{8}+−(20440276081116032+ O(7^{2}0))∗a+3827795802700185+O(7^{2}0)q^{9}+−(11678943798541983+O(7^{2}0))∗a+35581260741374843+ O(7^{2}0)q^{10}+(21274710360332768+O(7^{2}0))∗a−31856268635061651+O(7^{2}0)q^{11}+(30888815272483575+ O(7^{2}0)) ∗ a − 1454140763125364 ∗ 7 + O(7^{2}0)q^{12} + 371293q^{13} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −37062850254546356 + O(7^{2}0)x + 25336027171837149 + O(7^{2}0). The slope of f_{0} is 1.

9. Consider f = q+z^{2}3q^{2}+z^{2}7q^{3}+z^{4}4q^{4}+z^{2}1q^{5}+z^{2}q^{6}+z^{3}3q^{8}+z^{4}q^{9}+z^{4}4q^{10}+z^{3}8q^{11}+z^{2}3q^{12}+6q^{13}+⋯ ∈ S_{12}(Γ_{0}(13); \overline{F}_{7}). Also, z satisfies the equation x^{2} + 6x + 3 = 0. We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-4 newform g = q + z^{7}q^{2} + z^{1}9q^{3} + z^{1}2q^{4} + z^{2}9q^{5} + z^{2}6q^{6} + z^{1}9q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}9)ω^{4} & 0 \\ 0 & u(z^{1}9)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(28021382744190732+ O(7^{2}0))∗a+29075221695248335+O(7^{2}0)q^{3}+(37062850254546356+O(7^{2}0))∗a−25336027171839197+ O(7^{2}0)q^{4} + (9137506709279255 + O(7^{2}0)) ∗ a + 21130329446418833 + O(7^{2}0)q^{5} + (27050151177395587 + O(7^{2}0))∗a−8588893950849347+O(7^{2}0)q^{6}+−(679107457631557∗7+O(7^{2}0))∗a−1532777315004389∗7+ O(7^{2}0)q^{7}+−(3196942254541653+O(7^{2}0))∗a+3685562706407208+O(7^{2}0)q^{8}+−(20440276081116032+ O(7^{2}0))∗a+3827795802700185+O(7^{2}0)q^{9}+−(11678943798541983+O(7^{2}0))∗a+35581260741374843+ O(7^{2}0)q^{10}+(21274710360332768+O(7^{2}0))∗a−31856268635061651+O(7^{2}0)q^{11}+(30888815272483575+ O(7^{2}0)) ∗ a − 1454140763125364 ∗ 7 + O(7^{2}0)q^{12} + 371293q^{13} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −37062850254546356 + O(7^{2}0)x + 25336027171837149 + O(7^{2}0). The slope of f_{0} is 1.

10. Consider f = q + 4q^{2} + 5q^{4} + 4q^{8} + 2q^{9} + q^{11} + q^{13} + ⋯ ∈ S_{12}(Γ_{0}(13); \overline{F}_{7}). Also, z satisfies the equation x^{6} + x^{4} + 5x^{3} + 4x^{2} + 6x + 3 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 2q^{2} + 3q^{4} + q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−16445726030701936+ O(7^{2}0)q^{2} +4189107182921369∗7+O(7^{2}0)q^{3} +−36544267013703472+O(7^{2}0)q^{4} +−1673173333360570∗ 7+O(7^{2}0)q^{5}+3818825363746450∗7+O(7^{2}0)q^{6}+−3763255715045811∗7+O(7^{2}0)q^{7}+−39519223426377040+ O(7^{2}0)q^{8}+−17828849065823878+O(7^{2}0)q^{9}+−1231661461880160∗7+O(7^{2}0)q^{10}+34753324923820259+ O(7^{2}0)q^{11} + −492489757187607 ∗ 7 + O(7^{2}0)q^{12} + −371293q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

11. Consider f = q + z^{4}1667q^{2} + z^{5}1471q^{3} + z^{4}9020q^{4} + z^{6}6177q^{5} + z^{9}3138q^{6} + z^{9}5589q^{8} + z^{6}8628q^{9} + z^{1}07844q^{10} + z^{6}3726q^{11} + z^{1}00491q^{12} + q^{13} + ⋯ ∈ S_{12}(Γ_{0}(13); \overline{F}_{7}). Also, z satisfies the equation x^{6} + x^{4} + 5x^{3} + 4x^{2} + 6x + 3 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + zq^{2} + z^{3}7q^{3} + z^{3}6q^{4} + z^{1}1q^{5} + z^{3}8q^{6} + z^{3}7q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}1)ω^{1} & 0 \\ 0 & u(z^{3}7)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(24858059536616482+ O(7^{2}0))∗a+27577079197165171+O(7^{2}0)q^{3} +(37448322846586752+O(7^{2}0))∗a−3235560941186296+ O(7^{2}0)q^{4}+−(26981113215177841+O(7^{2}0))∗a−27410988609564893+O(7^{2}0)q^{5}+−(21933428747839745+ O(7^{2}0))∗a−36218700187818853+O(7^{2}0)q^{6} +(1487600004904868∗7+O(7^{2}0))∗a+5181527171766606∗ 7+O(7^{2}0)q^{7}+(9098703365898928+O(7^{2}0))∗a−14135833915540886+O(7^{2}0)q^{8}+(33372596272073750+ O(7^{2}0))∗a−32780654801961352+O(7^{2}0)q^{9} +(2912222458089271+O(7^{2}0))∗a+14381002453515553+ O(7^{2}0)q^{10}+−(39721507954084261+O(7^{2}0))∗a−6004586478496232+O(7^{2}0)q^{11}+−(21394170555465912+ O(7^{2}0)) ∗ a + 902583619638053 ∗ 7 + O(7^{2}0)q^{12} + −371293q^{13} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −37448322846586752 + O(7^{2}0)x + 3235560941184248 + O(7^{2}0). The slope of f_{0} is 1.

12. Consider f = q + z^{5}6373q^{2} + z^{7}353q^{3} + z^{1}07844q^{4} + z^{1}10295q^{5} + z^{6}3726q^{6} + z^{8}0883q^{8} + z^{9}804q^{9} + z^{4}9020q^{10} + z^{9}3138q^{11} + z^{1}15197q^{12} + q^{13} + ⋯ ∈ S_{12}(Γ_{0}(13); \overline{F}_{7}). Also, z satisfies the equation x^{6} + x^{4} + 5x^{3} + 4x^{2} + 6x + 3 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + z^{7}q^{2} + z^{1}9q^{3} + z^{1}2q^{4} + z^{2}9q^{5} + z^{2}6q^{6} + z^{1}9q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}9)ω^{1} & 0 \\ 0 & u(z^{1}9)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(24858059536616482+ O(7^{2}0))∗a+27577079197165171+O(7^{2}0)q^{3} +(37448322846586752+O(7^{2}0))∗a−3235560941186296+ O(7^{2}0)q^{4}+−(26981113215177841+O(7^{2}0))∗a−27410988609564893+O(7^{2}0)q^{5}+−(21933428747839745+ O(7^{2}0))∗a−36218700187818853+O(7^{2}0)q^{6} +(1487600004904868∗7+O(7^{2}0))∗a+5181527171766606∗ 7+O(7^{2}0)q^{7}+(9098703365898928+O(7^{2}0))∗a−14135833915540886+O(7^{2}0)q^{8}+(33372596272073750+ O(7^{2}0))∗a−32780654801961352+O(7^{2}0)q^{9} +(2912222458089271+O(7^{2}0))∗a+14381002453515553+ O(7^{2}0)q^{10}+−(39721507954084261+O(7^{2}0))∗a−6004586478496232+O(7^{2}0)q^{11}+−(21394170555465912+ O(7^{2}0)) ∗ a + 902583619638053 ∗ 7 + O(7^{2}0)q^{12} + −371293q^{13} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −37448322846586752 + O(7^{2}0)x + 3235560941184248 + O(7^{2}0). The slope of f_{0} is 1.

13. Consider f = q+z^{1}90q^{2}+z^{1}55q^{3}+z^{1}32q^{4}+z^{2}27q^{5}+z^{3}q^{6}+z^{2}02q^{8}+z^{2}90q^{9}+z^{7}5q^{10}+z^{1}18q^{11}+z^{2}87q^{12}+ 6q^{13} + ⋯ ∈ S_{14}(Γ_{0}(13); \overline{F}_{7}). Also, z satisfies the equation x^{3} + 6x^{2} + 4 = 0. We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{7}6q^{2} + z^{9}8q^{3} + z^{2}46q^{4} + z^{2}84q^{5} + z^{1}74q^{6} + z^{1}55q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}87)ω^{0} & 0 \\ 0 & u(z^{1}55)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(23564704238604447+ O(7^{2}0)) ∗ a^{2} − (37477698527152809 + O(7^{2}0)) ∗ a + 10063242979415074 + O(7^{2}0)q^{3} + a^{2} − 8192q^{4} + (20136000030785542 + O(7^{2}0)) ∗ a^{2} − (19990868132239370 + O(7^{2}0)) ∗ a + 3454282104617609 ∗ 7 + O(7^{2}0)q^{5}+(20777838790460260+O(7^{2}0))∗a^{2}+(27504583131820866+O(7^{2}0))∗a−31443928158816109+ O(7^{2}0)q^{6}+−(5555028395258113∗7+O(7^{2}0))∗a^{2}+(891278119877658∗7+O(7^{2}0))∗a−2890923861855912∗ 7+O(7^{2}0)q^{7}+(506870939569256∗7^{2}+O(7^{2}0))∗a^{2}+(11773863848622179+O(7^{2}0))∗a−38691889953301651+ O(7^{2}0)q^{8}+−(33263274134552832+O(7^{2}0))∗a^{2}+(27430308738465673+O(7^{2}0))∗a+11920500666346210+ O(7^{2}0)q^{9}+(32202090275955164+O(7^{2}0))∗a^{2}−(375819806480880∗7+O(7^{2}0))∗a−7329482726004120+ O(7^{2}0)q^{10}+−(21396170670304244+O(7^{2}0))∗a^{2}−(16517401540426420+O(7^{2}0))∗a+17840341826496872+ O(7^{2}0)q^{11}+−(22102840798298102+O(7^{2}0))∗a^{2}+(34772712092014713+O(7^{2}0))∗a−16618992163264095+ O(7^{2}0)q^{12} + −4826809q^{13} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + −506870939569256 ∗ 7^{2} + O(7^{2}0)x^{2} + −1681980549805509 ∗ 7 + O(7^{2}0)x + 38691889953301651 + O(7^{2}0). The slope of f_{0} is 1.

14. Consider f = q+z^{3}04q^{2}+z^{5}9q^{3}+z^{2}40q^{4}+z^{2}21q^{5}+z^{2}1q^{6}+z^{4}6q^{8}+z^{3}20q^{9}+z^{1}83q^{10}+z^{1}42q^{11}+z^{2}99q^{12}+ 6q^{13} + ⋯ ∈ S_{14}(Γ_{0}(13); \overline{F}_{7}). Also, z satisfies the equation x^{3} + 6x^{2} + 4 = 0. We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{1}90q^{2} + z^{2}q^{3} + z^{1}2q^{4} + z^{2}78q^{5} + z^{1}92q^{6} + z^{5}9q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}83)ω^{0} & 0 \\ 0 & u(z^{5}9)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(23564704238604447+ O(7^{2}0)) ∗ a^{2} − (37477698527152809 + O(7^{2}0)) ∗ a + 10063242979415074 + O(7^{2}0)q^{3} + a^{2} − 8192q^{4} + (20136000030785542 + O(7^{2}0)) ∗ a^{2} − (19990868132239370 + O(7^{2}0)) ∗ a + 3454282104617609 ∗ 7 + O(7^{2}0)q^{5}+(20777838790460260+O(7^{2}0))∗a^{2}+(27504583131820866+O(7^{2}0))∗a−31443928158816109+ O(7^{2}0)q^{6}+−(5555028395258113∗7+O(7^{2}0))∗a^{2}+(891278119877658∗7+O(7^{2}0))∗a−2890923861855912∗ 7+O(7^{2}0)q^{7}+(506870939569256∗7^{2}+O(7^{2}0))∗a^{2}+(11773863848622179+O(7^{2}0))∗a−38691889953301651+ O(7^{2}0)q^{8}+−(33263274134552832+O(7^{2}0))∗a^{2}+(27430308738465673+O(7^{2}0))∗a+11920500666346210+ O(7^{2}0)q^{9}+(32202090275955164+O(7^{2}0))∗a^{2}−(375819806480880∗7+O(7^{2}0))∗a−7329482726004120+ O(7^{2}0)q^{10}+−(21396170670304244+O(7^{2}0))∗a^{2}−(16517401540426420+O(7^{2}0))∗a+17840341826496872+ O(7^{2}0)q^{11}+−(22102840798298102+O(7^{2}0))∗a^{2}+(34772712092014713+O(7^{2}0))∗a−16618992163264095+ O(7^{2}0)q^{12} + −4826809q^{13} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + −506870939569256 ∗ 7^{2} + O(7^{2}0)x^{2} + −1681980549805509 ∗ 7 + O(7^{2}0)x + 38691889953301651 + O(7^{2}0). The slope of f_{0} is 1.

15. Consider f = q+z^{7}6q^{2}+z^{7}1q^{3}+z^{3}12q^{4}+z^{1}79q^{5}+z^{1}47q^{6}+z^{3}22q^{8}+z^{1}88q^{9}+z^{2}55q^{10}+z^{3}10q^{11}+z^{4}1q^{12}+ 6q^{13} + ⋯ ∈ S_{14}(Γ_{0}(13); \overline{F}_{7}). Also, z satisfies the equation x^{3} + 6x^{2} + 4 = 0. We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{3}04q^{2} + z^{1}4q^{3} + z^{8}4q^{4} + z^{2}36q^{5} + z^{3}18q^{6} + z^{7}1q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}71)ω^{0} & 0 \\ 0 & u(z^{7}1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(23564704238604447+ O(7^{2}0)) ∗ a^{2} − (37477698527152809 + O(7^{2}0)) ∗ a + 10063242979415074 + O(7^{2}0)q^{3} + a^{2} − 8192q^{4} + (20136000030785542 + O(7^{2}0)) ∗ a^{2} − (19990868132239370 + O(7^{2}0)) ∗ a + 3454282104617609 ∗ 7 + O(7^{2}0)q^{5}+(20777838790460260+O(7^{2}0))∗a^{2}+(27504583131820866+O(7^{2}0))∗a−31443928158816109+ O(7^{2}0)q^{6}+−(5555028395258113∗7+O(7^{2}0))∗a^{2}+(891278119877658∗7+O(7^{2}0))∗a−2890923861855912∗ 7+O(7^{2}0)q^{7}+(506870939569256∗7^{2}+O(7^{2}0))∗a^{2}+(11773863848622179+O(7^{2}0))∗a−38691889953301651+ O(7^{2}0)q^{8}+−(33263274134552832+O(7^{2}0))∗a^{2}+(27430308738465673+O(7^{2}0))∗a+11920500666346210+ O(7^{2}0)q^{9}+(32202090275955164+O(7^{2}0))∗a^{2}−(375819806480880∗7+O(7^{2}0))∗a−7329482726004120+ O(7^{2}0)q^{10}+−(21396170670304244+O(7^{2}0))∗a^{2}−(16517401540426420+O(7^{2}0))∗a+17840341826496872+ O(7^{2}0)q^{11}+−(22102840798298102+O(7^{2}0))∗a^{2}+(34772712092014713+O(7^{2}0))∗a−16618992163264095+ O(7^{2}0)q^{12} + −4826809q^{13} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + −506870939569256 ∗ 7^{2} + O(7^{2}0)x^{2} + −1681980549805509 ∗ 7 + O(7^{2}0)x + 38691889953301651 + O(7^{2}0). The slope of f_{0} is 1.

16. Consider f = q +z^{7}3530q^{2} +z^{4}9020q^{3} +z^{8}5785q^{4} +z^{1}07844q^{5} +z^{4}902q^{6} +z^{1}7157q^{8} +2q^{9} +z^{6}3726q^{10} + 4q^{11}+z^{1}7157q^{12}+q^{13}+⋯ ∈ S_{14}(Γ_{0}(13); \overline{F}_{7}). Also, z satisfies the equation x^{6}+x^{4}+5x^{3}+4x^{2}+6x+3 = 0. We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{1}4q^{2} + z^{1}2q^{3} + z^{3}q^{4} + z^{4}q^{5} + z^{2}6q^{6} + 3q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{0} & 0 \\ 0 & u(3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(7717665980667858+ O(7^{2}0))∗a−10691423681325054+O(7^{2}0)q^{3}+−(5879378750302886+O(7^{2}0))∗a−28878309303152417+ O(7^{2}0)q^{4}+−(9136772978480194+O(7^{2}0))∗a+33793478753135503+O(7^{2}0)q^{5}+−(32752493017622662+ O(7^{2}0))∗a+21561179967463383+O(7^{2}0)q^{6}+−(763521939201171∗7^{2}+O(7^{2}0))∗a+4651957060170795∗ 7+O(7^{2}0)q^{7}+(33099161618614809+O(7^{2}0))∗a+12146217013474267+O(7^{2}0)q^{8}+−(542242030239460∗ 7^{2}+O(7^{2}0))∗a+39470250908885288+O(7^{2}0)q^{9}+−(35481560194173971+O(7^{2}0))∗a−17763112971085554+ O(7^{2}0)q^{10}+−(105368760990832∗7^{3}+O(7^{2}0))∗a+31695680287411368+O(7^{2}0)q^{11}+−(5297203016040876+ O(7^{2}0)) ∗ a − 28978451731769733 + O(7^{2}0)q^{12} + 4826809q^{13} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 5879378750302886 + O(7^{2}0)x + 28878309303144225 + O(7^{2}0). The slope of f_{0} is 1.

17. Consider f = q + 3q^{2} + 4q^{3} + 6q^{5} + 5q^{6} + q^{8} + 6q^{9} + 4q^{10} + 5q^{11} + q^{13} + ⋯ ∈ S_{14}(Γ_{0}(13); \overline{F}_{7}). Also, z satisfies the equation x^{6} + x^{4} + 5x^{3} + 4x^{2} + 6x + 3 = 0. We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + 3q^{2} + 4q^{3} + 6q^{5} + 5q^{6} + q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+10743948263694852+ O(7^{2}0)q^{2}+−20063411012620230+O(7^{2}0)q^{3}+−4333152687900405∗7+O(7^{2}0)q^{4}+27724218787470519+ O(7^{2}0)q^{5}+4130640570972466+O(7^{2}0)q^{6}+−4459075454800968∗7+O(7^{2}0)q^{7}+−25704989492975778+ O(7^{2}0)q^{8} +−8413093252907344+O(7^{2}0)q^{9} +28850283146458778+O(7^{2}0)q^{10} +−17125835957858259+ O(7^{2}0)q^{11} + −1847911340498411 ∗ 7 + O(7^{2}0)q^{12} + 4826809q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

18. Consider f = q +z^{4}4118q^{2} +z^{1}07844q^{3} +z^{1}2255q^{4} +z^{4}9020q^{5} +z^{3}4314q^{6} +z^{2}451q^{8} +2q^{9} +z^{9}3138q^{10} + 4q^{11}+z^{2}451q^{12}+q^{13}+⋯ ∈ S_{14}(Γ_{0}(13); \overline{F}_{7}). Also, z satisfies the equation x^{6}+x^{4}+5x^{3}+4x^{2}+6x+3 = 0. We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{2}q^{2} + z^{3}6q^{3} + z^{2}1q^{4} + z^{2}8q^{5} + z^{3}8q^{6} + 3q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{0} & 0 \\ 0 & u(3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(7717665980667858+ O(7^{2}0))∗a−10691423681325054+O(7^{2}0)q^{3}+−(5879378750302886+O(7^{2}0))∗a−28878309303152417+ O(7^{2}0)q^{4}+−(9136772978480194+O(7^{2}0))∗a+33793478753135503+O(7^{2}0)q^{5}+−(32752493017622662+ O(7^{2}0))∗a+21561179967463383+O(7^{2}0)q^{6}+−(763521939201171∗7^{2}+O(7^{2}0))∗a+4651957060170795∗ 7+O(7^{2}0)q^{7}+(33099161618614809+O(7^{2}0))∗a+12146217013474267+O(7^{2}0)q^{8}+−(542242030239460∗ 7^{2}+O(7^{2}0))∗a+39470250908885288+O(7^{2}0)q^{9}+−(35481560194173971+O(7^{2}0))∗a−17763112971085554+ O(7^{2}0)q^{10}+−(105368760990832∗7^{3}+O(7^{2}0))∗a+31695680287411368+O(7^{2}0)q^{11}+−(5297203016040876+ O(7^{2}0)) ∗ a − 28978451731769733 + O(7^{2}0)q^{12} + 4826809q^{13} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 5879378750302886 + O(7^{2}0)x + 28878309303144225 + O(7^{2}0). The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

19. Consider f = q + 2q2 + 3q4 + q7 + 4q8 + q9 + 2q11 + 6q13 + ⋯ ∈ S10(Γ0(13); \overline{F}7). Also, z satisfies the equation x2 + 6x + 3 = 0. Since f is ordinary, we do not continue our analysis.

20. Consider f = q +z2451q2 +z90687q3 +z88236q4 +z26961q5 +z93138q6 +z90687q7 +z95589q8 +z29412q9 +z29412q10 +z102942q11 +z61275q12 + q13 + ⋯ ∈ S10(Γ0(13); \overline{F}7). Also, z satisfies the equation x6 + x4 + 5x3 + 4x2 + 6x + 3 = 0. Since f is ordinary, we do not continue our analysis.

21. Consider f = q +z17157q2 +z46569q3 +z29412q4 +z71079q5 +z63726q6 +z46569q7 +z80883q8 +z88236q9 +z88236q10 +z14706q11 +z75981q12 + q13 + ⋯ ∈ S10(Γ0(13); \overline{F}7). Also, z satisfies the equation x6 + x4 + 5x3 + 4x2 + 6x + 3 = 0. Since f is ordinary, we do not continue our analysis.

22. Consider f = q + z2q2 + z36q3 + z21q4 + z28q5 + z38q6 + 3q7 + zq8 + q9 + z30q10 + q11 + z9q12 + 6q13 + ⋯ ∈ S12(Γ0(13); \overline{F}7). Also, z satisfies the equation x2 + 6x + 3 = 0. Since f is ordinary, we do not continue our analysis.

23. Consider f = q + z14q2 + z12q3 + z3q4 + z4q5 + z26q6 + 3q7 + z7q8 + q9 + z18q10 + q11 + z15q12 + 6q13 + ⋯ ∈ S12(Γ0(13); \overline{F}7). Also, z satisfies the equation x2 + 6x + 3 = 0. Since f is ordinary, we do not continue our analysis.

24. Consider f = q + z65360q2 + z688q3 + z4128q4 + z95632q5 + z66048q6 + z20296q7 + z15824q8 + z70864q9 + z43344q10 + z88064q11 + z4816q12 + q13 + ⋯ ∈ S12(Γ0(13); \overline{F}7). Also, z satisfies the equation x6 + x4 + 5x3 + 4x2 + 6x + 3 = 0. Since f is ordinary, we do not continue our analysis.

25. Consider f = q + z104576q2 + z4816q3 + z28896q4 + z81184q5 + z109392q6 + z24424q7 + z110768q8 + z25456q9 + z68112q10 + z28208q11 + z33712q12 + q13 + ⋯ ∈ S12(Γ0(13); \overline{F}7). Also, z satisfies the equation x6 + x4 + 5x3 + 4x2 + 6x + 3 = 0. Since f is ordinary, we do not continue our analysis.

26. Consider f = q + z26144q2 + z33712q3 + z84624q4 + z97696q5 + z59856q6 + z53320q7 + z69488q8 + z60544q9 + z6192q10 + z79808q11 + z688q12 + q13 + ⋯ ∈ S12(Γ0(13); \overline{F}7). Also, z satisfies the equation x6 + x4 + 5x3 + 4x2 + 6x + 3 = 0. Since f is ordinary, we do not continue our analysis.

27. Consider f = q + 4q2 + 3q3 + 5q6 + q7 + 6q8 + 6q9 + 2q11 + 6q13 + ⋯ ∈ S14(Γ0(13); \overline{F}7). Also, z satisfies the equation x3 + 6x2 + 4 = 0. Since f is ordinary, we do not continue our analysis.

28. Consider f = q + 5q2 + 2q4 + 4q5 + 6q7 + 4q9 + 6q10 + q11 + 6q13 + ⋯ ∈ S14(Γ0(13); \overline{F}7). Also, z satisfies the equation x3 + 6x2 + 4 = 0. Since f is ordinary, we do not continue our analysis.

29. Consider f = q + 3q2 + 4q3 + 6q5 + 5q6 + q7 + q8 + 6q9 + 4q10 + 5q11 + 6q13 + ⋯ ∈ S14(Γ0(13); \overline{F}7). Also, z satisfies the equation x3 + 6x2 + 4 = 0. Since f is ordinary, we do not continue our analysis.

30. Consider f = q + 5q3 + 5q4 + 4q5 + q7 + q9 + 4q12 + q13 + ⋯ ∈ S14(Γ0(13); \overline{F}7). Also, z satisfies the equation x6 + x4 + 5x3 + 4x2 + 6x + 3 = 0. Since f is ordinary, we do not continue our analysis.

31. Consider f = q +z80152q2 +z72240q3 +z65016q4 +z113176q5 +z34744q6 +6q7 +z65016q8 +z38872q9 +z75680q10 +z13416q11 +3q12 +q13 +⋯ ∈ S14(Γ0(13); \overline{F}7). Also, z satisfies the equation x6 + x4 + 5x3 + 4x2 + 6x + 3 = 0. Since f is ordinary, we do not continue our analysis.

32. Consider f = q +z90472q2 +z35088q3 +z102168q4 +z86344q5 +z7912q6 +6q7 +z102168q8 +z36808q9 +z59168q10 +z93912q11 +3q12 +q13 +⋯ ∈ S14(Γ0(13); \overline{F}7). Also, z satisfies the equation x6 + x4 + 5x3 + 4x2 + 6x + 3 = 0. Since f is ordinary, we do not continue our analysis.

33. Consider f = q + z45064q2 + z10320q3 + z9288q4 + z16168q5 + z55384q6 + 6q7 + z9288q8 + z22360q9 + z61232q10 + z69144q11 + 3q12 + q13 + ⋯ ∈ S14(Γ0(13); \overline{F}7). Also, z satisfies the equation x6 + x4 + 5x3 + 4x2 + 6x + 3 = 0. Since f is ordinary, we do not continue our analysis.

1. Consider f = q+4q^{3} +6q^{4} +5q^{5} +2q^{9} +3q^{12} +6q^{13} +⋯ ∈ S_{10}(Γ_{0}(15); \overline{F}_{7}). We have ρ_{f} ⊗χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 2q^{3} + 3q^{4} + 3q^{5} + 5q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(3)ω^{1} & 0 \\ 0 & u(5)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 4724792588377961 ∗ 7 + O(7^{2}0)q^{2} + 81q^{3} + −9940716126626615 + O(7^{2}0)q^{4} + −625q^{5} + −4854236644072021 ∗ 7 + O(7^{2}0)q^{6} + −25211586056330585∗7+O(7^{2}1)q^{7}+−2067322582271221∗7+O(7^{2}0)q^{8}+6561q^{9}+−681514724581588∗ 7+O(7^{2}0)q^{10}+−1088927546902001∗7+O(7^{2}0)q^{11}+−7275343280635805+O(7^{2}0)q^{12}+26970258623687392+ O(7^{2}0)q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q+5q^{2}+4q^{3}+3q^{4}+5q^{5}+6q^{6}+3q^{8}+2q^{9}+4q^{10}+5q^{11}+5q^{12}+2q^{13}+⋯ ∈ S_{10}(Γ_{0}(15); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-2 newform g = q + 6q^{2} + 6q^{3} + 6q^{4} + q^{5} + q^{6} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{9}. The form f_{0} = q + −33073548118645708 + O(7^{2}0)q^{2} + 81q^{3} + 9940716126628136 + O(7^{2}0)q^{4} + −625q^{5} + 33979656508505686+O(7^{2}0)q^{6} +3601655150904127∗7^{2} +O(7^{2}1)q^{7} +14471258075948194+O(7^{2}0)q^{8} + 6561q^{9}+4770603072059241+O(7^{2}0)q^{10}+7622492828349495+O(7^{2}0)q^{11}+7275343280759006+O(7^{2}0)q^{12}+ −26970258623543716 + O(7^{2}0)q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 2.

3. Consider f = q+z^{9}q^{2} +3q^{3} +z^{1}7q^{4} +2q^{5} +z^{1}7q^{6} +z^{1}3q^{8} +2q^{9} +z^{2}5q^{10} +z^{3}1q^{11} +z^{2}5q^{12} +z^{4}5q^{13} +⋯ ∈ S_{10}(Γ_{0}(15); \overline{F}_{7}). Also, z satisfies the equation x^{2} + 6x + 3 = 0. We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{2}5q^{2} + 5q^{3} + zq^{4} + 4q^{5} + z^{1}7q^{6} + z^{4}q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{4}4)ω^{1} & 0 \\ 0 & u(z^{4})ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + −81q^{3} + 31 ∗ a − 30 ∗ 7q^{4} + 625q^{5} + −81 ∗ aq^{6} + 32 ∗ 7 ∗ a + 512 ∗ 7q^{7} + 239 ∗ a + 9362q^{8} + 6561q^{9} + 625 ∗ aq^{10} + −2368 ∗ a + 25948q^{11} + −2511 ∗ a + 2430 ∗ 7q^{12} + −5344 ∗ a + 94974q^{13} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −31x + −302. The slope of f_{0} is 1.

4. Consider f = q+z^{1}5q^{2}+3q^{3}+z^{2}3q^{4}+2q^{5}+z^{2}3q^{6}+z^{4}3q^{8}+2q^{9}+z^{3}1q^{10}+z^{2}5q^{11}+z^{3}1q^{12}+z^{2}7q^{13}+⋯ ∈ S_{10}(Γ_{0}(15); \overline{F}_{7}). Also, z satisfies the equation x^{2} + 6x + 3 = 0. We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{3}1q^{2} + 5q^{3} + z^{7}q^{4} + 4q^{5} + z^{2}3q^{6} + z^{2}8q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}0)ω^{1} & 0 \\ 0 & u(z^{2}8)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + −81q^{3} + 31 ∗ a − 30 ∗ 7q^{4} + 625q^{5} + −81 ∗ aq^{6} + 32 ∗ 7 ∗ a + 512 ∗ 7q^{7} + 239 ∗ a + 9362q^{8} + 6561q^{9} + 625 ∗ aq^{10} + −2368 ∗ a + 25948q^{11} + −2511 ∗ a + 2430 ∗ 7q^{12} + −5344 ∗ a + 94974q^{13} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −31x + −302. The slope of f_{0} is 1.

5. Consider f = q + 6q^{2} + 2q^{3} + 4q^{4} + 4q^{5} + 5q^{6} + 4q^{9} + 3q^{10} + 2q^{11} + q^{12} + 4q^{13} + ⋯ ∈ S_{12}(Γ_{0}(15); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 3q^{2} + 4q^{3} + q^{4} + 2q^{5} + 5q^{6} + 6q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{1} & 0 \\ 0 & u(6)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +6986710225391042+ O(7^{2}0)q^{2} + −243q^{3} + −11034966632473185 + O(7^{2}0)q^{4} + −3125q^{5} + −22132992520171185 + O(7^{2}0)q^{6} + −13456347992322657∗7+O(7^{2}1)q^{7}+−1459323355226542∗7+O(7^{2}0)q^{8}+59049q^{9}+29611511198682024+ O(7^{2}0)q^{10}+23709742589519408+O(7^{2}0)q^{11}+−31440162427824079+O(7^{2}0)q^{12}+15498634927858149+ O(7^{2}0)q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

6. Consider f = q + 2q^{2} + 2q^{3} + 4q^{5} + 4q^{6} + 6q^{8} + 4q^{9} + q^{10} + 5q^{11} + 6q^{13} + ⋯ ∈ S_{12}(Γ_{0}(15); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-4 newform g = q + q^{2} + 3q^{3} + 5q^{5} + 3q^{6} + 4q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{4} & 0 \\ 0 & u(4)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−6986710225391055+ O(7^{2}0)q^{2} + −243q^{3} + 1576423804638582 ∗ 7 + O(7^{2}0)q^{4} + −3125q^{5} + 22132992520174344 + O(7^{2}0)q^{6} + 13456347992323769∗7+O(7^{2}1)q^{7} +10215263486620933+O(7^{2}0)q^{8} +59049q^{9} +−29611511198641399+ O(7^{2}0)q^{10}+−23709742589223840+O(7^{2}0)q^{11}+4491451775511436∗7+O(7^{2}0)q^{12}+−15498634927200657+ O(7^{2}0)q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

7. Consider f = q + 6q^{2} + 5q^{3} + 4q^{4} + 3q^{5} + 2q^{6} + 4q^{9} + 4q^{10} + 2q^{11} + 6q^{12} + 3q^{13} + ⋯ ∈ S_{12}(Γ_{0}(15); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-4 newform g = q + 3q^{2} + 4q^{3} + q^{4} + 2q^{5} + 5q^{6} + 6q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{4} & 0 \\ 0 & u(6)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+17016513340204580+ O(7^{2}0)q^{2} + 243q^{3} + −29082468024135704 + O(7^{2}0)q^{4} + 3125q^{5} + −14185105806111112 + O(7^{2}0)q^{6} + 5396370061046494 ∗ 7 + O(7^{2}0)q^{7} + 193399401046111 ∗ 7^{2} + O(7^{2}0)q^{8} + 59049q^{9} + 34954833929719834 + O(7^{2}0)q^{10} +34266114840756507+O(7^{2}0)q^{11} +34471970622492017+O(7^{2}0)q^{12} +17887534118948637+ O(7^{2}0)q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

8. Consider f = q + 2q^{2} + 5q^{3} + 3q^{5} + 3q^{6} + 6q^{8} + 4q^{9} + 6q^{10} + 5q^{11} + q^{13} + ⋯ ∈ S_{12}(Γ_{0}(15); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + q^{2} + 3q^{3} + 5q^{5} + 3q^{6} + 4q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(4)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +7331023606406831+ O(7^{2}0)q^{2} + 243q^{3} + 1536937681856897 ∗ 7 + O(7^{2}0)q^{4} + 3125q^{5} + 26008877809395911 + O(7^{2}0)q^{6} + −4044974932987831 ∗ 7 + O(7^{2}0)q^{7} + 21206378397953475 + O(7^{2}0)q^{8} + 59049q^{9} + 9068342606702588 + O(7^{2}0)q^{10}+11333354898510447+O(7^{2}0)q^{11}+−2687684426087748∗7+O(7^{2}0)q^{12}+−14865760551194923+ O(7^{2}0)q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

9. Consider f = q +z^{4}1q^{2} +q^{3} +z^{3}3q^{4} +6q^{5} +z^{4}1q^{6} +z^{1}3q^{8} +q^{9} +z^{1}7q^{10} +z^{4}7q^{11} +z^{3}3q^{12} +z^{2}1q^{13} +⋯ ∈ S_{14}(Γ_{0}(15); \overline{F}_{7}). Also, z satisfies the equation x^{2} + 6x + 3 = 0. We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{2}5q^{2} + 5q^{3} + zq^{4} + 4q^{5} + z^{1}7q^{6} + z^{4}q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{4}4)ω^{0} & 0 \\ 0 & u(z^{4})ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + 729q^{3} + 131 ∗ a − 780 ∗ 7q^{4} + −15625q^{5} + 729 ∗ aq^{6} + 232 ∗ 7 ∗ a + 20252 ∗ 7q^{7} + 3509 ∗ a + 357892q^{8} + 531441q^{9} + −15625∗aq^{10} +35552∗a+794288q^{11} +95499∗a−568620∗7q^{12} +3016∗a+683034q^{13} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} +−131x+−2732. The slope of f_{0} is 1.

10. Consider f = q +z^{4}7q^{2} +q^{3} +z^{3}9q^{4} +6q^{5} +z^{4}7q^{6} +z^{4}3q^{8} +q^{9} +z^{2}3q^{10} +z^{4}1q^{11} +z^{3}9q^{12} +z^{3}q^{13} +⋯ ∈ S_{14}(Γ_{0}(15); \overline{F}_{7}). Also, z satisfies the equation x^{2} + 6x + 3 = 0. We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{3}1q^{2} + 5q^{3} + z^{7}q^{4} + 4q^{5} + z^{2}3q^{6} + z^{2}8q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}0)ω^{0} & 0 \\ 0 & u(z^{2}8)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + 729q^{3} + 131 ∗ a − 780 ∗ 7q^{4} + −15625q^{5} + 729 ∗ aq^{6} + 232 ∗ 7 ∗ a + 20252 ∗ 7q^{7} + 3509 ∗ a + 357892q^{8} + 531441q^{9} + −15625∗aq^{10} +35552∗a+794288q^{11} +95499∗a−568620∗7q^{12} +3016∗a+683034q^{13} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} +−131x+−2732. The slope of f_{0} is 1.

11. Consider f = q + 6q^{3} + 5q^{4} + q^{5} + q^{9} + 2q^{12} + q^{13} + ⋯ ∈ S_{14}(Γ_{0}(15); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-6 newform g = q + 2q^{3} + 3q^{4} + 3q^{5} + 5q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(3)ω^{0} & 0 \\ 0 & u(5)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 5189973582128632 ∗ 7 + O(7^{2}0)q^{2} + −729q^{3} + 45309890105238015 + O(7^{2}1)q^{4} + 15625q^{5} + 942460172110748 ∗ 7 + O(7^{2}0)q^{6} + −24826402549611∗7+O(7^{1}9)q^{7}+1872658279172179∗7+O(7^{2}0)q^{8}+531441q^{9}+1596872015335698∗7+ O(7^{2}0)q^{10}+−335192499846327∗7+O(7^{2}0)q^{11}+−76703905804756522+O(7^{2}1)q^{12}+−31450250938702742+ O(7^{2}0)q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

12. Consider f = q + 6q^{2} + 6q^{3} + 6q^{4} + q^{5} + q^{6} + 3q^{8} + q^{9} + 6q^{10} + 3q^{11} + q^{12} + 5q^{13} + ⋯ ∈ S_{14}(Γ_{0}(15); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-2 newform g = q + 6q^{2} + 6q^{3} + 6q^{4} + q^{5} + q^{6} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{1}. The form f_{0} = q + aq^{2} + −729q^{3} + −(36329815074900489 + O(7^{2}0)) ∗ a − 12979881045403891 + O(7^{2}0)q^{4} + 15625q^{5} +−729∗aq^{6} +−(21798591536378883∗7^{−}1+O(7^{1}9))∗a−3273065291826409∗7^{−}1+O(7^{1}9)q^{7} + −(32330009059826591 + O(7^{2}0)) ∗ a − 34022661322911448 + O(7^{2}0)q^{8} + 531441q^{9} + 15625 ∗ aq^{10} + (21332298747079424+O(7^{2}0))∗a+20758949022325523+O(7^{2}0)q^{11}+−(6597221204727851+O(7^{2}0))∗a− 32946407316391580+O(7^{2}0)q^{12}+(20924590739614594+O(7^{2}0))∗a−15244769172240830+O(7^{2}0)q^{13}+ ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 36329815074900489 + O(7^{2}0)x + 12979881045395699 + O(7^{2}0). The slope of f_{0} is 1/2. —————————————————————————————————- (Uninteresting cases below.)

13. Consider f = q + 3q2 + 4q3 + q4 + 2q5 + 5q6 + 6q7 + 2q9 + 6q10 + 4q11 + 4q12 + 4q13 + ⋯ ∈ S10(Γ0(15); \overline{F}7). Since f is ordinary, we do not continue our analysis.

14. Consider f = q + q2 + 3q3 + 5q5 + 3q6 + 4q7 + 6q8 + 2q9 + 5q10 + 3q11 + q13 + ⋯ ∈ S10(Γ0(15); \overline{F}7). Since f is ordinary, we do not continue our analysis.

15. Consider f = q + 2q3 + 3q4 + 3q5 + 5q7 + 4q9 + 6q12 + 6q13 + ⋯ ∈ S12(Γ0(15); \overline{F}7). Since f is ordinary, we do not continue our analysis.

16. Consider f = q + z25q2 + 5q3 + zq4 + 4q5 + z17q6 + z4q7 + z13q8 + 4q9 + z9q10 + z15q11 + z41q12 + z45q13 + ⋯ ∈ S12(Γ0(15); \overline{F}7). Also, z satisfies the equation x2 + 6x + 3 = 0. Since f is ordinary, we do not continue our analysis.

17. Consider f = q + z31q2 + 5q3 + z7q4 + 4q5 + z23q6 + z28q7 + z43q8 + 4q9 + z15q10 + z9q11 + z47q12 + z27q13 + ⋯ ∈ S12(Γ0(15); \overline{F}7). Also, z satisfies the equation x2 + 6x + 3 = 0. Since f is ordinary, we do not continue our analysis.

18. Consider f = q + 5q2 + 5q3 + 3q5 + 4q6 + q7 + q8 + 4q9 + q10 + 3q11 + ⋯ ∈ S12(Γ0(15); \overline{F}7). Since f is ordinary, we do not continue our analysis.

19. Consider f = q + q2 + q3 + 6q4 + q5 + q6 + q7 + 4q8 + q9 + q10 + 6q12 + q13 + ⋯ ∈ S14(Γ0(15); \overline{F}7). Since f is ordinary, we do not continue our analysis.

20. Consider f = q + z44q2 + 6q3 + 3q4 + 6q5 + z20q6 + q7 + z44q8 + q9 + z20q10 + z11q11 + 4q12 + z36q13 + ⋯ ∈ S14(Γ0(15); \overline{F}7). Also, z satisfies the equation x2 + 6x + 3 = 0. Since f is ordinary, we do not continue our analysis.

21. Consider f = q + z20q2 + 6q3 + 3q4 + 6q5 + z44q6 + q7 + z20q8 + q9 + z44q10 + z29q11 + 4q12 + z12q13 + ⋯ ∈ S14(Γ0(15); \overline{F}7). Also, z satisfies the equation x2 + 6x + 3 = 0. Since f is ordinary, we do not continue our analysis.

7.13 N = 16, ℓ = 7 (15 forms)

1. Consider f = q + 2q^{3} + 5q^{5} + 5q^{9} + 4q^{11} + 2q^{13} + ⋯ ∈ S_{10}(Γ_{0}(16); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + q^{3} + 3q^{5} + 3q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{1} & 0 \\ 0 & u(3)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −68q^{3} + 1510q^{5} + −1464 ∗ 7q^{7} + −15059q^{9} + −3916q^{11} + −176594q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 3q^{3} + 6q^{5} + 3q^{9} + 5q^{11} + 2q^{13} + ⋯ ∈ S_{10}(Γ_{0}(16); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 5q^{3} + 5q^{5} + 4q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(4)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −228q^{3} + −666q^{5} + 904 ∗ 7q^{7} + 32301q^{9} + 30420q^{11} + −32338q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q + 5q^{3} + 4q^{5} + 6q^{9} + q^{11} + 6q^{13} + ⋯ ∈ S_{12}(Γ_{0}(16); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-4 newform g = q + 4q^{3} + 5q^{5} + 4q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{4} & 0 \\ 0 & u(4)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−26125321683010552+ O(7^{2}0)q^{3} +5665204994325650+O(7^{2}0)q^{5} +2832602497154354∗7+O(7^{2}1)q^{7} +−212619578000992576+ O(7^{2}1)q^{9} + 231380275661125368 + O(7^{2}1)q^{11} + 31202403234781680 + O(7^{2}0)q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

4. Consider f = q + 2q^{3} + 3q^{5} + 6q^{9} + q^{11} + q^{13} + ⋯ ∈ S_{12}(Γ_{0}(16); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 4q^{3} + 5q^{5} + 4q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(4)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+26125321683010496+ O(7^{2}0)q^{3}+−5665204994317782+O(7^{2}0)q^{5}+−2832602497167362∗7+O(7^{2}1)q^{7}+212619578001532778+ O(7^{2}1)q^{9} + −231380275661284448 + O(7^{2}1)q^{11} + −31202403233731204 + O(7^{2}0)q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

5. Consider f = q + q^{3} + 4q^{5} + 5q^{9} + 3q^{11} + 5q^{13} + ⋯ ∈ S_{14}(Γ_{0}(16); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-6 newform g = q + 5q^{3} + 5q^{5} + 4q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{0} & 0 \\ 0 & u(4)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −468q^{3} + 56214q^{5} + −47576 ∗ 7q^{7} + −1375299q^{9} + 6397380q^{11} + 15199742q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

6. Consider f = q + 3q^{3} + q^{5} + 6q^{9} + q^{11} + 5q^{13} + ⋯ ∈ S_{14}(Γ_{0}(16); \overline{F}_{7}). We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-6 newform g = q + q^{3} + 3q^{5} + 3q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{0} & 0 \\ 0 & u(3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−19813083833991809+ O(7^{2}0)q^{3}+−1619792884930289+O(7^{2}0)q^{5}+1418566396767041∗7+O(7^{2}0)q^{7}+−37912683339532212+ O(7^{2}0)q^{9} + 11831635427128049 + O(7^{2}0)q^{11} + −25421190565315715 + O(7^{2}0)q^{13} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

7. Consider f = q + 4q3 + 5q5 + 4q7 + 3q9 + 2q11 + q13 + ⋯ ∈ S10(Γ0(16); \overline{F}7). Since f is ordinary, we do not continue our analysis.

8. Consider f = q + 2q3 + 2q5 + 5q9 + q11 + ⋯ ∈ S10(Γ0(16); \overline{F}7). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

9. Consider f = q + q3 + 3q5 + 3q7 + 3q9 + 2q11 + 2q13 + ⋯ ∈ S12(Γ0(16); \overline{F}7). Since f is ordinary, we do not continue our analysis.

10. Consider f = q + 2q9 + 6q11 + ⋯ ∈ S12(Γ0(16); \overline{F}7). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

11. Consider f = q + 5q3 + 5q5 + 4q7 + 6q9 + 6q11 + 2q13 + ⋯ ∈ S12(Γ0(16); \overline{F}7). Since f is ordinary, we do not continue our analysis.

12. Consider f = q + 5q3 + 3q5 + 6q7 + q9 + ⋯ ∈ S14(Γ0(16); \overline{F}7). Since f is ordinary, we do not continue our analysis.

13. Consider f = q + 3q3 + 6q5 + 6q9 + 2q11 + ⋯ ∈ S14(Γ0(16); \overline{F}7). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

14. Consider f = q + 2q3 + 6q7 + q9 + 3q13 + ⋯ ∈ S14(Γ0(16); \overline{F}7). Since f is ordinary, we do not continue our analysis.

15. Consider f = q + 2q5 + q7 + 4q9 + 4q11 + 2q13 + ⋯ ∈ S14(Γ0(16); \overline{F}7). Since f is ordinary, we do not continue our analysis.

11 2 ≤ N ≤ 16, ℓ = 11

11.1 N = 2, ℓ = 11 (8 forms)

1. Consider f = q + 9q^{2} + 4q^{3} + 4q^{4} + 3q^{5} + 3q^{6} + 3q^{7} + 3q^{8} + 5q^{10} + 5q^{12} + 5q^{13} + 5q^{14} + q^{15} + ⋯ ∈ S_{14}(Γ_{0}(2); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 5q^{2} + 9q^{3} + 3q^{4} + q^{5} + q^{6} + 5q^{7} + 4q^{8} + 5q^{10} + 7q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{1} & 0 \\ 0 & u(7)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 64q^{2} + 1236q^{3} + 4096q^{4} + −57450q^{5} + 79104q^{6} + 64232q^{7} + 262144q^{8} + −6057 ∗ 11q^{9} + −3676800q^{10} + 224052 ∗ 11q^{11} + 5062656q^{12} + 8032766q^{13} + 4110848q^{14} + −71008200q^{15} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 4q^{2} + 4q^{3} + 5q^{4} + 2q^{5} + 5q^{6} + q^{7} + 9q^{8} + 4q^{9} + 8q^{10} + 9q^{12} + 2q^{13} + 4q^{14} + 8q^{15} + ⋯ ∈ S_{16}(Γ_{0}(2); \overline{F}_{11}). We have ρ_{f} ⊗χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q +3q^{2} +q^{3} +9q^{4} +10q^{5} + 3q^{6} + 4q^{7} + 5q^{8} + 3q^{9} + 8q^{10} + 3q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{1} & 0 \\ 0 & u(3)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −128q^{2} + 6252q^{3} + 16384q^{4} +90510q^{5} +−800256q^{6} +56q^{7} +−2097152q^{8} +24738597q^{9} +−11585280q^{10} +−8717268∗11q^{11} + 102432768q^{12} + −59782138q^{13} + −7168q^{14} + 565868520q^{15} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q + 6q^{2} + 3q^{3} + 3q^{4} + 6q^{5} + 7q^{6} + 6q^{7} + 7q^{8} + 5q^{9} + 3q^{10} + 9q^{12} + 3q^{13} + 3q^{14} + 7q^{15} + ⋯ ∈ S_{20}(Γ_{0}(2); \overline{F}_{11}). We have ρ_{f} ⊗χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q +3q^{2} +q^{3} +9q^{4} +10q^{5} + 3q^{6} + 4q^{7} + 5q^{8} + 3q^{9} + 8q^{10} + 3q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{8} & 0 \\ 0 & u(3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 512q^{2} + −53028q^{3} + 262144q^{4}+−5556930q^{5}+−27150336q^{6}+−44496424q^{7}+134217728q^{8}+1649707317q^{9}+−2845148160q^{10}+ 574606812 ∗ 11q^{11} + −13900972032q^{12} + −33124973098q^{13} + −22782169088q^{14} + 294672884040q^{15} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

4. Consider f = q + 10q^{2} + 5q^{3} + q^{4} + 5q^{5} + 6q^{6} + 2q^{7} + 10q^{8} + 6q^{10} + 5q^{12} + 8q^{13} + 9q^{14} + 3q^{15} + ⋯ ∈ S_{22}(Γ_{0}(2); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + 5q^{2} + 9q^{3} + 3q^{4} + q^{5} + q^{6} + 5q^{7} + 4q^{8} + 5q^{10} + 7q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{0} & 0 \\ 0 & u(7)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −1024q^{2} + 71604q^{3} + 1048576q^{4} + −28693770q^{5} + −73322496q^{6} + −853202392q^{7} + −1073741824q^{8} + −484838217 ∗ 11q^{9}+29382420480q^{10}+7884652692∗11q^{11}+75082235904q^{12}+−895323442786q^{13}+873679249408q^{14}+ −2054588707080q^{15} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

5. Consider f = q + 2q2 + q3 + 4q4 + 8q5 + 2q6 + q7 + 8q8 + 7q9 + 5q10 + 4q12 + 6q13 + 2q14 + 8q15 + ⋯ ∈ S14(Γ0(2); \overline{F}11). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

6. Consider f = q + 3q2 + q3 + 9q4 + 10q5 + 3q6 + 4q7 + 5q8 + 3q9 + 8q10 + 3q11 + 9q12 + 7q13 + q14 + 10q15 + ⋯ ∈ S18(Γ0(2); \overline{F}11). Since f is ordinary, we do not continue our analysis.

7. Consider f = q + 5q2 + 9q3 + 3q4 + q5 + q6 + 5q7 + 4q8 + 5q10 + 7q11 + 5q12 + 4q13 + 3q14 + 9q15 + ⋯ ∈ S20(Γ0(2); \overline{F}11). Since f is ordinary, we do not continue our analysis.

8. Consider f = q + q2 + 4q3 + q4 + 6q5 + 4q6 + 8q7 + q8 + 2q9 + 6q10 + 4q12 + 3q13 + 8q14 + 2q15 + ⋯ ∈ S22(Γ0(2); \overline{F}11). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

11.2 N = 3, ℓ = 11 (13 forms)

1. Consider f = q + 10q^{2} + 8q^{3} + 4q^{4} + 7q^{5} + 3q^{6} + 7q^{7} + 4q^{8} + 9q^{9} + 4q^{10} + 10q^{12} + q^{13} + 4q^{14} + q^{15} + 8q^{16} + 8q^{17} + ⋯ ∈ S_{14}(Γ_{0}(3); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q+8q^{2}+7q^{3}+3q^{4}+6q^{5}+q^{6}+8q^{7}+9q^{8}+5q^{9}+4q^{10}+2q^{11}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{1} & 0 \\ 0 & u(2)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −12q^{2} + −729q^{3} + −8048q^{4} + −30210q^{5} + 8748q^{6} + 235088q^{7} + 194880q^{8} + 531441q^{9} + 362520q^{10} + −1016628 ∗ 11q^{11} + 5866992q^{12} + 8049614q^{13} + −2821056q^{14} + 22023090q^{15} + 63590656q^{16} + −117494622q^{17} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 6q^{2} + 3q^{3} + 6q^{4} + 8q^{5} + 7q^{6} + q^{7} + 10q^{8} + 9q^{9} + 4q^{10} + 7q^{12} + 6q^{13} + 6q^{14} + 2q^{15} + q^{16} + 9q^{17} + ⋯ ∈ S_{14}(Γ_{0}(3); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 7q^{2} + 4q^{3} + 10q^{4} + 10q^{5} + 6q^{6} + 9q^{7} + 6q^{8} + 5q^{9} + 4q^{10} + q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+729q^{3}+−54∗a+ 800∗11q^{4}+128∗a+23814q^{5}+729∗aq^{6}+−3456∗a−103816q^{7}+3524∗a−917568q^{8}+531441q^{9}+16902∗a+ 2174976q^{10}+−3328∗11∗a−59292∗11q^{11}+−39366∗a+583200∗11q^{12}+89856∗a+11192414q^{13}+7528∗11∗ a − 58724352q^{14} + 93312 ∗ a + 17360406q^{15} + −665496 ∗ a − 12209792q^{16} + 72960 ∗ a + 43889202q^{17} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 54x + −16992. The slope of f_{0} is 1.

3. Consider f = q + 8q^{2} + 2q^{3} + 10q^{4} + q^{5} + 5q^{6} + 6q^{7} + 4q^{9} + 8q^{10} + 9q^{12} + 8q^{13} + 4q^{14} + 2q^{15} + 10q^{16} + 10q^{17} + ⋯ ∈ S_{16}(Γ_{0}(3); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q +6q^{2} +6q^{3} +7q^{4} +5q^{5} +3q^{6} +2q^{7} +3q^{9} +8q^{10} +9q^{11} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{1} & 0 \\ 0 & u(9)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −234q^{2} + −2187q^{3} +21988q^{4} +280710q^{5} +511758q^{6} +−1373344q^{7} +229320∗11q^{8} +4782969q^{9} +−65686140q^{10} + 3093732 ∗ 11q^{11} + −48087756q^{12} + 384022262q^{13} + 321362496q^{14} + −613912770q^{15} + −1310772464q^{16} + 1259207586q^{17} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

4. Consider f = q+10q^{2}+5q^{3}+5q^{4}+8q^{5}+6q^{6}+6q^{7}+2q^{8}+3q^{9}+3q^{10}+3q^{12}+5q^{14}+7q^{15}+7q^{16}+q^{17}+⋯ ∈ S_{18}(Γ_{0}(3); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-6 newform g = q + 5q^{2} + 9q^{3} + 4q^{4} + 6q^{5} + q^{6} + 4q^{7} + 3q^{8} + 4q^{9} + 8q^{10} + 8q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(7)ω^{6} & 0 \\ 0 & u(8)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+295774179405406427098+ O(11^{2}0)q^{2}+6561q^{3}+−1916136095384424821012+O(11^{2}1)q^{4}+2310130633369952122661+O(11^{2}1)q^{5}+ −309344301564058354907+O(11^{2}0)q^{6}+−96109659206662639477+O(11^{2}0)q^{7}+−48487063722068966619+ O(11^{2}0)q^{8}+43046721q^{9}+−30106657499102359755+O(11^{2}0)q^{10}+24917664534207370931∗11+O(11^{2}1)q^{11}+ 1255733477402761297757+O(11^{2}1)q^{12}+−96133497373011434857∗11+O(11^{2}1)q^{13}+−3004827105011682913199+ O(11^{2}1)q^{14}+1055199699543989498693+O(11^{2}1)q^{15}+1852375435989639745917+O(11^{2}1)q^{16}+233102293014980454352+ O(11^{2}0)q^{17} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

5. Consider f = q+q^{2}+5q^{3}+5q^{4}+8q^{5}+5q^{6}+5q^{7}+9q^{8}+3q^{9}+8q^{10}+3q^{12}+5q^{14}+7q^{15}+7q^{16}+10q^{17}+⋯ ∈ S_{18}(Γ_{0}(3); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 5q^{2} + 9q^{3} + 4q^{4} + 6q^{5} + q^{6} + 4q^{7} + 3q^{8} + 4q^{9} + 8q^{10} + 8q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(7)ω^{1} & 0 \\ 0 & u(8)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−295774179405406426504+ O(11^{2}0)q^{2}+6561q^{3}+1916136095384424997528+O(11^{2}1)q^{4}+−2310130633369951739801+O(11^{2}1)q^{5}+ 309344301564062252141 + O(11^{2}0)q^{6} + 96109659206687111045 + O(11^{2}0)q^{7} + 48487063722199306851 + O(11^{2}0)q^{8}+43046721q^{9}+30106657498292977335+O(11^{2}0)q^{10}+−24917664534297148523∗11+O(11^{2}1)q^{11}+ −1255733477401603176281+O(11^{2}1)q^{12}+96133497372782398653∗11+O(11^{2}1)q^{13}+3004827105011247347903+ O(11^{2}1)q^{14} + −1055199699541477554233 + O(11^{2}1)q^{15} + −1852375435939028421997 + O(11^{2}1)q^{16} + −233102293049293580716 + O(11^{2}0)q^{17} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

6. Consider f = q+q^{2}+7q^{3}+6q^{4}+3q^{5}+7q^{6}+3q^{7}+5q^{9}+3q^{10}+9q^{12}+q^{13}+3q^{14}+10q^{15}+8q^{16}+3q^{17}+⋯ ∈ S_{20}(Γ_{0}(3); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q + 6q^{2} + 6q^{3} + 7q^{4} + 5q^{5} + 3q^{6} + 2q^{7} + 3q^{9} + 8q^{10} + 9q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{8} & 0 \\ 0 & u(9)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−191290808516859524879+ O(11^{2}0)q^{2} + −19683q^{3} + 263851407676615514982 + O(11^{2}0)q^{4} + 93480575082868936053 + O(11^{2}0)q^{5} + −204737600192343304640+O(11^{2}0)q^{6}+−225774540010305222868+O(11^{2}0)q^{7}+−7227616423302553478∗ 11 + O(11^{2}0)q^{8} + 387420489q^{9} + 230222509989842585614 + O(11^{2}0)q^{10} + −4652872219213217838 ∗ 11 + O(11^{2}0)q^{11} + 242703580540089641014 + O(11^{2}0)q^{12} + 201833677077681461484 + O(11^{2}0)q^{13} + 318976276482900556072+O(11^{2}0)q^{14}+−6923215557643166464+O(11^{2}0)q^{15}+33972979971878919738+ O(11^{2}0)q^{16} + 134283332744169480116 + O(11^{2}0)q^{17} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

7. Consider f = q + 5q^{2} + 10q^{3} + q^{4} + 8q^{5} + 6q^{6} + q^{7} + 6q^{8} + q^{9} + 7q^{10} + 10q^{12} + 6q^{13} + 5q^{14} + 3q^{15} + 6q^{16} + 5q^{17} + ⋯ ∈ S_{22}(Γ_{0}(3); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q +8q^{2} +7q^{3} +3q^{4} +6q^{5} +q^{6} +8q^{7} +9q^{8} +5q^{9} +4q^{10} +2q^{11} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{0} & 0 \\ 0 & u(2)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −2844q^{2} + −59049q^{3} + 5991184q^{4} + 3109950q^{5} + 167935356q^{6} + 363303920q^{7} + −11074627008q^{8} + 3486784401q^{9} + −8844697800q^{10} + 1325621196 ∗ 11q^{11} + −353773424016q^{12} + 113350790702q^{13} + −1033236348480q^{14} + −183639437550q^{15} + 18931815702784q^{16} + −8589389597982q^{17} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

8. Consider f = q + 3q^{2} + q^{3} + 7q^{4} + 6q^{5} + 3q^{6} + 8q^{7} + 4q^{8} + q^{9} + 7q^{10} + 7q^{12} + 3q^{13} + 2q^{14} + 6q^{15} + 9q^{16} + 7q^{17} + ⋯ ∈ S_{22}(Γ_{0}(3); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q+7q^{2}+4q^{3}+10q^{4}+10q^{5}+6q^{6}+9q^{7}+6q^{8}+5q^{9}+4q^{10}+q^{11}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + 59049q^{3} + 666 ∗ a + 3040 ∗ 11^{2}q^{4} + 13568 ∗ a − 4019706q^{5} + 59049 ∗ aq^{6} + −182016 ∗ a + 400559384q^{7} + −1285756 ∗ a + 1641684672q^{8} + 3486784401q^{9} + 5016582 ∗ a + 33445011456q^{10} + 2842112 ∗ 11 ∗ a + 9047627748 ∗ 11q^{11} + 39326634 ∗ a + 179508960 ∗ 11^{2}q^{12} + −475236864 ∗ a + 134019420734q^{13} + 2308568 ∗ 11^{2} ∗ a − 448667983872q^{14}+801176832∗a−237359619594q^{15}+−611332056∗a−3940794645632q^{16}+1677304320∗ a − 6225306859278q^{17} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −666x + −2464992. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

9. Consider f = q + 5q2 + 9q3 + 4q4 + 6q5 + q6 + 4q7 + 3q8 + 4q9 + 8q10 + 8q11 + 3q12 + 9q14 + 10q15 + 8q16 + 2q17 + ⋯ ∈ S16(Γ0(3); \overline{F}11). Since f is ordinary, we do not continue our analysis.

10. Consider f = q + 6q2 + 6q3 + 7q4 + 5q5 + 3q6 + 2q7 + 3q9 + 8q10 + 9q11 + 9q12 + 6q13 + q14 + 8q15 + 6q16 + 6q17 + ⋯ ∈ S18(Γ0(3); \overline{F}11). Since f is ordinary, we do not continue our analysis.

11. Consider f = q + 7q2 + 4q3 + 10q4 + 10q5 + 6q6 + 9q7 + 6q8 + 5q9 + 4q10 + q11 + 7q12 + 7q13 + 8q14 + 7q15 + 4q16 + 3q17 + ⋯ ∈ S20(Γ0(3); \overline{F}11). Since f is ordinary, we do not continue our analysis.

12. Consider f = q + 8q2 + 7q3 + 3q4 + 6q5 + q6 + 8q7 + 9q8 + 5q9 + 4q10 + 2q11 + 10q12 + 3q13 + 9q14 + 9q15 + 10q16 + 10q17 + ⋯ ∈ S20(Γ0(3); \overline{F}11). Since f is ordinary, we do not continue our analysis.

13. Consider f = q + q2 + 10q3 + 10q4 + 9q5 + 10q6 + 4q7 + 8q8 + q9 + 9q10 + q11 + q12 + 9q13 + 4q14 + 2q15 + 10q16 + 9q17 + ⋯ ∈ S22(Γ0(3); \overline{F}11). Since f is ordinary, we do not continue our analysis.

11.3 N = 4, ℓ = 11 (6 forms)

1. Consider f = q + 6q^{3} + 4q^{5} + 7q^{7} + 9q^{9} + 8q^{13} + 2q^{15} + 2q^{17} + 3q^{19} + 9q^{21} + ⋯ ∈ S_{14}(Γ_{0}(4); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 8q^{3} + 5q^{5} + 8q^{7} + 5q^{9} + 6q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(6)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 468q^{3} + 56214q^{5} + 333032q^{7}+−1375299q^{9}+−581580∗11q^{11}+15199742q^{13}+26308152q^{15}+43114194q^{17}+−365115484q^{19}+ 155858976q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q +8q^{3} +6q^{5} +4q^{15} +⋯ ∈ S_{18}(Γ_{0}(4); \overline{F}_{11}). We have ρ_{f} ⊗χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 10q^{3} + 10q^{5} + q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{3} + −36 ∗ a + 196182q^{5} + 54 ∗ 11 ∗ a + 1311040 ∗ 11q^{7} + −5880 ∗ a + 207300141q^{9} + 315 ∗ 11^{2} ∗ a + 67443840 ∗ 11q^{11} + −14796∗11∗a−9409330∗11^{2}q^{13} +407862∗a−12111850944q^{15} +−15912∗11∗a−1296700650∗11q^{17} + −260145 ∗ 11 ∗ a + 3832157056 ∗ 11q^{19} + 90320 ∗ 11^{2} ∗ a + 18167776416 ∗ 11q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 5880x + −336440304. The slope of f_{0} is 1.

3. Consider f = q + 2q^{3} + 3q^{5} + q^{7} + q^{9} + 4q^{13} + 6q^{15} + 4q^{17} + 10q^{19} + 2q^{21} + ⋯ ∈ S_{22}(Γ_{0}(4); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + 8q^{3} + 5q^{5} + 8q^{7} + 5q^{9} + 6q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{0} & 0 \\ 0 & u(6)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−197815801794667479828+ O(11^{2}0)q^{3}+10576129841434816530+O(11^{2}0)q^{5}+−186431864848526734843+O(11^{2}0)q^{7}+118422391371438141362+ O(11^{2}0)q^{9} + −24976012745428922040 ∗ 11 + O(11^{2}0)q^{11} + 298761524669349109867 + O(11^{2}0)q^{13} + −310158248916494961548+O(11^{2}0)q^{15}+66120954489554434749+O(11^{2}0)q^{17}+39625584201932371499+ O(11^{2}0)q^{19} + −3351172969707456781248 + O(11^{2}1)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

4. Consider f = q + 10q3 + 10q5 + q11 + q15 + ⋯ ∈ S16(Γ0(4); \overline{F}11). Since f is ordinary, we do not continue our analysis.

5. Consider f = q + 8q3 + 5q5 + 8q7 + 5q9 + 6q11 + 2q13 + 7q15 + 8q17 + 4q19 + 9q21 + ⋯ ∈ S20(Γ0(4); \overline{F}11). Since f is ordinary, we do not continue our analysis.

6. Consider f = q + q3 + 8q5 + 2q7 + 9q9 + 10q11 + 7q13 + 8q15 + 6q17 + 8q19 + 2q21 + ⋯ ∈ S22(Γ0(4); \overline{F}11). Since f is ordinary, we do not continue our analysis.

11.4 N = 5, ℓ = 11 (29 forms)

1. Consider f = q + q^{2} + 8q^{3} + 4q^{4} + 6q^{5} + 8q^{6} + 2q^{7} + 7q^{8} + 4q^{9} + 6q^{10} + 10q^{12} + 2q^{13} + 2q^{14} + 4q^{15} + 8q^{16} + 10q^{17} + 4q^{18} + 3q^{19} + 2q^{20} + 5q^{21} + ⋯ ∈ S_{14}(Γ_{0}(5); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 3q^{2} + 7q^{3} + 3q^{4} + 2q^{5} + 10q^{6} + 7q^{7} + 2q^{8} + q^{9} + 6q^{10} + 3q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{1} & 0 \\ 0 & u(3)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−12620384221731986006+ O(11^{2}0)q^{2}+151444610660783831982+O(11^{2}0)q^{3}+−335869252126561139730+O(11^{2}0)q^{4}+−15625q^{5}+ −215734353985867754612+O(11^{2}0)q^{6}+208301256155830936705+O(11^{2}0)q^{7}+−313616457178931514788+ O(11^{2}0)q^{8}+−277202792719173270143+O(11^{2}0)q^{9}+77754949322198647857+O(11^{2}0)q^{10}+467179384276462574∗ 11 + O(11^{2}0)q^{11} + −235359686522384348915 + O(11^{2}0)q^{12} + −102850188749638604217 + O(11^{2}0)q^{13} + 217376909472117945156+O(11^{2}0)q^{14}+−260309396933822358833+O(11^{2}0)q^{15}+−44070817469399131788+ O(11^{2}0)q^{16}+197645923551384171082+O(11^{2}0)q^{17}+2766274810279464146062+O(11^{2}1)q^{18}+−93633860762009307517+ O(11^{2}0)q^{19} + −165645991382823495751 + O(11^{2}0)q^{20} + 172363870151720649110 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + z^{6}3q^{2} + z^{1}6q^{3} + z^{5}7q^{4} + 5q^{5} + z^{7}9q^{6} + z^{7}5q^{7} + z^{8}3q^{8} + z^{7}3q^{9} + z^{1}11q^{10} + z^{7}3q^{12} + z^{9}1q^{13} + z^{1}8q^{14} + z^{6}4q^{15} + z^{7}8q^{16} + z^{8}9q^{17} + z^{1}6q^{18} + z^{1}6q^{19} + z^{1}05q^{20} + z^{9}1q^{21} + ⋯ ∈ S_{14}(Γ_{0}(5); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{3}9q^{2} + z^{6}4q^{3} + z^{9}q^{4} + 9q^{5} + z^{1}03q^{6} + z^{2}7q^{7} + z^{1}1q^{8} + z^{4}9q^{9} + z^{1}11q^{10} + z^{2}6q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{9}4)ω^{1} & 0 \\ 0 & u(z^{2}6)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(53532297971390447749+ O(11^{2}0)) ∗ a + 14587874043124250453 + O(11^{2}0)q^{3} + −(321193787828342686396 + O(11^{2}0)) ∗ a − 87527244258745500566+O(11^{2}0)q^{4}+15625q^{5}+(2355421110741179700104+O(11^{2}1))∗a+641866457897467094284+ O(11^{2}1)q^{6} + −(2100552445090505776236 + O(11^{2}1)) ∗ a + 131290866388118275777 ∗ 11 + O(11^{2}1)q^{7} + (137481783789419152196+O(11^{2}0))∗a−319368775955780652738+O(11^{2}0)q^{8}+−(68795660465473189713+ O(11^{2}0)) ∗ a + 333174423502429869670 + O(11^{2}0)q^{9} + 15625 ∗ aq^{10} + (260468949276593413428 ∗ 11 + O(11^{2}1))∗a−235113773638832669599∗11+O(11^{2}1)q^{11} +−(2038949325205391461800+O(11^{2}1))∗a− 1888028016817827857410+O(11^{2}1)q^{12}+(26242871261719934174+O(11^{2}0))∗a−53383707803725881565+ O(11^{2}0)q^{13} + −(327120698677696605894 + O(11^{2}0)) ∗ a − 295435245628249738038 + O(11^{2}0)q^{14} + (213912101803654641282+O(11^{2}0))∗a−126716358321429791014+O(11^{2}0)q^{15}+−(1403595937020214583060+ O(11^{2}1)) ∗ a + 2786587190778092566760 + O(11^{2}1)q^{16} + −(2495539722731842335796 + O(11^{2}1)) ∗ a − 2606613883979187991749+O(11^{2}1)q^{17}+−(97111800611622346384+O(11^{2}0))∗a+99317520207522213159+ O(11^{2}0)q^{18} + −(168466499355167487220 + O(11^{2}0)) ∗ a − 243656292689409967035 + O(11^{2}0)q^{19} + (62027379043193701960+O(11^{2}0))∗a+87548154996052361883+O(11^{2}0)q^{20}+(1494225993331619927446+ O(11^{2}1)) ∗ a + 1936905577777111958077 + O(11^{2}1)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 321193787828342686396 + O(11^{2}0)x + 87527244258745492374 + O(11^{2}0). The slope of f_{0} is 1.

3. Consider f = q + z^{9}3q^{2} + z^{5}6q^{3} + z^{2}7q^{4} + 5q^{5} + z^{2}9q^{6} + z^{1}05q^{7} + z^{7}3q^{8} + z^{8}3q^{9} + z^{2}1q^{10} + z^{8}3q^{12} + z^{4}1q^{13} + z^{7}8q^{14} + z^{1}04q^{15} + z^{1}8q^{16} + z^{1}9q^{17} + z^{5}6q^{18} + z^{5}6q^{19} + z^{7}5q^{20} + z^{4}1q^{21} + ⋯ ∈ S_{14}(Γ_{0}(5); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{6}9q^{2} + z^{1}04q^{3} + z^{9}9q^{4} + 9q^{5} + z^{5}3q^{6} + z^{5}7q^{7} + zq^{8} + z^{5}9q^{9} + z^{2}1q^{10} + z^{4}6q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{7}4)ω^{1} & 0 \\ 0 & u(z^{4}6)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(53532297971390447749+ O(11^{2}0)) ∗ a + 14587874043124250453 + O(11^{2}0)q^{3} + −(321193787828342686396 + O(11^{2}0)) ∗ a − 87527244258745500566+O(11^{2}0)q^{4}+15625q^{5}+(2355421110741179700104+O(11^{2}1))∗a+641866457897467094284+ O(11^{2}1)q^{6} + −(2100552445090505776236 + O(11^{2}1)) ∗ a + 131290866388118275777 ∗ 11 + O(11^{2}1)q^{7} + (137481783789419152196+O(11^{2}0))∗a−319368775955780652738+O(11^{2}0)q^{8}+−(68795660465473189713+ O(11^{2}0)) ∗ a + 333174423502429869670 + O(11^{2}0)q^{9} + 15625 ∗ aq^{10} + (260468949276593413428 ∗ 11 + O(11^{2}1))∗a−235113773638832669599∗11+O(11^{2}1)q^{11} +−(2038949325205391461800+O(11^{2}1))∗a− 1888028016817827857410+O(11^{2}1)q^{12}+(26242871261719934174+O(11^{2}0))∗a−53383707803725881565+ O(11^{2}0)q^{13} + −(327120698677696605894 + O(11^{2}0)) ∗ a − 295435245628249738038 + O(11^{2}0)q^{14} + (213912101803654641282+O(11^{2}0))∗a−126716358321429791014+O(11^{2}0)q^{15}+−(1403595937020214583060+ O(11^{2}1)) ∗ a + 2786587190778092566760 + O(11^{2}1)q^{16} + −(2495539722731842335796 + O(11^{2}1)) ∗ a − 2606613883979187991749+O(11^{2}1)q^{17}+−(97111800611622346384+O(11^{2}0))∗a+99317520207522213159+ O(11^{2}0)q^{18} + −(168466499355167487220 + O(11^{2}0)) ∗ a − 243656292689409967035 + O(11^{2}0)q^{19} + (62027379043193701960+O(11^{2}0))∗a+87548154996052361883+O(11^{2}0)q^{20}+(1494225993331619927446+ O(11^{2}1)) ∗ a + 1936905577777111958077 + O(11^{2}1)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 321193787828342686396 + O(11^{2}0)x + 87527244258745492374 + O(11^{2}0). The slope of f_{0} is 1.

4. Consider f = q + 7q^{2} + 6q^{3} + 6q^{4} + 3q^{5} + 9q^{6} + 7q^{7} + 5q^{8} + 2q^{9} + 10q^{10} + 3q^{12} + 5q^{13} + 5q^{14} + 7q^{15} + 8q^{16} + 3q^{18} + 3q^{19} + 7q^{20} + 9q^{21} + ⋯ ∈ S_{16}(Γ_{0}(5); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + 8q^{2} + 7q^{3} + 2q^{4} + 4q^{5} + q^{6} + 6q^{7} + 4q^{8} + 7q^{9} + 10q^{10} + 7q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{1} & 0 \\ 0 & u(7)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+26863872109196382498+ O(11^{2}0)q^{2} + 53727744218392766176 + O(11^{2}0)q^{3} + −254800414660158492240 + O(11^{2}0)q^{4} + 78125q^{5} + 243268492633654005450+O(11^{2}0)q^{6}+222357481521239421474+O(11^{2}0)q^{7}+−2636757158843895280+ O(11^{2}0)q^{8}+−25974355622443124422+O(11^{2}0)q^{9}+−239975658619846050870+O(11^{2}0)q^{10}+7210046684414903056∗ 11 + O(11^{2}0)q^{11} + 13470731334194788998 + O(11^{2}0)q^{12} + −142311223279653905769 + O(11^{2}0)q^{13} + 32194819596725715198+O(11^{2}0)q^{14}+192798677692960094961+O(11^{2}0)q^{15}+55022258457214287643+ O(11^{2}0)q^{16}+890737154567952352∗11^{2}+O(11^{2}0)q^{17}+127778294715179727358+O(11^{2}0)q^{18}+25292277742610792565+ O(11^{2}0)q^{19} + −282795265364094001611 + O(11^{2}0)q^{20} + 73719810557565181326 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

5. Consider f = q + z^{6}9q^{2} + z^{8}9q^{3} + z^{3}2q^{4} + 8q^{5} + z^{3}8q^{6} + z^{5}8q^{7} + z^{8}6q^{8} + z^{2}7q^{9} + z^{1}05q^{10} + zq^{12} + z^{4}3q^{13} + z^{7}q^{14} + z^{5}q^{15} + z^{1}06q^{16} + z^{8}0q^{17} + 3q^{18} + z^{8}8q^{19} + z^{6}8q^{20} + z^{2}7q^{21} + ⋯ ∈ S_{16}(Γ_{0}(5); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{2}1q^{2} + z^{6}5q^{3} + z^{5}6q^{4} + 7q^{5} + z^{8}6q^{6} + z^{8}2q^{7} + z^{6}2q^{8} + z^{9}9q^{9} + z^{1}05q^{10} + z^{1}13q^{11} + · · · . We

deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{7})ω^{1} & 0 \\ 0 & u(z^{1}13)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(119332116899183789522+ O(11^{2}0))∗a−289963247195975422809+O(11^{2}0)q^{3}+(33938867221558239249+O(11^{2}0))∗a+73927404995762355109+ O(11^{2}0)q^{4}+−78125q^{5}+(101857504939825308548+O(11^{2}0))∗a−222769845645938376653+O(11^{2}0)q^{6}+ (102745151811532260156+O(11^{2}0))∗a+229822133225343899379+O(11^{2}0)q^{7}+(135755468886232950796+ O(11^{2}0))∗a+295709619983054959764+O(11^{2}0)q^{8}+(63030467333075780378+O(11^{2}0))∗a+312165537169705468046+ O(11^{2}0)q^{9} + −78125 ∗ aq^{10} + −(168765785960933627012 ∗ 11 + O(11^{2}1)) ∗ a + 311152888191809111534 ∗ 11 + O(11^{2}1)q^{11} + (125443157808560691236 + O(11^{2}0)) ∗ a − 87548066228650238690 + O(11^{2}0)q^{12} + −(197858812923529688945+O(11^{2}0))∗a−131588734994259574436+O(11^{2}0)q^{13}+−(93063758791842907936+ O(11^{2}0))∗a+93358383686554041597+O(11^{2}0)q^{14}+(147797026683051101208+O(11^{2}0))∗a−131892178513282872148+ O(11^{2}0)q^{15}+−(33015971656111501483+O(11^{2}0))∗a−250301083578597829776+O(11^{2}0)q^{16}+(175182350764656660804+ O(11^{2}0)) ∗ a − 203873197500330068637 + O(11^{2}0)q^{17} + (29283455725163503173 ∗ 11 + O(11^{2}0)) ∗ a − 73711312622803942424+O(11^{2}0)q^{18}+−(248255047869677621248+O(11^{2}0))∗a−234685703463819140721+ O(11^{2}0)q^{19}+−(166271655018445066984+O(11^{2}0))∗a−19808797906313900040+O(11^{2}0)q^{20}+(1577235821978196292390+ O(11^{2}1)) ∗ a − 3681766448181318667329 + O(11^{2}1)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −33938867221558239249 + O(11^{2}0)x + −73927404995762387877 + O(11^{2}0). The slope of f_{0} is 1.

6. Consider f = q + 3q^{2} + 6q^{3} + 10q^{4} + 8q^{5} + 7q^{6} + 9q^{7} + 2q^{9} + 2q^{10} + 5q^{12} + q^{13} + 5q^{14} + 4q^{15} + 10q^{16} + 2q^{17} + 6q^{18} + 8q^{19} + 3q^{20} + 10q^{21} + ⋯ ∈ S_{16}(Γ_{0}(5); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-4 newform g = q + 7q^{2} + 2q^{3} + 8q^{4} + 6q^{5} + 3q^{6} + 6q^{7} + 10q^{9} + 9q^{10} + 10q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(10)ω^{4} & 0 \\ 0 & u(10)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−33938867221558239245+ O(11^{2}0)q^{2}+188105742256150117779+O(11^{2}0)q^{3}+−209682873881995318289+O(11^{2}0)q^{4}+−78125q^{5}+ −154509316578261820091+O(11^{2}0)q^{6}+−136758374433501896649+O(11^{2}0)q^{7}+−28704830984146844140∗ 11+O(11^{2}0)q^{8}+38466444786103917783+O(11^{2}0)q^{9}+166271655018444754484+O(11^{2}0)q^{10}+49773419432827616160∗ 11+O(11^{2}1)q^{11}+98395801355221801117+O(11^{2}0)q^{12}+71978677365746137636+O(11^{2}0)q^{13}+101178225538479627081+ O(11^{2}0)q^{14}+−210224454887110497731+O(11^{2}0)q^{15}+−305864306441435764892+O(11^{2}0)q^{16}+161124223894243526842+ O(11^{2}0)q^{17}+−203378932655237365306+O(11^{2}0)q^{18}+−307582526485900806407+O(11^{2}0)q^{19}+12145423048017283775+ O(11^{2}0)q^{20} + 188631173229543487486 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

7. Consider f = q + z^{3}9q^{2} + z^{1}9q^{3} + z^{1}12q^{4} + 8q^{5} + z^{5}8q^{6} + z^{3}8q^{7} + z^{1}06q^{8} + z^{5}7q^{9} + z^{7}5q^{10} + z^{1}1q^{12} + z^{1}13q^{13} + z^{7}7q^{14} + z^{5}5q^{15} + z^{8}6q^{16} + z^{4}0q^{17} + 3q^{18} + z^{8}q^{19} + z^{2}8q^{20} + z^{5}7q^{21} + ⋯ ∈ S_{16}(Γ_{0}(5); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{1}11q^{2} + z^{1}15q^{3} + z^{1}6q^{4} + 7q^{5} + z^{1}06q^{6} + z^{6}2q^{7} + z^{8}2q^{8} + z^{9}q^{9} + z^{7}5q^{10} + z^{4}3q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{7}7)ω^{1} & 0 \\ 0 & u(z^{4}3)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(119332116899183789522+ O(11^{2}0))∗a−289963247195975422809+O(11^{2}0)q^{3}+(33938867221558239249+O(11^{2}0))∗a+73927404995762355109+ O(11^{2}0)q^{4}+−78125q^{5}+(101857504939825308548+O(11^{2}0))∗a−222769845645938376653+O(11^{2}0)q^{6}+ (102745151811532260156+O(11^{2}0))∗a+229822133225343899379+O(11^{2}0)q^{7}+(135755468886232950796+ O(11^{2}0))∗a+295709619983054959764+O(11^{2}0)q^{8}+(63030467333075780378+O(11^{2}0))∗a+312165537169705468046+ O(11^{2}0)q^{9} + −78125 ∗ aq^{10} + −(168765785960933627012 ∗ 11 + O(11^{2}1)) ∗ a + 311152888191809111534 ∗ 11 + O(11^{2}1)q^{11} + (125443157808560691236 + O(11^{2}0)) ∗ a − 87548066228650238690 + O(11^{2}0)q^{12} + −(197858812923529688945+O(11^{2}0))∗a−131588734994259574436+O(11^{2}0)q^{13}+−(93063758791842907936+ O(11^{2}0))∗a+93358383686554041597+O(11^{2}0)q^{14}+(147797026683051101208+O(11^{2}0))∗a−131892178513282872148+ O(11^{2}0)q^{15}+−(33015971656111501483+O(11^{2}0))∗a−250301083578597829776+O(11^{2}0)q^{16}+(175182350764656660804+ O(11^{2}0)) ∗ a − 203873197500330068637 + O(11^{2}0)q^{17} + (29283455725163503173 ∗ 11 + O(11^{2}0)) ∗ a − 73711312622803942424+O(11^{2}0)q^{18}+−(248255047869677621248+O(11^{2}0))∗a−234685703463819140721+ O(11^{2}0)q^{19}+−(166271655018445066984+O(11^{2}0))∗a−19808797906313900040+O(11^{2}0)q^{20}+(1577235821978196292390+ O(11^{2}1)) ∗ a − 3681766448181318667329 + O(11^{2}1)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −33938867221558239249 + O(11^{2}0)x + −73927404995762387877 + O(11^{2}0). The slope of f_{0} is 1.

8. Consider f = q + 7q^{2} + 10q^{3} + 9q^{4} + 4q^{5} + 4q^{6} + 9q^{7} + 3q^{8} + 3q^{9} + 6q^{10} + 2q^{12} + 8q^{14} + 7q^{15} + 2q^{16} + 3q^{17} + 10q^{18} + q^{19} + 3q^{20} + 2q^{21} + ⋯ ∈ S_{18}(Γ_{0}(5); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 2q^{2} + 7q^{3} + 5q^{4} + 3q^{5} + 3q^{6} + 5q^{7} + q^{8} + 4q^{9} + 6q^{10} + 6q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(6)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−41838085238737211681+ O(11^{2}0)q^{2}+−125868660557771057862+O(11^{2}0)q^{3}+−116028672200071462526+O(11^{2}0)q^{4}+390625q^{5}+ −191689911494874837601+O(11^{2}0)q^{6}+135217260697586253646+O(11^{2}0)q^{7}+184115446319870555834+ O(11^{2}0)q^{8}+−219923773394547572468+O(11^{2}0)q^{9}+113580514956990629268+O(11^{2}0)q^{10}+−369373046933865619∗ 11 + O(11^{2}0)q^{11} + 84033159749155394923 + O(11^{2}0)q^{12} + 1077580229623046632 ∗ 11^{2} + O(11^{2}0)q^{13} + 317822314725039383938+O(11^{2}0)q^{14}+−184900728103764897866+O(11^{2}0)q^{15}+−41578797403087840344+ O(11^{2}0)q^{16}+276267507177063493545+O(11^{2}0)q^{17}+329743213651131547816+O(11^{2}0)q^{18}+−180073874627271031835+ O(11^{2}0)q^{19} + 139830448585330661821 + O(11^{2}0)q^{20} + 42398176670751595751 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

9. Consider f = q + 4q^{2} + 10q^{3} + 9q^{4} + 4q^{5} + 7q^{6} + 2q^{7} + 8q^{8} + 3q^{9} + 5q^{10} + 2q^{12} + 8q^{14} + 7q^{15} + 2q^{16} + 8q^{17} + q^{18} + 10q^{19} + 3q^{20} + 9q^{21} + ⋯ ∈ S_{18}(Γ_{0}(5); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-6 newform g = q + 2q^{2} + 7q^{3} + 5q^{4} + 3q^{5} + 3q^{6} + 5q^{7} + q^{8} + 4q^{9} + 6q^{10} + 6q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{6} & 0 \\ 0 & u(6)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−52517281607646393853+ O(11^{2}0)q^{2} +313596772736366085416+O(11^{2}0)q^{3} +−63601593711270688482+O(11^{2}0)q^{4} +390625q^{5} + 88084623947363806297+O(11^{2}0)q^{6}+−137852184341504600933+O(11^{2}0)q^{7}+24751372135818747680+ O(11^{2}0)q^{8}+96587932484229413098+O(11^{2}0)q^{9}+275217486612321747169+O(11^{2}0)q^{10}+−11097555990614619965∗ 11+O(11^{2}0)q^{11} +−273444966375397611408+O(11^{2}0)q^{12} +19880528258029213040∗11+O(11^{2}0)q^{13} + −331339842428920666333+O(11^{2}0)q^{14}+−288977142052279757487+O(11^{2}0)q^{15}+184699018207861145089+ O(11^{2}0)q^{16}+12968987894830027519+O(11^{2}0)q^{17}+175467291678770841234+O(11^{2}0)q^{18}+−31065610046332008357+ O(11^{2}0)q^{19} + 284769394328451511680 + O(11^{2}0)q^{20} + 237673583155104514515 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

10. Consider f = q+5q^{2}+10q^{3}+8q^{4}+9q^{5}+6q^{6}+9q^{7}+10q^{8}+8q^{9}+q^{10}+3q^{12}+2q^{13}+q^{14}+2q^{15}+2q^{16}+7q^{18}+ 6q^{19}+6q^{20}+2q^{21}+⋯ ∈ S_{20}(Γ_{0}(5); \overline{F}_{11}). Also, z satisfies the equation x^{2}+7x+2 = 0. We have ρ_{f} ⊗χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q+8q^{2}+7q^{3}+2q^{4}+4q^{5}+q^{6}+6q^{7}+4q^{8}+7q^{9}+10q^{10}+7q^{11}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{8} & 0 \\ 0 & u(7)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−143279868018007443674+ O(11^{2}0)q^{2}+−40999177994245764811+O(11^{2}0)q^{3}+−218618070134126456632+O(11^{2}0)q^{4}+1953125q^{5}+ −57975920002408264268+O(11^{2}0)q^{6}+16788600972432273422+O(11^{2}0)q^{7}+275074162722638796637+ O(11^{2}0)q^{8}+−19902633163916402870+O(11^{2}0)q^{9}+323169426198601558720+O(11^{2}0)q^{10}+−23704511305214108100∗ 11 + O(11^{2}0)q^{11} + 187764657299851644950 + O(11^{2}0)q^{12} + −76209357487108445169 + O(11^{2}0)q^{13} + −194874073895433378533+O(11^{2}0)q^{14}+239626816425938701454+O(11^{2}0)q^{15}+−148088487879668702716+ O(11^{2}0)q^{16}+14199686208170960925∗11+O(11^{2}0)q^{17}+123182197907097793779+O(11^{2}0)q^{18}+35027518924252861100+ O(11^{2}0)q^{19} + −51196974290809583109 + O(11^{2}0)q^{20} + −49579012172535498816 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

11. Consider f = q+10q^{2}+10q^{3}+6q^{4}+2q^{5}+q^{6}+10q^{7}+8q^{9}+9q^{10}+5q^{12}+7q^{13}+q^{14}+9q^{15}+8q^{16}+5q^{17}+3q^{18}+ 5q^{19} +q^{20} +q^{21} +⋯ ∈ S_{20}(Γ_{0}(5); \overline{F}_{11}). Also, z satisfies the equation x^{2} +7x+2 = 0. We have ρ_{f} ⊗χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q +7q^{2} +2q^{3} +8q^{4} +6q^{5} +3q^{6} +6q^{7} +10q^{9} +9q^{10} +10q^{11} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(10)ω^{1} & 0 \\ 0 & u(10)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−11419817576029337091+ O(11^{2}0)q^{2}+−138376541028004409032+O(11^{2}0)q^{3}+323426180412476008467+O(11^{2}0)q^{4}+−1953125q^{5}+ −90270594417415877178 + O(11^{2}0)q^{6} + 157757266900772466223 + O(11^{2}0)q^{7} + 5208622346228947969 ∗ 11 + O(11^{2}0)q^{8} + −186723589698142585377 + O(11^{2}0)q^{9} + −22128811795539190579 + O(11^{2}0)q^{10} + 20440361386437411159∗11+O(11^{2}0)q^{11}+160731782944841853761+O(11^{2}0)q^{12}+−70264113356716138039+ O(11^{2}0)q^{13}+292942586750316565426+O(11^{2}0)q^{14}+135231084048654270466+O(11^{2}0)q^{15}+−196056989340825550809+ O(11^{2}0)q^{16}+170872454165080198723+O(11^{2}0)q^{17}+262717483627521215120+O(11^{2}0)q^{18}+−312701035815224350872+ O(11^{2}0)q^{19} + 304123708667802353595 + O(11^{2}0)q^{20} + −82318538399599896987 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

12. Consider f = q + z^{3}q^{2} + z^{9}1q^{3} + z^{4}0q^{4} + 2q^{5} + z^{9}4q^{6} + z^{2}6q^{7} + z^{1}18q^{8} + z^{8}1q^{9} + z^{1}5q^{10} + z^{1}1q^{12} + z^{7}7q^{13} + z^{2}9q^{14} + z^{1}03q^{15} + z^{6}2q^{16} + z^{7}6q^{17} + 7q^{18} + z^{2}0q^{19} + z^{5}2q^{20} + z^{1}17q^{21} + ⋯ ∈ S_{20}(Γ_{0}(5); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{1}11q^{2} + z^{1}15q^{3} + z^{1}6q^{4} + 7q^{5} + z^{1}06q^{6} + z^{6}2q^{7} + z^{8}2q^{8} + z^{9}q^{9} + z^{7}5q^{10} + z^{4}3q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{7}7)ω^{8} & 0 \\ 0 & u(z^{4}3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(272356213029084940251+ O(11^{2}0)) ∗ a + 12992438387570805400 + O(11^{2}0)q^{3} + −(282552299218640417025 + O(11^{2}0)) ∗ a − 147420816882237526411+O(11^{2}0)q^{4}+−1953125q^{5}+(321349146368741906997+O(11^{2}0))∗a+199460733169605884512+ O(11^{2}0)q^{6}+(281585915160954213610+O(11^{2}0))∗a+175450113922492945138+O(11^{2}0)q^{7}+−(296779586376047968571+ O(11^{2}0))∗a+156351965950057937003+O(11^{2}0)q^{8}+(218611007891324290149+O(11^{2}0))∗a+56007533017502634031+ O(11^{2}0)q^{9} + −1953125 ∗ aq^{10} + −(8054580602308956735 ∗ 11 + O(11^{2}0)) ∗ a − 12515340148482855933 ∗ 11 + O(11^{2}0)q^{11} + −(9262577459088346980 + O(11^{2}0)) ∗ a + 88426330248628911181 + O(11^{2}0)q^{12} + −(156323926578217706212+O(11^{2}0))∗a+64522289621963107047+O(11^{2}0)q^{13}+(268816343727391025066+ O(11^{2}0)) ∗ a − 267669210612741564193 + O(11^{2}0)q^{14} + (289329271526972476072 + O(11^{2}0)) ∗ a + 273583131934250186720+O(11^{2}0)q^{15}+−(238570378441371333164+O(11^{2}0))∗a+17570412699848378530+ O(11^{2}0)q^{16}+(123463712972720713231+O(11^{2}0))∗a+126277729342156318947+O(11^{2}0)q^{17}+(16670092660012589484∗ 11 + O(11^{2}0)) ∗ a − 62066153661114141103 + O(11^{2}0)q^{18} + (153053537731965260672 + O(11^{2}0)) ∗ a + 152088341463358382960+O(11^{2}0)q^{19}+−(225181746573845673180+O(11^{2}0))∗a−332858056054686470017+ O(11^{2}0)q^{20} + (136494026686099183435 + O(11^{2}0)) ∗ a − 228067542290854489548 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 282552299218640417025 + O(11^{2}0)x + 147420816882237002123 + O(11^{2}0). The slope of f_{0} is 1.

13. Consider f = q + z^{3}3q^{2} + z^{4}1q^{3} + z^{8}0q^{4} + 2q^{5} + z^{7}4q^{6} + z^{4}6q^{7} + z^{9}8q^{8} + z^{5}1q^{9} + z^{4}5q^{10} + zq^{12} + z^{7}q^{13} + z^{7}9q^{14} + z^{5}3q^{15} + z^{8}2q^{16} + z^{1}16q^{17} + 7q^{18} + z^{1}00q^{19} + z^{9}2q^{20} + z^{8}7q^{21} + ⋯ ∈ S_{20}(Γ_{0}(5); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{2}1q^{2} + z^{6}5q^{3} + z^{5}6q^{4} + 7q^{5} + z^{8}6q^{6} + z^{8}2q^{7} + z^{6}2q^{8} + z^{9}9q^{9} + z^{1}05q^{10} + z^{1}13q^{11} + · · · . We

(\begin{matrix}u(z^{7})ω^{8} & 0 \\ 0 & u(z^{1}13)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(272356213029084940251+ O(11^{2}0)) ∗ a + 12992438387570805400 + O(11^{2}0)q^{3} + −(282552299218640417025 + O(11^{2}0)) ∗ a − 147420816882237526411+O(11^{2}0)q^{4}+−1953125q^{5}+(321349146368741906997+O(11^{2}0))∗a+199460733169605884512+ O(11^{2}0)q^{6}+(281585915160954213610+O(11^{2}0))∗a+175450113922492945138+O(11^{2}0)q^{7}+−(296779586376047968571+ O(11^{2}0))∗a+156351965950057937003+O(11^{2}0)q^{8}+(218611007891324290149+O(11^{2}0))∗a+56007533017502634031+ O(11^{2}0)q^{9} + −1953125 ∗ aq^{10} + −(8054580602308956735 ∗ 11 + O(11^{2}0)) ∗ a − 12515340148482855933 ∗ 11 + O(11^{2}0)q^{11} + −(9262577459088346980 + O(11^{2}0)) ∗ a + 88426330248628911181 + O(11^{2}0)q^{12} + −(156323926578217706212+O(11^{2}0))∗a+64522289621963107047+O(11^{2}0)q^{13}+(268816343727391025066+ O(11^{2}0)) ∗ a − 267669210612741564193 + O(11^{2}0)q^{14} + (289329271526972476072 + O(11^{2}0)) ∗ a + 273583131934250186720+O(11^{2}0)q^{15}+−(238570378441371333164+O(11^{2}0))∗a+17570412699848378530+ O(11^{2}0)q^{16}+(123463712972720713231+O(11^{2}0))∗a+126277729342156318947+O(11^{2}0)q^{17}+(16670092660012589484∗ 11 + O(11^{2}0)) ∗ a − 62066153661114141103 + O(11^{2}0)q^{18} + (153053537731965260672 + O(11^{2}0)) ∗ a + 152088341463358382960+O(11^{2}0)q^{19}+−(225181746573845673180+O(11^{2}0))∗a−332858056054686470017+ O(11^{2}0)q^{20} + (136494026686099183435 + O(11^{2}0)) ∗ a − 228067542290854489548 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 282552299218640417025 + O(11^{2}0)x + 147420816882237002123 + O(11^{2}0). The slope of f_{0} is 1.

14. Consider f = q + 6q^{2} + 10q^{3} + q^{4} + 10q^{5} + 5q^{6} + 5q^{7} + 5q^{8} + 9q^{9} + 5q^{10} + 10q^{12} + q^{13} + 8q^{14} + q^{15} + 6q^{16} + 9q^{17} +10q^{18} +10q^{19} +10q^{20} +6q^{21} +⋯ ∈ S_{22}(Γ_{0}(5); \overline{F}_{11}). Also, z satisfies the equation x^{2} +7x+2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + 3q^{2} + 7q^{3} + 3q^{4} + 2q^{5} + 10q^{6} + 7q^{7} + 2q^{8} + q^{9} + 6q^{10} + 3q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{0} & 0 \\ 0 & u(3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+275144605214625748443+ O(11^{2}0)q^{2}+240721355629404215796+O(11^{2}0)q^{3}+−131467123738830936627+O(11^{2}0)q^{4}+−9765625q^{5}+ 283213755417585405286+O(11^{2}0)q^{6}+133978987453599872842+O(11^{2}0)q^{7}+−281812026498844332035+ O(11^{2}0)q^{8}+−224440976202003808645+O(11^{2}0)q^{9}+−264788399425809942282+O(11^{2}0)q^{10}+13025707697585856402∗ 11 + O(11^{2}0)q^{11} + 37097954646704543734 + O(11^{2}0)q^{12} + 178663291681943713261 + O(11^{2}0)q^{13} + −159334207672755924141+O(11^{2}0)q^{14}+−144775586509371702994+O(11^{2}0)q^{15}+−197323897962062355786+ O(11^{2}0)q^{16}+−154795404961822146590+O(11^{2}0)q^{17}+15993416421861507529+O(11^{2}0)q^{18}+25608834240951565526+ O(11^{2}0)q^{19} + 31432591590499097701 + O(11^{2}0)q^{20} + 203305657421868531140 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

15. Consider f = q + z^{8}1q^{2} + z^{8}0q^{3} + z^{3}q^{4} + q^{5} + z^{4}1q^{6} + z^{2}1q^{7} + z^{3}7q^{8} + z^{1}1q^{9} + z^{8}1q^{10} + z^{8}3q^{12} + z^{2}9q^{13} + z^{1}02q^{14} + z^{8}0q^{15} + z^{9}0q^{16} + z^{3}1q^{17} + z^{9}2q^{18} + z^{2}0q^{19} + z^{3}q^{20} + z^{1}01q^{21} + ⋯ ∈ S_{22}(Γ_{0}(5); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{6}9q^{2} + z^{1}04q^{3} + z^{9}9q^{4} + 9q^{5} + z^{5}3q^{6} + z^{5}7q^{7} + zq^{8} + z^{5}9q^{9} + z^{2}1q^{10} + z^{4}6q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{7}4)ω^{0} & 0 \\ 0 & u(z^{4}6)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(201253886205836665501+ O(11^{2}0)) ∗ a + 282917300157211736178 + O(11^{2}0)q^{3} + −(147063205062835800033 + O(11^{2}0)) ∗ a − 118397767261058874098+O(11^{2}0)q^{4}+9765625q^{5}+−(138770785959858363375+O(11^{2}0))∗a−80976933408913351870+ O(11^{2}0)q^{6} + −(289621283629817684546 + O(11^{2}0)) ∗ a + 1984080566642884678 ∗ 11^{2} + O(11^{2}0)q^{7} + (270148319435878129057+O(11^{2}0))∗a+193812161616154491890+O(11^{2}0)q^{8}+(147557283009907406001+ O(11^{2}0)) ∗ a − 227804422840343248626 + O(11^{2}0)q^{9} + 9765625 ∗ aq^{10} + −(9722534301335260794 ∗ 11 + O(11^{2}0)) ∗ a − 3669875388785746732 ∗ 11 + O(11^{2}0)q^{11} + (48394546349081985981 + O(11^{2}0)) ∗ a + 280878321188874515515+O(11^{2}0)q^{12}+−(249644661096362926151+O(11^{2}0))∗a+75358202062485679197+ O(11^{2}0)q^{13}+−(76514543094535245394+O(11^{2}0))∗a−4676499495859155688+O(11^{2}0)q^{14}+−(165532877673333429328+ O(11^{2}0)) ∗ a − 220341155100993587972 + O(11^{2}0)q^{15} + −(238597242031777567192 + O(11^{2}0)) ∗ a + 74558103200983251026+O(11^{2}0)q^{16}+(179995478818842509264+O(11^{2}0))∗a−247679175938471278396+ O(11^{2}0)q^{17}+−(247138044754070174766+O(11^{2}0))∗a−74938610889099021873+O(11^{2}0)q^{18}+(58363019741151765335+ O(11^{2}0)) ∗ a + 330052848558615776465 + O(11^{2}0)q^{19} + −(296259554459095283659 + O(11^{2}0)) ∗ a + 310382015518050109410+O(11^{2}0)q^{20}+(15286801533937813988+O(11^{2}0))∗a−180525604980275283537+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 147063205062835800033 + O(11^{2}0)x + 118397767261056776946 + O(11^{2}0). The slope of f_{0} is 1.

16. Consider f = q + z^{5}1q^{2} + z^{4}0q^{3} + z^{3}3q^{4} + q^{5} + z^{9}1q^{6} + z^{1}11q^{7} + z^{4}7q^{8} + zq^{9} + z^{5}1q^{10} + z^{7}3q^{12} + z^{7}9q^{13} + z^{4}2q^{14} + z^{4}0q^{15} + z^{3}0q^{16} + z^{1}01q^{17} + z^{5}2q^{18} + z^{1}00q^{19} + z^{3}3q^{20} + z^{3}1q^{21} + ⋯ ∈ S_{22}(Γ_{0}(5); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{3}9q^{2} + z^{6}4q^{3} + z^{9}q^{4} + 9q^{5} + z^{1}03q^{6} + z^{2}7q^{7} + z^{1}1q^{8} + z^{4}9q^{9} + z^{1}11q^{10} + z^{2}6q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{9}4)ω^{0} & 0 \\ 0 & u(z^{2}6)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(201253886205836665501+ O(11^{2}0)) ∗ a + 282917300157211736178 + O(11^{2}0)q^{3} + −(147063205062835800033 + O(11^{2}0)) ∗ a − 118397767261058874098+O(11^{2}0)q^{4}+9765625q^{5}+−(138770785959858363375+O(11^{2}0))∗a−80976933408913351870+ O(11^{2}0)q^{6} + −(289621283629817684546 + O(11^{2}0)) ∗ a + 1984080566642884678 ∗ 11^{2} + O(11^{2}0)q^{7} + (270148319435878129057+O(11^{2}0))∗a+193812161616154491890+O(11^{2}0)q^{8}+(147557283009907406001+ O(11^{2}0)) ∗ a − 227804422840343248626 + O(11^{2}0)q^{9} + 9765625 ∗ aq^{10} + −(9722534301335260794 ∗ 11 + O(11^{2}0)) ∗ a − 3669875388785746732 ∗ 11 + O(11^{2}0)q^{11} + (48394546349081985981 + O(11^{2}0)) ∗ a + 280878321188874515515+O(11^{2}0)q^{12}+−(249644661096362926151+O(11^{2}0))∗a+75358202062485679197+ O(11^{2}0)q^{13}+−(76514543094535245394+O(11^{2}0))∗a−4676499495859155688+O(11^{2}0)q^{14}+−(165532877673333429328+ O(11^{2}0)) ∗ a − 220341155100993587972 + O(11^{2}0)q^{15} + −(238597242031777567192 + O(11^{2}0)) ∗ a + 74558103200983251026+O(11^{2}0)q^{16}+(179995478818842509264+O(11^{2}0))∗a−247679175938471278396+ O(11^{2}0)q^{17}+−(247138044754070174766+O(11^{2}0))∗a−74938610889099021873+O(11^{2}0)q^{18}+(58363019741151765335+ O(11^{2}0)) ∗ a + 330052848558615776465 + O(11^{2}0)q^{19} + −(296259554459095283659 + O(11^{2}0)) ∗ a + 310382015518050109410+O(11^{2}0)q^{20}+(15286801533937813988+O(11^{2}0))∗a−180525604980275283537+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 147063205062835800033 + O(11^{2}0)x + 118397767261056776946 + O(11^{2}0). The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

17. Consider f = q + 7q2 + 2q3 + 8q4 + 6q5 + 3q6 + 6q7 + 10q9 + 9q10 + 10q11 + 5q12 + 6q13 + 9q14 + q15 + 2q16 + 4q17 + 4q18 + q19 + 4q20 + q21 + ⋯ ∈ S14(Γ0(5); \overline{F}11). Since f is ordinary, we do not continue our analysis.

18. Consider f = q + 6q2 + q3 + 6q4 + 5q5 + 6q6 + q7 + 10q8 + 7q9 + 8q10 + 6q12 + 6q13 + 6q14 + 5q15 + q16 + 9q17 + 9q18 + 6q19 + 8q20 + q21 + ⋯ ∈ S14(Γ0(5); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

19. Consider f = q + 2q2 + 7q3 + 5q4 + 3q5 + 3q6 + 5q7 + q8 + 4q9 + 6q10 + 6q11 + 2q12 + 10q14 + 10q15 + 7q16 + 5q17 + 8q18 + 4q19 + 4q20 + 2q21 + ⋯ ∈ S16(Γ0(5); \overline{F}11). Since f is ordinary, we do not continue our analysis.

20. Consider f = q + z111q2 + z115q3 + z16q4 + 7q5 + z106q6 + z62q7 + z82q8 + z9q9 + z75q10 + z43q11 + z11q12 + z65q13 + z53q14 + z79q15 + z14q16 + z88q17 + q18 + z104q19 + z100q20 + z57q21 + ⋯ ∈ S18(Γ0(5); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

21. Consider f = q + z21q2 + z65q3 + z56q4 + 7q5 + z86q6 + z82q7 + z62q8 + z99q9 + z105q10 + z113q11 + zq12 + z115q13 + z103q14 + z29q15 + z34q16 + z8q17 + q18 + z64q19 + z20q20 + z27q21 + ⋯ ∈ S18(Γ0(5); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

22. Consider f = q + 8q2 + 7q3 + 2q4 + 4q5 + q6 + 6q7 + 4q8 + 7q9 + 10q10 + 7q11 + 3q12 + q13 + 4q14 + 6q15 + 7q16 + q18 + 9q19 + 8q20 + 9q21 + ⋯ ∈ S18(Γ0(5); \overline{F}11). Since f is ordinary, we do not continue our analysis.

23. Consider f = q + z69q2 + z104q3 + z99q4 + 9q5 + z53q6 + z57q7 + zq8 + z59q9 + z21q10 + z46q11 + z83q12 + z17q13 + z6q14 + z56q15 + z42q16 + z43q17 + z8q18 + z104q19 + z51q20 + z41q21 + ⋯ ∈ S20(Γ0(5); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

24. Consider f = q + z39q2 + z64q3 + z9q4 + 9q5 + z103q6 + z27q7 + z11q8 + z49q9 + z111q10 + z26q11 + z73q12 + z67q13 + z66q14 + z16q15 + z102q16 + z113q17 + z88q18 + z64q19 + z81q20 + z91q21 + ⋯ ∈ S20(Γ0(5); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

25. Consider f = q+3q2 +7q3 +3q4 +2q5 +10q6 +7q7 +2q8 +q9 +6q10 +3q11 +10q12 +6q13 +10q14 +3q15 +10q16 +7q17 +3q18 +4q19 +6q20 +5q21 +⋯ ∈ S20(Γ0(5); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

26. Consider f = q + z5q2 + z78q3 + z32q4 + 10q5 + z83q6 + 9q7 + z57q8 + 5q9 + z65q10 + q11 + z110q12 + z73q13 + z77q14 + z18q15 + 3q16 + z23q17 + z53q18 + z92q20 + z30q21 + ⋯ ∈ S22(Γ0(5); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

27. Consider f = q + z55q2 + z18q3 + z112q4 + 10q5 + z73q6 + 9q7 + z27q8 + 5q9 + z115q10 + q11 + z10q12 + z83q13 + z7q14 + z78q15 + 3q16 + z13q17 + z103q18 + z52q20 + z90q21 + ⋯ ∈ S22(Γ0(5); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

28. Consider f = q + 3q2 + 4q3 + 7q4 + q5 + q6 + 8q7 + 4q8 + 2q9 + 3q10 + 6q12 + 3q13 + 2q14 + 4q15 + 9q16 + 7q17 + 6q18 + 9q19 + 7q20 + 10q21 + ⋯ ∈ S22(Γ0(5); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

29. Consider f = q + q2 + 10q4 + q5 + 8q8 + 8q9 + q10 + 10q11 + 2q13 + 10q16 + 6q17 + 8q18 + 7q19 + 10q20 + ⋯ ∈ S22(Γ0(5); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

11.5 N = 6, ℓ = 11 (13 forms)

1. Consider f = q + 7q^{2} + 9q^{3} + 5q^{4} + 8q^{5} + 8q^{6} + 2q^{7} + 2q^{8} + 4q^{9} + q^{10} + q^{12} + 9q^{13} + 3q^{14} + 6q^{15} + 3q^{16} + q^{17} + 6q^{18} + 7q^{20} + 7q^{21} + ⋯ ∈ S_{16}(Γ_{0}(6); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + 8q^{2} + 5q^{3} + 9q^{4} + 7q^{5} + 7q^{6} + 8q^{7} + 6q^{8} + 3q^{9} + q^{10} + 6q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(6)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 128q^{2} + 2187q^{3} + 16384q^{4} + 77646q^{5} + 279936q^{6} + 762104q^{7} + 2097152q^{8} + 4782969q^{9} + 9938688q^{10} + 4364652 ∗ 11q^{11} + 35831808q^{12}+285130118q^{13}+97549312q^{14}+169811802q^{15}+268435456q^{16}+−3173671566q^{17}+612220032q^{18}+ −535919660∗11q^{19} +1272152064q^{20} +1666721448q^{21} +⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 7q^{2} + 2q^{3} + 5q^{4} + 8q^{5} + 3q^{6} + 9q^{7} + 2q^{8} + 4q^{9} + q^{10} + 10q^{12} + 10q^{13} + 8q^{14} + 5q^{15} + 3q^{16} + 3q^{17} + 6q^{18} + 6q^{19} + 7q^{20} + 7q^{21} + ⋯ ∈ S_{16}(Γ_{0}(6); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-4 newform g = q + 9q^{2} + 8q^{3} + 4q^{4} + 6q^{5} + 6q^{6} + 6q^{7} + 3q^{8} + 9q^{9} + 10q^{10} + q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{4} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 128q^{2} + −2187q^{3} + 16384q^{4} +−114810q^{5} +−279936q^{6} +−3034528q^{7} +2097152q^{8} +4782969q^{9} +−14695680q^{10} +−9404700∗ 11q^{11}+−35831808q^{12}+−104365834q^{13}+−388419584q^{14}+251089470q^{15}+268435456q^{16}+997689762q^{17}+ 612220032q^{18}+4934015444q^{19}+−1881047040q^{20}+6636512736q^{21}+⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q + 3q^{2} + 6q^{3} + 9q^{4} + 7q^{6} + 5q^{8} + 3q^{9} + 10q^{12} + 7q^{13} + 4q^{16} + 9q^{17} + 9q^{18} + 8q^{19} + ⋯ ∈ S_{18}(Γ_{0}(6); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 4q^{2} + 2q^{3} + 5q^{4} + 8q^{6} + 9q^{8} + 4q^{9} + 6q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(6)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 256q^{2} + −6561q^{3} + 65536q^{4} + −150 ∗ 11^{3}q^{5} + −1679616q^{6} + 2269024 ∗ 11q^{7} + 16777216q^{8} + 43046721q^{9} + −38400 ∗ 11^{3}q^{10} + 11414220∗11q^{11}+−429981696q^{12}+4227195518q^{13}+580870144∗11q^{14}+984150∗11^{3}q^{15}+4294967296q^{16}+ 35551782594q^{17} +11019960576q^{18} +−64354589764q^{19} +−9830400∗11^{3}q^{20} +−14887066464∗11q^{21} +⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

4. Consider f = q + 8q^{2} + 6q^{3} + 9q^{4} + 4q^{6} + 6q^{8} + 3q^{9} + 10q^{12} + 4q^{13} + 4q^{16} + 2q^{17} + 2q^{18} + 3q^{19} + ⋯ ∈ S_{18}(Γ_{0}(6); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-6 newform g = q + 4q^{2} + 2q^{3} + 5q^{4} + 8q^{6} + 9q^{8} + 4q^{9} + 6q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{6} & 0 \\ 0 & u(6)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +−256q^{2} +−6561q^{3} + 65536q^{4} + 58650 ∗ 11q^{5} + 1679616q^{6} + 361312 ∗ 11q^{7} + −16777216q^{8} + 43046721q^{9} + −15014400 ∗ 11q^{10} + −45460788 ∗ 11q^{11} + −429981696q^{12} + −5425661314q^{13} + −92495872 ∗ 11q^{14} + −384802650 ∗ 11q^{15}+4294967296q^{16}+−5466992958q^{17}+−11019960576q^{18}+−53889877060q^{19}+3843686400∗11q^{20}+ −2370568032 ∗ 11q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

5. Consider f = q + 5q^{2} + 7q^{3} + 3q^{4} + 2q^{5} + 2q^{6} + 10q^{7} + 4q^{8} + 5q^{9} + 10q^{10} + 10q^{12} + 4q^{13} + 6q^{14} + 3q^{15} + 9q^{16} + 2q^{17} + 3q^{18} + q^{19} + 6q^{20} + 4q^{21} + ⋯ ∈ S_{20}(Γ_{0}(6); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 9q^{2} + 8q^{3} + 4q^{4} + 6q^{5} + 6q^{6} + 6q^{7} + 3q^{8} + 9q^{9} + 10q^{10} + q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−512q^{2}+−19683q^{3}+ 262144q^{4}+−3732474q^{5}+10077696q^{6}+−149672656q^{7}+−134217728q^{8}+387420489q^{9}+1911026688q^{10}+ −678152028∗11q^{11}+−5159780352q^{12}+59238459878q^{13}+76632399872q^{14}+73466285742q^{15}+68719476736q^{16}+ 523110429954q^{17}+−198359290368q^{18}+969502037780q^{19}+−978445664256q^{20}+2946006888048q^{21}+⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

6. Consider f = q + 5q^{2} + 4q^{3} + 3q^{4} + 2q^{5} + 9q^{6} + q^{7} + 4q^{8} + 5q^{9} + 10q^{10} + q^{12} + 8q^{13} + 5q^{14} + 8q^{15} + 9q^{16} + 8q^{17} + 3q^{18} + 6q^{20} + 4q^{21} + ⋯ ∈ S_{20}(Γ_{0}(6); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q + 8q^{2} + 5q^{3} + 9q^{4} + 7q^{5} + 7q^{6} + 8q^{7} + 6q^{8} + 3q^{9} + q^{10} + 6q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{8} & 0 \\ 0 & u(6)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −512q^{2} + 19683q^{3} + 262144q^{4}+−5849490q^{5}+−10077696q^{6}+173530952q^{7}+−134217728q^{8}+387420489q^{9}+2994938880q^{10}+ −664802580 ∗ 11q^{11} + 5159780352q^{12} + −41845065034q^{13} + −88847847424q^{14} + −115135511670q^{15} + 68719476736q^{16}+−95834399598q^{17}+−198359290368q^{18}+−219915683756∗11q^{19}+−1533408706560q^{20}+ 3415609728216q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

7. Consider f = q+9q2 +8q3 +4q4 +6q5 +6q6 +6q7 +3q8 +9q9 +10q10 +q11 +10q12 +5q13 +10q14 +4q15 +5q16 +6q17 +4q18 +9q19 +2q20 +4q21 +⋯ ∈ S14(Γ0(6); \overline{F}11). Since f is ordinary, we do not continue our analysis.

8. Consider f = q + 4q2 + 2q3 + 5q4 + 8q6 + 9q8 + 4q9 + 6q11 + 10q12 + 2q13 + 3q16 + 4q17 + 5q18 + 10q19 + ⋯ ∈ S16(Γ0(6); \overline{F}11). Since f is ordinary, we do not continue our analysis.

9. Consider f = q + 8q2 + 5q3 + 9q4 + 7q5 + 7q6 + 8q7 + 6q8 + 3q9 + q10 + 6q11 + q12 + 4q13 + 9q14 + 2q15 + 4q16 + 5q17 + 2q18 + 8q20 + 7q21 + ⋯ ∈ S18(Γ0(6); \overline{F}11). Since f is ordinary, we do not continue our analysis.

10. Consider f = q+6q2 +4q3 +3q4 +10q5 +2q6 +9q7 +7q8 +5q9 +5q10 +q11 +q12 +7q13 +10q14 +7q15 +9q16 +3q17 +8q18 +8q19 +8q20 +3q21 +⋯ ∈ S20(Γ0(6); \overline{F}11). Since f is ordinary, we do not continue our analysis.

11. Consider f = q +q2 +10q3 +q4 +2q5 +10q6 +7q7 +q8 +q9 +2q10 +10q11 +10q12 +5q13 +7q14 +9q15 +q16 +2q17 +q18 +4q19 +2q20 +4q21 +⋯ ∈ S22(Γ0(6); \overline{F}11). Since f is ordinary, we do not continue our analysis.

12. Consider f = q +q2 +q3 +q4 +7q5 +q6 +9q7 +q8 +q9 +7q10 +q11 +q12 +4q13 +9q14 +7q15 +q16 +9q17 +q18 +7q20 +9q21 +⋯ ∈ S22(Γ0(6); \overline{F}11). Since f is ordinary, we do not continue our analysis.

13. Consider f = q + 10q2 + q3 + q4 + 10q6 + 2q7 + 10q8 + q9 + 10q11 + q12 + 7q13 + 9q14 + q16 + 5q17 + 10q18 + 7q19 + 2q21 + ⋯ ∈ S22(Γ0(6); \overline{F}11). Since f is ordinary, we do not continue our analysis.

1. Consider f = q + z^{9}2q^{2} + 9q^{3} + z^{8}1q^{4} + z^{1}06q^{5} + z^{4}4q^{6} + 7q^{7} + zq^{8} + 10q^{9} + z^{7}8q^{10} + z^{3}3q^{12} + z^{7}9q^{13} + z^{5}6q^{14} + z^{5}8q^{15} + z^{9}q^{16} + z^{4}4q^{17} + z^{3}2q^{18} + z^{4}5q^{19} + z^{6}7q^{20} + 8q^{21} + ⋯ ∈ S_{14}(Γ_{0}(7); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{6}8q^{2} + q^{3} + z^{3}3q^{4} + z^{1}0q^{5} + z^{6}8q^{6} + 8q^{7} + z^{4}9q^{8} + 8q^{9} + z^{7}8q^{10} + z^{1}10q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}0)ω^{1} & 0 \\ 0 & u(z^{1}10)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(27405027656081507142∗ 11 + O(11^{2}0)) ∗ a + 76192257249220610122 + O(11^{2}0)q^{3} + (253713675939958896649 + O(11^{2}0)) ∗ a − 241557092058087312783+O(11^{2}0)q^{4}+−(231615917718129726741+O(11^{2}0))∗a+228576771747661826352+ O(11^{2}0)q^{5}+(14184447333117365038+O(11^{2}0))∗a+8413809707846420873∗11+O(11^{2}0)q^{6}+−117649q^{7}+ −(130944374886668774744+O(11^{2}0))∗a−225734439117230427277+O(11^{2}0)q^{8}+(18354764540037106737∗ 11 + O(11^{2}0)) ∗ a − 177113812277559718573 + O(11^{2}0)q^{9} + −(324148185104364567748 + O(11^{2}0)) ∗ a + 223271753433761017613+O(11^{2}0)q^{10}+−(32736093721667194711∗11+O(11^{2}1))∗a+111753888953832168033∗ 11 + O(11^{2}1)q^{11} + −(329669831102922188292 + O(11^{2}0)) ∗ a − 77279977617895811042 + O(11^{2}0)q^{12} + (32129796828429361965 + O(11^{2}0)) ∗ a − 107967458427797648454 + O(11^{2}0)q^{13} + −117649 ∗ aq^{14} + (102588930153388913867+O(11^{2}0))∗a+259897339387239936865+O(11^{2}0)q^{15}+−(774779600808462352443+ O(11^{2}1)) ∗ a − 201138724867539031094 + O(11^{2}1)q^{16} + −(107497656441820267977 + O(11^{2}0)) ∗ a + 5538118393333420979∗11+O(11^{2}0)q^{17}+−(129745026733968502319+O(11^{2}0))∗a−12556083353382694743∗ 11 + O(11^{2}0)q^{18} + −(132369483296703034297 + O(11^{2}0)) ∗ a − 317560857072964718418 + O(11^{2}0)q^{19} + −(295202169116159997185+O(11^{2}0))∗a−122898259699358589179+O(11^{2}0)q^{20}+−(10831549184817619780∗ 11 + O(11^{2}0)) ∗ a − 221940632125997649054 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −253713675939958896649 + O(11^{2}0)x + 241557092058087304591 + O(11^{2}0). The slope of f_{0} is 1.

2. Consider f = q + z^{5}2q^{2} + 9q^{3} + z^{5}1q^{4} + z^{8}6q^{5} + z^{4}q^{6} + 7q^{7} + z^{1}1q^{8} + 10q^{9} + z^{1}8q^{10} + z^{3}q^{12} + z^{2}9q^{13} + z^{1}6q^{14} + z^{3}8q^{15} + z^{9}9q^{16} + z^{4}q^{17} + z^{1}12q^{18} + z^{1}5q^{19} + z^{1}7q^{20} + 8q^{21} + ⋯ ∈ S_{14}(Γ_{0}(7); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{2}8q^{2} + q^{3} + z^{3}q^{4} + z^{1}10q^{5} + z^{2}8q^{6} + 8q^{7} + z^{5}9q^{8} + 8q^{9} + z^{1}8q^{10} + z^{1}0q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}10)ω^{1} & 0 \\ 0 & u(z^{1}0)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(27405027656081507142∗ 11 + O(11^{2}0)) ∗ a + 76192257249220610122 + O(11^{2}0)q^{3} + (253713675939958896649 + O(11^{2}0)) ∗ a − 241557092058087312783+O(11^{2}0)q^{4}+−(231615917718129726741+O(11^{2}0))∗a+228576771747661826352+ O(11^{2}0)q^{5}+(14184447333117365038+O(11^{2}0))∗a+8413809707846420873∗11+O(11^{2}0)q^{6}+−117649q^{7}+ −(130944374886668774744+O(11^{2}0))∗a−225734439117230427277+O(11^{2}0)q^{8}+(18354764540037106737∗ 11 + O(11^{2}0)) ∗ a − 177113812277559718573 + O(11^{2}0)q^{9} + −(324148185104364567748 + O(11^{2}0)) ∗ a + 223271753433761017613+O(11^{2}0)q^{10}+−(32736093721667194711∗11+O(11^{2}1))∗a+111753888953832168033∗ 11 + O(11^{2}1)q^{11} + −(329669831102922188292 + O(11^{2}0)) ∗ a − 77279977617895811042 + O(11^{2}0)q^{12} + (32129796828429361965 + O(11^{2}0)) ∗ a − 107967458427797648454 + O(11^{2}0)q^{13} + −117649 ∗ aq^{14} + (102588930153388913867+O(11^{2}0))∗a+259897339387239936865+O(11^{2}0)q^{15}+−(774779600808462352443+ O(11^{2}1)) ∗ a − 201138724867539031094 + O(11^{2}1)q^{16} + −(107497656441820267977 + O(11^{2}0)) ∗ a + 5538118393333420979∗11+O(11^{2}0)q^{17}+−(129745026733968502319+O(11^{2}0))∗a−12556083353382694743∗ 11 + O(11^{2}0)q^{18} + −(132369483296703034297 + O(11^{2}0)) ∗ a − 317560857072964718418 + O(11^{2}0)q^{19} + −(295202169116159997185+O(11^{2}0))∗a−122898259699358589179+O(11^{2}0)q^{20}+−(10831549184817619780∗ 11 + O(11^{2}0)) ∗ a − 221940632125997649054 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −253713675939958896649 + O(11^{2}0)x + 241557092058087304591 + O(11^{2}0). The slope of f_{0} is 1.

3. Consider f = q + z^{8}50q^{2} + z^{4}08q^{3} + z^{5}52q^{4} + z^{5}75q^{5} + z^{1}258q^{6} + 4q^{7} + z^{1}313q^{8} + z^{4}96q^{9} + z^{9}5q^{10} + z^{9}60q^{12} +z^{1}87q^{13} +z^{1}116q^{14} +z^{9}83q^{15} +z^{1}12q^{16} +z^{3}2q^{17} +z^{1}6q^{18} +z^{9}65q^{19} +z^{1}127q^{20} +z^{6}74q^{21} +⋯ ∈ S_{14}(Γ_{0}(7); \overline{F}_{11}). Also, z satisfies the equation x^{3} + 2x + 9 = 0. We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q +z^{5}84q^{2} +z^{9}40q^{3} +z^{2}0q^{4} +z^{8}41q^{5} +z^{1}94q^{6} +3q^{7} +z^{5}15q^{8} +z^{2}30q^{9} +z^{9}5q^{10} + z^{4}60q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{8}70)ω^{1} & 0 \\ 0 & u(z^{4}60)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(73062537695835581310+ O(11^{2}0)) ∗ a^{2} + (20182199046657130027 + O(11^{2}0)) ∗ a − 19085458489416885708 ∗ 11 + O(11^{2}0)q^{3} + a^{2} − 8192q^{4} + (307437306453382102643 + O(11^{2}0)) ∗ a^{2} + (100910995233285650727 + O(11^{2}0)) ∗ a + 295799772947191376720+O(11^{2}0)q^{5}+−(325497702777785185789+O(11^{2}0))∗a^{2}+(134568047065692216710+ O(11^{2}0))∗a−185847300703495734115+O(11^{2}0)q^{6} +117649q^{7} +−(82503212057148747198+O(11^{2}0))∗ a^{2}−(322945264878481109917+O(11^{2}0))∗a−143463775974642050278+O(11^{2}0)q^{8}+(60338053741629420285+ O(11^{2}0))∗a^{2}−(13745573973452655995+O(11^{2}0))∗a−304333433843260696454+O(11^{2}0)q^{9}+−(294712822499175941568+ O(11^{2}0)) ∗ a^{2} − (107347620306490141933 + O(11^{2}0)) ∗ a + 218473704279657731649 + O(11^{2}0)q^{10} + (20859038665989026243∗11+O(11^{2}0))∗a^{2}−(2107930898399128340∗11^{2}+O(11^{2}0))∗a+16734318217629676135∗ 11+O(11^{2}0)q^{11}+(3025117775428787397912+O(11^{2}1))∗a^{2}−(2959173509638679583292+O(11^{2}1))∗a+ 2793588470984168460627+O(11^{2}1)q^{12}+(119443844231161382957+O(11^{2}0))∗a^{2}+(111468075218233833164+ O(11^{2}0)) ∗ a + 19906582757566308358 + O(11^{2}0)q^{13} + 117649 ∗ aq^{14} + (1156754406851822243 ∗ 11 + O(11^{2}0)) ∗ a^{2} + (107437860702390453820 + O(11^{2}0)) ∗ a + 17980889278896392910 ∗ 11 + O(11^{2}0)q^{15} + (19030612795030557951∗11+O(11^{2}0))∗a^{2}−(26227782404289787462+O(11^{2}0))∗a−162978018279961939940+ O(11^{2}0)q^{16} + (29140007820784670465 + O(11^{2}0)) ∗ a^{2} − (263391835261160920759 + O(11^{2}0)) ∗ a + 8711997465235413189∗11+O(11^{2}0)q^{17}+−(156376686294056584119+O(11^{2}0))∗a^{2}+(2411879672265064857∗ 11 + O(11^{2}0)) ∗ a − 59034091519339306820 + O(11^{2}0)q^{18} + −(187579369185804595599 + O(11^{2}0)) ∗ a^{2} − (67253420233139422598+O(11^{2}0))∗a−15519192751975591322∗11+O(11^{2}0)q^{19}+(264743144774869977495+ O(11^{2}0)) ∗ a^{2} + (91856352343470977542 + O(11^{2}0)) ∗ a + 307210597158629948359 + O(11^{2}0)q^{20} + −(7812124041067979013+O(11^{2}0))∗a^{2}+(280803523160418076194+O(11^{2}0))∗a+9740901684465867282∗ 11 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 82503212057148747198 + O(11^{2}0)x^{2} + 322945264878481093533 + O(11^{2}0)x + 143463775974642050278 + O(11^{2}0). The slope of f_{0} is 1.

4. Consider f = q + z^{4}0q^{2} + z^{4}98q^{3} + z^{7}52q^{4} + z^{1}005q^{5} + z^{5}38q^{6} + 4q^{7} + z^{1}143q^{8} + z^{1}36q^{9} + z^{1}045q^{10} + z^{1}250q^{12}+z^{7}27q^{13}+z^{3}06q^{14}+z^{1}73q^{15}+z^{1}232q^{16}+z^{3}52q^{17}+z^{1}76q^{18}+z^{1}305q^{19}+z^{4}27q^{20}+z^{7}64q^{21}+⋯ ∈ S_{14}(Γ_{0}(7); \overline{F}_{11}). Also, z satisfies the equation x^{3} + 2x + 9 = 0. We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{1}104q^{2} + z^{1}030q^{3} + z^{2}20q^{4} + z^{1}271q^{5} + z^{8}04q^{6} + 3q^{7} + z^{3}45q^{8} + z^{1}200q^{9} + z^{1}045q^{10} + z^{1}070q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}60)ω^{1} & 0 \\ 0 & u(z^{1}070)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(73062537695835581310+ O(11^{2}0)) ∗ a^{2} + (20182199046657130027 + O(11^{2}0)) ∗ a − 19085458489416885708 ∗ 11 + O(11^{2}0)q^{3} + a^{2} − 8192q^{4} + (307437306453382102643 + O(11^{2}0)) ∗ a^{2} + (100910995233285650727 + O(11^{2}0)) ∗ a + 295799772947191376720+O(11^{2}0)q^{5}+−(325497702777785185789+O(11^{2}0))∗a^{2}+(134568047065692216710+ O(11^{2}0))∗a−185847300703495734115+O(11^{2}0)q^{6} +117649q^{7} +−(82503212057148747198+O(11^{2}0))∗ a^{2}−(322945264878481109917+O(11^{2}0))∗a−143463775974642050278+O(11^{2}0)q^{8}+(60338053741629420285+ O(11^{2}0))∗a^{2}−(13745573973452655995+O(11^{2}0))∗a−304333433843260696454+O(11^{2}0)q^{9}+−(294712822499175941568+ O(11^{2}0)) ∗ a^{2} − (107347620306490141933 + O(11^{2}0)) ∗ a + 218473704279657731649 + O(11^{2}0)q^{10} + (20859038665989026243∗11+O(11^{2}0))∗a^{2}−(2107930898399128340∗11^{2}+O(11^{2}0))∗a+16734318217629676135∗ 11+O(11^{2}0)q^{11}+(3025117775428787397912+O(11^{2}1))∗a^{2}−(2959173509638679583292+O(11^{2}1))∗a+ 2793588470984168460627+O(11^{2}1)q^{12}+(119443844231161382957+O(11^{2}0))∗a^{2}+(111468075218233833164+ O(11^{2}0)) ∗ a + 19906582757566308358 + O(11^{2}0)q^{13} + 117649 ∗ aq^{14} + (1156754406851822243 ∗ 11 + O(11^{2}0)) ∗ a^{2} + (107437860702390453820 + O(11^{2}0)) ∗ a + 17980889278896392910 ∗ 11 + O(11^{2}0)q^{15} + (19030612795030557951∗11+O(11^{2}0))∗a^{2}−(26227782404289787462+O(11^{2}0))∗a−162978018279961939940+ O(11^{2}0)q^{16} + (29140007820784670465 + O(11^{2}0)) ∗ a^{2} − (263391835261160920759 + O(11^{2}0)) ∗ a + 8711997465235413189∗11+O(11^{2}0)q^{17}+−(156376686294056584119+O(11^{2}0))∗a^{2}+(2411879672265064857∗ 11 + O(11^{2}0)) ∗ a − 59034091519339306820 + O(11^{2}0)q^{18} + −(187579369185804595599 + O(11^{2}0)) ∗ a^{2} − (67253420233139422598+O(11^{2}0))∗a−15519192751975591322∗11+O(11^{2}0)q^{19}+(264743144774869977495+ O(11^{2}0)) ∗ a^{2} + (91856352343470977542 + O(11^{2}0)) ∗ a + 307210597158629948359 + O(11^{2}0)q^{20} + −(7812124041067979013+O(11^{2}0))∗a^{2}+(280803523160418076194+O(11^{2}0))∗a+9740901684465867282∗ 11 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 82503212057148747198 + O(11^{2}0)x^{2} + 322945264878481093533 + O(11^{2}0)x + 143463775974642050278 + O(11^{2}0). The slope of f_{0} is 1.

5. Consider f = q + z^{4}40q^{2} + z^{1}58q^{3} + z^{2}92q^{4} + z^{4}15q^{5} + z^{5}98q^{6} + 4q^{7} + z^{6}03q^{8} + z^{1}66q^{9} + z^{8}55q^{10} + z^{4}50q^{12}+z^{1}7q^{13}+z^{7}06q^{14}+z^{5}73q^{15}+z^{2}52q^{16}+z^{1}212q^{17}+z^{6}06q^{18}+z^{1}055q^{19}+z^{7}07q^{20}+z^{4}24q^{21}+⋯ ∈ S_{14}(Γ_{0}(7); \overline{F}_{11}). Also, z satisfies the equation x^{3} + 2x + 9 = 0. We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{1}74q^{2} + z^{6}90q^{3} + z^{1}090q^{4} + z^{6}81q^{5} + z^{8}64q^{6} + 3q^{7} + z^{1}135q^{8} + z^{1}230q^{9} + z^{8}55q^{10} + z^{1}130q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}00)ω^{1} & 0 \\ 0 & u(z^{1}130)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(73062537695835581310+ O(11^{2}0)) ∗ a^{2} + (20182199046657130027 + O(11^{2}0)) ∗ a − 19085458489416885708 ∗ 11 + O(11^{2}0)q^{3} + a^{2} − 8192q^{4} + (307437306453382102643 + O(11^{2}0)) ∗ a^{2} + (100910995233285650727 + O(11^{2}0)) ∗ a + 295799772947191376720+O(11^{2}0)q^{5}+−(325497702777785185789+O(11^{2}0))∗a^{2}+(134568047065692216710+ O(11^{2}0))∗a−185847300703495734115+O(11^{2}0)q^{6} +117649q^{7} +−(82503212057148747198+O(11^{2}0))∗ a^{2}−(322945264878481109917+O(11^{2}0))∗a−143463775974642050278+O(11^{2}0)q^{8}+(60338053741629420285+ O(11^{2}0))∗a^{2}−(13745573973452655995+O(11^{2}0))∗a−304333433843260696454+O(11^{2}0)q^{9}+−(294712822499175941568+ O(11^{2}0)) ∗ a^{2} − (107347620306490141933 + O(11^{2}0)) ∗ a + 218473704279657731649 + O(11^{2}0)q^{10} + (20859038665989026243∗11+O(11^{2}0))∗a^{2}−(2107930898399128340∗11^{2}+O(11^{2}0))∗a+16734318217629676135∗ 11+O(11^{2}0)q^{11}+(3025117775428787397912+O(11^{2}1))∗a^{2}−(2959173509638679583292+O(11^{2}1))∗a+ 2793588470984168460627+O(11^{2}1)q^{12}+(119443844231161382957+O(11^{2}0))∗a^{2}+(111468075218233833164+ O(11^{2}0)) ∗ a + 19906582757566308358 + O(11^{2}0)q^{13} + 117649 ∗ aq^{14} + (1156754406851822243 ∗ 11 + O(11^{2}0)) ∗ a^{2} + (107437860702390453820 + O(11^{2}0)) ∗ a + 17980889278896392910 ∗ 11 + O(11^{2}0)q^{15} + (19030612795030557951∗11+O(11^{2}0))∗a^{2}−(26227782404289787462+O(11^{2}0))∗a−162978018279961939940+ O(11^{2}0)q^{16} + (29140007820784670465 + O(11^{2}0)) ∗ a^{2} − (263391835261160920759 + O(11^{2}0)) ∗ a + 8711997465235413189∗11+O(11^{2}0)q^{17}+−(156376686294056584119+O(11^{2}0))∗a^{2}+(2411879672265064857∗ 11 + O(11^{2}0)) ∗ a − 59034091519339306820 + O(11^{2}0)q^{18} + −(187579369185804595599 + O(11^{2}0)) ∗ a^{2} − (67253420233139422598+O(11^{2}0))∗a−15519192751975591322∗11+O(11^{2}0)q^{19}+(264743144774869977495+ O(11^{2}0)) ∗ a^{2} + (91856352343470977542 + O(11^{2}0)) ∗ a + 307210597158629948359 + O(11^{2}0)q^{20} + −(7812124041067979013+O(11^{2}0))∗a^{2}+(280803523160418076194+O(11^{2}0))∗a+9740901684465867282∗ 11 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 82503212057148747198 + O(11^{2}0)x^{2} + 322945264878481093533 + O(11^{2}0)x + 143463775974642050278 + O(11^{2}0). The slope of f_{0} is 1.

6. Consider f = q + 9q^{2} + 5q^{3} + 5q^{4} + 3q^{5} + q^{6} + 6q^{7} + 10q^{8} + 2q^{9} + 5q^{10} + 3q^{12} + q^{13} + 10q^{14} + 4q^{15} + 7q^{16} + 5q^{17} + 7q^{18} + 4q^{20} + 8q^{21} + ⋯ ∈ S_{16}(Γ_{0}(7); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-4 newform g = q + 10q^{2} + 9q^{3} + 4q^{4} + 5q^{5} + 2q^{6} + 4q^{7} + 4q^{8} + 10q^{9} + 6q^{10} + 3q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{4} & 0 \\ 0 & u(3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+28722719146595434090+ O(11^{2}0)q^{2}+328822576687956964057+O(11^{2}0)q^{3}+31478740502248114311+O(11^{2}0)q^{4}+157196323713483771511+ O(11^{2}0)q^{5} + 290698969731361434893 + O(11^{2}0)q^{6} + 823543q^{7} + 93987852129612995916 + O(11^{2}0)q^{8} + −301215984273656613203+O(11^{2}0)q^{9}+102993401635051646823+O(11^{2}0)q^{10}+−8289955520052000907∗ 11 + O(11^{2}0)q^{11} + −183253795895146278794 + O(11^{2}0)q^{12} + −325662347394605096303 + O(11^{2}0)q^{13} + −168277679098906735491+O(11^{2}0)q^{14}+−300685747036274395693+O(11^{2}0)q^{15}+101176278126498764735+ O(11^{2}0)q^{16}+317297279640967835951+O(11^{2}0)q^{17}+−303999164762280200327+O(11^{2}0)q^{18}+1145298134780832843∗ 11 + O(11^{2}0)q^{19} + 188519264192668841920 + O(11^{2}0)q^{20} + 166813106491786752225 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

7. Consider f = q + 3q^{2} + 8q^{3} + 10q^{4} + 3q^{5} + 2q^{6} + 6q^{7} + 8q^{9} + 9q^{10} + 3q^{12} + 2q^{13} + 7q^{14} + 2q^{15} + 10q^{16} + 10q^{17} + 2q^{18} + 6q^{19} + 8q^{20} + 4q^{21} + ⋯ ∈ S_{16}(Γ_{0}(7); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + 5q^{2} + 2q^{3} + 7q^{4} + 4q^{5} + 10q^{6} + 2q^{7} + 6q^{9} + 9q^{10} + 9q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{1} & 0 \\ 0 & u(9)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−138614148576207886975+ O(11^{2}0)q^{2}+239332666431501511602+O(11^{2}0)q^{3}+−121585747547483008593+O(11^{2}0)q^{4}+−335326179801247452914+ O(11^{2}0)q^{5}+277971151856717771124+O(11^{2}0)q^{6}+823543q^{7}+8459690774548121497∗11+O(11^{2}0)q^{8}+ −210837693083463794197+O(11^{2}0)q^{9}+209301275982400929908+O(11^{2}0)q^{10}+16528367446492011499∗ 11+O(11^{2}0)q^{11}+5081085829827769622+O(11^{2}0)q^{12}+221497299944659678788+O(11^{2}0)q^{13}+198379240540738210059+ O(11^{2}0)q^{14}+334474909313402187794+O(11^{2}0)q^{15}+168787108538498063129+O(11^{2}0)q^{16}+−35601025975750472326+ O(11^{2}0)q^{17}+180708856832748527439+O(11^{2}0)q^{18}+199129865421488667573+O(11^{2}0)q^{19}+32513237723261157981+ O(11^{2}0)q^{20} + −205904353517006444692 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

8. Consider f = q + z^{7}7q^{2} + z^{5}q^{3} + z^{9}3q^{4} + z^{1}02q^{5} + z^{8}2q^{6} + 5q^{7} + 6q^{8} + z^{1}7q^{9} + z^{5}9q^{10} + z^{9}8q^{12} + z^{2}1q^{13} + z^{5}q^{14} + z^{1}07q^{15} + z^{5}1q^{16} + z^{9}5q^{17} + z^{9}4q^{18} + 5q^{19} + z^{7}5q^{20} + z^{5}3q^{21} + ⋯ ∈ S_{16}(Γ_{0}(7); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{2}9q^{2} + z^{1}01q^{3} + z^{1}17q^{4} + z^{3}0q^{5} + z^{1}0q^{6} + 9q^{7} + 7q^{8} + z^{8}9q^{9} + z^{5}9q^{10} + z^{1}0q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}10)ω^{1} & 0 \\ 0 & u(z^{1}0)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(50463253003821933039+ O(11^{2}0)) ∗ a + 220316573912364198 ∗ 11^{2} + O(11^{2}0)q^{3} + (216077803275256990261 + O(11^{2}0)) ∗ a + 261776061909737877606+O(11^{2}0)q^{4}+−(326676203302520720535+O(11^{2}0))∗a+83513661171343584101+ O(11^{2}0)q^{5}+−(7234437342046056580+O(11^{2}0))∗a−175204820635375767623+O(11^{2}0)q^{6}+−823543q^{7}+ (864428157559304036∗11^{2}+O(11^{2}0))∗a−226914130103445350626+O(11^{2}0)q^{8}+(38385989983966513943+ O(11^{2}0)) ∗ a − 22598052696525107668 ∗ 11 + O(11^{2}0)q^{9} + −(180636882297727091904 + O(11^{2}0)) ∗ a − 231151081512365883589+O(11^{2}0)q^{10}+(6969265337691883746∗11+O(11^{2}0))∗a+19861206531078616213∗ 11 + O(11^{2}0)q^{11} + (143592526238306783875 + O(11^{2}0)) ∗ a − 104710372170636463212 + O(11^{2}0)q^{12} + −(64713494215686064318 + O(11^{2}0)) ∗ a + 174373707336937170362 + O(11^{2}0)q^{13} + −823543 ∗ aq^{14} + (335052002151537678088+O(11^{2}0))∗a+275442710746747539592+O(11^{2}0)q^{15}+(162250831955272120249+ O(11^{2}0)) ∗ a + 324199508623692953102 + O(11^{2}0)q^{16} + −(235426534357400757006 + O(11^{2}0)) ∗ a − 92968995082271008793+O(11^{2}0)q^{17}+−(107669562496056579982+O(11^{2}0))∗a+34554866955531112604+ O(11^{2}0)q^{18} + −(13864543600948089143 ∗ 11 + O(11^{2}0)) ∗ a + 100026418944953616406 + O(11^{2}0)q^{19} + −(251310441243754095833+O(11^{2}0))∗a+123480929389221313105+O(11^{2}0)q^{20}+(200581562564192354603+ O(11^{2}0)) ∗ a + 2169378995333437440 ∗ 11^{2} + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −216077803275256990261 + O(11^{2}0)x + −261776061909737910374 + O(11^{2}0). The slope of f_{0} is 1.

9. Consider f = q + z^{7}q^{2} + z^{5}5q^{3} + z^{6}3q^{4} + z^{4}2q^{5} + z^{6}2q^{6} + 5q^{7} + 6q^{8} + z^{6}7q^{9} + z^{4}9q^{10} + z^{1}18q^{12} + z^{1}11q^{13} + z^{5}5q^{14} + z^{9}7q^{15} + z^{8}1q^{16} + z^{8}5q^{17} + z^{7}4q^{18} + 5q^{19} + z^{1}05q^{20} + z^{1}03q^{21} + ⋯ ∈ S_{16}(Γ_{0}(7); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{7}9q^{2} + z^{3}1q^{3} + z^{8}7q^{4} + z^{9}0q^{5} + z^{1}10q^{6} + 9q^{7} + 7q^{8} + z^{1}9q^{9} + z^{4}9q^{10} + z^{1}10q^{11} + · · · . We deduce

that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}0)ω^{1} & 0 \\ 0 & u(z^{1}10)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(50463253003821933039+ O(11^{2}0)) ∗ a + 220316573912364198 ∗ 11^{2} + O(11^{2}0)q^{3} + (216077803275256990261 + O(11^{2}0)) ∗ a + 261776061909737877606+O(11^{2}0)q^{4}+−(326676203302520720535+O(11^{2}0))∗a+83513661171343584101+ O(11^{2}0)q^{5}+−(7234437342046056580+O(11^{2}0))∗a−175204820635375767623+O(11^{2}0)q^{6}+−823543q^{7}+ (864428157559304036∗11^{2}+O(11^{2}0))∗a−226914130103445350626+O(11^{2}0)q^{8}+(38385989983966513943+ O(11^{2}0)) ∗ a − 22598052696525107668 ∗ 11 + O(11^{2}0)q^{9} + −(180636882297727091904 + O(11^{2}0)) ∗ a − 231151081512365883589+O(11^{2}0)q^{10}+(6969265337691883746∗11+O(11^{2}0))∗a+19861206531078616213∗ 11 + O(11^{2}0)q^{11} + (143592526238306783875 + O(11^{2}0)) ∗ a − 104710372170636463212 + O(11^{2}0)q^{12} + −(64713494215686064318 + O(11^{2}0)) ∗ a + 174373707336937170362 + O(11^{2}0)q^{13} + −823543 ∗ aq^{14} + (335052002151537678088+O(11^{2}0))∗a+275442710746747539592+O(11^{2}0)q^{15}+(162250831955272120249+ O(11^{2}0)) ∗ a + 324199508623692953102 + O(11^{2}0)q^{16} + −(235426534357400757006 + O(11^{2}0)) ∗ a − 92968995082271008793+O(11^{2}0)q^{17}+−(107669562496056579982+O(11^{2}0))∗a+34554866955531112604+ O(11^{2}0)q^{18} + −(13864543600948089143 ∗ 11 + O(11^{2}0)) ∗ a + 100026418944953616406 + O(11^{2}0)q^{19} + −(251310441243754095833+O(11^{2}0))∗a+123480929389221313105+O(11^{2}0)q^{20}+(200581562564192354603+ O(11^{2}0)) ∗ a + 2169378995333437440 ∗ 11^{2} + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −216077803275256990261 + O(11^{2}0)x + −261776061909737910374 + O(11^{2}0). The slope of f_{0} is 1.

10. Consider f = q + z^{6}1q^{2} + z^{1}7q^{3} + z^{2}1q^{4} + z^{9}4q^{5} + z^{7}8q^{6} + 2q^{7} + 8q^{8} + z^{3}1q^{9} + z^{3}5q^{10} + z^{3}8q^{12} + z^{4}6q^{13} + z^{7}3q^{14} + z^{1}11q^{15} + z^{6}3q^{16} + z^{9}4q^{17} + z^{9}2q^{18} + z^{1}05q^{19} + z^{1}15q^{20} + z^{2}9q^{21} + ⋯ ∈ S_{18}(Γ_{0}(7); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{4}9q^{2} + z^{4}1q^{3} + z^{1}17q^{4} + z^{4}6q^{5} + z^{9}0q^{6} + 5q^{7} + q^{8} + z^{7}9q^{9} + z^{9}5q^{10} + z^{3}0q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{9}0)ω^{6} & 0 \\ 0 & u(z^{3}0)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(83549459453171255905+ O(11^{2}0))∗a−203315027988359294298+O(11^{2}0)q^{3}+(70364874143680768604+O(11^{2}0))∗a+321347652210027060895+ O(11^{2}0)q^{4}+−(141487581123530331+O(11^{2}0))∗a+1563061403830260042+O(11^{2}0)q^{5}+(136168343641187347774+ O(11^{2}0)) ∗ a + 11825848568837789669 + O(11^{2}0)q^{6} + −5764801q^{7} + −(221008866026044798 ∗ 11^{3} + O(11^{2}0))∗a+266948884840890161995+O(11^{2}0)q^{8}+(267533274523222212799+O(11^{2}0))∗a−62168635256483435100+ O(11^{2}0)q^{9}+(319434250632006509729+O(11^{2}0))∗a+264335284056890048425+O(11^{2}0)q^{10}+(12582759249248148287∗ 11 + O(11^{2}0)) ∗ a − 5694512302716534655 ∗ 11 + O(11^{2}0)q^{11} + (41809132172570526897 + O(11^{2}0)) ∗ a − 304096322625365345267+O(11^{2}0)q^{12}+(317100315838742281422+O(11^{2}0))∗a+193093552342265079022+ O(11^{2}0)q^{13} + −5764801 ∗ aq^{14} + (330303627519030369318 + O(11^{2}0)) ∗ a + 280456658996624934281 + O(11^{2}0)q^{15}+−(141465265040415260480+O(11^{2}0))∗a+15090179863218615944+O(11^{2}0)q^{16}+−(273088038020356915738+ O(11^{2}0)) ∗ a − 309778491326470733298 + O(11^{2}0)q^{17} + −(205665703061335634504 + O(11^{2}0)) ∗ a + 202214938946437019125+O(11^{2}0)q^{18}+−(231234716325588962311+O(11^{2}0))∗a+33315839557885297284+ O(11^{2}0)q^{19}+(61369041587171186914+O(11^{2}0))∗a+322348265485277671542+O(11^{2}0)q^{20}+(67033063826465072769+ O(11^{2}0)) ∗ a + 253490856139618348890 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −70364874143680768604 + O(11^{2}0)x + −321347652210027191967 + O(11^{2}0). The slope of f_{0} is 1.

11. Consider f = q + z^{7}1q^{2} + z^{6}7q^{3} + z^{1}11q^{4} + z^{7}4q^{5} + z^{1}8q^{6} + 2q^{7} + 8q^{8} + z^{1}01q^{9} + z^{2}5q^{10} + z^{5}8q^{12} + z^{2}6q^{13} + z^{8}3q^{14} + z^{2}1q^{15} + z^{9}3q^{16} + z^{7}4q^{17} + z^{5}2q^{18} + z^{7}5q^{19} + z^{6}5q^{20} + z^{7}9q^{21} + ⋯ ∈ S_{18}(Γ_{0}(7); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{5}9q^{2} + z^{9}1q^{3} + z^{8}7q^{4} + z^{2}6q^{5} + z^{3}0q^{6} + 5q^{7} + q^{8} + z^{2}9q^{9} + z^{8}5q^{10} + z^{9}0q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{3}0)ω^{6} & 0 \\ 0 & u(z^{9}0)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(83549459453171255905+ O(11^{2}0))∗a−203315027988359294298+O(11^{2}0)q^{3}+(70364874143680768604+O(11^{2}0))∗a+321347652210027060895+ O(11^{2}0)q^{4}+−(141487581123530331+O(11^{2}0))∗a+1563061403830260042+O(11^{2}0)q^{5}+(136168343641187347774+ O(11^{2}0)) ∗ a + 11825848568837789669 + O(11^{2}0)q^{6} + −5764801q^{7} + −(221008866026044798 ∗ 11^{3} + O(11^{2}0))∗a+266948884840890161995+O(11^{2}0)q^{8}+(267533274523222212799+O(11^{2}0))∗a−62168635256483435100+ O(11^{2}0)q^{9}+(319434250632006509729+O(11^{2}0))∗a+264335284056890048425+O(11^{2}0)q^{10}+(12582759249248148287∗ 11 + O(11^{2}0)) ∗ a − 5694512302716534655 ∗ 11 + O(11^{2}0)q^{11} + (41809132172570526897 + O(11^{2}0)) ∗ a − 304096322625365345267+O(11^{2}0)q^{12}+(317100315838742281422+O(11^{2}0))∗a+193093552342265079022+ O(11^{2}0)q^{13} + −5764801 ∗ aq^{14} + (330303627519030369318 + O(11^{2}0)) ∗ a + 280456658996624934281 + O(11^{2}0)q^{15}+−(141465265040415260480+O(11^{2}0))∗a+15090179863218615944+O(11^{2}0)q^{16}+−(273088038020356915738+ O(11^{2}0)) ∗ a − 309778491326470733298 + O(11^{2}0)q^{17} + −(205665703061335634504 + O(11^{2}0)) ∗ a + 202214938946437019125+O(11^{2}0)q^{18}+−(231234716325588962311+O(11^{2}0))∗a+33315839557885297284+ O(11^{2}0)q^{19}+(61369041587171186914+O(11^{2}0))∗a+322348265485277671542+O(11^{2}0)q^{20}+(67033063826465072769+ O(11^{2}0)) ∗ a + 253490856139618348890 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −70364874143680768604 + O(11^{2}0)x + −321347652210027191967 + O(11^{2}0). The slope of f_{0} is 1.

12. Consider f = q + 9q^{2} + 2q^{3} + 8q^{4} + 6q^{5} + 7q^{6} + 2q^{7} + 9q^{8} + 6q^{9} + 10q^{10} + 5q^{12} + 5q^{13} + 7q^{14} + q^{15} + 3q^{16} + 3q^{17} + 10q^{18} + 3q^{19} + 4q^{20} + 4q^{21} + ⋯ ∈ S_{18}(Γ_{0}(7); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + q^{2} + 8q^{3} + 2q^{4} + 10q^{5} + 8q^{6} + 6q^{7} + 3q^{8} + 8q^{9} + 10q^{10} + q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+24131187095258917320+ O(11^{2}0)q^{2}+−79896737664886242989+O(11^{2}0)q^{3}+46288417973476442046+O(11^{2}0)q^{4}+85755590572600813497+ O(11^{2}0)q^{5}+−36583156555439521361+O(11^{2}0)q^{6}+−5764801q^{7}+−132655106482342549514+O(11^{2}0)q^{8}+ 303993345420337750942+O(11^{2}0)q^{9}+−288588688614718979180+O(11^{2}0)q^{10}+−879555530415950837∗ 11 + O(11^{2}0)q^{11} + −150762461044817885214 + O(11^{2}0)q^{12} + −269483676632790469867 + O(11^{2}0)q^{13} + −247545780943122670540+O(11^{2}0)q^{14}+48221566315362850840+O(11^{2}0)q^{15}+222144230469488194343+ O(11^{2}0)q^{16}+−192004602532611321703+O(11^{2}0)q^{17}+134054476507041136529+O(11^{2}0)q^{18}+66671652416033454144+ O(11^{2}0)q^{19} + 142999127060917988062 + O(11^{2}0)q^{20} + −72343374275990105647 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

13. Consider f = q + zq^{2} + z^{1}7q^{3} + z^{2}1q^{4} + z^{9}4q^{5} + z^{1}8q^{6} + 9q^{7} + 3q^{8} + z^{3}1q^{9} + z^{9}5q^{10} + z^{3}8q^{12} + z^{1}06q^{13} + z^{7}3q^{14} + z^{1}11q^{15} + z^{6}3q^{16} + z^{3}4q^{17} + z^{3}2q^{18} + z^{4}5q^{19} + z^{1}15q^{20} + z^{8}9q^{21} + ⋯ ∈ S_{18}(Γ_{0}(7); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{4}9q^{2} + z^{4}1q^{3} + z^{1}17q^{4} + z^{4}6q^{5} + z^{9}0q^{6} + 5q^{7} + q^{8} + z^{7}9q^{9} + z^{9}5q^{10} + z^{3}0q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{9}0)ω^{1} & 0 \\ 0 & u(z^{3}0)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(157598079783608786137+ O(11^{2}0)) ∗ a + 155162673315769183015 + O(11^{2}0)q^{3} + −(96164516882920856377 + O(11^{2}0)) ∗ a + 51101753542662540511+O(11^{2}0)q^{4}+−(125411260536092387068+O(11^{2}0))∗a+287324528713869070836+ O(11^{2}0)q^{5}+−(214080350725389220371+O(11^{2}0))∗a+102950832826561535050+O(11^{2}0)q^{6}+5764801q^{7}+ −(10546240007240075843∗11+O(11^{2}0))∗a+63340971931890538702+O(11^{2}0)q^{8}+−(206871177026949503725+ O(11^{2}0)) ∗ a + 230751259263087511561 + O(11^{2}0)q^{9} + −(16294471694123876018 + O(11^{2}0)) ∗ a − 140817166446489228602+O(11^{2}0)q^{10}+−(45293301924904850∗11+O(11^{2}0))∗a−26509563813878221576∗ 11 + O(11^{2}0)q^{11} + (216185369467914843743 + O(11^{2}0)) ∗ a + 99121465448923341474 + O(11^{2}0)q^{12} + (64867884180449544749 + O(11^{2}0)) ∗ a + 254306766073100318588 + O(11^{2}0)q^{13} + 5764801 ∗ aq^{14} + (266201747921539143842+O(11^{2}0))∗a−318334688771765979196+O(11^{2}0)q^{15}+(235961190883791110746+ O(11^{2}0)) ∗ a − 24200298886686916678 + O(11^{2}0)q^{16} + −(164381232457909388501 + O(11^{2}0)) ∗ a + 332689496964561037065+O(11^{2}0)q^{17}+(254841016044715280882+O(11^{2}0))∗a−176688589371178515395+ O(11^{2}0)q^{18}+(276717176101804375470+O(11^{2}0))∗a+217499832924514295812+O(11^{2}0)q^{19}+(126601664666266708973+ O(11^{2}0)) ∗ a − 146298366840380031243 + O(11^{2}0)q^{20} + −(282527997656465790478 + O(11^{2}0)) ∗ a + 268531037039380497425+O(11^{2}0)q^{21} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} +96164516882920856377+O(11^{2}0)x+−51101753542662671583+ O(11^{2}0). The slope of f_{0} is 1.

14. Consider f = q + z^{1}1q^{2} + z^{6}7q^{3} + z^{1}11q^{4} + z^{7}4q^{5} + z^{7}8q^{6} + 9q^{7} + 3q^{8} + z^{1}01q^{9} + z^{8}5q^{10} + z^{5}8q^{12} + z^{8}6q^{13} + z^{8}3q^{14} + z^{2}1q^{15} + z^{9}3q^{16} + z^{1}4q^{17} + z^{1}12q^{18} + z^{1}5q^{19} + z^{6}5q^{20} + z^{1}9q^{21} + ⋯ ∈ S_{18}(Γ_{0}(7); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{5}9q^{2} + z^{9}1q^{3} + z^{8}7q^{4} + z^{2}6q^{5} + z^{3}0q^{6} + 5q^{7} + q^{8} + z^{2}9q^{9} + z^{8}5q^{10} + z^{9}0q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{3}0)ω^{1} & 0 \\ 0 & u(z^{9}0)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(157598079783608786137+ O(11^{2}0)) ∗ a + 155162673315769183015 + O(11^{2}0)q^{3} + −(96164516882920856377 + O(11^{2}0)) ∗ a + 51101753542662540511+O(11^{2}0)q^{4}+−(125411260536092387068+O(11^{2}0))∗a+287324528713869070836+ O(11^{2}0)q^{5}+−(214080350725389220371+O(11^{2}0))∗a+102950832826561535050+O(11^{2}0)q^{6}+5764801q^{7}+ −(10546240007240075843∗11+O(11^{2}0))∗a+63340971931890538702+O(11^{2}0)q^{8}+−(206871177026949503725+ O(11^{2}0)) ∗ a + 230751259263087511561 + O(11^{2}0)q^{9} + −(16294471694123876018 + O(11^{2}0)) ∗ a − 140817166446489228602+O(11^{2}0)q^{10}+−(45293301924904850∗11+O(11^{2}0))∗a−26509563813878221576∗ 11 + O(11^{2}0)q^{11} + (216185369467914843743 + O(11^{2}0)) ∗ a + 99121465448923341474 + O(11^{2}0)q^{12} + (64867884180449544749 + O(11^{2}0)) ∗ a + 254306766073100318588 + O(11^{2}0)q^{13} + 5764801 ∗ aq^{14} + (266201747921539143842+O(11^{2}0))∗a−318334688771765979196+O(11^{2}0)q^{15}+(235961190883791110746+ O(11^{2}0)) ∗ a − 24200298886686916678 + O(11^{2}0)q^{16} + −(164381232457909388501 + O(11^{2}0)) ∗ a + 332689496964561037065+O(11^{2}0)q^{17}+(254841016044715280882+O(11^{2}0))∗a−176688589371178515395+ O(11^{2}0)q^{18}+(276717176101804375470+O(11^{2}0))∗a+217499832924514295812+O(11^{2}0)q^{19}+(126601664666266708973+ O(11^{2}0)) ∗ a − 146298366840380031243 + O(11^{2}0)q^{20} + −(282527997656465790478 + O(11^{2}0)) ∗ a + 268531037039380497425+O(11^{2}0)q^{21} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} +96164516882920856377+O(11^{2}0)x+−51101753542662671583+ O(11^{2}0). The slope of f_{0} is 1.

15. Consider f = q+2q^{2}+2q^{3}+8q^{4}+6q^{5}+4q^{6}+9q^{7}+2q^{8}+6q^{9}+q^{10}+5q^{12}+6q^{13}+7q^{14}+q^{15}+3q^{16}+8q^{17}+q^{18}+ 8q^{19}+4q^{20}+7q^{21}+⋯ ∈ S_{18}(Γ_{0}(7); \overline{F}_{11}). Also, z satisfies the equation x^{2}+7x+2 = 0. We have ρ_{f} ⊗χ^{9} ∼ ρ_{g}, where g is the weight-6 newform g = q+q^{2}+8q^{3}+2q^{4}+10q^{5}+8q^{6}+6q^{7}+3q^{8}+8q^{9}+10q^{10}+q^{11}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{6} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−213723143574164042641+ O(11^{2}0)q^{2}+311769228536200123680+O(11^{2}0)q^{3}+−35861127409056105115+O(11^{2}0)q^{4}+−9959985508365214580+ O(11^{2}0)q^{5} +110518168045823772139+O(11^{2}0)q^{6} +5764801q^{7} +204522071995854136559+O(11^{2}0)q^{8} + 243723328102204421609+O(11^{2}0)q^{9}+−233714260169754892439+O(11^{2}0)q^{10}+18449583309355997603∗ 11+O(11^{2}0)q^{11}+990016171475452258+O(11^{2}0)q^{12}+99215669991482183574+O(11^{2}0)q^{13}+257920026223429795155+ O(11^{2}0)q^{14}+18032029895007034083+O(11^{2}0)q^{15}+−313644292870144868904+O(11^{2}0)q^{16}+107072577410306374743+ O(11^{2}0)q^{17}+−287981081348848320905+O(11^{2}0)q^{18}+−82521241588880729319+O(11^{2}0)q^{19}+−42727239208129246030+ O(11^{2}0)q^{20} + 293222649518929628527 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

16. Consider f = q + z^{9}1q^{2} + z^{7}q^{3} + z^{1}11q^{4} + z^{1}8q^{5} + z^{9}8q^{6} + 8q^{7} + q^{8} + z^{9}1q^{9} + z^{1}09q^{10} + z^{1}18q^{12} + z^{7}5q^{13} + z^{7}q^{14} + z^{2}5q^{15} + z^{5}7q^{16} + zq^{17} + z^{6}2q^{18} + 10q^{19} + z^{9}q^{20} + z^{4}3q^{21} + ⋯ ∈ S_{20}(Γ_{0}(7); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{7}9q^{2} + z^{3}1q^{3} + z^{8}7q^{4} + z^{9}0q^{5} + z^{1}10q^{6} + 9q^{7} + 7q^{8} + z^{1}9q^{9} + z^{4}9q^{10} + z^{1}10q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}0)ω^{8} & 0 \\ 0 & u(z^{1}10)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(63650806289416642299+ O(11^{2}0)) ∗ a − 13714945158597712626 ∗ 11 + O(11^{2}0)q^{3} + (47604408625681466157 + O(11^{2}0)) ∗ a − 63963179489442255822+O(11^{2}0)q^{4}+(220962963433041813684+O(11^{2}0))∗a+134882930305930869812+ O(11^{2}0)q^{5}+−(46821037972846985764+O(11^{2}0))∗a−202064088140581323629+O(11^{2}0)q^{6}+40353607q^{7}+ −(8558322368393945210∗11+O(11^{2}0))∗a−189545800946950964922+O(11^{2}0)q^{8}+(84205600406603548027+ O(11^{2}0)) ∗ a − 516468892227729230 ∗ 11^{2} + O(11^{2}0)q^{9} + (28230145255149748150 + O(11^{2}0)) ∗ a − 96959255885936012981+O(11^{2}0)q^{10}+(13230592897817609764∗11+O(11^{2}0))∗a+18682018154584057522∗ 11 + O(11^{2}0)q^{11} + −(256252680459523242546 + O(11^{2}0)) ∗ a + 120001759021678492263 + O(11^{2}0)q^{12} + (24147729573368140423 + O(11^{2}0)) ∗ a + 258214624012930926923 + O(11^{2}0)q^{13} + 40353607 ∗ aq^{14} + −(118473110036973660985+O(11^{2}0))∗a−35088158028986371118+O(11^{2}0)q^{15}+−(271745214864993537257+ O(11^{2}0)) ∗ a − 37917102360983444837 + O(11^{2}0)q^{16} + −(218124924667851948619 + O(11^{2}0)) ∗ a − 179436235600640133653+O(11^{2}0)q^{17}+−(295710641337425829004+O(11^{2}0))∗a−30867121417501933995+ O(11^{2}0)q^{18} + −(14380888506915062089 ∗ 11 + O(11^{2}0)) ∗ a + 79095834150493342324 + O(11^{2}0)q^{19} + −(64684527799523893104+O(11^{2}0))∗a−170499175204220164189+O(11^{2}0)q^{20}+−(324083581304264280523+ O(11^{2}0)) ∗ a + 670047023073520937 ∗ 11 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −47604408625681466157 + O(11^{2}0)x + 63963179489441731534 + O(11^{2}0). The slope of f_{0} is 1.

17. Consider f = q + z^{4}1q^{2} + z^{7}7q^{3} + z^{2}1q^{4} + z^{7}8q^{5} + z^{1}18q^{6} + 8q^{7} + q^{8} + z^{4}1q^{9} + z^{1}19q^{10} + z^{9}8q^{12} + z^{1}05q^{13} + z^{7}7q^{14} + z^{3}5q^{15} + z^{2}7q^{16} + z^{1}1q^{17} + z^{8}2q^{18} + 10q^{19} + z^{9}9q^{20} + z^{1}13q^{21} + ⋯ ∈ S_{20}(Γ_{0}(7); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{2}9q^{2} + z^{1}01q^{3} + z^{1}17q^{4} + z^{3}0q^{5} + z^{1}0q^{6} + 9q^{7} + 7q^{8} + z^{8}9q^{9} + z^{5}9q^{10} + z^{1}0q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}10)ω^{8} & 0 \\ 0 & u(z^{1}0)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(63650806289416642299+ O(11^{2}0)) ∗ a − 13714945158597712626 ∗ 11 + O(11^{2}0)q^{3} + (47604408625681466157 + O(11^{2}0)) ∗ a − 63963179489442255822+O(11^{2}0)q^{4}+(220962963433041813684+O(11^{2}0))∗a+134882930305930869812+ O(11^{2}0)q^{5}+−(46821037972846985764+O(11^{2}0))∗a−202064088140581323629+O(11^{2}0)q^{6}+40353607q^{7}+ −(8558322368393945210∗11+O(11^{2}0))∗a−189545800946950964922+O(11^{2}0)q^{8}+(84205600406603548027+ O(11^{2}0)) ∗ a − 516468892227729230 ∗ 11^{2} + O(11^{2}0)q^{9} + (28230145255149748150 + O(11^{2}0)) ∗ a − 96959255885936012981+O(11^{2}0)q^{10}+(13230592897817609764∗11+O(11^{2}0))∗a+18682018154584057522∗ 11 + O(11^{2}0)q^{11} + −(256252680459523242546 + O(11^{2}0)) ∗ a + 120001759021678492263 + O(11^{2}0)q^{12} + (24147729573368140423 + O(11^{2}0)) ∗ a + 258214624012930926923 + O(11^{2}0)q^{13} + 40353607 ∗ aq^{14} + −(118473110036973660985+O(11^{2}0))∗a−35088158028986371118+O(11^{2}0)q^{15}+−(271745214864993537257+ O(11^{2}0)) ∗ a − 37917102360983444837 + O(11^{2}0)q^{16} + −(218124924667851948619 + O(11^{2}0)) ∗ a − 179436235600640133653+O(11^{2}0)q^{17}+−(295710641337425829004+O(11^{2}0))∗a−30867121417501933995+ O(11^{2}0)q^{18} + −(14380888506915062089 ∗ 11 + O(11^{2}0)) ∗ a + 79095834150493342324 + O(11^{2}0)q^{19} + −(64684527799523893104+O(11^{2}0))∗a−170499175204220164189+O(11^{2}0)q^{20}+−(324083581304264280523+ O(11^{2}0)) ∗ a + 670047023073520937 ∗ 11 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −47604408625681466157 + O(11^{2}0)x + 63963179489441731534 + O(11^{2}0). The slope of f_{0} is 1.

18. Consider f = q + 10q^{2} + 6q^{3} + 6q^{4} + 9q^{5} + 5q^{6} + 3q^{7} + 10q^{9} + 2q^{10} + 3q^{12} + 3q^{13} + 8q^{14} + 10q^{15} + 8q^{16} + 3q^{17} + q^{18} + q^{19} + 10q^{20} + 7q^{21} + ⋯ ∈ S_{20}(Γ_{0}(7); \overline{F}_{11}). Also, z satisfies the equation x^{3} + 2x + 9 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q + 5q^{2} + 2q^{3} + 7q^{4} + 4q^{5} + 10q^{6} + 2q^{7} + 6q^{9} + 9q^{10} + 9q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{8} & 0 \\ 0 & u(9)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−231211139539057666297+ O(11^{2}0)q^{2}+−77034244987525121851+O(11^{2}0)q^{3}+−58200419591501482861+O(11^{2}0)q^{4}+278354185510559692939+ O(11^{2}0)q^{5} + −321189171921997818412 + O(11^{2}0)q^{6} + −40353607q^{7} + 14030118723786888339 ∗ 11 + O(11^{2}0)q^{8}+36761587830696740153+O(11^{2}0)q^{9}+28764661846920058820+O(11^{2}0)q^{10}+4428561558230320354∗ 11 + O(11^{2}0)q^{11} + −80172733265751448825 + O(11^{2}0)q^{12} + −283854260540937371194 + O(11^{2}0)q^{13} + −51489632209200113074+O(11^{2}0)q^{14}+171827902724384092999+O(11^{2}0)q^{15}+53447487331063155085+ O(11^{2}0)q^{16}+−146131003072360743687+O(11^{2}0)q^{17}+−11697569339195017805+O(11^{2}0)q^{18}+−86517437311746740056+ O(11^{2}0)q^{19} + −259386062405783112801 + O(11^{2}0)q^{20} + 108683682007286845807 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

19. Consider f = q+8q^{2}+q^{3}+3q^{4}+9q^{5}+8q^{6}+3q^{7}+9q^{8}+8q^{9}+6q^{10}+3q^{12}+7q^{13}+2q^{14}+9q^{15}+10q^{16}+7q^{17}+ 9q^{18}+5q^{20}+3q^{21}+⋯ ∈ S_{20}(Γ_{0}(7); \overline{F}_{11}). Also, z satisfies the equation x^{3}+2x+9 = 0. We have ρ_{f} ⊗χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q+10q^{2}+9q^{3}+4q^{4}+5q^{5}+2q^{6}+4q^{7}+4q^{8}+10q^{9}+6q^{10}+3q^{11}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{1} & 0 \\ 0 & u(3)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+159688429475667623000+ O(11^{2}0)q^{2}+−120566352222240894507+O(11^{2}0)q^{3}+294997228400355812620+O(11^{2}0)q^{4}+216973890736726035350+ O(11^{2}0)q^{5}+−124877248115583828351+O(11^{2}0)q^{6}+−40353607q^{7}+−2239765413685705193+O(11^{2}0)q^{8}+ 166491426698578621351+O(11^{2}0)q^{9}+−322623372507349228276+O(11^{2}0)q^{10}+10058491843486235771∗ 11 + O(11^{2}0)q^{11} + 276162620853913445402 + O(11^{2}0)q^{12} + 36448223358161613819 + O(11^{2}0)q^{13} + 321452701848086556002+O(11^{2}0)q^{14}+−252962986110195526882+O(11^{2}0)q^{15}+−218780709015277011494+ O(11^{2}0)q^{16}+10646878445570297608+O(11^{2}0)q^{17}+−326382961113593897878+O(11^{2}0)q^{18}+22824646337837639986∗ 11 + O(11^{2}0)q^{19} + 295682211349000052549 + O(11^{2}0)q^{20} + 269397302615882866010 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

20. Consider f = q+z^{7}17q^{2}+z^{6}74q^{3}+z^{2}86q^{4}+z^{4}3q^{5}+z^{6}1q^{6}+10q^{7}+z^{9}14q^{8}+z^{1}028q^{9}+z^{7}60q^{10}+z^{9}60q^{12}+ z^{5}4q^{13}+z^{5}2q^{14}+z^{7}17q^{15}+z^{9}10q^{16}+z^{1}65q^{17}+z^{4}15q^{18}+z^{5}66q^{19}+z^{3}29q^{20}+z^{9}q^{21}+⋯ ∈ S_{22}(Γ_{0}(7); \overline{F}_{11}). Also, z satisfies the equation x^{3} + 2x + 9 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{5}84q^{2} + z^{9}40q^{3} + z^{2}0q^{4} + z^{8}41q^{5} + z^{1}94q^{6} + 3q^{7} + z^{5}15q^{8} + z^{2}30q^{9} + z^{9}5q^{10} + z^{4}60q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{8}70)ω^{0} & 0 \\ 0 & u(z^{4}60)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(31661743643353139695+ O(11^{2}0)) ∗ a^{2} + (316000893364366672092 + O(11^{2}0)) ∗ a + 15972496321807558330 ∗ 11 + O(11^{2}0)q^{3} + a^{2} − 2097152q^{4} + (140132178330796586431 + O(11^{2}0)) ∗ a^{2} − (295145357767755056168 + O(11^{2}0)) ∗ a − 8348824864292780635+O(11^{2}0)q^{5}+−(28962804087306400823+O(11^{2}0))∗a^{2}−(105209807630771820175+ O(11^{2}0))∗a−218862079324714696825+O(11^{2}0)q^{6}+−282475249q^{7}+(53589855169818165317+O(11^{2}0))∗ a^{2}+(219600516926201824268+O(11^{2}0))∗a+6817775513041859439+O(11^{2}0)q^{8}+−(186951769925361874028+ O(11^{2}0)) ∗ a^{2} − (278045242069704139535 + O(11^{2}0)) ∗ a − 278816808866691125620 + O(11^{2}0)q^{9} + −(110953986250322069784+O(11^{2}0))∗a^{2}+(9081258750047912291+O(11^{2}0))∗a+164788606814319855579+ O(11^{2}0)q^{10} +(15471912630176697369∗11+O(11^{2}0))∗a^{2} −(23701510831976800056∗11+O(11^{2}0))∗a− 23327100012336069053∗11+O(11^{2}0)q^{11}+(306473781009761053261+O(11^{2}0))∗a^{2}+(215294955741384913794+ O(11^{2}0)) ∗ a + 211831800795252434128 + O(11^{2}0)q^{12} + −(144592111549391845291 + O(11^{2}0)) ∗ a^{2} − (77341095632517178760 + O(11^{2}0)) ∗ a + 223310345751536819058 + O(11^{2}0)q^{13} + −282475249 ∗ aq^{14} + −(12386429415790948611∗11+O(11^{2}0))∗a^{2}+(242885730649114384437+O(11^{2}0))∗a−30073564064299869065∗ 11 + O(11^{2}0)q^{15} + (931841605152376 ∗ 11^{5} + O(11^{2}0)) ∗ a^{2} + (296695343657291870327 + O(11^{2}0)) ∗ a − 157133299523256026332+O(11^{2}0)q^{16}+(22618431954197105555+O(11^{2}0))∗a^{2}−(11526264815621578944+ O(11^{2}0)) ∗ a + 15668050800309374568 ∗ 11 + O(11^{2}0)q^{17} + −(319170368555686317555 + O(11^{2}0)) ∗ a^{2} − (2309231249489196357∗11^{2}+O(11^{2}0))∗a−213824047714356489644+O(11^{2}0)q^{18}+(335398715144540592278+ O(11^{2}0)) ∗ a^{2} + (311790150275739254953 + O(11^{2}0)) ∗ a − 19525242245031416232 ∗ 11 + O(11^{2}0)q^{19} + −(50602760664477573252+O(11^{2}0))∗a^{2}−(287002380627607310275+O(11^{2}0))∗a−245582865097477214232+ O(11^{2}0)q^{20} + −(64547799639717168076 + O(11^{2}0)) ∗ a^{2} + (29315361939286459208 + O(11^{2}0)) ∗ a + 27769851271572784676∗11+O(11^{2}0)q^{21}+⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3}+−53589855169818165317+O(11^{2}0)x^{2}+−219600516926206018572+ O(11^{2}0)x + −6817775513041859439 + O(11^{2}0). The slope of f_{0} is 1.

21. Consider f = q + z^{1}237q^{2} + z^{7}64q^{3} + z^{4}86q^{4} + z^{4}73q^{5} + z^{6}71q^{6} + 10q^{7} + z^{7}44q^{8} + z^{6}68q^{9} + z^{3}80q^{10} + z^{1}250q^{12}+z^{5}94q^{13}+z^{5}72q^{14}+z^{1}237q^{15}+z^{7}00q^{16}+z^{4}85q^{17}+z^{5}75q^{18}+z^{9}06q^{19}+z^{9}59q^{20}+z^{9}9q^{21}+⋯ ∈ S_{22}(Γ_{0}(7); \overline{F}_{11}). Also, z satisfies the equation x^{3} + 2x + 9 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{1}104q^{2} + z^{1}030q^{3} + z^{2}20q^{4} + z^{1}271q^{5} + z^{8}04q^{6} + 3q^{7} + z^{3}45q^{8} + z^{1}200q^{9} + z^{1}045q^{10} + z^{1}070q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}60)ω^{0} & 0 \\ 0 & u(z^{1}070)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(31661743643353139695+ O(11^{2}0)) ∗ a^{2} + (316000893364366672092 + O(11^{2}0)) ∗ a + 15972496321807558330 ∗ 11 + O(11^{2}0)q^{3} + a^{2} − 2097152q^{4} + (140132178330796586431 + O(11^{2}0)) ∗ a^{2} − (295145357767755056168 + O(11^{2}0)) ∗ a − 8348824864292780635+O(11^{2}0)q^{5}+−(28962804087306400823+O(11^{2}0))∗a^{2}−(105209807630771820175+ O(11^{2}0))∗a−218862079324714696825+O(11^{2}0)q^{6}+−282475249q^{7}+(53589855169818165317+O(11^{2}0))∗ a^{2}+(219600516926201824268+O(11^{2}0))∗a+6817775513041859439+O(11^{2}0)q^{8}+−(186951769925361874028+ O(11^{2}0)) ∗ a^{2} − (278045242069704139535 + O(11^{2}0)) ∗ a − 278816808866691125620 + O(11^{2}0)q^{9} + −(110953986250322069784+O(11^{2}0))∗a^{2}+(9081258750047912291+O(11^{2}0))∗a+164788606814319855579+ O(11^{2}0)q^{10} +(15471912630176697369∗11+O(11^{2}0))∗a^{2} −(23701510831976800056∗11+O(11^{2}0))∗a− 23327100012336069053∗11+O(11^{2}0)q^{11}+(306473781009761053261+O(11^{2}0))∗a^{2}+(215294955741384913794+ O(11^{2}0)) ∗ a + 211831800795252434128 + O(11^{2}0)q^{12} + −(144592111549391845291 + O(11^{2}0)) ∗ a^{2} − (77341095632517178760 + O(11^{2}0)) ∗ a + 223310345751536819058 + O(11^{2}0)q^{13} + −282475249 ∗ aq^{14} + −(12386429415790948611∗11+O(11^{2}0))∗a^{2}+(242885730649114384437+O(11^{2}0))∗a−30073564064299869065∗ 11 + O(11^{2}0)q^{15} + (931841605152376 ∗ 11^{5} + O(11^{2}0)) ∗ a^{2} + (296695343657291870327 + O(11^{2}0)) ∗ a − 157133299523256026332+O(11^{2}0)q^{16}+(22618431954197105555+O(11^{2}0))∗a^{2}−(11526264815621578944+ O(11^{2}0)) ∗ a + 15668050800309374568 ∗ 11 + O(11^{2}0)q^{17} + −(319170368555686317555 + O(11^{2}0)) ∗ a^{2} − (2309231249489196357∗11^{2}+O(11^{2}0))∗a−213824047714356489644+O(11^{2}0)q^{18}+(335398715144540592278+ O(11^{2}0)) ∗ a^{2} + (311790150275739254953 + O(11^{2}0)) ∗ a − 19525242245031416232 ∗ 11 + O(11^{2}0)q^{19} + −(50602760664477573252+O(11^{2}0))∗a^{2}−(287002380627607310275+O(11^{2}0))∗a−245582865097477214232+ O(11^{2}0)q^{20} + −(64547799639717168076 + O(11^{2}0)) ∗ a^{2} + (29315361939286459208 + O(11^{2}0)) ∗ a + 27769851271572784676∗11+O(11^{2}0)q^{21}+⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3}+−53589855169818165317+O(11^{2}0)x^{2}+−219600516926206018572+ O(11^{2}0)x + −6817775513041859439 + O(11^{2}0). The slope of f_{0} is 1.

22. Consider f = q+z^{3}07q^{2}+z^{4}24q^{3}+z^{2}6q^{4}+z^{1}213q^{5}+z^{7}31q^{6}+10q^{7}+z^{2}04q^{8}+z^{6}98q^{9}+z^{1}90q^{10}+z^{4}50q^{12}+ z^{1}214q^{13} + z^{9}72q^{14} + z^{3}07q^{15} + z^{1}050q^{16} + z^{1}5q^{17} + z^{1}005q^{18} + z^{6}56q^{19} + z^{1}239q^{20} + z^{1}089q^{21} + ⋯ ∈ S_{22}(Γ_{0}(7); \overline{F}_{11}). Also, z satisfies the equation x^{3} + 2x + 9 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{1}74q^{2} + z^{6}90q^{3} + z^{1}090q^{4} + z^{6}81q^{5} + z^{8}64q^{6} + 3q^{7} + z^{1}135q^{8} + z^{1}230q^{9} + z^{8}55q^{10} + z^{1}130q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}00)ω^{0} & 0 \\ 0 & u(z^{1}130)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(31661743643353139695+ O(11^{2}0)) ∗ a^{2} + (316000893364366672092 + O(11^{2}0)) ∗ a + 15972496321807558330 ∗ 11 + O(11^{2}0)q^{3} + a^{2} − 2097152q^{4} + (140132178330796586431 + O(11^{2}0)) ∗ a^{2} − (295145357767755056168 + O(11^{2}0)) ∗ a − 8348824864292780635+O(11^{2}0)q^{5}+−(28962804087306400823+O(11^{2}0))∗a^{2}−(105209807630771820175+ O(11^{2}0))∗a−218862079324714696825+O(11^{2}0)q^{6}+−282475249q^{7}+(53589855169818165317+O(11^{2}0))∗ a^{2}+(219600516926201824268+O(11^{2}0))∗a+6817775513041859439+O(11^{2}0)q^{8}+−(186951769925361874028+ O(11^{2}0)) ∗ a^{2} − (278045242069704139535 + O(11^{2}0)) ∗ a − 278816808866691125620 + O(11^{2}0)q^{9} + −(110953986250322069784+O(11^{2}0))∗a^{2}+(9081258750047912291+O(11^{2}0))∗a+164788606814319855579+ O(11^{2}0)q^{10} +(15471912630176697369∗11+O(11^{2}0))∗a^{2} −(23701510831976800056∗11+O(11^{2}0))∗a− 23327100012336069053∗11+O(11^{2}0)q^{11}+(306473781009761053261+O(11^{2}0))∗a^{2}+(215294955741384913794+ O(11^{2}0)) ∗ a + 211831800795252434128 + O(11^{2}0)q^{12} + −(144592111549391845291 + O(11^{2}0)) ∗ a^{2} − (77341095632517178760 + O(11^{2}0)) ∗ a + 223310345751536819058 + O(11^{2}0)q^{13} + −282475249 ∗ aq^{14} + −(12386429415790948611∗11+O(11^{2}0))∗a^{2}+(242885730649114384437+O(11^{2}0))∗a−30073564064299869065∗ 11 + O(11^{2}0)q^{15} + (931841605152376 ∗ 11^{5} + O(11^{2}0)) ∗ a^{2} + (296695343657291870327 + O(11^{2}0)) ∗ a − 157133299523256026332+O(11^{2}0)q^{16}+(22618431954197105555+O(11^{2}0))∗a^{2}−(11526264815621578944+ O(11^{2}0)) ∗ a + 15668050800309374568 ∗ 11 + O(11^{2}0)q^{17} + −(319170368555686317555 + O(11^{2}0)) ∗ a^{2} − (2309231249489196357∗11^{2}+O(11^{2}0))∗a−213824047714356489644+O(11^{2}0)q^{18}+(335398715144540592278+ O(11^{2}0)) ∗ a^{2} + (311790150275739254953 + O(11^{2}0)) ∗ a − 19525242245031416232 ∗ 11 + O(11^{2}0)q^{19} + −(50602760664477573252+O(11^{2}0))∗a^{2}−(287002380627607310275+O(11^{2}0))∗a−245582865097477214232+ O(11^{2}0)q^{20} + −(64547799639717168076 + O(11^{2}0)) ∗ a^{2} + (29315361939286459208 + O(11^{2}0)) ∗ a + 27769851271572784676∗11+O(11^{2}0)q^{21}+⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3}+−53589855169818165317+O(11^{2}0)x^{2}+−219600516926206018572+ O(11^{2}0)x + −6817775513041859439 + O(11^{2}0). The slope of f_{0} is 1.

23. Consider f = q + z^{8}0q^{2} + 3q^{3} + z^{5}7q^{4} + z^{5}8q^{5} + z^{5}6q^{6} + q^{7} + z^{8}5q^{8} + 6q^{9} + z^{1}8q^{10} + z^{3}3q^{12} + z^{6}7q^{13} + z^{8}0q^{14} + z^{3}4q^{15} + z^{8}1q^{16} + z^{5}6q^{17} + z^{6}8q^{18} + z^{9}q^{19} + z^{1}15q^{20} + 3q^{21} + ⋯ ∈ S_{22}(Γ_{0}(7); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{6}8q^{2} + q^{3} + z^{3}3q^{4} + z^{1}0q^{5} + z^{6}8q^{6} + 8q^{7} + z^{4}9q^{8} + 8q^{9} + z^{7}8q^{10} + z^{1}10q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}0)ω^{0} & 0 \\ 0 & u(z^{1}10)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(27083839005499675565∗ 11 + O(11^{2}0)) ∗ a − 271291718417836041021 + O(11^{2}0)q^{3} + (162292480428689336571 + O(11^{2}0)) ∗ a + 188072936790349862776+O(11^{2}0)q^{4}+(328002264841053264650+O(11^{2}0))∗a−169306127708261495964+ O(11^{2}0)q^{5} + −(17522993953437238439 + O(11^{2}0)) ∗ a − 29862709479915471587 ∗ 11 + O(11^{2}0)q^{6} + 282475249q^{7}+−(204817257092035286065+O(11^{2}0))∗a+75006667467933851464+O(11^{2}0)q^{8}+−(2600856497265273682∗ 11 + O(11^{2}0)) ∗ a − 328908677083270862528 + O(11^{2}0)q^{9} + (326908494271874684189 + O(11^{2}0)) ∗ a + 322829047537154051827+O(11^{2}0)q^{10}+(5193493114207165986∗11+O(11^{2}0))∗a+16679766549346378529∗ 11 + O(11^{2}0)q^{11} + −(66280093307907788092 + O(11^{2}0)) ∗ a − 29024485236007722877 + O(11^{2}0)q^{12} + −(201658518870666665014 + O(11^{2}0)) ∗ a − 37550823497101441392 + O(11^{2}0)q^{13} + 282475249 ∗ aq^{14} + −(321917918696564702047+O(11^{2}0))∗a−298625959286463341745+O(11^{2}0)q^{15}+(266939708313810257645+ O(11^{2}0))∗a+269800052534467704941+O(11^{2}0)q^{16}+(12714531000015928471+O(11^{2}0))∗a+4153171437614408898∗ 11 + O(11^{2}0)q^{17} + (198562101608725710433 + O(11^{2}0)) ∗ a + 8797109329197787109 ∗ 11 + O(11^{2}0)q^{18} + (180626245645746674248+O(11^{2}0))∗a+279751405607502672069+O(11^{2}0)q^{19}+(11703271630058028030+ O(11^{2}0)) ∗ a − 306518262898986637781 + O(11^{2}0)q^{20} + −(1399551831717892865 ∗ 11 + O(11^{2}0)) ∗ a − 253450045493771948467 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −162292480428689336571 + O(11^{2}0)x + −188072936790351959928 + O(11^{2}0). The slope of f_{0} is 1.

24. Consider f = q + z^{4}0q^{2} + 3q^{3} + z^{2}7q^{4} + z^{3}8q^{5} + z^{1}6q^{6} + q^{7} + z^{9}5q^{8} + 6q^{9} + z^{7}8q^{10} + z^{3}q^{12} + z^{1}7q^{13} + z^{4}0q^{14} + z^{1}4q^{15} + z^{5}1q^{16} + z^{1}6q^{17} + z^{2}8q^{18} + z^{9}9q^{19} + z^{6}5q^{20} + 3q^{21} + ⋯ ∈ S_{22}(Γ_{0}(7); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{2}8q^{2} + q^{3} + z^{3}q^{4} + z^{1}10q^{5} + z^{2}8q^{6} + 8q^{7} + z^{5}9q^{8} + 8q^{9} + z^{1}8q^{10} + z^{1}0q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}10)ω^{0} & 0 \\ 0 & u(z^{1}0)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(27083839005499675565∗ 11 + O(11^{2}0)) ∗ a − 271291718417836041021 + O(11^{2}0)q^{3} + (162292480428689336571 + O(11^{2}0)) ∗ a + 188072936790349862776+O(11^{2}0)q^{4}+(328002264841053264650+O(11^{2}0))∗a−169306127708261495964+ O(11^{2}0)q^{5} + −(17522993953437238439 + O(11^{2}0)) ∗ a − 29862709479915471587 ∗ 11 + O(11^{2}0)q^{6} + 282475249q^{7}+−(204817257092035286065+O(11^{2}0))∗a+75006667467933851464+O(11^{2}0)q^{8}+−(2600856497265273682∗ 11 + O(11^{2}0)) ∗ a − 328908677083270862528 + O(11^{2}0)q^{9} + (326908494271874684189 + O(11^{2}0)) ∗ a + 322829047537154051827+O(11^{2}0)q^{10}+(5193493114207165986∗11+O(11^{2}0))∗a+16679766549346378529∗ 11 + O(11^{2}0)q^{11} + −(66280093307907788092 + O(11^{2}0)) ∗ a − 29024485236007722877 + O(11^{2}0)q^{12} + −(201658518870666665014 + O(11^{2}0)) ∗ a − 37550823497101441392 + O(11^{2}0)q^{13} + 282475249 ∗ aq^{14} + −(321917918696564702047+O(11^{2}0))∗a−298625959286463341745+O(11^{2}0)q^{15}+(266939708313810257645+ O(11^{2}0))∗a+269800052534467704941+O(11^{2}0)q^{16}+(12714531000015928471+O(11^{2}0))∗a+4153171437614408898∗ 11 + O(11^{2}0)q^{17} + (198562101608725710433 + O(11^{2}0)) ∗ a + 8797109329197787109 ∗ 11 + O(11^{2}0)q^{18} + (180626245645746674248+O(11^{2}0))∗a+279751405607502672069+O(11^{2}0)q^{19}+(11703271630058028030+ O(11^{2}0)) ∗ a − 306518262898986637781 + O(11^{2}0)q^{20} + −(1399551831717892865 ∗ 11 + O(11^{2}0)) ∗ a − 253450045493771948467 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −162292480428689336571 + O(11^{2}0)x + −188072936790351959928 + O(11^{2}0). The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

25. Consider f = q + 6q2 + q3 + 6q4 + 8q5 + 6q6 + 7q7 + 10q8 + 7q9 + 4q10 + 6q12 + 6q13 + 9q14 + 8q15 + q16 + 9q17 + 9q18 + 6q19 + 4q20 + 7q21 + ⋯ ∈ S14(Γ0(7); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

26. Consider f = q + 10q2 + 9q3 + 4q4 + 5q5 + 2q6 + 4q7 + 4q8 + 10q9 + 6q10 + 3q11 + 3q12 + 6q13 + 7q14 + q15 + 8q16 + 10q17 + q18 + 9q20 + 3q21 + ⋯ ∈ S14(Γ0(7); \overline{F}11). Also, z satisfies the equation x3 + 2x + 9 = 0. Since f is ordinary, we do not continue our analysis.

27. Consider f = q +q2 +8q3 +2q4 +10q5 +8q6 +6q7 +3q8 +8q9 +10q10 +q11 +5q12 +3q13 +6q14 +3q15 +5q16 +5q17 +8q18 +q19 +9q20 +4q21 +⋯ ∈ S16(Γ0(7); \overline{F}11). Since f is ordinary, we do not continue our analysis.

28. Consider f = q + z49q2 + z41q3 + z117q4 + z46q5 + z90q6 + 5q7 + q8 + z79q9 + z95q10 + z30q11 + z38q12 + z34q13 + z97q14 + z87q15 + z15q16 + z106q17 + z8q18 + z69q19 + z43q20 + z89q21 + ⋯ ∈ S16(Γ0(7); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

29. Consider f = q + z59q2 + z91q3 + z87q4 + z26q5 + z30q6 + 5q7 + q8 + z29q9 + z85q10 + z90q11 + z58q12 + z14q13 + z107q14 + z117q15 + z45q16 + z86q17 + z88q18 + z39q19 + z113q20 + z19q21 + ⋯ ∈ S16(Γ0(7); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

30. Consider f = q + 5q2 + 2q3 + 7q4 + 4q5 + 10q6 + 2q7 + 6q9 + 9q10 + 9q11 + 3q12 + 7q13 + 10q14 + 8q15 + 6q16 + 6q17 + 8q18 + 7q19 + 6q20 + 4q21 + ⋯ ∈ S18(Γ0(7); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

31. Consider f = q + z79q2 + z31q3 + z87q4 + z90q5 + z110q6 + 9q7 + 7q8 + z19q9 + z49q10 + z110q11 + z118q12 + z63q13 + z31q14 + zq15 + z9q16 + z13q17 + z98q18 + 4q19 + z57q20 + z103q21 + ⋯ ∈ S18(Γ0(7); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

32. Consider f = q + z29q2 + z101q3 + z117q4 + z30q5 + z10q6 + 9q7 + 7q8 + z89q9 + z59q10 + z10q11 + z98q12 + z93q13 + z101q14 + z11q15 + z99q16 + z23q17 + z118q18 + 4q19 + z27q20 + z53q21 + ⋯ ∈ S18(Γ0(7); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

33. Consider f = q + z68q2 + q3 + z33q4 + z10q5 + z68q6 + 8q7 + z49q8 + 8q9 + z78q10 + z110q11 + z33q12 + z55q13 + z104q14 + z10q15 + z33q16 + z68q17 + z104q18 + z93q19 + z43q20 + 8q21 + ⋯ ∈ S20(Γ0(7); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

34. Consider f = q + z28q2 + q3 + z3q4 + z110q5 + z28q6 + 8q7 + z59q8 + 8q9 + z18q10 + z10q11 + z3q12 + z5q13 + z64q14 + z110q15 + z3q16 + z28q17 + z64q18 + z63q19 + z113q20 + 8q21 + ⋯ ∈ S20(Γ0(7); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

35. Consider f = q +z1104q2 +z1030q3 +z220q4 +z1271q5 +z804q6 +3q7 +z345q8 +z1200q9 +z1045q10 +z1070q11 +z1250q12 +z461q13 +z838q14 + z971q15 + z168q16 + z618q17 + z974q18 + z507q19 + z161q20 + z764q21 + ⋯ ∈ S20(Γ0(7); \overline{F}11). Also, z satisfies the equation x3 + 2x + 9 = 0. Since f is ordinary, we do not continue our analysis.

36. Consider f = q + z174q2 + z690q3 + z1090q4 + z681q5 + z864q6 + 3q7 + z1135q8 + z1230q9 + z855q10 + z1130q11 + z450q12 + z1081q13 + z1238q14 + z41q15 + z518q16 + z148q17 + z74q18 + z257q19 + z441q20 + z424q21 + ⋯ ∈ S20(Γ0(7); \overline{F}11). Also, z satisfies the equation x3 + 2x + 9 = 0. Since f is ordinary, we do not continue our analysis.

37. Consider f = q + z584q2 + z940q3 + z20q4 + z841q5 + z194q6 + 3q7 + z515q8 + z230q9 + z95q10 + z460q11 + z960q12 + z1251q13 + z318q14 + z451q15 + z378q16 + z298q17 + z814q18 + z167q19 + z861q20 + z674q21 + ⋯ ∈ S20(Γ0(7); \overline{F}11). Also, z satisfies the equation x3 + 2x + 9 = 0. Since f is ordinary, we do not continue our analysis.

38. Consider f = q + 8q3 + 9q4 + 10q5 + 10q7 + 6q9 + 10q11 + 6q12 + 7q13 + 3q15 + 4q16 + 2q17 + 5q19 + 2q20 + 3q21 + ⋯ ∈ S22(Γ0(7); \overline{F}11). Also, z satisfies the equation x3 + 2x + 9 = 0. Since f is ordinary, we do not continue our analysis.

39. Consider f = q + q2 + 2q3 + 10q4 + 9q5 + 2q6 + 10q7 + 8q8 + q9 + 9q10 + q11 + 9q12 + 4q13 + 10q14 + 7q15 + 10q16 + 4q17 + q18 + 2q20 + 9q21 + ⋯ ∈ S22(Γ0(7); \overline{F}11). Also, z satisfies the equation x3 + 2x + 9 = 0. Since f is ordinary, we do not continue our analysis.

40. Consider f = q + q3 + 9q4 + 3q5 + q7 + 9q9 + 10q11 + 9q12 + 7q13 + 3q15 + 4q16 + 5q17 + 2q19 + 5q20 + q21 + ⋯ ∈ S22(Γ0(7); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

41. Consider f = q+7q2+5q3+3q4+9q5+2q6+q7+7q8+8q10+10q11+4q12+8q13+7q14+q15+10q16+3q17+5q19+5q20+5q21+⋯ ∈ S22(Γ0(7); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

42. Consider f = q + 3q2 + 4q3 + 7q4 + 6q5 + q6 + q7 + 4q8 + 2q9 + 7q10 + 6q12 + 3q13 + 3q14 + 2q15 + 9q16 + 7q17 + 6q18 + 9q19 + 9q20 + 4q21 + ⋯ ∈ S22(Γ0(7); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

43. Consider f = q+4q2 +8q3 +3q4 +9q5 +10q6 +q7 +4q8 +6q9 +3q10 +10q11 +2q12 +5q13 +4q14 +6q15 +10q16 +6q17 +2q18 +10q19 +5q20 +8q21 +⋯ ∈ S22(Γ0(7); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

11.7 N = 8, ℓ = 11 (19 forms)

1. Consider f = q + 10q^{3} + 4q^{5} + 5q^{7} + 7q^{9} + 4q^{13} + 7q^{15} + 10q^{17} + 3q^{19} + 6q^{21} + ⋯ ∈ S_{14}(Γ_{0}(8); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 6q^{3} + 5q^{5} + q^{7} + 10q^{9} + q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −12q^{3} + −4330q^{5} + −139992q^{7}+−1594179q^{9}+−589484∗11q^{11}+−22588034q^{13}+51960q^{15}+−23732270q^{17}+325344836q^{19}+ 1679904q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q+7q^{3}+9q^{5}+2q^{7}+8q^{15}+6q^{17}+3q^{21}+⋯ ∈ S_{14}(Γ_{0}(8); \overline{F}_{11}). We have ρ_{f} ⊗χ^{10} ∼ ρ_{g}, where g is the weight-4 newform g = q +7q^{3} +9q^{5} +2q^{7} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{3}. The form f_{0} = q + aq^{3} + 12 ∗ a + 4006q^{5} + −222 ∗ a + 152256q^{7} + 872∗a+1414557q^{9} +531∗a+8005504q^{11} +−10452∗a+13929358q^{13} +14470∗a+36106560q^{15} +38184∗ a−93545038q^{17} +−8529∗11∗a−1678976∗11q^{19} +−41328∗a−667971360q^{21} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −872x + −3008880. The slope of f_{0} is 1/2.

3. Consider f = q + 5q^{3} + 10q^{5} + 7q^{7} + 2q^{9} + 10q^{13} + 6q^{15} + 3q^{17} + 7q^{19} + 2q^{21} + ⋯ ∈ S_{16}(Γ_{0}(8); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + 4q^{3} + 6q^{5} + 6q^{7} + 7q^{9} + 6q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(6)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+2700q^{3}+−251890q^{5}+ 1374072q^{7} + −7058907q^{9} + −3935156 ∗ 11q^{11} + −323161466q^{13} + −680103000q^{15} + −191653646q^{17} + −6515456644q^{19} + 3709994400q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

4. Consider f = q + 10q^{3} + q^{5} + 3q^{7} + 10q^{15} + 3q^{17} + 8q^{21} + ⋯ ∈ S_{16}(Γ_{0}(8); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-4 newform g = q +7q^{3} +9q^{5} +2q^{7} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{15}. The form f_{0} = q + −3444q^{3} + 313358q^{5} + −2324616q^{7} + −226161∗11q^{9}+−456604∗11^{2}q^{11}+−10023598∗11q^{13}+−1079204952q^{15}+−2601428750q^{17}+177465844∗ 11q^{19} + 8005977504q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 2.

5. Consider f = q + 9q^{5} + 2q^{7} + 10q^{9} + 10q^{13} + 2q^{17} + 8q^{19} + ⋯ ∈ S_{16}(Γ_{0}(8); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q+q^{5}+8q^{7}+2q^{9}+4q^{11}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(3)ω^{1} & 0 \\ 0 & u(4)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+29659132878511889008∗ 11 + O(11^{2}0)q^{3} + 308266706037187893917 + O(11^{2}0)q^{5} + −238458227305870040588 + O(11^{2}0)q^{7} + 189360097501490988236+O(11^{2}0)q^{9}+2366594861980184581∗11+O(11^{2}0)q^{11}+−145163955559332709737+ O(11^{2}0)q^{13} + 29743906027638592201 ∗ 11 + O(11^{2}0)q^{15} + −96112956299898508947 + O(11^{2}0)q^{17} + 3473901814486932800936 + O(11^{2}1)q^{19} + 6385330492892579432 ∗ 11 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

(\begin{matrix}u(4)ω^{1} & 0 \\ 0 & u(3)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+237853258544855010238+ O(11^{2}0)q^{3}+143445619107540422809+O(11^{2}0)q^{5}+−151011389797562871838+O(11^{2}0)q^{7}+280446157570764627678+ O(11^{2}0)q^{9}+−6128292804387041005∗11+O(11^{2}0)q^{11}+136748187992773101625+O(11^{2}0)q^{13}+−181024828818751938484+ O(11^{2}0)q^{15}+−335098537127383232363+O(11^{2}0)q^{17}+779378992030693016008+O(11^{2}1)q^{19}+306402192853309311070+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

7. Consider f = q + 5q^{3} + 4q^{5} + 8q^{7} + 5q^{9} + 10q^{13} + 9q^{15} + 6q^{17} + 9q^{19} + 7q^{21} + ⋯ ∈ S_{18}(Γ_{0}(8); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-6 newform g = q + 9q^{3} + 3q^{5} + 9q^{7} + 3q^{9} + 3q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{6} & 0 \\ 0 & u(3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−27120212785074008683+ O(11^{2}0)q^{3}+−174075515412647438993+O(11^{2}0)q^{5}+−151823049696759149382+O(11^{2}0)q^{7}+−203258971072395800392+ O(11^{2}0)q^{9}+−1912073832274955453∗11+O(11^{2}0)q^{11}+318101605297731572008+O(11^{2}0)q^{13}+321580757079223955300+ O(11^{2}0)q^{15}+269735114044000647664+O(11^{2}0)q^{17}+220005465942469156519+O(11^{2}0)q^{19}+−173162390089023029182+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

8. Consider f = q+5q^{5} +q^{7} +7q^{9} +4q^{13} +5q^{17} +5q^{19} +⋯ ∈ S_{20}(Γ_{0}(8); \overline{F}_{11}). We have ρ_{f} ⊗χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q + q^{5} + 8q^{7} + 2q^{9} + 4q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(3)ω^{8} & 0 \\ 0 & u(4)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+28762896031311307819∗ 11 + O(11^{2}0)q^{3} + 879258209361647217272 + O(11^{2}1)q^{5} + −2856029319604017868398 + O(11^{2}1)q^{7} + 59689194141760551716+O(11^{2}0)q^{9}+−17682337531174718495∗11+O(11^{2}0)q^{11}+−292846419252694870801+ O(11^{2}0)q^{13}+30133086800404770211∗11+O(11^{2}0)q^{15}+88133675594113824405+O(11^{2}0)q^{17}+−246354975700501317689+ O(11^{2}0)q^{19} + 8799218661165635436 ∗ 11 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

9. Consider f = q + 2q^{3} + 3q^{5} + 7q^{7} + 6q^{15} + 2q^{17} + 3q^{21} + ⋯ ∈ S_{20}(Γ_{0}(8); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 7q^{3} + 9q^{5} + 2q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{99}. The form f_{0} = q + aq^{3} + (71739913100518356081 + O(11^{2}0)) ∗ a − 286771297198932320206 + O(11^{2}0)q^{5} + −(89298272145247099096 + O(11^{2}0)) ∗ a − 319324010414393401097+O(11^{2}0)q^{7}+−(3012953843206723610+O(11^{2}0))∗a+40305095844010507117+ O(11^{2}0)q^{9}+−(261868908749327877655+O(11^{2}0))∗a+306917766934413219031+O(11^{2}0)q^{11}+−(146960528936824745075+ O(11^{2}0))∗a+283648462680551198426+O(11^{2}0)q^{13}+(49429855242292176837+O(11^{2}1))∗a+2114529052840512092740+ O(11^{2}1)q^{15}+−(5222660501278696115+O(11^{2}0))∗a−183128850258256009838+O(11^{2}0)q^{17}+(9252652515355046232∗ 11 + O(11^{2}0)) ∗ a + 526357107575210889 ∗ 11^{2} + O(11^{2}0)q^{19} + (47007052107242204452 + O(11^{2}0)) ∗ a + 61024949400868555464 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 3012953843206723610 + O(11^{2}0)x + −40305095845172768584 + O(11^{2}0). The slope of f_{0} is 1/2.

10. Consider f = q + q^{3} + 8q^{5} + 9q^{7} + 8q^{9} + 4q^{13} + 8q^{15} + 2q^{17} + 3q^{19} + 9q^{21} + ⋯ ∈ S_{20}(Γ_{0}(8); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q + 4q^{3} + 6q^{5} + 6q^{7} + 7q^{9} + 6q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{8} & 0 \\ 0 & u(6)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+3012953843206747342+ O(11^{2}0)q^{3}+237341441956642283587+O(11^{2}0)q^{5}+272316958307207048365+O(11^{2}0)q^{7}+151616048286617080251+ O(11^{2}0)q^{9}+13055498409929419156∗11+O(11^{2}0)q^{11}+293985161745674099474+O(11^{2}0)q^{13}+−6080281819353922040+ O(11^{2}0)q^{15}+−106390654215567499567+O(11^{2}0)q^{17}+133460784114714809888+O(11^{2}0)q^{19}+241850928540593178288+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

11. Consider f = q+6q^{3} +4q^{5} +5q^{7} +2q^{15} +q^{17} +8q^{21} +⋯ ∈ S_{22}(Γ_{0}(8); \overline{F}_{11}). We have ρ_{f} ⊗χ^{1} ∼ ρ_{g}, where g is the weight-4 newform g = q +7q^{3} +9q^{5} +2q^{7} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{111}. The form f_{0} = q + 192011347013165958526 + O(11^{2}0)q^{3} + 196273029332040884436 + O(11^{2}0)q^{5} + −8163195851865990623 + O(11^{2}0)q^{7} + 22955383498403253067 ∗ 11+O(11^{2}0)q^{9}+−650348885411941718∗11^{2}+O(11^{2}0)q^{11}+−117066435239381991577∗11+O(11^{2}1)q^{13}+ 94228476522828356579+O(11^{2}0)q^{15}+−162433658798202972476+O(11^{2}0)q^{17}+26058816584734320422∗ 11 + O(11^{2}0)q^{19} + −235635184637186998876 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 2.

12. Consider f = q + 7q^{3} + 3q^{5} + 7q^{7} + 2q^{9} + 2q^{13} + 10q^{15} + 9q^{17} + 10q^{19} + 5q^{21} + ⋯ ∈ S_{22}(Γ_{0}(8); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + 6q^{3} + 5q^{5} + q^{7} + 10q^{9} + q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{0} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−54482294323998791782+ O(11^{2}0)q^{3}+325033101411364126993+O(11^{2}0)q^{5}+−36661258535087029210+O(11^{2}0)q^{7}+165475433795445166417+ O(11^{2}0)q^{9} + 16179817711094748369 ∗ 11 + O(11^{2}0)q^{11} + −322845341485774539144 + O(11^{2}0)q^{13} + −329062144376499511957+O(11^{2}0)q^{15}+267123841190985321431+O(11^{2}0)q^{17}+70862335466729695272+ O(11^{2}0)q^{19} + 180401648425834546243 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

13. Consider f = q + 9q3 + 3q5 + 9q7 + 3q9 + 3q11 + 5q13 + 5q15 + q17 + 8q19 + 4q21 + ⋯ ∈ S16(Γ0(8); \overline{F}11). Since f is ordinary, we do not continue our analysis.

14. Consider f = q + q5 + 8q7 + 2q9 + 4q11 + 2q13 + 10q17 + 2q19 + ⋯ ∈ S18(Γ0(8); \overline{F}11). Since f is ordinary, we do not continue our analysis.

15. Consider f = q + 4q3 + 6q5 + 6q7 + 7q9 + 6q11 + 2q13 + 2q15 + 4q17 + 10q19 + 2q21 + ⋯ ∈ S18(Γ0(8); \overline{F}11). Since f is ordinary, we do not continue our analysis.

16. Consider f = q + 6q3 + 5q5 + q7 + 10q9 + q11 + q13 + 8q15 + 7q17 + 4q19 + 6q21 + ⋯ ∈ S20(Γ0(8); \overline{F}11). Since f is ordinary, we do not continue our analysis.

17. Consider f = q + 8q3 + 8q5 + 9q7 + 6q9 + 10q11 + 9q15 + 5q17 + 4q19 + 6q21 + ⋯ ∈ S22(Γ0(8); \overline{F}11). Since f is ordinary, we do not continue our analysis.

18. Consider f = q + z97q3 + z116q5 + z49q7 + z76q9 + 10q11 + z59q13 + z93q15 + 2q17 + 7q19 + z26q21 + ⋯ ∈ S22(Γ0(8); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

19. Consider f = q + z107q3 + z76q5 + z59q7 + z116q9 + 10q11 + z49q13 + z63q15 + 2q17 + 7q19 + z46q21 + ⋯ ∈ S22(Γ0(8); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

11.8 N = 9, ℓ = 11 (28 forms)

1. Consider f = q + q^{2} + 4q^{4} + 4q^{5} + 7q^{7} + 7q^{8} + 4q^{10} + q^{13} + 7q^{14} + 8q^{16} + 3q^{17} + 6q^{19} + 5q^{20} + ⋯ ∈ S_{14}(Γ_{0}(9); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 3q^{2} + 3q^{4} + 5q^{5} + 8q^{7} + 2q^{8} + 4q^{10} + 9q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{1} & 0 \\ 0 & u(9)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 12q^{2} + −8048q^{4} + 30210q^{5}+235088q^{7}+−194880q^{8}+362520q^{10}+1016628∗11q^{11}+8049614q^{13}+2821056q^{14}+63590656q^{16}+ 117494622q^{17} + −214061380q^{19} + −243130080q^{20} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 3q^{4} + 9q^{7} + 7q^{13} + 9q^{16} + q^{19} + ⋯ ∈ S_{14}(Γ_{0}(9); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{10} ∼ ρ_{g}, where g is the weight-4 newform g = q + 3q^{4} + 9q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{3}. The form f_{0} = q + aq^{2} + −272q^{4} + −520 ∗ aq^{5} + −133300q^{7} + −8464 ∗ aq^{8} + −374400 ∗ 11q^{10} + 90400 ∗ aq^{11} + −30477550q^{13} + −133300 ∗ aq^{14} + −64806656q^{16} + −58992 ∗ 11 ∗ aq^{17} + 81793064q^{19} + 141440 ∗ aq^{20} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −720 ∗ 11. The slope of f_{0} is 1/2.

3. Consider f = q+5q^{2}+6q^{4}+3q^{5}+q^{7}+q^{8}+4q^{10}+6q^{13}+5q^{14}+q^{16}+2q^{17}+6q^{19}+7q^{20}+⋯ ∈ S_{14}(Γ_{0}(9); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 4q^{2} + 10q^{4} + q^{5} + 9q^{7} + 5q^{8} + 4q^{10} + 10q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(10)ω^{1} & 0 \\ 0 & u(10)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + 54 ∗ a + 800 ∗ 11q^{4} +128∗a−23814q^{5} +3456∗a−103816q^{7} +3524∗a+917568q^{8} +−16902∗a+2174976q^{10} +−3328∗ 11∗a+59292∗11q^{11}+−89856∗a+11192414q^{13}+7528∗11∗a+58724352q^{14}+665496∗a−12209792q^{16}+ 72960 ∗ a − 43889202q^{17} + −767232 ∗ a + 148862036q^{19} + 213692 ∗ a − 92114496q^{20} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} +−54x+−16992. The slope of f_{0} is 1.

4. Consider f = q + q^{4} + 8q^{7} + 3q^{13} + q^{16} + 8q^{19} + ⋯ ∈ S_{16}(Γ_{0}(9); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-4 newform g = q + 3q^{4} + 9q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{15}. The form f_{0} = q + −32768q^{4} + 1244900q^{7} + 397771850q^{13} + 1073741824q^{16} + 7700827736q^{19} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is Infinity.

5. Consider f = q+3q^{2}+10q^{4}+10q^{5}+6q^{7}+8q^{10}+8q^{13}+7q^{14}+10q^{16}+q^{17}+5q^{19}+q^{20}+⋯ ∈ S_{16}(Γ_{0}(9); \overline{F}_{11}). We have ρ_{f} ⊗χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q +5q^{2} +7q^{4} +6q^{5} +2q^{7} +8q^{10} +2q^{11} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{1} & 0 \\ 0 & u(2)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 234q^{2} + 21988q^{4} + −280710q^{5} + −1373344q^{7} + −229320 ∗ 11q^{8} + −65686140q^{10} + −3093732 ∗ 11q^{11} + 384022262q^{13} + −321362496q^{14} + −1310772464q^{16} + −1259207586q^{17} + −2499071020q^{19} + −6172251480q^{20} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

6. Consider f = q + z^{6}6q^{2} + 3q^{4} + z^{7}8q^{5} + 10q^{7} + z^{9}0q^{8} + 4q^{10} + 9q^{13} + z^{6}q^{14} + z^{1}8q^{17} + z^{5}4q^{20} + ⋯ ∈ S_{16}(Γ_{0}(9); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{1}8q^{2} + q^{4} + z^{6}q^{5} + 7q^{7} + z^{6}6q^{8} + 4q^{10} + z^{7}8q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{4}2)ω^{1} & 0 \\ 0 & u(z^{7}8)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +aq^{2} +87112q^{4} +464∗ aq^{5} + −2591260q^{7} + 54344 ∗ aq^{8} + 55624320q^{10} + 9920 ∗ 11 ∗ aq^{11} + −77911990q^{13} + −2591260 ∗ aq^{14} + 332752064 ∗ 11q^{16} + 217248 ∗ aq^{17} + −88778744 ∗ 11q^{19} + 40419968 ∗ aq^{20} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −119880. The slope of f_{0} is 1.

7. Consider f = q + z^{6}q^{2} + 3q^{4} + z^{1}8q^{5} + 10q^{7} + z^{3}0q^{8} + 4q^{10} + 9q^{13} + z^{6}6q^{14} + z^{7}8q^{17} + z^{1}14q^{20} + ⋯ ∈ S_{16}(Γ_{0}(9); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{7}8q^{2} + q^{4} + z^{6}6q^{5} + 7q^{7} + z^{6}q^{8} + 4q^{10} + z^{1}8q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}02)ω^{1} & 0 \\ 0 & u(z^{1}8)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +aq^{2} +87112q^{4} +464∗ aq^{5} + −2591260q^{7} + 54344 ∗ aq^{8} + 55624320q^{10} + 9920 ∗ 11 ∗ aq^{11} + −77911990q^{13} + −2591260 ∗ aq^{14} + 332752064 ∗ 11q^{16} + 217248 ∗ aq^{17} + −88778744 ∗ 11q^{19} + 40419968 ∗ aq^{20} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −119880. The slope of f_{0} is 1.

8. Consider f = q+10q^{2}+5q^{4}+3q^{5}+5q^{7}+2q^{8}+8q^{10}+6q^{14}+7q^{16}+q^{17}+4q^{19}+4q^{20}+⋯ ∈ S_{18}(Γ_{0}(9); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 6q^{2} + 4q^{4} + 5q^{5} + 4q^{7} + 8q^{8} + 8q^{10} + 3q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{1} & 0 \\ 0 & u(3)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+295774179405406426504+ O(11^{2}0)q^{2}+1916136095384424997528+O(11^{2}1)q^{4}+2310130633369951739801+O(11^{2}1)q^{5}+96109659206687111045+ O(11^{2}0)q^{7}+−48487063722199306851+O(11^{2}0)q^{8}+30106657498292977335+O(11^{2}0)q^{10}+24917664534297148523∗ 11+O(11^{2}1)q^{11}+96133497372782398653∗11+O(11^{2}1)q^{13}+−3004827105011247347903+O(11^{2}1)q^{14}+ −1852375435939028421997+O(11^{2}1)q^{16}+233102293049293580716+O(11^{2}0)q^{17}+−298325460763110472501+ O(11^{2}0)q^{19} + −1920776631590943762797 + O(11^{2}1)q^{20} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

9. Consider f = q+q^{2}+5q^{4}+3q^{5}+6q^{7}+9q^{8}+3q^{10}+6q^{14}+7q^{16}+10q^{17}+7q^{19}+4q^{20}+⋯ ∈ S_{18}(Γ_{0}(9); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-6 newform g = q + 6q^{2} + 4q^{4} + 5q^{5} + 4q^{7} + 8q^{8} + 8q^{10} + 3q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{6} & 0 \\ 0 & u(3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−295774179405406427098+ O(11^{2}0)q^{2}+−1916136095384424821012+O(11^{2}1)q^{4}+−2310130633369952122661+O(11^{2}1)q^{5}+−96109659206662639477+ O(11^{2}0)q^{7}+48487063722068966619+O(11^{2}0)q^{8}+−30106657499102359755+O(11^{2}0)q^{10}+−24917664534207370931∗ 11+O(11^{2}1)q^{11}+−96133497373011434857∗11+O(11^{2}1)q^{13}+3004827105011682913199+O(11^{2}1)q^{14}+ 1852375435989639745917+O(11^{2}1)q^{16}+−233102293014980454352+O(11^{2}0)q^{17}+298325460843164014685+ O(11^{2}0)q^{19} + 1920776632105469857877 + O(11^{2}1)q^{20} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

10. Consider f = q+10q^{2}+6q^{4}+8q^{5}+3q^{7}+3q^{10}+q^{13}+8q^{14}+8q^{16}+8q^{17}+10q^{19}+4q^{20}+⋯ ∈ S_{20}(Γ_{0}(9); \overline{F}_{11}). We have ρ_{f} ⊗χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q +5q^{2} +7q^{4} +6q^{5} +2q^{7} +8q^{10} +2q^{11} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{8} & 0 \\ 0 & u(2)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+191290808516859524879+ O(11^{2}0)q^{2}+263851407676615514982+O(11^{2}0)q^{4}+−93480575082868936053+O(11^{2}0)q^{5}+−225774540010305222868+ O(11^{2}0)q^{7}+7227616423302553478∗11+O(11^{2}0)q^{8}+230222509989842585614+O(11^{2}0)q^{10}+4652872219213217838∗ 11 + O(11^{2}0)q^{11} + 201833677077681461484 + O(11^{2}0)q^{13} + −318976276482900556072 + O(11^{2}0)q^{14} + 33972979971878919738+O(11^{2}0)q^{16}+−134283332744169480116+O(11^{2}0)q^{17}+−204062495327584792427+ O(11^{2}0)q^{19} + 2416328696856668277127 + O(11^{2}1)q^{20} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

11. Consider f = q + 5q^{4} + 4q^{7} + 10q^{13} + 3q^{16} + 5q^{19} + ⋯ ∈ S_{20}(Γ_{0}(9); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 3q^{4} + 9q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{99}. The form f_{0} = q + aq^{2} + O(11^{2}0) ∗ a − 9127356037703374411 + O(11^{2}0)q^{4} + −(50228065718999398322 + O(11^{2}0)) ∗ a + O(11^{2}1)q^{5} + O(11^{2}0) ∗ a + 119759928955378298679 + O(11^{2}0)q^{7} + −(9127356037703898699 + O(11^{2}0)) ∗ a + O(11^{2}1)q^{8} + O(11^{2}0) ∗ a − 24712502156210551871 ∗ 11 + O(11^{2}0)q^{10} + −(167518910768740612545 + O(11^{2}0)) ∗ a + O(11^{2}1)q^{11}+O(11^{2}0)∗a−280505939484285814916+O(11^{2}0)q^{13}+(119759928955378298679+O(11^{2}0))∗ a + O(11^{2}1)q^{14} + O(11^{2}0) ∗ a − 136574039047327621814 + O(11^{2}0)q^{16} + (28070630099113566267 ∗ 11 + O(11^{2}0))∗a+O(11^{2}1)q^{17}+O(11^{2}0)∗a+87865861684397344905+O(11^{2}0)q^{19}+(247230513841791219412+ O(11^{2}0)) ∗ a + O(11^{2}1)q^{20} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + O(11^{2}0)x + 829759639791168193 ∗ 11 + O(11^{2}0). The slope of f_{0} is 1/2.

12. Consider f = q + z^{3}0q^{2} + 4q^{4} + z^{5}4q^{5} + 5q^{7} + z^{1}02q^{8} + 7q^{10} + 8q^{13} + z^{7}8q^{14} + z^{5}4q^{17} + z^{7}8q^{20} + ⋯ ∈ S_{20}(Γ_{0}(9); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{1}8q^{2} + q^{4} + z^{6}q^{5} + 7q^{7} + z^{6}6q^{8} + 4q^{10} + z^{7}8q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{4}2)ω^{8} & 0 \\ 0 & u(z^{7}8)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + O(11^{2}0) ∗ a + 9127356037703772675+O(11^{2}0)q^{4} +(50228065718999424381+O(11^{2}0))∗a+O(11^{2}0)q^{5} +O(11^{2}0)∗a− 119759928955295162639 + O(11^{2}0)q^{7} + (9127356037703248387 + O(11^{2}0)) ∗ a + O(11^{2}0)q^{8} + O(11^{2}0) ∗ a + 271837523716429532341 + O(11^{2}0)q^{10} + (15228991888071561415 ∗ 11 + O(11^{2}0)) ∗ a + O(11^{2}0)q^{11} + O(11^{2}0) ∗ a + 280505939510864653176 + O(11^{2}0)q^{13} + −(119759928955295162639 + O(11^{2}0)) ∗ a + O(11^{2}0)q^{14} + O(11^{2}0) ∗ a + 12415821745236316322 ∗ 11 + O(11^{2}0)q^{16} + −(308776931088733071411 + O(11^{2}0))∗a+O(11^{2}0)q^{17}+O(11^{2}0)∗a−7987805436370603243∗11+O(11^{2}0)q^{19}+−(247230513857340178644+ O(11^{2}0)) ∗ a + O(11^{2}0)q^{20} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + O(11^{2}0)x + −9127356037704296963 + O(11^{2}0). The slope of f_{0} is 1.

13. Consider f = q + 9q^{4} + 6q^{7} + 9q^{13} + 4q^{16} + 7q^{19} + ⋯ ∈ S_{22}(Γ_{0}(9); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{1} ∼ ρ_{g}, where g is the weight-4 newform g = q + 3q^{4} + 9q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{111}. The form f_{0} = q + −2097152q^{4} + 1123983020q^{7} + −370076825230q^{13} + 4398046511104q^{16} + −35540635313176q^{19} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is Infinity.

14. Consider f = q + 6q^{2} + q^{4} + 3q^{5} + q^{7} + 5q^{8} + 7q^{10} + 6q^{13} + 6q^{14} + 6q^{16} + 6q^{17} + 9q^{19} + 3q^{20} + ⋯ ∈ S_{22}(Γ_{0}(9); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + 3q^{2} + 3q^{4} + 5q^{5} + 8q^{7} + 2q^{8} + 4q^{10} + 9q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{0} & 0 \\ 0 & u(9)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 2844q^{2} + 5991184q^{4} + −3109950q^{5} + 363303920q^{7} + 11074627008q^{8} + −8844697800q^{10} + −1325621196 ∗ 11q^{11} + 113350790702q^{13}+1033236348480q^{14}+18931815702784q^{16}+8589389597982q^{17}+−29202939273796q^{19}+ −18632282680800q^{20} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

15. Consider f = q + 8q^{2} + 7q^{4} + 5q^{5} + 8q^{7} + 7q^{8} + 7q^{10} + 3q^{13} + 9q^{14} + 9q^{16} + 4q^{17} + 9q^{19} + 2q^{20} + ⋯ ∈ S_{22}(Γ_{0}(9); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + 4q^{2} + 10q^{4} + q^{5} + 9q^{7} + 5q^{8} + 4q^{10} + 10q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(10)ω^{0} & 0 \\ 0 & u(10)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−666∗a+3040∗ 11^{2}q^{4} +13568∗a+4019706q^{5} +182016∗a+400559384q^{7} +−1285756∗a−1641684672q^{8} +−5016582∗a+ 33445011456q^{10} + 2842112 ∗ 11 ∗ a − 9047627748 ∗ 11q^{11} + 475236864 ∗ a + 134019420734q^{13} + 2308568 ∗ 11^{2} ∗ a + 448667983872q^{14} + 611332056 ∗ a − 3940794645632q^{16} + 1677304320 ∗ a + 6225306859278q^{17} + 22384332288 ∗ a + 13434275157716q^{19} + 8331896732 ∗ a − 20795768974656q^{20} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 666x + −2464992. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

16. Consider f = q + 6q2 + 4q4 + 5q5 + 4q7 + 8q8 + 8q10 + 3q11 + 2q14 + 8q16 + 9q17 + 5q19 + 9q20 + ⋯ ∈ S16(Γ0(9); \overline{F}11). Since f is ordinary, we do not continue our analysis.

17. Consider f = q + 4q2 + 6q4 + 8q5 + 10q7 + 6q8 + 10q10 + 7q13 + 7q14 + 8q16 + 7q17 + q19 + 4q20 + ⋯ ∈ S16(Γ0(9); \overline{F}11). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

18. Consider f = q + 5q2 + 7q4 + 6q5 + 2q7 + 8q10 + 2q11 + 6q13 + 10q14 + 6q16 + 5q17 + 4q19 + 9q20 + ⋯ ∈ S18(Γ0(9); \overline{F}11). Since f is ordinary, we do not continue our analysis.

19. Consider f = q + 4q4 + q5 + 5q16 + 4q20 + ⋯ ∈ S18(Γ0(9); \overline{F}11). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

20. Consider f = q + z78q2 + q4 + z66q5 + 7q7 + z6q8 + 4q10 + z18q11 + 4q13 + z42q14 + z6q17 + z66q20 + ⋯ ∈ S18(Γ0(9); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

21. Consider f = q + z18q2 + q4 + z6q5 + 7q7 + z66q8 + 4q10 + z78q11 + 4q13 + z102q14 + z66q17 + z6q20 + ⋯ ∈ S18(Γ0(9); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

22. Consider f = q + 6q2 + 8q4 + 2q5 + 5q7 + q8 + q10 + 5q13 + 8q14 + 2q16 + q17 + 2q19 + 5q20 + ⋯ ∈ S20(Γ0(9); \overline{F}11). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

23. Consider f = q + 4q2 + 10q4 + q5 + 9q7 + 5q8 + 4q10 + 10q11 + 7q13 + 3q14 + 4q16 + 8q17 + 8q19 + 10q20 + ⋯ ∈ S20(Γ0(9); \overline{F}11). Since f is ordinary, we do not continue our analysis.

24. Consider f = q + 3q2 + 3q4 + 5q5 + 8q7 + 2q8 + 4q10 + 9q11 + 3q13 + 2q14 + 10q16 + q17 + 8q19 + 4q20 + ⋯ ∈ S20(Γ0(9); \overline{F}11). Since f is ordinary, we do not continue our analysis.

25. Consider f = q + 2q2 + 2q4 + 10q5 + 9q7 + 9q10 + 10q11 + 4q13 + 7q14 + 7q16 + 2q17 + 9q20 + ⋯ ∈ S22(Γ0(9); \overline{F}11). Since f is ordinary, we do not continue our analysis.

26. Consider f = q + 10q2 + 10q4 + 2q5 + 4q7 + 3q8 + 9q10 + 10q11 + 9q13 + 7q14 + 10q16 + 2q17 + 9q20 + ⋯ ∈ S22(Γ0(9); \overline{F}11). Since f is ordinary, we do not continue our analysis.

27. Consider f = q + 10q2 + 10q4 + 7q5 + 9q7 + 3q8 + 4q10 + 10q11 + 9q13 + 2q14 + 10q16 + 2q17 + 5q19 + 4q20 + ⋯ ∈ S22(Γ0(9); \overline{F}11). Since f is ordinary, we do not continue our analysis.

28. Consider f = q + q2 + 10q4 + 4q5 + 9q7 + 8q8 + 4q10 + q11 + 9q13 + 9q14 + 10q16 + 9q17 + 5q19 + 7q20 + ⋯ ∈ S22(Γ0(9); \overline{F}11). Since f is ordinary, we do not continue our analysis.

11.9 N = 10, ℓ = 11 (24 forms)

1. Consider f = q + 2q^{2} + 7q^{3} + 4q^{4} + 6q^{5} + 3q^{6} + 8q^{8} + q^{10} + 6q^{12} + 2q^{13} + 9q^{15} + 5q^{16} + 2q^{17} + 3q^{19} + 2q^{20} + ⋯ ∈ S_{14}(Γ_{0}(10); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 6q^{2} + 2q^{3} + 3q^{4} + 2q^{5} + q^{6} + 7q^{8} + q^{10} + 9q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{1} & 0 \\ 0 & u(9)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −64q^{2} + −26q^{3} + 4096q^{4} + −15625q^{5} + 1664q^{6} + 48958 ∗ 11q^{7} + −262144q^{8} + −144877 ∗ 11q^{9} + 1000000q^{10} + −363168 ∗ 11q^{11} + −106496q^{12} + −23834446q^{13} + −3133312 ∗ 11q^{14} + 406250q^{15} + 16777216q^{16} + −192273222q^{17} + 9272128 ∗ 11q^{18} + 166485740q^{19} + −64000000q^{20} + −1272908 ∗ 11q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 9q^{2} + 4q^{3} + 4q^{4} + 6q^{5} + 3q^{6} + 3q^{7} + 3q^{8} + 10q^{10} + 5q^{12} + 5q^{13} + 5q^{14} + 2q^{15} + 5q^{16} + 9q^{17} + 2q^{19} + 2q^{20} + q^{21} + ⋯ ∈ S_{14}(Γ_{0}(10); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 5q^{2} + 9q^{3} + 3q^{4} + 2q^{5} + q^{6} + 5q^{7} + 4q^{8} + 10q^{10} + 7q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{1} & 0 \\ 0 & u(7)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 64q^{2} + −2394q^{3} + 4096q^{4}+−15625q^{5}+−153216q^{6}+438122q^{7}+262144q^{8}+376083∗11q^{9}+−1000000q^{10}+−146208∗11q^{11}+ −9805824q^{12} + 2653106q^{13} + 28039808q^{14} + 37406250q^{15} + 16777216q^{16} + 108907962q^{17} + 24069312 ∗ 11q^{18} + −63937300q^{19} + −64000000q^{20} + −1048864068q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q + 4q^{2} + 9q^{3} + 5q^{4} + 3q^{5} + 3q^{6} + 5q^{7} + 9q^{8} + 3q^{9} + q^{10} + q^{12} + 5q^{13} + 9q^{14} + 5q^{15} + 3q^{16} + q^{18} + 3q^{19} + 4q^{20} + q^{21} + ⋯ ∈ S_{16}(Γ_{0}(10); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-4 newform g = q + 2q^{2} + 3q^{3} + 4q^{4} + 5q^{5} + 6q^{6} + 7q^{7} + 8q^{8} + 4q^{9} + 10q^{10} + q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{4} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +−128q^{2} +−5568q^{3} + 16384q^{4} + 78125q^{5} + 712704q^{6} + 2564996q^{7} + −2097152q^{8} + 16653717q^{9} + −10000000q^{10} + −7369788 ∗ 11q^{11}+−91226112q^{12}+351412022q^{13}+−328319488q^{14}+−435000000q^{15}+268435456q^{16}+−196165914∗ 11q^{17} + −2131675776q^{18} + −5107458100q^{19} + 1280000000q^{20} + −14281897728q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

4. Consider f = q + 7q^{2} + 2q^{3} + 5q^{4} + 3q^{5} + 3q^{6} + 4q^{7} + 2q^{8} + 3q^{9} + 10q^{10} + 10q^{12} + 6q^{14} + 6q^{15} + 3q^{16} + 9q^{17} + 10q^{18} + 5q^{19} + 4q^{20} + 8q^{21} + ⋯ ∈ S_{16}(Γ_{0}(10); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{10} ∼ ρ_{g}, where g is the weight-6 newform g = q + 7q^{2} + 2q^{3} + 5q^{4} + 3q^{5} + 3q^{6} + 4q^{7} + 2q^{8} + 3q^{9} + 10q^{10} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{5}. The form f_{0} = q + 128q^{2} + aq^{3} + 16384q^{4} + 78125q^{5} + 128 ∗ aq^{6} + 423 ∗ a − 102460q^{7} + 2097152q^{8} + −1844 ∗ a + 8757909q^{9} + 10000000q^{10} + −12006 ∗ a + 44706540q^{11} + 16384 ∗ aq^{12} + −28548 ∗ a + 118335542q^{13} + 54144 ∗ a − 13114880q^{14} + 78125 ∗ aq^{15} + 268435456q^{16} + −13428 ∗ a + 698489058q^{17} + −236032 ∗ a + 1121012352q^{18} + −134964 ∗ a + 2955266252q^{19} + 1280000000q^{20} + −882472 ∗ a + 9774183168q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 1844x + −23106816. The slope of f_{0} is 1/2.

5. Consider f = q + 3q^{2} + 6q^{3} + 9q^{4} + 4q^{5} + 7q^{6} + 6q^{7} + 5q^{8} + 5q^{9} + q^{10} + 10q^{12} + 7q^{14} + 2q^{15} + 4q^{16} + 10q^{17} +4q^{18} +7q^{19} +3q^{20} +3q^{21} +⋯ ∈ S_{18}(Γ_{0}(10); \overline{F}_{11}). We have ρ_{f} ⊗χ^{9} ∼ ρ_{g}, where g is the weight-6 newform g = q +7q^{2} +2q^{3} +5q^{4} +3q^{5} +3q^{6} +4q^{7} +2q^{8} +3q^{9} +10q^{10} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{17}. The form f_{0} = q + 256q^{2} + −14976q^{3} + 65536q^{4}+390625q^{5}+−3833856q^{6}+14808668q^{7}+16777216q^{8}+95140413q^{9}+100000000q^{10}+−8967228∗ 11^{2}q^{11} + −981467136q^{12} + −417754886 ∗ 11q^{13} + 3791019008q^{14} + −5850000000q^{15} + 4294967296q^{16} + −16104698622q^{17} + 24355945728q^{18} + 48093117860q^{19} + 25600000000q^{20} + −221774611968q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 2.

6. Consider f = q + 8q^{2} + 6q^{3} + 9q^{4} + 4q^{5} + 4q^{6} + 5q^{7} + 6q^{8} + 5q^{9} + 10q^{10} + 10q^{12} + 7q^{14} + 2q^{15} + 4q^{16} + q^{17} + 7q^{18} + 4q^{19} + 3q^{20} + 8q^{21} + ⋯ ∈ S_{18}(Γ_{0}(10); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 7q^{2} + 2q^{3} + 5q^{4} + 3q^{5} + 3q^{6} + 4q^{7} + 2q^{8} + 3q^{9} + 10q^{10} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{77}. The form f_{0} = q +−256q^{2} +aq^{3} +65536q^{4} +390625q^{5} +−256∗aq^{6} +2907∗a+2203100q^{7} +−16777216q^{8} +−1308∗a− 54614979q^{9} + −100000000q^{10} + −5238 ∗ a − 239166300q^{11} + 65536 ∗ aq^{12} + −87228 ∗ a − 827964226q^{13} + −744192∗a−563993600q^{14}+390625∗aq^{15}+4294967296q^{16}+4460868∗a+18987357954q^{17}+334848∗a+ 13981434624q^{18}+−7351668∗a+59528273828q^{19}+25600000000q^{20}+−1599256∗a+216644709888q^{21}+⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 1308x + −74525184. The slope of f_{0} is 1/2.

7. Consider f = q + 8q^{2} + 10q^{3} + 9q^{4} + 7q^{5} + 3q^{6} + 6q^{8} + 3q^{9} + q^{10} + 2q^{12} + 4q^{13} + 4q^{15} + 4q^{16} + 9q^{17} + 2q^{18} + 8q^{20} + ⋯ ∈ S_{18}(Γ_{0}(10); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q +7q^{2} +7q^{3} +5q^{4} +8q^{5} +5q^{6} +2q^{8} +4q^{9} +q^{10} +2q^{11} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{1} & 0 \\ 0 & u(2)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −256q^{2} + −188521619038109729967+O(11^{2}0)q^{3}+65536q^{4}+−390625q^{5}+−176465161388229790920+O(11^{2}0)q^{6}+ 14066004094758254895 ∗ 11 + O(11^{2}0)q^{7} + −16777216q^{8} + −227618148369827933011 + O(11^{2}0)q^{9} + 100000000q^{10} + −335713506950885276839 ∗ 11 + O(11^{2}1)q^{11} + 100831654905305859053 + O(11^{2}0)q^{12} + 114613112986167305954+O(11^{2}0)q^{13}+7489288198344978049∗11+O(11^{2}0)q^{14}+24741458796981190312+ O(11^{2}0)q^{15}+4294967296q^{16}+78774514299817932803+O(11^{2}0)q^{17}+−259003576456769949671+O(11^{2}0)q^{18}+ −9292373287189110378∗11+O(11^{2}0)q^{19}+−25600000000q^{20}+2667967109634342535∗11+O(11^{2}0)q^{21}+ ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

8. Consider f = q + 8q^{2} + 7q^{3} + 9q^{4} + 7q^{5} + q^{6} + 10q^{7} + 6q^{8} + 7q^{9} + q^{10} + 8q^{12} + q^{13} + 3q^{14} + 5q^{15} + 4q^{16} + 7q^{17} + q^{18} + 3q^{19} + 8q^{20} + 4q^{21} + ⋯ ∈ S_{18}(Γ_{0}(10); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-6 newform g = q + 4q^{2} + 6q^{3} + 5q^{4} + 8q^{5} + 2q^{6} + 3q^{7} + 9q^{8} + 2q^{9} + 10q^{10} + 5q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(9)ω^{6} & 0 \\ 0 & u(5)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −256q^{2} + 188521619038109723659 + O(11^{2}0)q^{3} + 65536q^{4} + −390625q^{5} + 176465161388231405768 + O(11^{2}0)q^{6} + −154726045042334260001+O(11^{2}0)q^{7}+−16777216q^{8}+227618148370051128917+O(11^{2}0)q^{9}+100000000q^{10}+ 335713506950993404903∗11+O(11^{2}1)q^{11}+−100831654905719260141+O(11^{2}0)q^{12}+−114613112988185225182+ O(11^{2}0)q^{13}+−82382170183469982603+O(11^{2}0)q^{14}+−24741458794517127812+O(11^{2}0)q^{15}+4294967296q^{16}+ −78774514318573572239+O(11^{2}0)q^{17}+259003576399631797735+O(11^{2}0)q^{18}+102216106022375383558+ O(11^{2}0)q^{19} +−25600000000q^{20} +−29347638615729666261+O(11^{2}0)q^{21} +⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

9. Consider f = q + 3q^{2} + 10q^{3} + 9q^{4} + 7q^{5} + 8q^{6} + 5q^{8} + 3q^{9} + 10q^{10} + 2q^{12} + 7q^{13} + 4q^{15} + 4q^{16} + 2q^{17} + 9q^{18} + 8q^{20} + ⋯ ∈ S_{18}(Γ_{0}(10); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-6 newform g = q +7q^{2} +7q^{3} +5q^{4} +8q^{5} +5q^{6} +2q^{8} +4q^{9} +q^{10} +2q^{11} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{6} & 0 \\ 0 & u(2)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+256q^{2}+123401660539921798077+ O(11^{2}0)q^{3}+65536q^{4}+−390625q^{5}+−28424663610340124735+O(11^{2}0)q^{6}+28204035417009362968∗11+ O(11^{2}0)q^{7}+16777216q^{8}+323738380774909329364+O(11^{2}0)q^{9}+−100000000q^{10}+−328277058005406018879∗ 11 + O(11^{2}1)q^{11} + 123536060011088169051 + O(11^{2}0)q^{12} + 32459729709241225145 + O(11^{2}0)q^{13} + 3460393841480457470 ∗ 11 + O(11^{2}0)q^{14} + 108988500837405441927 + O(11^{2}0)q^{15} + 4294967296q^{16} + −121218826518552859215+O(11^{2}0)q^{17}+128776101671907185461+O(11^{2}0)q^{18}+−15008898930489994616∗ 11 + O(11^{2}0)q^{19} + −25600000000q^{20} + −14341529735651646387 ∗ 11 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

10. Consider f = q + 3q^{2} + 7q^{3} + 9q^{4} + 7q^{5} + 10q^{6} + q^{7} + 5q^{8} + 7q^{9} + 10q^{10} + 8q^{12} + 10q^{13} + 3q^{14} + 5q^{15} + 4q^{16} + 4q^{17} + 10q^{18} + 8q^{19} + 8q^{20} + 7q^{21} + ⋯ ∈ S_{18}(Γ_{0}(10); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 4q^{2} + 6q^{3} + 5q^{4} + 8q^{5} + 2q^{6} + 3q^{7} + 9q^{8} + 2q^{9} + 10q^{10} + 5q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(9)ω^{1} & 0 \\ 0 & u(5)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+256q^{2}+−123401660539921780449+ O(11^{2}0)q^{3} + 65536q^{4} + −390625q^{5} + 28424663610344637503 + O(11^{2}0)q^{6} + −310244389587075308452 + O(11^{2}0)q^{7}+16777216q^{8}+−323738380774673433298+O(11^{2}0)q^{9}+−100000000q^{10}+328277058005411883903∗ 11 + O(11^{2}1)q^{11} + −123536060009932900443 + O(11^{2}0)q^{12} + −32459729706345386677 + O(11^{2}0)q^{13} + −38064332249197877994 + O(11^{2}0)q^{14} + −108988500844291379427 + O(11^{2}0)q^{15} + 4294967296q^{16} + 121218826520133071811+O(11^{2}0)q^{17}+−128776101611517792565+O(11^{2}0)q^{18}+165097888279603653536+ O(11^{2}0)q^{19} +−25600000000q^{20} +157756827375816588201+O(11^{2}0)q^{21} +⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

11. Consider f = q + 5q^{2} + 7q^{3} + 3q^{4} + 9q^{5} + 2q^{6} + 2q^{7} + 4q^{8} + q^{9} + q^{10} + 10q^{12} + 10q^{14} + 8q^{15} + 9q^{16} + 6q^{17} + 5q^{18} + 10q^{19} + 5q^{20} + 3q^{21} + ⋯ ∈ S_{20}(Γ_{0}(10); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{3} ∼ ρ_{g}, where g is the weight-6 newform g = q + 7q^{2} + 2q^{3} + 5q^{4} + 3q^{5} + 3q^{6} + 4q^{7} + 2q^{8} + 3q^{9} + 10q^{10} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{89}. The form f_{0} = q + −512q^{2} + 38628q^{3} + 262144q^{4} + 1953125q^{5} + −19777536q^{6} + −144185776q^{7} + −134217728q^{8} + 329860917q^{9} + −1000000000q^{10} + −42615708 ∗ 11^{2}q^{11} + 10126098432q^{12} + 312319618 ∗ 11q^{13} + 73823117312q^{14} + 75445312500q^{15} + 68719476736q^{16} + 366347849874q^{17} + −168888789504q^{18} + −1604379002260q^{19} + 512000000000q^{20} + −5569608155328q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 2.

12. Consider f = q + 6q^{2} + 4q^{3} + 3q^{4} + 9q^{5} + 2q^{6} + 8q^{7} + 7q^{8} + q^{9} + 10q^{10} + q^{12} + 2q^{13} + 4q^{14} + 3q^{15} + 9q^{16} + 6q^{18} + 6q^{19} + 5q^{20} + 10q^{21} + ⋯ ∈ S_{20}(Γ_{0}(10); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 2q^{2} + 3q^{3} + 4q^{4} + 5q^{5} + 6q^{6} + 7q^{7} + 8q^{8} + 4q^{9} + 10q^{10} + q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+512q^{2}+−62360943767820225117+ O(11^{2}0)q^{3}+262144q^{4}+1953125q^{5}+−309553447293634827457+O(11^{2}0)q^{6}+−232736251188757922950+ O(11^{2}0)q^{7}+134217728q^{8}+−43983466786959330993+O(11^{2}0)q^{9}+1000000000q^{10}+25541891217659863966∗ 11 + O(11^{2}0)q^{11} + 277633789743130513452 + O(11^{2}0)q^{12} + −181191391336971433350 + O(11^{2}0)q^{13} + −84211505580934921823 + O(11^{2}0)q^{14} + −22713963617755836379 + O(11^{2}0)q^{15} + 68719476736q^{16} + 23681820091478238212∗11+O(11^{2}0)q^{17}+−318785162148697164783+O(11^{2}0)q^{18}+−74596591293062429409+ O(11^{2}0)q^{19} + 512000000000q^{20} + −163052511567518303094 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

13. Consider f = q+q^{2}+6q^{3}+q^{4}+10q^{5}+6q^{6}+q^{8}+10q^{10}+6q^{12}+q^{13}+5q^{15}+q^{16}+4q^{17}+10q^{19}+10q^{20}+⋯ ∈ S_{22}(Γ_{0}(10); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + 6q^{2} + 2q^{3} + 3q^{4} + 2q^{5} + q^{6} + 7q^{8} + q^{10} + 9q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{0} & 0 \\ 0 & u(9)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 1024q^{2} + 121796719666687454351+O(11^{2}0)q^{3}+1048576q^{4}+−9765625q^{5}+261091876164351553239+O(11^{2}0)q^{6}+ −16408784589677705241 ∗ 11 + O(11^{2}0)q^{7} + 1073741824q^{8} + 1477520080877283846 ∗ 11^{2} + O(11^{2}0)q^{9} + −10000000000q^{10} + 6188617856735597439 ∗ 11 + O(11^{2}0)q^{11} + 276333204069666863939 + O(11^{2}0)q^{12} + 227145338352581703539+O(11^{2}0)q^{13}+16154453484030063241∗11+O(11^{2}0)q^{14}+227795776492310874424+ O(11^{2}0)q^{15} +1099511627776q^{16} +−67864864861926321427+O(11^{2}0)q^{17} +683053548451695472∗11^{2} + O(11^{2}0)q^{18}+−1496707798305994099398+O(11^{2}1)q^{19}+−10240000000000q^{20}+28737293916594244754∗ 11 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

14. Consider f = q + 10q^{2} + 5q^{3} + q^{4} + 10q^{5} + 6q^{6} + 2q^{7} + 10q^{8} + q^{10} + 5q^{12} + 8q^{13} + 9q^{14} + 6q^{15} + q^{16} + 7q^{17} + 3q^{19} + 10q^{20} + 10q^{21} + ⋯ ∈ S_{22}(Γ_{0}(10); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + 5q^{2} + 9q^{3} + 3q^{4} + 2q^{5} + q^{6} + 5q^{7} + 4q^{8} + 10q^{10} + 7q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{0} & 0 \\ 0 & u(7)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −1024q^{2} + −67199857022868093464+O(11^{2}0)q^{3}+1048576q^{4}+−9765625q^{5}+192154108295806768634+O(11^{2}0)q^{6}+ 78892459375844053183+O(11^{2}0)q^{7}+−1073741824q^{8}+273554294673459834∗11^{2}+O(11^{2}0)q^{9}+10000000000q^{10}+ −20453552651828856666∗11+O(11^{2}0)q^{11}+−322808374598608394524+O(11^{2}0)q^{12}+−62805754801694191730+ O(11^{2}0)q^{13}+−55879008957109355272+O(11^{2}0)q^{14}+−179317907952060942872+O(11^{2}0)q^{15}+1099511627776q^{16}+ 141506906831414809956+O(11^{2}0)q^{17}+−2123732071011295966∗11^{2}+O(11^{2}0)q^{18}+87716077063810466142+ O(11^{2}0)q^{19} + −10240000000000q^{20} + −86525995233260843327 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

15. Consider f = q + 2q2 + 3q3 + 4q4 + 5q5 + 6q6 + 7q7 + 8q8 + 4q9 + 10q10 + q11 + q12 + 8q13 + 3q14 + 4q15 + 5q16 + 8q18 + 10q19 + 9q20 + 10q21 + ⋯ ∈ S14(Γ0(10); \overline{F}11). Since f is ordinary, we do not continue our analysis.

16. Consider f = q+4q2 +6q3 +5q4 +8q5 +2q6 +3q7 +9q8 +2q9 +10q10 +5q11 +8q12 +6q13 +q14 +4q15 +3q16 +3q17 +8q18 +10q19 +7q20 +7q21 +⋯ ∈ S16(Γ0(10); \overline{F}11). Since f is ordinary, we do not continue our analysis.

17. Consider f = q + 7q2 + 7q3 + 5q4 + 8q5 + 5q6 + 2q8 + 4q9 + q10 + 2q11 + 2q12 + 9q13 + q15 + 3q16 + 4q17 + 6q18 + 7q20 + ⋯ ∈ S16(Γ0(10); \overline{F}11). Since f is ordinary, we do not continue our analysis.

18. Consider f = q + 6q2 + 2q3 + 3q4 + 2q5 + q6 + 7q8 + q10 + 9q11 + 6q12 + 6q13 + 4q15 + 9q16 + 8q17 + 4q19 + 6q20 + ⋯ ∈ S20(Γ0(10); \overline{F}11). Since f is ordinary, we do not continue our analysis.

19. Consider f = q + 5q2 + 9q3 + 3q4 + 2q5 + q6 + 5q7 + 4q8 + 10q10 + 7q11 + 5q12 + 4q13 + 3q14 + 7q15 + 9q16 + 3q17 + 10q19 + 6q20 + q21 + ⋯ ∈ S20(Γ0(10); \overline{F}11). Since f is ordinary, we do not continue our analysis.

20. Consider f = q+6q2 +5q3 +3q4 +9q5 +8q6 +9q7 +7q8 +10q9 +10q10 +q11 +4q12 +7q13 +10q14 +q15 +9q16 +3q17 +5q18 +8q19 +5q20 +q21 +⋯ ∈ S20(Γ0(10); \overline{F}11). Since f is ordinary, we do not continue our analysis.

21. Consider f = q + q2 + 10q3 + q4 + q5 + 10q6 + 3q7 + q8 + 9q9 + q10 + q11 + 10q12 + 5q13 + 3q14 + 10q15 + q16 + 4q17 + 9q18 + 5q19 + q20 + 8q21 + ⋯ ∈ S22(Γ0(10); \overline{F}11). Since f is ordinary, we do not continue our analysis.

22. Consider f = q+q2 +q3 +q4 +10q5 +q6 +10q7 +q8 +9q9 +10q10 +10q11 +q12 +2q13 +10q14 +10q15 +q16 +8q17 +9q18 +10q19 +10q20 +10q21 +⋯ ∈ S22(Γ0(10); \overline{F}11). Since f is ordinary, we do not continue our analysis.

23. Consider f = q + 10q2 + 5q3 + q4 + q5 + 6q6 + 6q7 + 10q8 + 10q10 + 10q11 + 5q12 + 2q13 + 5q14 + 5q15 + q16 + 4q17 + 9q19 + q20 + 8q21 + ⋯ ∈ S22(Γ0(10); \overline{F}11). Since f is ordinary, we do not continue our analysis.

24. Consider f = q+10q2 +q3 +q4 +10q5 +10q6 +5q7 +10q8 +9q9 +q10 +q11 +q12 +2q13 +6q14 +10q15 +q16 +3q17 +2q18 +4q19 +10q20 +5q21 +⋯ ∈ S22(Γ0(10); \overline{F}11). Since f is ordinary, we do not continue our analysis.

11.10 N = 12, ℓ = 11 (14 forms)

1. Consider f = q + 8q^{3} + 2q^{7} + 9q^{9} + 4q^{19} + 5q^{21} + ⋯ ∈ S_{14}(Γ_{0}(12); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 7q^{3} + 7q^{7} + 5q^{9} + 9q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{1} & 0 \\ 0 & u(9)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −729q^{3} + −1350 ∗ 11q^{5} + −62896q^{7} + 531441q^{9} + 464076 ∗ 11q^{11} + 94910 ∗ 11^{2}q^{13} + 984150 ∗ 11q^{15} + 10905894 ∗ 11q^{17} + 332601020q^{19} + 45851184q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 2q^{3} + 8q^{5} + q^{7} + 4q^{9} + q^{13} + 5q^{15} + 10q^{17} + 8q^{19} + 2q^{21} + ⋯ ∈ S_{16}(Γ_{0}(12); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + 6q^{3} + 7q^{5} + 4q^{7} + 3q^{9} + 2q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{1} & 0 \\ 0 & u(2)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −2187q^{3} + 45702q^{5}+1217888q^{7}+4782969q^{9}+−2445084∗11q^{11}+−162581770q^{13}+−99950274q^{15}+−743272542q^{17}+ −4003014700q^{19} + −2663521056q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q + 9q^{3} + 10q^{5} + 3q^{7} + 4q^{9} + 8q^{13} + 2q^{15} + 10q^{17} + 6q^{19} + 5q^{21} + ⋯ ∈ S_{16}(Γ_{0}(12); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + 5q^{3} + 6q^{5} + q^{7} + 3q^{9} + 7q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{1} & 0 \\ 0 & u(7)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 2187q^{3} + 99368425518821242871+O(11^{2}0)q^{5}+−221565834736829617416+O(11^{2}0)q^{7}+4782969q^{9}+−107420916240889308462∗ 11 + O(11^{2}1)q^{11} + −56612114814419702090 + O(11^{2}0)q^{13} + 20498246445175186954 + O(11^{2}0)q^{15} + 33375331517442434101+O(11^{2}0)q^{17}+−35379829084844793433+O(11^{2}0)q^{19}+−184484218003166664072+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

(\begin{matrix}u(4)ω^{4} & 0 \\ 0 & u(3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 2187q^{3} + −99368425518821173211+O(11^{2}0)q^{5}+221565834736832108920+O(11^{2}0)q^{7}+4782969q^{9}+107420916240887947014∗ 11 + O(11^{2}1)q^{11} + 56612114814431459670 + O(11^{2}0)q^{13} + −20498246445022840534 + O(11^{2}0)q^{15} + −33375331513186356817+O(11^{2}0)q^{17}+35379829094086482833+O(11^{2}0)q^{19}+184484218008615583320+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

5. Consider f = q + 7q^{3} + 2q^{5} + 6q^{7} + 5q^{9} + 7q^{13} + 3q^{15} + 3q^{17} + 5q^{19} + 9q^{21} + ⋯ ∈ S_{20}(Γ_{0}(12); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q + 6q^{3} + 7q^{5} + 4q^{7} + 3q^{9} + 2q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{8} & 0 \\ 0 & u(2)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −19683q^{3} + 295581888853621095563+O(11^{2}0)q^{5}+−90943751437688053778+O(11^{2}0)q^{7}+387420489q^{9}+−7682885593580008699∗ 11 + O(11^{2}0)q^{11} + −225220569095124000237 + O(11^{2}0)q^{13} + 3637870954935603719 + O(11^{2}0)q^{15} + −281462457965651956925+O(11^{2}0)q^{17}+195456618287351333896+O(11^{2}0)q^{19}+−141876967528221971487+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

6. Consider f = q + 4q^{3} + 8q^{5} + 7q^{7} + 5q^{9} + q^{13} + 10q^{15} + 3q^{17} + q^{19} + 6q^{21} + ⋯ ∈ S_{20}(Γ_{0}(12); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q + 5q^{3} + 6q^{5} + q^{7} + 3q^{9} + 7q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{8} & 0 \\ 0 & u(7)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 19683q^{3} + −189034548010531130878+O(11^{2}0)q^{5}+−144518240427715935048+O(11^{2}0)q^{7}+387420489q^{9}+−222581537913649827424∗ 11 + O(11^{2}1)q^{11} + −79660889634454646484 + O(11^{2}0)q^{13} + 213213480705161819057 + O(11^{2}0)q^{15} + 11395749718213555159+O(11^{2}0)q^{17}+−139993093474241427892+O(11^{2}0)q^{19}+−165547763869030647956+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

7. Consider f = q + 4q^{3} + 5q^{5} + 6q^{7} + 5q^{9} + 3q^{13} + 9q^{15} + 6q^{17} + 6q^{19} + 2q^{21} + ⋯ ∈ S_{20}(Γ_{0}(12); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 3q^{3} + 4q^{5} + 8q^{7} + 9q^{9} + 3q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{1} & 0 \\ 0 & u(3)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 19683q^{3} + 189034548010527863770+O(11^{2}0)q^{5}+144518240427688911064+O(11^{2}0)q^{7}+387420489q^{9}+222581537914620915256∗ 11 + O(11^{2}1)q^{11} + 79660889668122616384 + O(11^{2}0)q^{13} + −213213480769468305821 + O(11^{2}0)q^{15} + −11395749980939700403+O(11^{2}0)q^{17}+139993093909559256812+O(11^{2}0)q^{19}+165547763337117570884+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

8. Consider f = q + 10q^{3} + 5q^{7} + q^{9} + 6q^{19} + 6q^{21} + ⋯ ∈ S_{22}(Γ_{0}(12); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + 7q^{3} + 7q^{7} + 5q^{9} + 9q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{0} & 0 \\ 0 & u(9)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −59049q^{3} + −1188113947537070116 ∗ 11^{2} + O(11^{2}0)q^{5} + −300638051359036864468 + O(11^{2}0)q^{7} + 3486784401q^{9} + −4711345065774705928∗11+O(11^{2}0)q^{11}+135293671852073098∗11^{2}+O(11^{2}0)q^{13}+1903826471476452426∗ 11^{2} +O(11^{2}0)q^{15} +−1175244497119713399∗11^{2} +O(11^{2}0)q^{17} +116266404642431588474+O(11^{2}0)q^{19} + −150571580625712825056 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

9. Consider f = q + 3q3 + 4q5 + 8q7 + 9q9 + 3q11 + q13 + q15 + 7q17 + 10q19 + 2q21 + ⋯ ∈ S14(Γ0(12); \overline{F}11). Since f is ordinary, we do not continue our analysis.

10. Consider f = q + 5q3 + 6q5 + q7 + 3q9 + 7q11 + 6q13 + 8q15 + 6q17 + 7q19 + 5q21 + ⋯ ∈ S18(Γ0(12); \overline{F}11). Since f is ordinary, we do not continue our analysis.

11. Consider f = q + 6q3 + 7q5 + 4q7 + 3q9 + 2q11 + 9q13 + 9q15 + 6q17 + 2q19 + 2q21 + ⋯ ∈ S18(Γ0(12); \overline{F}11). Since f is ordinary, we do not continue our analysis.

12. Consider f = q + 7q3 + 7q7 + 5q9 + 9q11 + 9q19 + 5q21 + ⋯ ∈ S20(Γ0(12); \overline{F}11). Since f is ordinary, we do not continue our analysis.

13. Consider f = q + q3 + 2q5 + 9q7 + q9 + q11 + 9q13 + 2q15 + 4q17 + 5q19 + 9q21 + ⋯ ∈ S22(Γ0(12); \overline{F}11). Since f is ordinary, we do not continue our analysis.

14. Consider f = q + 10q3 + 2q5 + 2q7 + q9 + 10q11 + 6q13 + 9q15 + 7q17 + 9q19 + 9q21 + ⋯ ∈ S22(Γ0(12); \overline{F}11). Since f is ordinary, we do not continue our analysis.

11.11 N = 13, ℓ = 11 (83 forms)

1. Consider f = q + z^{3}45q^{2} + z^{8}73q^{3} + z^{8}85q^{4} + z^{4}13q^{5} + z^{1}218q^{6} + z^{1}009q^{7} + z^{9}00q^{8} + z^{5}33q^{9} + z^{7}58q^{10} + z^{4}28q^{12} + 2q^{13} + z^{2}4q^{14} + z^{1}286q^{15} + z^{2}68q^{16} + z^{4}21q^{17} + z^{8}78q^{18} + z^{7}77q^{19} + z^{1}298q^{20} + z^{5}52q^{21} + ⋯ ∈ S_{14}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{3} + 2x + 9 = 0. We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{7}9q^{2} + z^{7}5q^{3} + z^{3}53q^{4} + z^{6}79q^{5} + z^{1}54q^{6} + z^{4}77q^{7} + z^{1}02q^{8} + z^{2}67q^{9} + z^{7}58q^{10} + z^{3}04q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}026)ω^{1} & 0 \\ 0 & u(z^{3}04)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(16005551973115477565+ O(11^{1}9))∗a^{2}+(8243041758427855253+O(11^{1}9))∗a+14465006709606521∗11^{2}+O(11^{1}9)q^{3}+a^{2}−8192q^{4}+ −(9291558526035859107+O(11^{1}9))∗a^{2}−(18103262747053753326+O(11^{1}9))∗a+12047729521719131160+ O(11^{1}9)q^{5} + −(16112201963386724258 + O(11^{1}9)) ∗ a^{2} − (29143970486797715143 + O(11^{1}9)) ∗ a + 22730914427755667242+O(11^{1}9)q^{6}+(25443959180343833208+O(11^{1}9))∗a^{2}−(18930615978107892288+ O(11^{1}9))∗a+16123129440440084632+O(11^{1}9)q^{7}+(683299281562938609+O(11^{2}0))∗a^{2}+(107502242810120141802+ O(11^{2}0))∗a+201966393462920176474+O(11^{2}0)q^{8}+(21659806673762192790+O(11^{1}9))∗a^{2}+(23771163889408222101+ O(11^{1}9))∗a−18647078376837362026+O(11^{1}9)q^{9}+(21455651864168806428+O(11^{1}9))∗a^{2}−(28335397182186992009+ O(11^{1}9)) ∗ a − 10605109664198985297 + O(11^{1}9)q^{10} + (2175340566621712294 ∗ 11 + O(11^{1}9)) ∗ a^{2} + (201722484649890868∗11^{2}+O(11^{1}9))∗a+1420064505915144472∗11+O(11^{1}9)q^{11}+(2816608497017965885+ O(11^{1}9))∗a^{2}−(17112447624288523339+O(11^{1}9))∗a+27859442041690006778+O(11^{1}9)q^{12}+−4826809q^{13}+ (24220199749171452769+O(11^{1}9))∗a^{2}−(25785290802068765213+O(11^{1}9))∗a+28734640566894471713+ O(11^{1}9)q^{14}+(1655834563361292579+O(11^{1}9))∗a^{2}−(12436641021993104489+O(11^{1}9))∗a−30161350824026259327+ O(11^{1}9)q^{15} + (214186507981966739522 + O(11^{2}0)) ∗ a^{2} − (323849865490893615113 + O(11^{2}0)) ∗ a + 131999550178696667116+O(11^{2}0)q^{16}+(20274037688345734416+O(11^{1}9))∗a^{2}−(2374970993421464829∗ 11 + O(11^{1}9)) ∗ a + 16707157294076172168 + O(11^{1}9)q^{17} + −(5218876177048932641 + O(11^{1}9)) ∗ a^{2} + (1879662416312519028∗11+O(11^{1}9))∗a+5012582914944999991+O(11^{1}9)q^{18}+−(1255269117748431068∗ 11 + O(11^{2}0)) ∗ a^{2} + (230995534170507427749 + O(11^{2}0)) ∗ a + 142221369404082563368 + O(11^{2}0)q^{19} + −(4449672850706400853+O(11^{1}9))∗a^{2}−(15501463390119753492+O(11^{1}9))∗a−29766122654957670508+ O(11^{1}9)q^{20}+(18612607575652097868+O(11^{1}9))∗a^{2}−(7683860872507135854+O(11^{1}9))∗a−16066917021893636873+ O(11^{1}9)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + −683299281562938609 + O(11^{2}0)x^{2} + −107502242810120158186 + O(11^{2}0)x + −201966393462920176474 + O(11^{2}0). The slope of f_{0} is 1.

2. Consider f = q+3q^{2}+7q^{3}+q^{4}+4q^{5}+10q^{6}+q^{8}+q^{10}+7q^{12}+2q^{13}+6q^{15}+6q^{16}+2q^{17}+10q^{19}+4q^{20}+⋯ ∈ S_{14}(Γ_{0}(13); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 9q^{2} + 2q^{3} + 9q^{4} + 5q^{5} + 7q^{6} + 5q^{8} + q^{10} + 3q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{1} & 0 \\ 0 & u(3)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+304462516611205180781+ O(11^{2}0)q^{2}+23695381738779418664+O(11^{1}9)q^{3}+−256162498512043596802+O(11^{2}0)q^{4}+3406677579710665814+ O(11^{1}9)q^{5}+−1054325337858528737+O(11^{1}9)q^{6}+−1265540813087045007∗11+O(11^{1}9)q^{7}+33042100503307375899+ O(11^{2}0)q^{8}+87385427198050635∗11^{2}+O(11^{1}9)q^{9}+28065407621175705232+O(11^{1}9)q^{10}+1462014581555199542∗ 11 + O(11^{1}9)q^{11} + 7257639367369664412 + O(11^{1}9)q^{12} + −4826809q^{13} + 745061798067827230 ∗ 11 + O(11^{1}9)q^{14}+−12562532401668894518+O(11^{1}9)q^{15}+184477290541173625584+O(11^{2}0)q^{16}+−25517247239265291673+ O(11^{1}9)q^{17} + 179126587456794036 ∗ 11^{2} + O(11^{1}9)q^{18} + −156584102871477226089 + O(11^{2}0)q^{19} + 20486026408557701138 + O(11^{1}9)q^{20} + −720670146443391721 ∗ 11 + O(11^{1}9)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q+4∗z^{9}+10∗z^{8}+7∗z^{7}+7∗z^{6}+6∗z^{5}+10∗z^{4}+8∗z^{3}+8∗z^{2}+10∗z+7q^{2}+4∗z^{9}+4∗z^{8}+9∗ z^{7}+9∗z^{6}+z^{4}+2∗z^{3}+5∗z^{2}+2∗z+10q^{3}+10∗z^{8}+9∗z^{7}+z^{6}+3∗z^{5}+z^{4}+7∗z^{3}+7∗z^{2}+zq^{4}+10∗z^{9}+9∗z^{8}+ 7∗z^{7}+4∗z^{5}+6∗z^{4}+3∗z^{3}+10∗z^{2}+5∗z+10q^{5}+z^{9}+10∗z^{8}+9∗z^{7}+10∗z^{6}+9∗z^{5}+2∗z^{4}+4∗z^{3}+7∗z^{2}+8∗ z+7q^{6}+8∗z^{9}+5∗z^{8}+3∗z^{7}+7∗z^{6}+8∗z^{5}+z^{3}+8∗z^{2}+4∗z+10q^{7}+2∗z^{9}+9∗z^{8}+4∗z^{7}+z^{6}+10∗z^{5}+z^{4}+ 3∗z^{3}+3∗z^{2}+5∗z+9q^{8}+10∗z^{9}+4∗z^{8}+6∗z^{7}+2∗z^{6}+4∗z^{5}+7∗z^{4}+8∗z^{3}+7∗z^{2}+6∗z+6q^{9}+9∗z^{9}+8∗z^{8}+ 10∗z^{7}+10∗z^{6}+6∗z^{5}+7∗z^{4}+z^{3}+7∗z^{2}+2∗zq^{10}+8∗z^{9}+5∗z^{8}+2∗z^{7}+9∗z^{6}+8∗z^{5}+6∗z^{4}+z^{3}+9∗z^{2}+7∗ z+3q^{12}+9q^{13}+5∗z^{9}+2∗z^{8}+5∗z^{6}+4∗z^{5}+2∗z^{4}+8∗z^{3}+10∗z^{2}+6∗zq^{14}+6∗z^{9}+10∗z^{8}+4∗z^{7}+7∗z^{6}+9∗ z^{5}+5∗z^{4}+6∗z^{3}+9∗z^{2}+5∗z+2q^{15}+3∗z^{8}+8∗z^{7}+8∗z^{6}+9∗z^{5}+2∗z^{4}+4∗z^{3}+3∗z^{2}+3∗z+4q^{16}+8∗z^{9}+ 6∗z^{8}+8∗z^{7}+8∗z^{6}+8∗z^{5}+4∗z^{4}+6∗z^{2}+7∗z+2q^{17}+2∗z^{9}+10∗z^{8}+5∗z^{7}+2∗z^{6}+7∗z^{5}+6∗z^{4}+7∗z^{3}+8∗ z^{2}+7∗z+7q^{18}+7∗z^{9}+5∗z^{8}+z^{7}+3∗z^{6}+z^{5}+7∗z^{4}+2∗z^{3}+6∗z+3q^{19}+z^{9}+10∗z^{8}+8∗z^{7}+z^{6}+9∗z^{5}+8∗ z^{4}+4∗z^{3}+8∗z^{2}q^{20}+5∗z^{9}+4∗z^{8}+5∗z^{7}+6∗z^{6}+4∗z^{5}+7∗z^{4}+6∗z^{3}+5∗z+5q^{21}+⋯ ∈ S_{14}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{10} +7x^{5} +8x^{4} +10x^{3} +6x^{2} +6x+2 = 0. We have ρ_{f} ⊗χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q +z^{1}42133q^{2} +z^{4}0440q^{3} +z^{1}41494q^{4} +z^{1}03104q^{5} +z^{2}1523q^{6} +z^{1}53127q^{7} + z^{9}5697q^{8} + z^{9}9297q^{9} + z^{8}4187q^{10} + z^{6}6312q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{9}4738)ω^{1} & 0 \\ 0 & u(z^{6}6312)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(21387163858956230833+ O(11^{2}0)) ∗ a^{4} − (59404096813310740182 + O(11^{2}0)) ∗ a^{3} + (285434393449487570636 + O(11^{2}0)) ∗ a^{2} + (258382076893934702347+O(11^{2}0))∗a+325427090450479554154+O(11^{2}0)q^{3}+a^{2}−8192q^{4}+(149048911017397375406+ O(11^{2}0)) ∗ a^{4} − (212751438041340910991 + O(11^{2}0)) ∗ a^{3} + (169062434483015614325 + O(11^{2}0)) ∗ a^{2} − (93869867883927977009+O(11^{2}0))∗a+252179276043442089627+O(11^{2}0)q^{5}+(245758664138838808658+ O(11^{2}0)) ∗ a^{4} + (127053753102524643367 + O(11^{2}0)) ∗ a^{3} + (121601149095455710971 + O(11^{2}0)) ∗ a^{2} + (138425283053563602671+O(11^{2}0))∗a+212168488396872087078+O(11^{2}0)q^{6}+(216634659326732592466+ O(11^{2}0)) ∗ a^{4} + (17173076027588612481 ∗ 11 + O(11^{2}0)) ∗ a^{3} − (79383888001421231121 + O(11^{2}0)) ∗ a^{2} −(168778358570009154338+O(11^{2}0))∗a−274123072264624336910+O(11^{2}0)q^{7} +a^{3} −16384∗aq^{8} + −(292340748728231565075+O(11^{2}0))∗a^{4}−(311822893053470544917+O(11^{2}0))∗a^{3}−(157885957550417846730+ O(11^{2}0))∗a^{2}+(228415951778337570761+O(11^{2}0))∗a−49953877595870578364+O(11^{2}0)q^{9}+−(19312600920288775336+ O(11^{2}0)) ∗ a^{4} − (131022188173597099255 + O(11^{2}0)) ∗ a^{3} + (273898172937446128592 + O(11^{2}0)) ∗ a^{2} − (58808135793201766318+O(11^{2}0))∗a+47512973853813980874+O(11^{2}0)q^{10}+(12633064790096194464∗ 11 + O(11^{2}0)) ∗ a^{4} − (25533115118981745577 ∗ 11 + O(11^{2}0)) ∗ a^{3} + (28779378768876882704 ∗ 11 + O(11^{2}0))∗a^{2} −(20699063029377833440∗11+O(11^{2}0))∗a−24144466969759816095∗11+O(11^{2}0)q^{11} + −(63849125614334263838+O(11^{2}0))∗a^{4}+(226173802006618836353+O(11^{2}0))∗a^{3}−(38968999222507754879+ O(11^{2}0))∗a^{2}−(333800080229330717099+O(11^{2}0))∗a−1967432049382362515+O(11^{2}0)q^{12}+4826809q^{13}+ −(293667292172346286834+O(11^{2}0))∗a^{4}+(25674949162002874488+O(11^{2}0))∗a^{3}−(48914422817466958908+ O(11^{2}0)) ∗ a^{2} − (187333631007220538067 + O(11^{2}0)) ∗ a + 164801161638778764219 + O(11^{2}0)q^{14} + (180266126201550111797+O(11^{2}0))∗a^{4}−(14235042070596246561+O(11^{2}0))∗a^{3}−(273381884940763051701+ O(11^{2}0)) ∗ a^{2} + (175548833054252498936 + O(11^{2}0)) ∗ a − 117119760516970272401 + O(11^{2}0)q^{15} + a^{4} − 24576 ∗ a^{2} + 67108864q^{16} + −(263085504903913715282 + O(11^{2}0)) ∗ a^{4} + (86811378675964927121 + O(11^{2}0)) ∗ a^{3} + (59772151561100280682 + O(11^{2}0)) ∗ a^{2} − (278490724226963341212 + O(11^{2}0)) ∗ a + 89678309788211630767+O(11^{2}0)q^{17}+(292960356135929954295+O(11^{2}0))∗a^{4}+(126788244543683647480+ O(11^{2}0)) ∗ a^{3} + (269630641206179273049 + O(11^{2}0)) ∗ a^{2} + (63933904165625109292 + O(11^{2}0)) ∗ a − 40656570550301511864+O(11^{2}0)q^{18}+−(129342595797067967202+O(11^{2}0))∗a^{4}+(92833397036767088011+ O(11^{2}0)) ∗ a^{3} − (153708565622632542186 + O(11^{2}0)) ∗ a^{2} − (131759468696263651596 + O(11^{2}0)) ∗ a + 208519509260609272666+O(11^{2}0)q^{19}+(155132964924173791298+O(11^{2}0))∗a^{4}+(164927169563368200405+ O(11^{2}0)) ∗ a^{3} − (207290028010266993472 + O(11^{2}0)) ∗ a^{2} + (88270279220978549808 + O(11^{2}0)) ∗ a + 19657352748313031426+O(11^{2}0)q^{20}+−(95906054232732768975+O(11^{2}0))∗a^{4}+(66408531782613740267+ O(11^{2}0)) ∗ a^{3} + (143191402289721066832 + O(11^{2}0)) ∗ a^{2} + (141901363757969604243 + O(11^{2}0)) ∗ a − 102563181908536027402+O(11^{2}0)q^{21} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{5}+−18405356653286492917+O(11^{2}0)x^{4}+−45435746779326533209+ O(11^{2}0)x^{3}+197200980531799123243+O(11^{2}0)x^{2}+−64509515765869694494+O(11^{2}0)x+−76188913029531034323+ O(11^{2}0). The slope of f_{0} is 1.

4. Consider f = q+10∗z^{8}+5∗z^{7}+5∗z^{6}+7∗z^{5}+8∗z^{4}+4∗z^{3}+7∗z^{2}+z+6q^{2}+8∗z^{8}+4∗z^{7}+6∗z^{6}+8∗z^{5}+ 5∗z^{4}+8∗z^{3}+z^{2}+9∗z+6q^{3}+9∗z^{9}+9∗z^{8}+10∗z^{7}+4∗z^{6}+z^{5}+10∗z^{4}+z^{3}+4∗z^{2}+2∗z+1q^{4}+7∗z^{9}+4∗ z^{8}+3∗z^{6}+3∗z^{5}+9∗z^{4}+7∗z^{3}+7∗z^{2}+7∗z+1q^{5}+5∗z^{9}+2∗z^{8}+4∗z^{7}+z^{5}+10∗z^{4}+2∗z^{3}+9∗z^{2}+3∗z+ 9q^{6}+9∗z^{9}+3∗z^{8}+8∗z^{7}+7∗z^{6}+2∗z^{5}+9∗z^{4}+9∗z^{3}+9∗z^{2}+7∗zq^{7}+8∗z^{9}+8∗z^{8}+2∗z^{7}+7∗z^{6}+10∗z^{5}+ 6∗z^{4}+2∗z^{3}+5∗z^{2}+3q^{8}+9∗z^{9}+8∗z^{8}+8∗z^{7}+10∗z^{6}+10∗z^{5}+2∗z^{4}+9∗z^{3}+5∗z^{2}+7∗z+3q^{9}+6∗z^{9}+3∗ z^{8}+5∗z^{7}+8∗z^{6}+6∗z^{5}+2∗z^{4}+7∗z^{3}+z^{2}+6∗z+1q^{10}+2∗z^{9}+5∗z^{8}+7∗z^{7}+7∗z^{6}+9∗z^{5}+8∗z^{4}+5∗z^{3}+ 4∗z+5q^{12}+9q^{13}+9∗z^{9}+5∗z^{8}+10∗z^{7}+6∗z^{6}+10∗z^{5}+9∗z^{4}+z^{3}+5∗z^{2}+2∗z+2q^{14}+2∗z^{9}+8∗z^{8}+10∗ z^{7}+9∗z^{6}+z^{5}+5∗z^{4}+8∗z^{3}+3∗z^{2}+7∗z+1q^{15}+2∗z^{9}+z^{8}+9∗z^{7}+8∗z^{6}+4∗z^{5}+5∗z^{4}+10∗z^{3}+2∗z^{2}+7∗ z+9q^{16}+7∗z^{9}+8∗z^{8}+9∗z^{7}+10∗z^{6}+2∗z^{4}+8∗z^{3}+2∗z^{2}+6∗z+8q^{17}+3∗z^{9}+z^{8}+2∗z^{7}+3∗z^{6}+5∗z^{5}+6∗ z^{4}+8∗z^{3}+3∗z^{2}+6∗z+1q^{18}+2∗z^{7}+5∗z^{6}+2∗z^{5}+7∗z^{4}+4∗z^{3}+7∗z^{2}+10∗z+2q^{19}+9∗z^{9}+4∗z^{8}+3∗z^{7}+ 8∗z^{5}+9∗z^{4}+9∗z^{3}+3q^{20}+3∗z^{9}+3∗z^{8}+7∗z^{7}+z^{6}+8∗z^{5}+z^{4}+z^{3}+z^{2}+7∗zq^{21}+⋯ ∈ S_{14}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{10} + 7x^{5} + 8x^{4} + 10x^{3} + 6x^{2} + 6x + 2 = 0. We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q +z^{1}06323q^{2} +z^{3}4940q^{3} +z^{6}1064q^{4} +z^{1}6824q^{5} +z^{1}41263q^{6} +z^{8}3787q^{7} + z^{1}43207q^{8} + z^{1}03307q^{9} + z^{1}23147q^{10} + z^{5}872q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}55178)ω^{1} & 0 \\ 0 & u(z^{5}872)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(21387163858956230833+ O(11^{2}0)) ∗ a^{4} − (59404096813310740182 + O(11^{2}0)) ∗ a^{3} + (285434393449487570636 + O(11^{2}0)) ∗ a^{2} + (258382076893934702347+O(11^{2}0))∗a+325427090450479554154+O(11^{2}0)q^{3}+a^{2}−8192q^{4}+(149048911017397375406+ O(11^{2}0)) ∗ a^{4} − (212751438041340910991 + O(11^{2}0)) ∗ a^{3} + (169062434483015614325 + O(11^{2}0)) ∗ a^{2} − (93869867883927977009+O(11^{2}0))∗a+252179276043442089627+O(11^{2}0)q^{5}+(245758664138838808658+ O(11^{2}0)) ∗ a^{4} + (127053753102524643367 + O(11^{2}0)) ∗ a^{3} + (121601149095455710971 + O(11^{2}0)) ∗ a^{2} + (138425283053563602671+O(11^{2}0))∗a+212168488396872087078+O(11^{2}0)q^{6}+(216634659326732592466+ O(11^{2}0)) ∗ a^{4} + (17173076027588612481 ∗ 11 + O(11^{2}0)) ∗ a^{3} − (79383888001421231121 + O(11^{2}0)) ∗ a^{2} −(168778358570009154338+O(11^{2}0))∗a−274123072264624336910+O(11^{2}0)q^{7} +a^{3} −16384∗aq^{8} + −(292340748728231565075+O(11^{2}0))∗a^{4}−(311822893053470544917+O(11^{2}0))∗a^{3}−(157885957550417846730+ O(11^{2}0))∗a^{2}+(228415951778337570761+O(11^{2}0))∗a−49953877595870578364+O(11^{2}0)q^{9}+−(19312600920288775336+ O(11^{2}0)) ∗ a^{4} − (131022188173597099255 + O(11^{2}0)) ∗ a^{3} + (273898172937446128592 + O(11^{2}0)) ∗ a^{2} − (58808135793201766318+O(11^{2}0))∗a+47512973853813980874+O(11^{2}0)q^{10}+(12633064790096194464∗ 11 + O(11^{2}0)) ∗ a^{4} − (25533115118981745577 ∗ 11 + O(11^{2}0)) ∗ a^{3} + (28779378768876882704 ∗ 11 + O(11^{2}0))∗a^{2} −(20699063029377833440∗11+O(11^{2}0))∗a−24144466969759816095∗11+O(11^{2}0)q^{11} + −(63849125614334263838+O(11^{2}0))∗a^{4}+(226173802006618836353+O(11^{2}0))∗a^{3}−(38968999222507754879+ O(11^{2}0))∗a^{2}−(333800080229330717099+O(11^{2}0))∗a−1967432049382362515+O(11^{2}0)q^{12}+4826809q^{13}+ −(293667292172346286834+O(11^{2}0))∗a^{4}+(25674949162002874488+O(11^{2}0))∗a^{3}−(48914422817466958908+ O(11^{2}0)) ∗ a^{2} − (187333631007220538067 + O(11^{2}0)) ∗ a + 164801161638778764219 + O(11^{2}0)q^{14} + (180266126201550111797+O(11^{2}0))∗a^{4}−(14235042070596246561+O(11^{2}0))∗a^{3}−(273381884940763051701+ O(11^{2}0)) ∗ a^{2} + (175548833054252498936 + O(11^{2}0)) ∗ a − 117119760516970272401 + O(11^{2}0)q^{15} + a^{4} − 24576 ∗ a^{2} + 67108864q^{16} + −(263085504903913715282 + O(11^{2}0)) ∗ a^{4} + (86811378675964927121 + O(11^{2}0)) ∗ a^{3} + (59772151561100280682 + O(11^{2}0)) ∗ a^{2} − (278490724226963341212 + O(11^{2}0)) ∗ a + 89678309788211630767+O(11^{2}0)q^{17}+(292960356135929954295+O(11^{2}0))∗a^{4}+(126788244543683647480+ O(11^{2}0)) ∗ a^{3} + (269630641206179273049 + O(11^{2}0)) ∗ a^{2} + (63933904165625109292 + O(11^{2}0)) ∗ a − 40656570550301511864+O(11^{2}0)q^{18}+−(129342595797067967202+O(11^{2}0))∗a^{4}+(92833397036767088011+ O(11^{2}0)) ∗ a^{3} − (153708565622632542186 + O(11^{2}0)) ∗ a^{2} − (131759468696263651596 + O(11^{2}0)) ∗ a + 208519509260609272666+O(11^{2}0)q^{19}+(155132964924173791298+O(11^{2}0))∗a^{4}+(164927169563368200405+ O(11^{2}0)) ∗ a^{3} − (207290028010266993472 + O(11^{2}0)) ∗ a^{2} + (88270279220978549808 + O(11^{2}0)) ∗ a + 19657352748313031426+O(11^{2}0)q^{20}+−(95906054232732768975+O(11^{2}0))∗a^{4}+(66408531782613740267+ O(11^{2}0)) ∗ a^{3} + (143191402289721066832 + O(11^{2}0)) ∗ a^{2} + (141901363757969604243 + O(11^{2}0)) ∗ a − 102563181908536027402+O(11^{2}0)q^{21} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{5}+−18405356653286492917+O(11^{2}0)x^{4}+−45435746779326533209+ O(11^{2}0)x^{3}+197200980531799123243+O(11^{2}0)x^{2}+−64509515765869694494+O(11^{2}0)x+−76188913029531034323+ O(11^{2}0). The slope of f_{0} is 1.

5. Consider f = q+9∗z^{9}+7∗z^{8}+10∗z^{7}+10∗z^{6}+6∗z^{5}+6∗z^{4}+2∗z^{3}+4∗z^{2}+6∗zq^{2}+9∗z^{9}+2∗z^{8}+7∗z^{7}+ 6∗z^{6}+5∗z^{5}+4∗z^{4}+5∗z^{3}+4∗z^{2}+5∗zq^{3}+9∗z^{9}+7∗z^{8}+3∗z^{7}+5∗z^{6}+3∗z^{5}+3∗z^{4}+7∗z^{3}+3∗z^{2}+3∗z+ 7q^{4}+2∗z^{9}+4∗z^{8}+5∗z^{7}+5∗z^{6}+4∗z^{5}+10∗z^{4}+7∗z^{3}+4∗z^{2}+7∗z+10q^{5}+10∗z^{9}+3∗z^{8}+4∗z^{7}+6∗z^{6}+ 4∗z^{5}+7∗z^{4}+9∗z^{3}+6∗z^{2}+5∗z+2q^{6}+5∗z^{9}+9∗z^{8}+9∗z^{7}+4∗z^{6}+9∗z^{5}+8∗z^{4}+6∗z^{2}+2∗z+10q^{7}+7∗ z^{9}+7∗z^{7}+z^{6}+2∗z^{5}+3∗z^{4}+9∗z^{3}+5∗z+10q^{8}+4∗z^{9}+z^{8}+z^{7}+3∗z^{6}+2∗z^{4}+5∗z^{3}+4∗z^{2}+5∗z+3q^{9}+ 7∗z^{9}+z^{8}+5∗z^{7}+10∗z^{6}+8∗z^{5}+7∗z^{4}+9∗z^{3}+10∗z^{2}+9q^{10}+9∗z^{9}+5∗z^{8}+5∗z^{7}+5∗z^{6}+10∗z^{5}+6∗z^{4}+ z^{3}+10∗z^{2}+6∗z+4q^{12}+9q^{13}+z^{9}+9∗z^{8}+5∗z^{7}+8∗z^{6}+8∗z^{5}+10∗z^{4}+3∗z^{3}+9∗z^{2}+5∗z+5q^{14}+10∗z^{9}+ 2∗z^{8}+2∗z^{6}+5∗z^{5}+z^{4}+z^{3}+6∗z^{2}+7∗z+5q^{15}+2∗z^{9}+z^{8}+10∗z^{7}+9∗z^{6}+z^{5}+9∗z^{4}+4∗z^{3}+4∗z^{2}+2∗z+ 7q^{16}+3∗z^{9}+9∗z^{8}+7∗z^{7}+4∗z^{6}+5∗z^{5}+6∗z^{4}+3∗zq^{17}+2∗z^{9}+8∗z^{6}+9∗z^{4}+3∗z^{3}+z^{2}+6∗z+2q^{18}+2∗ z^{9}+3∗z^{8}+6∗z^{6}+2∗z^{5}+9∗z^{4}+4∗z^{3}+8∗z^{2}+10∗z+7q^{19}+3∗z^{9}+10∗z^{8}+7∗z^{6}+5∗z^{5}+5∗z^{4}+10∗z^{3}+ 10∗z^{2} +9∗z +7q^{20} +6∗z^{9} +3∗z^{7} +2∗z^{6} +5∗z^{5} +7∗z^{4} +8∗z^{3} +6∗z^{2} +5∗zq^{21} +⋯ ∈ S_{14}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{10} + 7x^{5} + 8x^{4} + 10x^{3} + 6x^{2} + 6x + 2 = 0. We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q +z^{4}2203q^{2} +z^{6}2240q^{3} +z^{2}7504q^{4} +z^{2}4014q^{5} +z^{1}04443q^{6} +z^{1}16407q^{7} + z^{1}25827q^{8} + z^{9}027q^{9} + z^{6}6217q^{10} + z^{6}4592q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{9}6458)ω^{1} & 0 \\ 0 & u(z^{6}4592)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(21387163858956230833+ O(11^{2}0)) ∗ a^{4} − (59404096813310740182 + O(11^{2}0)) ∗ a^{3} + (285434393449487570636 + O(11^{2}0)) ∗ a^{2} + (258382076893934702347+O(11^{2}0))∗a+325427090450479554154+O(11^{2}0)q^{3}+a^{2}−8192q^{4}+(149048911017397375406+ O(11^{2}0)) ∗ a^{4} − (212751438041340910991 + O(11^{2}0)) ∗ a^{3} + (169062434483015614325 + O(11^{2}0)) ∗ a^{2} − (93869867883927977009+O(11^{2}0))∗a+252179276043442089627+O(11^{2}0)q^{5}+(245758664138838808658+ O(11^{2}0)) ∗ a^{4} + (127053753102524643367 + O(11^{2}0)) ∗ a^{3} + (121601149095455710971 + O(11^{2}0)) ∗ a^{2} + (138425283053563602671+O(11^{2}0))∗a+212168488396872087078+O(11^{2}0)q^{6}+(216634659326732592466+ O(11^{2}0)) ∗ a^{4} + (17173076027588612481 ∗ 11 + O(11^{2}0)) ∗ a^{3} − (79383888001421231121 + O(11^{2}0)) ∗ a^{2} −(168778358570009154338+O(11^{2}0))∗a−274123072264624336910+O(11^{2}0)q^{7} +a^{3} −16384∗aq^{8} + −(292340748728231565075+O(11^{2}0))∗a^{4}−(311822893053470544917+O(11^{2}0))∗a^{3}−(157885957550417846730+ O(11^{2}0))∗a^{2}+(228415951778337570761+O(11^{2}0))∗a−49953877595870578364+O(11^{2}0)q^{9}+−(19312600920288775336+ O(11^{2}0)) ∗ a^{4} − (131022188173597099255 + O(11^{2}0)) ∗ a^{3} + (273898172937446128592 + O(11^{2}0)) ∗ a^{2} − (58808135793201766318+O(11^{2}0))∗a+47512973853813980874+O(11^{2}0)q^{10}+(12633064790096194464∗ 11 + O(11^{2}0)) ∗ a^{4} − (25533115118981745577 ∗ 11 + O(11^{2}0)) ∗ a^{3} + (28779378768876882704 ∗ 11 + O(11^{2}0))∗a^{2} −(20699063029377833440∗11+O(11^{2}0))∗a−24144466969759816095∗11+O(11^{2}0)q^{11} + −(63849125614334263838+O(11^{2}0))∗a^{4}+(226173802006618836353+O(11^{2}0))∗a^{3}−(38968999222507754879+ O(11^{2}0))∗a^{2}−(333800080229330717099+O(11^{2}0))∗a−1967432049382362515+O(11^{2}0)q^{12}+4826809q^{13}+ −(293667292172346286834+O(11^{2}0))∗a^{4}+(25674949162002874488+O(11^{2}0))∗a^{3}−(48914422817466958908+ O(11^{2}0)) ∗ a^{2} − (187333631007220538067 + O(11^{2}0)) ∗ a + 164801161638778764219 + O(11^{2}0)q^{14} + (180266126201550111797+O(11^{2}0))∗a^{4}−(14235042070596246561+O(11^{2}0))∗a^{3}−(273381884940763051701+ O(11^{2}0)) ∗ a^{2} + (175548833054252498936 + O(11^{2}0)) ∗ a − 117119760516970272401 + O(11^{2}0)q^{15} + a^{4} − 24576 ∗ a^{2} + 67108864q^{16} + −(263085504903913715282 + O(11^{2}0)) ∗ a^{4} + (86811378675964927121 + O(11^{2}0)) ∗ a^{3} + (59772151561100280682 + O(11^{2}0)) ∗ a^{2} − (278490724226963341212 + O(11^{2}0)) ∗ a + 89678309788211630767+O(11^{2}0)q^{17}+(292960356135929954295+O(11^{2}0))∗a^{4}+(126788244543683647480+ O(11^{2}0)) ∗ a^{3} + (269630641206179273049 + O(11^{2}0)) ∗ a^{2} + (63933904165625109292 + O(11^{2}0)) ∗ a − 40656570550301511864+O(11^{2}0)q^{18}+−(129342595797067967202+O(11^{2}0))∗a^{4}+(92833397036767088011+ O(11^{2}0)) ∗ a^{3} − (153708565622632542186 + O(11^{2}0)) ∗ a^{2} − (131759468696263651596 + O(11^{2}0)) ∗ a + 208519509260609272666+O(11^{2}0)q^{19}+(155132964924173791298+O(11^{2}0))∗a^{4}+(164927169563368200405+ O(11^{2}0)) ∗ a^{3} − (207290028010266993472 + O(11^{2}0)) ∗ a^{2} + (88270279220978549808 + O(11^{2}0)) ∗ a + 19657352748313031426+O(11^{2}0)q^{20}+−(95906054232732768975+O(11^{2}0))∗a^{4}+(66408531782613740267+ O(11^{2}0)) ∗ a^{3} + (143191402289721066832 + O(11^{2}0)) ∗ a^{2} + (141901363757969604243 + O(11^{2}0)) ∗ a − 102563181908536027402+O(11^{2}0)q^{21} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{5}+−18405356653286492917+O(11^{2}0)x^{4}+−45435746779326533209+ O(11^{2}0)x^{3}+197200980531799123243+O(11^{2}0)x^{2}+−64509515765869694494+O(11^{2}0)x+−76188913029531034323+ O(11^{2}0). The slope of f_{0} is 1.

6. Consider f = q+10∗z^{9}+10∗z^{8}+9∗z^{7}+5∗z^{6}+6∗z^{5}+3∗z^{4}+6∗z^{3}+9∗z^{2}+2q^{2}+10∗z^{9}+4∗z^{8}+5∗z^{7}+6∗ z^{6}+2∗z^{5}+5∗z^{4}+4∗z^{3}+10∗z^{2}+4∗z+7q^{3}+8∗z^{9}+4∗z^{8}+5∗z^{7}+9∗z^{6}+2∗z^{5}+7∗z^{4}+z^{3}+9∗z^{2}+9∗z+ 7q^{4}+8∗z^{9}+z^{8}+7∗z^{7}+7∗z^{5}+z^{4}+3∗z^{3}+4∗z^{2}+1q^{5}+10∗z^{9}+8∗z^{8}+10∗z^{7}+7∗z^{6}+2∗z^{5}+z^{4}+6∗z^{2}+ 6∗z+2q^{6}+6∗z^{9}+8∗z^{8}+8∗z^{7}+2∗z^{6}+6∗z^{5}+2∗z^{3}+2∗z^{2}+10∗zq^{7}+6∗z^{9}+4∗z^{8}+5∗z^{7}+7∗z^{6}+7∗z^{5}+ 7∗z^{4}+8∗z^{3}+3∗z^{2}+10∗zq^{8}+z^{8}+8∗z^{7}+8∗z^{6}+9∗z^{5}+6∗z^{3}+8∗z^{2}+10q^{9}+4∗z^{9}+8∗z^{8}+4∗z^{7}+2∗z^{6}+ 6∗z^{5}+4∗z^{4}+10∗z^{3}+5∗z^{2}+5∗z+7q^{10}+z^{9}+2∗z^{8}+8∗z^{7}+z^{6}+9∗z^{5}+z^{3}+6∗z^{2}+10∗z+3q^{12}+9q^{13}+4∗ z^{9}+3∗z^{8}+5∗z^{7}+2∗z^{6}+4∗z^{5}+7∗z^{4}+2∗z^{2}+6∗z+9q^{14}+7∗z^{9}+4∗z^{8}+9∗z^{7}+10∗z^{6}+9∗z^{4}+z^{3}+z^{2}+ 2q^{15}+3∗z^{9}+3∗z^{8}+5∗z^{7}+6∗z^{6}+7∗z^{5}+9∗z^{4}+10∗z^{3}+4∗z^{2}+5∗z+9q^{16}+3∗z^{9}+6∗z^{8}+10∗z^{7}+3∗z^{6}+ 9∗z^{4}+4∗z^{3}+4∗z^{2}+10∗z+3q^{17}+4∗z^{9}+5∗z^{8}+8∗z^{6}+8∗z^{5}+6∗z^{3}+9∗z^{2}+z+1q^{18}+z^{9}+z^{8}+8∗z^{7}+7∗ z^{6}+3∗z^{5}+8∗z^{4}+z^{3}+9∗z^{2}+10∗z+9q^{19}+5∗z^{9}+2∗z^{8}+10∗z^{7}+6∗z^{6}+5∗z^{5}+z^{4}+10∗z^{3}+10∗z^{2}+6∗ z+5q^{20}+6∗z^{9}+2∗z^{8}+9∗z^{7}+10∗z^{6}+7∗z^{5}+3∗z^{4}+10∗z^{3}+4∗z^{2}+6∗z+9q^{21}+⋯ ∈ S_{14}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{10} + 7x^{5} + 8x^{4} + 10x^{3} + 6x^{2} + 6x + 2 = 0. We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{1}26793q^{2} + z^{6}1740q^{3} + z^{4}9474q^{4} + z^{7}4734q^{5} + z^{2}7483q^{6} + z^{7}617q^{7} + z^{1}44787q^{8} + z^{9}7237q^{9} + z^{4}0477q^{10} + z^{1}32302q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}8748)ω^{1} & 0 \\ 0 & u(z^{1}32302)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(21387163858956230833+ O(11^{2}0)) ∗ a^{4} − (59404096813310740182 + O(11^{2}0)) ∗ a^{3} + (285434393449487570636 + O(11^{2}0)) ∗ a^{2} + (258382076893934702347+O(11^{2}0))∗a+325427090450479554154+O(11^{2}0)q^{3}+a^{2}−8192q^{4}+(149048911017397375406+ O(11^{2}0)) ∗ a^{4} − (212751438041340910991 + O(11^{2}0)) ∗ a^{3} + (169062434483015614325 + O(11^{2}0)) ∗ a^{2} − (93869867883927977009+O(11^{2}0))∗a+252179276043442089627+O(11^{2}0)q^{5}+(245758664138838808658+ O(11^{2}0)) ∗ a^{4} + (127053753102524643367 + O(11^{2}0)) ∗ a^{3} + (121601149095455710971 + O(11^{2}0)) ∗ a^{2} + (138425283053563602671+O(11^{2}0))∗a+212168488396872087078+O(11^{2}0)q^{6}+(216634659326732592466+ O(11^{2}0)) ∗ a^{4} + (17173076027588612481 ∗ 11 + O(11^{2}0)) ∗ a^{3} − (79383888001421231121 + O(11^{2}0)) ∗ a^{2} −(168778358570009154338+O(11^{2}0))∗a−274123072264624336910+O(11^{2}0)q^{7} +a^{3} −16384∗aq^{8} + −(292340748728231565075+O(11^{2}0))∗a^{4}−(311822893053470544917+O(11^{2}0))∗a^{3}−(157885957550417846730+ O(11^{2}0))∗a^{2}+(228415951778337570761+O(11^{2}0))∗a−49953877595870578364+O(11^{2}0)q^{9}+−(19312600920288775336+ O(11^{2}0)) ∗ a^{4} − (131022188173597099255 + O(11^{2}0)) ∗ a^{3} + (273898172937446128592 + O(11^{2}0)) ∗ a^{2} − (58808135793201766318+O(11^{2}0))∗a+47512973853813980874+O(11^{2}0)q^{10}+(12633064790096194464∗ 11 + O(11^{2}0)) ∗ a^{4} − (25533115118981745577 ∗ 11 + O(11^{2}0)) ∗ a^{3} + (28779378768876882704 ∗ 11 + O(11^{2}0))∗a^{2} −(20699063029377833440∗11+O(11^{2}0))∗a−24144466969759816095∗11+O(11^{2}0)q^{11} + −(63849125614334263838+O(11^{2}0))∗a^{4}+(226173802006618836353+O(11^{2}0))∗a^{3}−(38968999222507754879+ O(11^{2}0))∗a^{2}−(333800080229330717099+O(11^{2}0))∗a−1967432049382362515+O(11^{2}0)q^{12}+4826809q^{13}+ −(293667292172346286834+O(11^{2}0))∗a^{4}+(25674949162002874488+O(11^{2}0))∗a^{3}−(48914422817466958908+ O(11^{2}0)) ∗ a^{2} − (187333631007220538067 + O(11^{2}0)) ∗ a + 164801161638778764219 + O(11^{2}0)q^{14} + (180266126201550111797+O(11^{2}0))∗a^{4}−(14235042070596246561+O(11^{2}0))∗a^{3}−(273381884940763051701+ O(11^{2}0)) ∗ a^{2} + (175548833054252498936 + O(11^{2}0)) ∗ a − 117119760516970272401 + O(11^{2}0)q^{15} + a^{4} − 24576 ∗ a^{2} + 67108864q^{16} + −(263085504903913715282 + O(11^{2}0)) ∗ a^{4} + (86811378675964927121 + O(11^{2}0)) ∗ a^{3} + (59772151561100280682 + O(11^{2}0)) ∗ a^{2} − (278490724226963341212 + O(11^{2}0)) ∗ a + 89678309788211630767+O(11^{2}0)q^{17}+(292960356135929954295+O(11^{2}0))∗a^{4}+(126788244543683647480+ O(11^{2}0)) ∗ a^{3} + (269630641206179273049 + O(11^{2}0)) ∗ a^{2} + (63933904165625109292 + O(11^{2}0)) ∗ a − 40656570550301511864+O(11^{2}0)q^{18}+−(129342595797067967202+O(11^{2}0))∗a^{4}+(92833397036767088011+ O(11^{2}0)) ∗ a^{3} − (153708565622632542186 + O(11^{2}0)) ∗ a^{2} − (131759468696263651596 + O(11^{2}0)) ∗ a + 208519509260609272666+O(11^{2}0)q^{19}+(155132964924173791298+O(11^{2}0))∗a^{4}+(164927169563368200405+ O(11^{2}0)) ∗ a^{3} − (207290028010266993472 + O(11^{2}0)) ∗ a^{2} + (88270279220978549808 + O(11^{2}0)) ∗ a + 19657352748313031426+O(11^{2}0)q^{20}+−(95906054232732768975+O(11^{2}0))∗a^{4}+(66408531782613740267+ O(11^{2}0)) ∗ a^{3} + (143191402289721066832 + O(11^{2}0)) ∗ a^{2} + (141901363757969604243 + O(11^{2}0)) ∗ a − 102563181908536027402+O(11^{2}0)q^{21} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{5}+−18405356653286492917+O(11^{2}0)x^{4}+−45435746779326533209+ O(11^{2}0)x^{3}+197200980531799123243+O(11^{2}0)x^{2}+−64509515765869694494+O(11^{2}0)x+−76188913029531034323+ O(11^{2}0). The slope of f_{0} is 1.

7. Consider f = q+10∗z^{9}+7∗z^{8}+2∗z^{7}+6∗z^{6}+8∗z^{5}+6∗z^{4}+2∗z^{3}+5∗z^{2}+5∗z+1q^{2}+10∗z^{9}+4∗z^{8}+8∗z^{7}+ 6∗z^{6}+7∗z^{5}+7∗z^{4}+3∗z^{3}+2∗z^{2}+2∗z+7q^{3}+7∗z^{9}+3∗z^{8}+6∗z^{7}+3∗z^{6}+2∗z^{5}+z^{4}+6∗z^{3}+10∗z^{2}+7∗z+ 8q^{4}+6∗z^{9}+4∗z^{8}+3∗z^{7}+3∗z^{6}+4∗z^{5}+7∗z^{4}+2∗z^{3}+8∗z^{2}+3∗z+8q^{5}+7∗z^{9}+10∗z^{8}+6∗z^{7}+10∗z^{6}+6∗ z^{5}+2∗z^{4}+7∗z^{3}+5∗z^{2}+9q^{6}+5∗z^{9}+8∗z^{8}+5∗z^{7}+2∗z^{6}+8∗z^{5}+5∗z^{4}+10∗z^{3}+8∗z^{2}+10∗z+7q^{7}+10∗z^{9}+ z^{8}+4∗z^{7}+6∗z^{6}+4∗z^{5}+5∗z^{4}+2∗z+8q^{8}+10∗z^{9}+8∗z^{8}+10∗z^{7}+10∗z^{6}+10∗z^{5}+5∗z^{3}+9∗z^{2}+4∗z+7q^{9}+ 7∗z^{9}+2∗z^{8}+9∗z^{7}+3∗z^{6}+7∗z^{5}+2∗z^{4}+6∗z^{3}+10∗z^{2}+9∗z+8q^{10}+2∗z^{9}+5∗z^{8}+8∗z^{5}+2∗z^{4}+3∗z^{3}+8∗ z^{2}+6∗z+8q^{12}+9q^{13}+3∗z^{9}+3∗z^{8}+2∗z^{7}+z^{6}+7∗z^{5}+5∗z^{4}+10∗z^{3}+7∗z^{2}+3∗z+7q^{14}+8∗z^{9}+9∗z^{8}+10∗ z^{7}+5∗z^{6}+7∗z^{5}+2∗z^{4}+6∗z^{3}+3∗z^{2}+3∗z+7q^{15}+4∗z^{9}+3∗z^{8}+z^{7}+2∗z^{6}+z^{5}+8∗z^{4}+5∗z^{3}+9∗z^{2}+5∗z+ 10q^{16}+z^{9}+4∗z^{8}+10∗z^{7}+8∗z^{6}+9∗z^{5}+z^{4}+10∗z^{3}+10∗z^{2}+7∗z+4q^{17}+6∗z^{8}+4∗z^{7}+z^{6}+2∗z^{5}+z^{4}+9∗ z^{3}+z^{2}+2∗z+6q^{18}+z^{9}+2∗z^{8}+z^{6}+3∗z^{5}+2∗z^{4}+9∗z^{2}+8∗z+6q^{19}+4∗z^{9}+7∗z^{8}+z^{7}+8∗z^{6}+6∗z^{5}+10∗z^{4}+ 5∗z^{2}+7∗z+10q^{20}+2∗z^{9}+2∗z^{8}+9∗z^{7}+3∗z^{6}+9∗z^{5}+4∗z^{4}+8∗z^{3}+10∗z+7q^{21}+⋯ ∈ S_{14}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{10} + 7x^{5} + 8x^{4} + 10x^{3} + 6x^{2} + 6x + 2 = 0. We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q +z^{1}14013q^{2} +z^{1}22740q^{3} +z^{1}06984q^{4} +z^{6}794q^{5} +z^{7}5703q^{6} +z^{7}3897q^{7} + z^{8}6367q^{8} + z^{1}25967q^{9} + z^{1}20807q^{10} + z^{8}5232q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{7}5818)ω^{1} & 0 \\ 0 & u(z^{8}5232)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(21387163858956230833+ O(11^{2}0)) ∗ a^{4} − (59404096813310740182 + O(11^{2}0)) ∗ a^{3} + (285434393449487570636 + O(11^{2}0)) ∗ a^{2} + (258382076893934702347+O(11^{2}0))∗a+325427090450479554154+O(11^{2}0)q^{3}+a^{2}−8192q^{4}+(149048911017397375406+ O(11^{2}0)) ∗ a^{4} − (212751438041340910991 + O(11^{2}0)) ∗ a^{3} + (169062434483015614325 + O(11^{2}0)) ∗ a^{2} − (93869867883927977009+O(11^{2}0))∗a+252179276043442089627+O(11^{2}0)q^{5}+(245758664138838808658+ O(11^{2}0)) ∗ a^{4} + (127053753102524643367 + O(11^{2}0)) ∗ a^{3} + (121601149095455710971 + O(11^{2}0)) ∗ a^{2} + (138425283053563602671+O(11^{2}0))∗a+212168488396872087078+O(11^{2}0)q^{6}+(216634659326732592466+ O(11^{2}0)) ∗ a^{4} + (17173076027588612481 ∗ 11 + O(11^{2}0)) ∗ a^{3} − (79383888001421231121 + O(11^{2}0)) ∗ a^{2} −(168778358570009154338+O(11^{2}0))∗a−274123072264624336910+O(11^{2}0)q^{7} +a^{3} −16384∗aq^{8} + −(292340748728231565075+O(11^{2}0))∗a^{4}−(311822893053470544917+O(11^{2}0))∗a^{3}−(157885957550417846730+ O(11^{2}0))∗a^{2}+(228415951778337570761+O(11^{2}0))∗a−49953877595870578364+O(11^{2}0)q^{9}+−(19312600920288775336+ O(11^{2}0)) ∗ a^{4} − (131022188173597099255 + O(11^{2}0)) ∗ a^{3} + (273898172937446128592 + O(11^{2}0)) ∗ a^{2} − (58808135793201766318+O(11^{2}0))∗a+47512973853813980874+O(11^{2}0)q^{10}+(12633064790096194464∗ 11 + O(11^{2}0)) ∗ a^{4} − (25533115118981745577 ∗ 11 + O(11^{2}0)) ∗ a^{3} + (28779378768876882704 ∗ 11 + O(11^{2}0))∗a^{2} −(20699063029377833440∗11+O(11^{2}0))∗a−24144466969759816095∗11+O(11^{2}0)q^{11} + −(63849125614334263838+O(11^{2}0))∗a^{4}+(226173802006618836353+O(11^{2}0))∗a^{3}−(38968999222507754879+ O(11^{2}0))∗a^{2}−(333800080229330717099+O(11^{2}0))∗a−1967432049382362515+O(11^{2}0)q^{12}+4826809q^{13}+ −(293667292172346286834+O(11^{2}0))∗a^{4}+(25674949162002874488+O(11^{2}0))∗a^{3}−(48914422817466958908+ O(11^{2}0)) ∗ a^{2} − (187333631007220538067 + O(11^{2}0)) ∗ a + 164801161638778764219 + O(11^{2}0)q^{14} + (180266126201550111797+O(11^{2}0))∗a^{4}−(14235042070596246561+O(11^{2}0))∗a^{3}−(273381884940763051701+ O(11^{2}0)) ∗ a^{2} + (175548833054252498936 + O(11^{2}0)) ∗ a − 117119760516970272401 + O(11^{2}0)q^{15} + a^{4} − 24576 ∗ a^{2} + 67108864q^{16} + −(263085504903913715282 + O(11^{2}0)) ∗ a^{4} + (86811378675964927121 + O(11^{2}0)) ∗ a^{3} + (59772151561100280682 + O(11^{2}0)) ∗ a^{2} − (278490724226963341212 + O(11^{2}0)) ∗ a + 89678309788211630767+O(11^{2}0)q^{17}+(292960356135929954295+O(11^{2}0))∗a^{4}+(126788244543683647480+ O(11^{2}0)) ∗ a^{3} + (269630641206179273049 + O(11^{2}0)) ∗ a^{2} + (63933904165625109292 + O(11^{2}0)) ∗ a − 40656570550301511864+O(11^{2}0)q^{18}+−(129342595797067967202+O(11^{2}0))∗a^{4}+(92833397036767088011+ O(11^{2}0)) ∗ a^{3} − (153708565622632542186 + O(11^{2}0)) ∗ a^{2} − (131759468696263651596 + O(11^{2}0)) ∗ a + 208519509260609272666+O(11^{2}0)q^{19}+(155132964924173791298+O(11^{2}0))∗a^{4}+(164927169563368200405+ O(11^{2}0)) ∗ a^{3} − (207290028010266993472 + O(11^{2}0)) ∗ a^{2} + (88270279220978549808 + O(11^{2}0)) ∗ a + 19657352748313031426+O(11^{2}0)q^{20}+−(95906054232732768975+O(11^{2}0))∗a^{4}+(66408531782613740267+ O(11^{2}0)) ∗ a^{3} + (143191402289721066832 + O(11^{2}0)) ∗ a^{2} + (141901363757969604243 + O(11^{2}0)) ∗ a − 102563181908536027402+O(11^{2}0)q^{21} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{5}+−18405356653286492917+O(11^{2}0)x^{4}+−45435746779326533209+ O(11^{2}0)x^{3}+197200980531799123243+O(11^{2}0)x^{2}+−64509515765869694494+O(11^{2}0)x+−76188913029531034323+ O(11^{2}0). The slope of f_{0} is 1.

8. Consider f = q + z^{6}7q^{2} + z^{4}q^{3} + z^{6}3q^{4} + q^{5} + z^{7}1q^{6} + z^{6}9q^{7} + 5q^{8} + z^{2}6q^{9} + z^{6}7q^{10} + z^{6}7q^{12} + 7q^{13} + z^{1}6q^{14} + z^{4}q^{15} + z^{8}1q^{16} + z^{1}07q^{17} + z^{9}3q^{18} + z^{1}01q^{19} + z^{6}3q^{20} + z^{7}3q^{21} + ⋯ ∈ S_{16}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{1}9q^{2} + z^{1}00q^{3} + z^{8}7q^{4} + 5q^{5} + z^{1}19q^{6} + z^{9}3q^{7} + 4q^{8} + z^{9}8q^{9} + z^{6}7q^{10} + z^{5}0q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{7}0)ω^{1} & 0 \\ 0 & u(z^{5}0)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(25415011363644582209+ O(11^{2}0)) ∗ a − 184760342222364118234 + O(11^{2}0)q^{3} + −(335474258096283468675 + O(11^{2}0)) ∗ a − 177645760487242438617+O(11^{2}0)q^{4}+−(2538309410536586017∗11+O(11^{2}0))∗a−81830498585879039578+ O(11^{2}0)q^{5}+(331673390413669917036+O(11^{2}0))∗a−169715914428692509177+O(11^{2}0)q^{6}+(50648189589551431646+ O(11^{2}0)) ∗ a + 186827113208490646372 + O(11^{2}0)q^{7} + −(3428561964719376846 ∗ 11 + O(11^{2}0)) ∗ a − 126226405244072606787+O(11^{2}0)q^{8}+(219078988456052067868+O(11^{2}0))∗a−243181394710002307672+ O(11^{2}0)q^{9} + −(272198639486647973593 + O(11^{2}0)) ∗ a − 19850484618106774163 ∗ 11 + O(11^{2}0)q^{10} + (14781856637584693792∗11+O(11^{2}0))∗a+12653974711060250344∗11+O(11^{2}0)q^{11}+(323315772427642468662+ O(11^{2}0)) ∗ a − 9945905551271317409 ∗ 11 + O(11^{2}0)q^{12} + 62748517q^{13} + (210111756423323978051 + O(11^{2}0)) ∗ a + 179351936869711050034 + O(11^{2}0)q^{14} + (321803059274870130355 + O(11^{2}0)) ∗ a − 86336895966490210604+O(11^{2}0)q^{15}+−(300751578085830773805+O(11^{2}0))∗a+286334523221002952547+ O(11^{2}0)q^{16}+(11607967999906930832+O(11^{2}0))∗a+215584749729566945000+O(11^{2}0)q^{17}+(226592800325333284729+ O(11^{2}0)) ∗ a + 336370467436025007270 + O(11^{2}0)q^{18} + (251460991733816605514 + O(11^{2}0)) ∗ a + 296513207093114043005+O(11^{2}0)q^{19}+(233611048864643133413+O(11^{2}0))∗a+178143539783441842043+ O(11^{2}0)q^{20} + (260457718896221044318 + O(11^{2}0)) ∗ a + 88069538065003986005 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 335474258096283468675 + O(11^{2}0)x + 177645760487242405849 + O(11^{2}0). The slope of f_{0} is 1.

9. Consider f = q + 6q^{2} + 5q^{3} + 4q^{4} + 7q^{5} + 8q^{6} + 5q^{7} + 8q^{8} + 2q^{9} + 9q^{10} + 9q^{12} + 7q^{13} + 8q^{14} + 2q^{15} + 8q^{16} + q^{18} + 5q^{19} + 6q^{20} + 3q^{21} + ⋯ ∈ S_{16}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + 10q^{2} + 4q^{3} + 5q^{4} + 2q^{5} + 7q^{6} + 9q^{7} + 2q^{8} + 7q^{9} + 9q^{10} + 10q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(10)ω^{1} & 0 \\ 0 & u(10)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+173844857799179963336+ O(11^{2}0)q^{2}+240073973805460162615+O(11^{2}0)q^{3}+−335724342202001875437+O(11^{2}0)q^{4}+80898634712599570588+ O(11^{2}0)q^{5}+287337641378173936041+O(11^{2}0)q^{6}+237598086791698877491+O(11^{2}0)q^{7}+96450322999256137997+ O(11^{2}0)q^{8}+106100770004383399401+O(11^{2}0)q^{9}+−100043751548761707491+O(11^{2}0)q^{10}+2997822834763267387∗ 11 + O(11^{2}0)q^{11} + 15015683595807241136 + O(11^{2}0)q^{12} + 62748517q^{13} + −293988603175990225715 + O(11^{2}0)q^{14}+203320461554941773352+O(11^{2}0)q^{15}+276666081118969051221+O(11^{2}0)q^{16}+−9683728259889591612∗ 11 + O(11^{2}0)q^{17} + 168731992359328732621 + O(11^{2}0)q^{18} + −50727812382215600154 + O(11^{2}0)q^{19} + −252020560701904513469 + O(11^{2}0)q^{20} + 198854047385826329786 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

10. Consider f = q + z^{1}09q^{2} + z^{4}6q^{3} + z^{1}17q^{4} + z^{1}04q^{5} + z^{3}5q^{6} + 7q^{7} + 10q^{8} + z^{7}7q^{9} + z^{9}3q^{10} + z^{4}3q^{12} + 7q^{13} + z^{7}3q^{14} + z^{3}0q^{15} + z^{1}5q^{16} + z^{6}7q^{17} + z^{6}6q^{18} + z^{9}q^{19} + z^{1}01q^{20} + z^{1}0q^{21} + ⋯ ∈ S_{16}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-4 newform g = q + z^{9}7q^{2} + z^{7}0q^{3} + z^{9}3q^{4} + z^{5}6q^{5} + z^{4}7q^{6} + q^{7} + 4q^{8} + z^{5}q^{9} + z^{3}3q^{10} + z^{7}6q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{4}4)ω^{4} & 0 \\ 0 & u(z^{7}6)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(24237401115059786149+ O(11^{2}0)) ∗ a + 292128984187267580101 + O(11^{2}0)q^{3} + −(169869830449523554579 + O(11^{2}0)) ∗ a + 264330951694383892236+O(11^{2}0)q^{4}+−(186494590566620281773+O(11^{2}0))∗a−113805390928469554456+ O(11^{2}0)q^{5}+(145288154855509583030+O(11^{2}0))∗a+69621407390455736942+O(11^{2}0)q^{6}+(9460327178388909489∗ 11 + O(11^{2}0)) ∗ a − 32314192975689651338 + O(11^{2}0)q^{7} + (29867268918349018364 ∗ 11 + O(11^{2}0)) ∗ a − 232169834773474077547+O(11^{2}0)q^{8}+(315777730859310409232+O(11^{2}0))∗a−72808936885031049852+ O(11^{2}0)q^{9}+(118308611458871685165+O(11^{2}0))∗a+139262634905023607831+O(11^{2}0)q^{10}+(5435952445308691654∗ 11 + O(11^{2}0)) ∗ a + 7647174326962571591 ∗ 11 + O(11^{2}0)q^{11} + −(204025216163668439451 + O(11^{2}0)) ∗ a + 64457689156898983676 + O(11^{2}0)q^{12} + 62748517q^{13} + (289802594587122485678 + O(11^{2}0)) ∗ a + 17043740203554914313∗11+O(11^{2}0)q^{14}+−(52194490642369756590+O(11^{2}0))∗a+295389411729825687425+ O(11^{2}0)q^{15} + −(237545632949919238867 + O(11^{2}0)) ∗ a − 148283632252491498834 + O(11^{2}0)q^{16} + −(191713586535369158025+O(11^{2}0))∗a−204377688102865711326+O(11^{2}0)q^{17}+−(184928893141485477532+ O(11^{2}0)) ∗ a − 149760491053600479538 + O(11^{2}0)q^{18} + −(200451390481843420982 + O(11^{2}0)) ∗ a + 131622306480198700706+O(11^{2}0)q^{19}+−(38293811984011086344+O(11^{2}0))∗a+109604457245176214781+ O(11^{2}0)q^{20} + −(196626535484987889527 + O(11^{2}0)) ∗ a − 214888379299636815975 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 169869830449523554579 + O(11^{2}0)x + −264330951694383925004 + O(11^{2}0). The slope of f_{0} is 1.

11. Consider f = q + z^{1}19q^{2} + z^{2}6q^{3} + z^{8}7q^{4} + z^{6}4q^{5} + z^{2}5q^{6} + 7q^{7} + 10q^{8} + z^{7}q^{9} + z^{6}3q^{10} + z^{1}13q^{12} + 7q^{13} + z^{8}3q^{14} + z^{9}0q^{15} + z^{4}5q^{16} + z^{1}7q^{17} + z^{6}q^{18} + z^{9}9q^{19} + z^{3}1q^{20} + z^{1}10q^{21} + ⋯ ∈ S_{16}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-4 newform g = q + z^{1}07q^{2} + z^{5}0q^{3} + z^{6}3q^{4} + z^{1}6q^{5} + z^{3}7q^{6} + q^{7} + 4q^{8} + z^{5}5q^{9} + z^{3}q^{10} + z^{1}16q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{4})ω^{4} & 0 \\ 0 & u(z^{1}16)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(24237401115059786149+ O(11^{2}0)) ∗ a + 292128984187267580101 + O(11^{2}0)q^{3} + −(169869830449523554579 + O(11^{2}0)) ∗ a + 264330951694383892236+O(11^{2}0)q^{4}+−(186494590566620281773+O(11^{2}0))∗a−113805390928469554456+ O(11^{2}0)q^{5}+(145288154855509583030+O(11^{2}0))∗a+69621407390455736942+O(11^{2}0)q^{6}+(9460327178388909489∗ 11 + O(11^{2}0)) ∗ a − 32314192975689651338 + O(11^{2}0)q^{7} + (29867268918349018364 ∗ 11 + O(11^{2}0)) ∗ a − 232169834773474077547+O(11^{2}0)q^{8}+(315777730859310409232+O(11^{2}0))∗a−72808936885031049852+ O(11^{2}0)q^{9}+(118308611458871685165+O(11^{2}0))∗a+139262634905023607831+O(11^{2}0)q^{10}+(5435952445308691654∗ 11 + O(11^{2}0)) ∗ a + 7647174326962571591 ∗ 11 + O(11^{2}0)q^{11} + −(204025216163668439451 + O(11^{2}0)) ∗ a + 64457689156898983676 + O(11^{2}0)q^{12} + 62748517q^{13} + (289802594587122485678 + O(11^{2}0)) ∗ a + 17043740203554914313∗11+O(11^{2}0)q^{14}+−(52194490642369756590+O(11^{2}0))∗a+295389411729825687425+ O(11^{2}0)q^{15} + −(237545632949919238867 + O(11^{2}0)) ∗ a − 148283632252491498834 + O(11^{2}0)q^{16} + −(191713586535369158025+O(11^{2}0))∗a−204377688102865711326+O(11^{2}0)q^{17}+−(184928893141485477532+ O(11^{2}0)) ∗ a − 149760491053600479538 + O(11^{2}0)q^{18} + −(200451390481843420982 + O(11^{2}0)) ∗ a + 131622306480198700706+O(11^{2}0)q^{19}+−(38293811984011086344+O(11^{2}0))∗a+109604457245176214781+ O(11^{2}0)q^{20} + −(196626535484987889527 + O(11^{2}0)) ∗ a − 214888379299636815975 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 169869830449523554579 + O(11^{2}0)x + −264330951694383925004 + O(11^{2}0). The slope of f_{0} is 1.

12. Consider f = q + z^{1}7q^{2} + z^{4}4q^{3} + z^{9}3q^{4} + q^{5} + z^{6}1q^{6} + z^{3}9q^{7} + 5q^{8} + z^{4}6q^{9} + z^{1}7q^{10} + z^{1}7q^{12} + 7q^{13} + z^{5}6q^{14} + z^{4}4q^{15} + z^{5}1q^{16} + z^{9}7q^{17} + z^{6}3q^{18} + z^{3}1q^{19} + z^{9}3q^{20} + z^{8}3q^{21} + ⋯ ∈ S_{16}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{8}9q^{2} + z^{2}0q^{3} + z^{1}17q^{4} + 5q^{5} + z^{1}09q^{6} + z^{6}3q^{7} + 4q^{8} + z^{1}18q^{9} + z^{1}7q^{10} + z^{7}0q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{5}0)ω^{1} & 0 \\ 0 & u(z^{7}0)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(25415011363644582209+ O(11^{2}0)) ∗ a − 184760342222364118234 + O(11^{2}0)q^{3} + −(335474258096283468675 + O(11^{2}0)) ∗ a − 177645760487242438617+O(11^{2}0)q^{4}+−(2538309410536586017∗11+O(11^{2}0))∗a−81830498585879039578+ O(11^{2}0)q^{5}+(331673390413669917036+O(11^{2}0))∗a−169715914428692509177+O(11^{2}0)q^{6}+(50648189589551431646+ O(11^{2}0)) ∗ a + 186827113208490646372 + O(11^{2}0)q^{7} + −(3428561964719376846 ∗ 11 + O(11^{2}0)) ∗ a − 126226405244072606787+O(11^{2}0)q^{8}+(219078988456052067868+O(11^{2}0))∗a−243181394710002307672+ O(11^{2}0)q^{9} + −(272198639486647973593 + O(11^{2}0)) ∗ a − 19850484618106774163 ∗ 11 + O(11^{2}0)q^{10} + (14781856637584693792∗11+O(11^{2}0))∗a+12653974711060250344∗11+O(11^{2}0)q^{11}+(323315772427642468662+ O(11^{2}0)) ∗ a − 9945905551271317409 ∗ 11 + O(11^{2}0)q^{12} + 62748517q^{13} + (210111756423323978051 + O(11^{2}0)) ∗ a + 179351936869711050034 + O(11^{2}0)q^{14} + (321803059274870130355 + O(11^{2}0)) ∗ a − 86336895966490210604+O(11^{2}0)q^{15}+−(300751578085830773805+O(11^{2}0))∗a+286334523221002952547+ O(11^{2}0)q^{16}+(11607967999906930832+O(11^{2}0))∗a+215584749729566945000+O(11^{2}0)q^{17}+(226592800325333284729+ O(11^{2}0)) ∗ a + 336370467436025007270 + O(11^{2}0)q^{18} + (251460991733816605514 + O(11^{2}0)) ∗ a + 296513207093114043005+O(11^{2}0)q^{19}+(233611048864643133413+O(11^{2}0))∗a+178143539783441842043+ O(11^{2}0)q^{20} + (260457718896221044318 + O(11^{2}0)) ∗ a + 88069538065003986005 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 335474258096283468675 + O(11^{2}0)x + 177645760487242405849 + O(11^{2}0). The slope of f_{0} is 1.

13. Consider f = q + z^{7}74q^{2} + z^{1}104q^{3} + z^{1}256q^{4} + z^{8}03q^{5} + z^{5}48q^{6} + z^{6}30q^{7} + z^{2}82q^{8} + z^{5}71q^{9} + z^{2}47q^{10} + z^{1}030q^{12} + 4q^{13} + z^{7}4q^{14} + z^{5}77q^{15} + z^{2}q^{16} + z^{1}190q^{17} + z^{1}5q^{18} + z^{6}81q^{19} + z^{7}29q^{20} + z^{4}04q^{21} + ⋯ ∈ S_{16}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{3} + 2x + 9 = 0. We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{2}42q^{2} + z^{8}38q^{3} + z^{1}92q^{4} + z^{5}q^{5} + z^{1}080q^{6} + z^{8}96q^{7} + z^{1}6q^{8} + z^{3}9q^{9} + z^{2}47q^{10} + z^{6}q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}324)ω^{1} & 0 \\ 0 & u(z^{6})ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(107392270277215435395+ O(11^{2}0)) ∗ a^{2} − (22348708376847180899 ∗ 11 + O(11^{2}0)) ∗ a + 176311994708273597588 + O(11^{2}0)q^{3} + a^{2} − 32768q^{4} + (323453728731794453944 + O(11^{2}0)) ∗ a^{2} − (268631194550805867401 + O(11^{2}0)) ∗ a − 122406606024983023102+O(11^{2}0)q^{5}+−(309116401493407520609+O(11^{2}0))∗a^{2}−(168628574716055392440+ O(11^{2}0)) ∗ a − 73220034978376798151 + O(11^{2}0)q^{6} + −(70232318676624297606 + O(11^{2}0)) ∗ a^{2} − (17773547368689671701+O(11^{2}0))∗a+243932781102322494862+O(11^{2}0)q^{7}+−(171610909919118756777+ O(11^{2}0)) ∗ a^{2} + (263757824938559686483 + O(11^{2}0)) ∗ a − 218293975610444480577 + O(11^{2}0)q^{8} + (248668873872975819408+O(11^{2}0))∗a^{2}−(104536185811181809923+O(11^{2}0))∗a+43493232672519273262+ O(11^{2}0)q^{9} + (24940688280074130633 ∗ 11 + O(11^{2}0)) ∗ a^{2} − (21592929638510053036 + O(11^{2}0)) ∗ a + 295326816074655749094+O(11^{2}0)q^{10}+−(26809171886242918035∗11+O(11^{2}0))∗a^{2}+(1404966355562807828∗ 11 + O(11^{2}0)) ∗ a − 25676307324502447280 ∗ 11 + O(11^{2}0)q^{11} + (260101884241989103767 + O(11^{2}0)) ∗ a^{2} +(302384616337052801014+O(11^{2}0))∗a−185204009850196427108+O(11^{2}0)q^{12} +−62748517q^{13} + (9826468108560345145∗11+O(11^{2}0))∗a^{2}+(176926083828782911517+O(11^{2}0))∗a−318838286544913824400+ O(11^{2}0)q^{14} + (175404692934669002270 + O(11^{2}0)) ∗ a^{2} + (156991144029098306333 + O(11^{2}0)) ∗ a + 44264371734777735144+O(11^{2}0)q^{15}+−(278923165780171530073+O(11^{2}0))∗a^{2}+(144993764922377329060+ O(11^{2}0)) ∗ a + 15382257327926584732 ∗ 11 + O(11^{2}0)q^{16} + −(96703876150915456301 + O(11^{2}0)) ∗ a^{2} + (13816540544235036757∗11+O(11^{2}0))∗a−312282454989481189918+O(11^{2}0)q^{17}+−(78636032597657674494+ O(11^{2}0)) ∗ a^{2} + (291822973381490966306 + O(11^{2}0)) ∗ a − 200834409958368412370 + O(11^{2}0)q^{18} + (331337894734084397923+O(11^{2}0))∗a^{2}−(163878108839801259555+O(11^{2}0))∗a−90398420624017575018+ O(11^{2}0)q^{19} + −(112149562426969479738 + O(11^{2}0)) ∗ a^{2} + (197186223990813612909 + O(11^{2}0)) ∗ a − 208423537557244500872+O(11^{2}0)q^{20}+(150697258076683180512+O(11^{2}0))∗a^{2}−(85957143302074125930+ O(11^{2}0)) ∗ a − 128066842017984476194 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 171610909919118756777 + O(11^{2}0)x^{2} + −263757824938559752019 + O(11^{2}0)x + 218293975610444480577 + O(11^{2}0). The slope of f_{0} is 1.

14. Consider f = q +7q^{2} +2q^{3} +6q^{4} +5q^{5} +3q^{6} +10q^{7} +5q^{8} +3q^{9} +2q^{10} +q^{12} +4q^{13} +4q^{14} +10q^{15} +8q^{16} + q^{17} + 10q^{18} + 10q^{19} + 8q^{20} + 9q^{21} + ⋯ ∈ S_{16}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{3} + 2x + 9 = 0. We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + 8q^{2} + 6q^{3} + 2q^{4} + 3q^{5} + 4q^{6} + 7q^{7} + 4q^{8} + 5q^{9} + 2q^{10} + 3q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{1} & 0 \\ 0 & u(3)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+120017050706114314430+ O(11^{2}0)q^{2}+222386774395346154420+O(11^{2}0)q^{3}+−186179288618642499713+O(11^{2}0)q^{4}+−72488432383327953262+ O(11^{2}0)q^{5}+−274853195572741581947+O(11^{2}0)q^{6}+−225508598224753970666+O(11^{2}0)q^{7}+218715252251246777284+ O(11^{2}0)q^{8}+−252571510513344599651+O(11^{2}0)q^{9}+129916529166247941262+O(11^{2}0)q^{10}+3584757830220004450∗ 11 + O(11^{2}0)q^{11} + 23944954636428774840 + O(11^{2}0)q^{12} + −62748517q^{13} + 62972509679491795651 + O(11^{2}0)q^{14}+284222649833340466533+O(11^{2}0)q^{15}+−108955160217259638769+O(11^{2}0)q^{16}+45307078439763058645+ O(11^{2}0)q^{17}+140119812713102471919+O(11^{2}0)q^{18}+327622480922146838779+O(11^{2}0)q^{19}+110236469566436168755+ O(11^{2}0)q^{20} + 214150103376108056100 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

15. Consider f = q+q^{2}+q^{3}+2q^{4}+9q^{5}+q^{6}+8q^{7}+3q^{8}+9q^{10}+2q^{12}+4q^{13}+8q^{14}+9q^{15}+5q^{16}+4q^{19}+7q^{20}+ 8q^{21} + ⋯ ∈ S_{16}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{3} + 2x + 9 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-4 newform g = q + 6q^{2} + 4q^{3} + 6q^{4} + 4q^{5} + 2q^{6} + 9q^{7} + 10q^{8} + 2q^{10} + 7q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{4} & 0 \\ 0 & u(7)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−107754626022217073184+ O(11^{2}0)q^{2}+−253497724721559854348+O(11^{2}0)q^{3}+−43024237534570777153+O(11^{2}0)q^{4}+−295364415211796863143+ O(11^{2}0)q^{5}+42312148037287521097+O(11^{2}0)q^{6}+−132578043065231485252+O(11^{2}0)q^{7}+108285930612604850768+ O(11^{2}0)q^{8}+−11025653945703364270∗11+O(11^{2}0)q^{9}+38254679475606266352+O(11^{2}0)q^{10}+−24010705195019425719∗ 11 + O(11^{2}0)q^{11} + 132950460521916639593 + O(11^{2}0)q^{12} + −62748517q^{13} + 105573664999695019828 + O(11^{2}0)q^{14}+252835307699693179459+O(11^{2}0)q^{15}+−247014496824987730471+O(11^{2}0)q^{16}+−10677820931093042125∗ 11+O(11^{2}0)q^{17}+−20833574399729661400∗11+O(11^{2}0)q^{18}+−278889103086255131840+O(11^{2}0)q^{19}+ −239110319563539755874+O(11^{2}0)q^{20}+−183357935691745192139+O(11^{2}0)q^{21}+⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

16. Consider f = q + z^{5}34q^{2} + z^{1}74q^{3} + z^{5}16q^{4} + z^{8}53q^{5} + z^{7}08q^{6} + z^{2}80q^{7} + z^{4}42q^{8} + z^{9}61q^{9} + z^{5}7q^{10} + z^{6}90q^{12} + 4q^{13} + z^{8}14q^{14} + z^{1}027q^{15} + z^{2}2q^{16} + z^{1}120q^{17} + z^{1}65q^{18} + z^{8}41q^{19} + z^{3}9q^{20} + z^{4}54q^{21} + ⋯ ∈ S_{16}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{3} + 2x + 9 = 0. We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{2}q^{2} + z^{1}238q^{3} + z^{7}82q^{4} + z^{5}5q^{5} + z^{1}240q^{6} + z^{5}46q^{7} + z^{1}76q^{8} + z^{4}29q^{9} + z^{5}7q^{10} + z^{6}6q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}264)ω^{1} & 0 \\ 0 & u(z^{6}6)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(107392270277215435395+ O(11^{2}0)) ∗ a^{2} − (22348708376847180899 ∗ 11 + O(11^{2}0)) ∗ a + 176311994708273597588 + O(11^{2}0)q^{3} + a^{2} − 32768q^{4} + (323453728731794453944 + O(11^{2}0)) ∗ a^{2} − (268631194550805867401 + O(11^{2}0)) ∗ a − 122406606024983023102+O(11^{2}0)q^{5}+−(309116401493407520609+O(11^{2}0))∗a^{2}−(168628574716055392440+ O(11^{2}0)) ∗ a − 73220034978376798151 + O(11^{2}0)q^{6} + −(70232318676624297606 + O(11^{2}0)) ∗ a^{2} − (17773547368689671701+O(11^{2}0))∗a+243932781102322494862+O(11^{2}0)q^{7}+−(171610909919118756777+ O(11^{2}0)) ∗ a^{2} + (263757824938559686483 + O(11^{2}0)) ∗ a − 218293975610444480577 + O(11^{2}0)q^{8} + (248668873872975819408+O(11^{2}0))∗a^{2}−(104536185811181809923+O(11^{2}0))∗a+43493232672519273262+ O(11^{2}0)q^{9} + (24940688280074130633 ∗ 11 + O(11^{2}0)) ∗ a^{2} − (21592929638510053036 + O(11^{2}0)) ∗ a + 295326816074655749094+O(11^{2}0)q^{10}+−(26809171886242918035∗11+O(11^{2}0))∗a^{2}+(1404966355562807828∗ 11 + O(11^{2}0)) ∗ a − 25676307324502447280 ∗ 11 + O(11^{2}0)q^{11} + (260101884241989103767 + O(11^{2}0)) ∗ a^{2} +(302384616337052801014+O(11^{2}0))∗a−185204009850196427108+O(11^{2}0)q^{12} +−62748517q^{13} + (9826468108560345145∗11+O(11^{2}0))∗a^{2}+(176926083828782911517+O(11^{2}0))∗a−318838286544913824400+ O(11^{2}0)q^{14} + (175404692934669002270 + O(11^{2}0)) ∗ a^{2} + (156991144029098306333 + O(11^{2}0)) ∗ a + 44264371734777735144+O(11^{2}0)q^{15}+−(278923165780171530073+O(11^{2}0))∗a^{2}+(144993764922377329060+ O(11^{2}0)) ∗ a + 15382257327926584732 ∗ 11 + O(11^{2}0)q^{16} + −(96703876150915456301 + O(11^{2}0)) ∗ a^{2} + (13816540544235036757∗11+O(11^{2}0))∗a−312282454989481189918+O(11^{2}0)q^{17}+−(78636032597657674494+ O(11^{2}0)) ∗ a^{2} + (291822973381490966306 + O(11^{2}0)) ∗ a − 200834409958368412370 + O(11^{2}0)q^{18} + (331337894734084397923+O(11^{2}0))∗a^{2}−(163878108839801259555+O(11^{2}0))∗a−90398420624017575018+ O(11^{2}0)q^{19} + −(112149562426969479738 + O(11^{2}0)) ∗ a^{2} + (197186223990813612909 + O(11^{2}0)) ∗ a − 208423537557244500872+O(11^{2}0)q^{20}+(150697258076683180512+O(11^{2}0))∗a^{2}−(85957143302074125930+ O(11^{2}0)) ∗ a − 128066842017984476194 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 171610909919118756777 + O(11^{2}0)x^{2} + −263757824938559752019 + O(11^{2}0)x + 218293975610444480577 + O(11^{2}0). The slope of f_{0} is 1.

17. Consider f = q + z^{5}54q^{2} + z^{5}84q^{3} + z^{3}56q^{4} + z^{7}3q^{5} + z^{1}138q^{6} + z^{4}20q^{7} + z^{8}72q^{8} + z^{1}261q^{9} + z^{6}27q^{10} + z^{9}40q^{12} +4q^{13} +z^{9}74q^{14} +z^{6}57q^{15} +z^{2}42q^{16} +z^{3}50q^{17} +z^{4}85q^{18} +z^{1}271q^{19} +z^{4}29q^{20} +z^{1}004q^{21} +⋯ ∈ S_{16}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{3} + 2x + 9 = 0. We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{2}2q^{2} + z^{3}18q^{3} + z^{6}22q^{4} + z^{6}05q^{5} + z^{3}40q^{6} + z^{6}86q^{7} + z^{6}06q^{8} + z^{7}29q^{9} + z^{6}27q^{10} + z^{7}26q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{6}04)ω^{1} & 0 \\ 0 & u(z^{7}26)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(107392270277215435395+ O(11^{2}0)) ∗ a^{2} − (22348708376847180899 ∗ 11 + O(11^{2}0)) ∗ a + 176311994708273597588 + O(11^{2}0)q^{3} + a^{2} − 32768q^{4} + (323453728731794453944 + O(11^{2}0)) ∗ a^{2} − (268631194550805867401 + O(11^{2}0)) ∗ a − 122406606024983023102+O(11^{2}0)q^{5}+−(309116401493407520609+O(11^{2}0))∗a^{2}−(168628574716055392440+ O(11^{2}0)) ∗ a − 73220034978376798151 + O(11^{2}0)q^{6} + −(70232318676624297606 + O(11^{2}0)) ∗ a^{2} − (17773547368689671701+O(11^{2}0))∗a+243932781102322494862+O(11^{2}0)q^{7}+−(171610909919118756777+ O(11^{2}0)) ∗ a^{2} + (263757824938559686483 + O(11^{2}0)) ∗ a − 218293975610444480577 + O(11^{2}0)q^{8} + (248668873872975819408+O(11^{2}0))∗a^{2}−(104536185811181809923+O(11^{2}0))∗a+43493232672519273262+ O(11^{2}0)q^{9} + (24940688280074130633 ∗ 11 + O(11^{2}0)) ∗ a^{2} − (21592929638510053036 + O(11^{2}0)) ∗ a + 295326816074655749094+O(11^{2}0)q^{10}+−(26809171886242918035∗11+O(11^{2}0))∗a^{2}+(1404966355562807828∗ 11 + O(11^{2}0)) ∗ a − 25676307324502447280 ∗ 11 + O(11^{2}0)q^{11} + (260101884241989103767 + O(11^{2}0)) ∗ a^{2} +(302384616337052801014+O(11^{2}0))∗a−185204009850196427108+O(11^{2}0)q^{12} +−62748517q^{13} + (9826468108560345145∗11+O(11^{2}0))∗a^{2}+(176926083828782911517+O(11^{2}0))∗a−318838286544913824400+ O(11^{2}0)q^{14} + (175404692934669002270 + O(11^{2}0)) ∗ a^{2} + (156991144029098306333 + O(11^{2}0)) ∗ a + 44264371734777735144+O(11^{2}0)q^{15}+−(278923165780171530073+O(11^{2}0))∗a^{2}+(144993764922377329060+ O(11^{2}0)) ∗ a + 15382257327926584732 ∗ 11 + O(11^{2}0)q^{16} + −(96703876150915456301 + O(11^{2}0)) ∗ a^{2} + (13816540544235036757∗11+O(11^{2}0))∗a−312282454989481189918+O(11^{2}0)q^{17}+−(78636032597657674494+ O(11^{2}0)) ∗ a^{2} + (291822973381490966306 + O(11^{2}0)) ∗ a − 200834409958368412370 + O(11^{2}0)q^{18} + (331337894734084397923+O(11^{2}0))∗a^{2}−(163878108839801259555+O(11^{2}0))∗a−90398420624017575018+ O(11^{2}0)q^{19} + −(112149562426969479738 + O(11^{2}0)) ∗ a^{2} + (197186223990813612909 + O(11^{2}0)) ∗ a − 208423537557244500872+O(11^{2}0)q^{20}+(150697258076683180512+O(11^{2}0))∗a^{2}−(85957143302074125930+ O(11^{2}0)) ∗ a − 128066842017984476194 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 171610909919118756777 + O(11^{2}0)x^{2} + −263757824938559752019 + O(11^{2}0)x + 218293975610444480577 + O(11^{2}0). The slope of f_{0} is 1.

18. Consider f = q+10∗z^{5}+8∗z^{4}+8∗z^{3}+5∗z^{2}+3q^{2}+4∗z^{5}+7∗z^{4}+7∗z^{3}+9∗z^{2}+5∗z+3q^{3}+3∗z^{5}+10∗z^{4}+ 10∗z^{3}+10∗z^{2}+10∗zq^{4}+7∗z^{5}+9∗z^{4}+9∗z^{3}+6∗z^{2}+zq^{5}+9∗z^{5}+4∗z^{4}+4∗z^{3}+7∗z^{2}+z+7q^{6}+6∗z^{5}+5∗ z^{4}+5∗z^{3}+8∗z^{2}+2∗z+1q^{7}+6∗z^{5}+10∗z^{4}+10∗z^{3}+z^{2}+8∗z+9q^{8}+5∗z^{5}+8∗z^{4}+8∗z^{3}+9∗z^{2}+7∗z+ 2q^{9}+7∗z^{5}+8∗z^{4}+8∗z^{3}+3∗z^{2}+2∗z+1q^{10}+4∗z^{5}+10∗z^{4}+10∗z^{3}+7∗z^{2}+2∗z+5q^{12}+8q^{13}+6∗z^{5}+4∗ z^{2}+7∗zq^{14}+4∗z^{5}+6∗z^{4}+6∗z^{3}+6∗z^{2}+6∗z+4q^{15}+7∗z^{5}+4∗z^{4}+4∗z^{3}+2∗z^{2}+6∗z+1q^{16}+3∗z^{5}+3∗z^{4}+ 3∗z^{3}+6∗z+3q^{17}+3∗z^{5}+6∗z^{4}+6∗z^{3}+9∗z^{2}+3∗z+2q^{18}+9∗z^{5}+4∗z^{4}+4∗z^{3}+7∗z^{2}+z+4q^{19}+6∗z^{5}+ 6∗z^{4}+6∗z^{3}+z+2q^{20}+10∗z^{5}+4∗z^{4}+4∗z^{3}+4∗z^{2}+4∗z+6q^{21}+⋯ ∈ S_{18}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{6} +3x^{4} +4x^{3} +6x^{2} +7x+2 = 0. We have ρ_{f} ⊗χ^{9} ∼ ρ_{g}, where g is the weight-6 newform g = q+z^{1}097q^{2}+z^{2}92q^{3}+z^{1}006q^{4}+z^{1}249q^{5}+z^{5}9q^{6}+z^{1}311q^{7}+z^{8}38q^{8}+z^{6}16q^{9}+z^{1}016q^{10}+z^{3}67q^{11}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{9}63)ω^{6} & 0 \\ 0 & u(z^{3}67)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(131446899915469254352+ O(11^{2}0)) ∗ a^{2} − (16668102967084971608 + O(11^{2}0)) ∗ a − 143028618626631968157 + O(11^{2}0)q^{3} + a^{2} − 131072q^{4}+(248431489477194627658+O(11^{2}0))∗a^{2}−(62985048310559850397+O(11^{2}0))∗a−107124721210528445277+ O(11^{2}0)q^{5} + −(112777766827640977646 + O(11^{2}0)) ∗ a^{2} + (115594248386970322716 + O(11^{2}0)) ∗ a − 65634966030451128723+O(11^{2}0)q^{6}+−(74876497582956721159+O(11^{2}0))∗a^{2}+(128308652523796125897+ O(11^{2}0)) ∗ a − 216030589636627076766 + O(11^{2}0)q^{7} + (179872375456932504195 + O(11^{2}0)) ∗ a^{2} + (41167414242319362385+O(11^{2}0))∗a−210841057517064670475+O(11^{2}0)q^{8}+−(196227769989181940335+ O(11^{2}0)) ∗ a^{2} − (104526655010805199407 + O(11^{2}0)) ∗ a + 197438649616604575264 + O(11^{2}0)q^{9} + −(71391365500713487726+O(11^{2}0))∗a^{2}+(237140037334375000868+O(11^{2}0))∗a+136642671148158180752+ O(11^{2}0)q^{10} +−(6471662588803686460∗11+O(11^{2}0))∗a^{2} +(21245108018379929106∗11+O(11^{2}0))∗a− 20686351859699962235∗11+O(11^{2}0)q^{11}+−(267173984533929290358+O(11^{2}0))∗a^{2}+(99754482923420751911+ O(11^{2}0)) ∗ a + 141597952696203725694 + O(11^{2}0)q^{12} + −815730721q^{13} + −(51096450356173876496 + O(11^{2}0)) ∗ a^{2} + (257257083675995671626 + O(11^{2}0)) ∗ a + 237408682042918616888 + O(11^{2}0)q^{14} + −(289934825349939680807+O(11^{2}0))∗a^{2}−(11000262763865428178∗11+O(11^{2}0))∗a+224147636419749071020+ O(11^{2}0)q^{15} + (295283955689600901776 + O(11^{2}0)) ∗ a^{2} − (20206118722874328192 + O(11^{2}0)) ∗ a + 300346790331365632049+O(11^{2}0)q^{16}+(127107690642767188535+O(11^{2}0))∗a^{2}−(290403863786268544102+ O(11^{2}0)) ∗ a − 206063594537721036723 + O(11^{2}0)q^{17} + (35828085643784992027 + O(11^{2}0)) ∗ a^{2} − (217264071391869846631+O(11^{2}0))∗a−126469075039791961352+O(11^{2}0)q^{18}+(21240962067465993280+ O(11^{2}0)) ∗ a^{2} − (14742874762373252012 + O(11^{2}0)) ∗ a + 113231826398361026205 + O(11^{2}0)q^{19} + −(65135236818980544173+O(11^{2}0))∗a^{2}−(278827988407146949560+O(11^{2}0))∗a−7831435695833978885+ O(11^{2}0)q^{20} + (97651087496701653768 + O(11^{2}0)) ∗ a^{2} − (5896933888367694243 ∗ 11 + O(11^{2}0)) ∗ a − 171489876588728548432+O(11^{2}0)q^{21} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3}+−179872375456932504195+O(11^{2}0)x^{2}+−41167414242319624529+ O(11^{2}0)x + 210841057517064670475 + O(11^{2}0). The slope of f_{0} is 1.

19. Consider f = q+7∗z^{5}+z^{2}+10∗z+4q^{2}+10∗z^{5}+7∗z^{4}+7∗z^{3}+2∗z^{2}+z+4q^{3}+3∗z^{4}+3∗z^{3}+9∗z^{2}+8∗ z+7q^{4}+7∗z^{5}+8∗z^{4}+8∗z^{3}+3∗z^{2}+2∗z+5q^{5}+4∗z^{5}+8∗z^{4}+8∗z^{3}+z^{2}+4∗z+10q^{6}+4∗z^{5}+5∗z^{4}+5∗ z^{3}+3∗z^{2}+7∗z+8q^{7}+10∗z^{5}+9∗z^{4}+9∗z^{3}+8∗z^{2}+10∗zq^{8}+5∗z^{5}+z^{4}+z^{3}+10∗z^{2}+3∗z+4q^{9}+8∗z^{5}+ 5∗z^{4}+5∗z^{3}+2∗z^{2}+8∗z+7q^{10}+7∗z^{5}+5∗z^{4}+5∗z^{3}+5∗z^{2}+5∗z+3q^{12}+8q^{13}+9∗z^{5}+z^{4}+z^{3}+9∗z^{2}+ 4∗z+1q^{14}+4∗z^{4}+4∗z^{3}+z^{2}+7∗z+6q^{15}+z^{5}+4∗z^{4}+4∗z^{3}+9∗z^{2}+10∗zq^{16}+7∗z^{5}+4∗z^{4}+4∗z^{3}+2∗ z^{2}+6∗z+6q^{17}+4∗z^{5}+2∗z^{4}+2∗z^{3}+5∗z^{2}+10∗z+2q^{18}+4∗z^{5}+8∗z^{4}+8∗z^{3}+z^{2}+4∗z+7q^{19}+3∗z^{5}+ 8∗z^{4} +8∗z^{3} +4∗z^{2} +z +8q^{20} +10∗z^{4} +10∗z^{3} +8∗z^{2} +zq^{21} +⋯ ∈ S_{18}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{6} +3x^{4} +4x^{3} +6x^{2} +7x+2 = 0. We have ρ_{f} ⊗χ^{9} ∼ ρ_{g}, where g is the weight-6 newform g = q +z^{9}7q^{2} +z^{5}52q^{3} +z^{4}26q^{4} +z^{4}39q^{5} +z^{6}49q^{6} +z^{1}121q^{7} +z^{1}238q^{8} +z^{1}26q^{9} +z^{5}36q^{10} +z^{4}7q^{11} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}283)ω^{6} & 0 \\ 0 & u(z^{4}7)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(131446899915469254352+ O(11^{2}0)) ∗ a^{2} − (16668102967084971608 + O(11^{2}0)) ∗ a − 143028618626631968157 + O(11^{2}0)q^{3} + a^{2} − 131072q^{4}+(248431489477194627658+O(11^{2}0))∗a^{2}−(62985048310559850397+O(11^{2}0))∗a−107124721210528445277+ O(11^{2}0)q^{5} + −(112777766827640977646 + O(11^{2}0)) ∗ a^{2} + (115594248386970322716 + O(11^{2}0)) ∗ a − 65634966030451128723+O(11^{2}0)q^{6}+−(74876497582956721159+O(11^{2}0))∗a^{2}+(128308652523796125897+ O(11^{2}0)) ∗ a − 216030589636627076766 + O(11^{2}0)q^{7} + (179872375456932504195 + O(11^{2}0)) ∗ a^{2} + (41167414242319362385+O(11^{2}0))∗a−210841057517064670475+O(11^{2}0)q^{8}+−(196227769989181940335+ O(11^{2}0)) ∗ a^{2} − (104526655010805199407 + O(11^{2}0)) ∗ a + 197438649616604575264 + O(11^{2}0)q^{9} + −(71391365500713487726+O(11^{2}0))∗a^{2}+(237140037334375000868+O(11^{2}0))∗a+136642671148158180752+ O(11^{2}0)q^{10} +−(6471662588803686460∗11+O(11^{2}0))∗a^{2} +(21245108018379929106∗11+O(11^{2}0))∗a− 20686351859699962235∗11+O(11^{2}0)q^{11}+−(267173984533929290358+O(11^{2}0))∗a^{2}+(99754482923420751911+ O(11^{2}0)) ∗ a + 141597952696203725694 + O(11^{2}0)q^{12} + −815730721q^{13} + −(51096450356173876496 + O(11^{2}0)) ∗ a^{2} + (257257083675995671626 + O(11^{2}0)) ∗ a + 237408682042918616888 + O(11^{2}0)q^{14} + −(289934825349939680807+O(11^{2}0))∗a^{2}−(11000262763865428178∗11+O(11^{2}0))∗a+224147636419749071020+ O(11^{2}0)q^{15} + (295283955689600901776 + O(11^{2}0)) ∗ a^{2} − (20206118722874328192 + O(11^{2}0)) ∗ a + 300346790331365632049+O(11^{2}0)q^{16}+(127107690642767188535+O(11^{2}0))∗a^{2}−(290403863786268544102+ O(11^{2}0)) ∗ a − 206063594537721036723 + O(11^{2}0)q^{17} + (35828085643784992027 + O(11^{2}0)) ∗ a^{2} − (217264071391869846631+O(11^{2}0))∗a−126469075039791961352+O(11^{2}0)q^{18}+(21240962067465993280+ O(11^{2}0)) ∗ a^{2} − (14742874762373252012 + O(11^{2}0)) ∗ a + 113231826398361026205 + O(11^{2}0)q^{19} + −(65135236818980544173+O(11^{2}0))∗a^{2}−(278827988407146949560+O(11^{2}0))∗a−7831435695833978885+ O(11^{2}0)q^{20} + (97651087496701653768 + O(11^{2}0)) ∗ a^{2} − (5896933888367694243 ∗ 11 + O(11^{2}0)) ∗ a − 171489876588728548432+O(11^{2}0)q^{21} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3}+−179872375456932504195+O(11^{2}0)x^{2}+−41167414242319624529+ O(11^{2}0)x + 210841057517064670475 + O(11^{2}0). The slope of f_{0} is 1.

20. Consider f = q+5∗z^{5}+3∗z^{4}+3∗z^{3}+5∗z^{2}+z+7q^{2}+8∗z^{5}+8∗z^{4}+8∗z^{3}+5∗z+6q^{3}+8∗z^{5}+9∗z^{4}+9∗ z^{3} +3∗z^{2} +4∗z+4q^{4} +8∗z^{5} +5∗z^{4} +5∗z^{3} +2∗z^{2} +8∗zq^{5} +9∗z^{5} +10∗z^{4} +10∗z^{3} +3∗z^{2} +6∗z+10q^{6} + z^{5}+z^{4}+z^{3}+2∗zq^{7}+6∗z^{5}+3∗z^{4}+3∗z^{3}+2∗z^{2}+4∗zq^{8}+z^{5}+2∗z^{4}+2∗z^{3}+3∗z^{2}+z+2q^{9}+7∗z^{5}+9∗z^{4}+ 9∗z^{3}+6∗z^{2}+z+7q^{10}+7∗z^{4}+7∗z^{3}+10∗z^{2}+4∗z+1q^{12}+8q^{13}+7∗z^{5}+10∗z^{4}+10∗z^{3}+9∗z^{2}+7q^{14}+7∗ z^{5}+z^{4}+z^{3}+4∗z^{2}+9∗z+2q^{15}+3∗z^{5}+3∗z^{4}+3∗z^{3}+6∗z+9q^{16}+z^{5}+4∗z^{4}+4∗z^{3}+9∗z^{2}+10∗z+5q^{17}+ 4∗z^{5} +3∗z^{4} +3∗z^{3} +8∗z^{2} +9∗z +8q^{18} +9∗z^{5} +10∗z^{4} +10∗z^{3} +3∗z^{2} +6∗z +7q^{19} +2∗z^{5} +8∗z^{4} +8∗ z^{3} +7∗z^{2} +9∗z +6q^{20} +z^{5} +8∗z^{4} +8∗z^{3} +10∗z^{2} +6∗z +1q^{21} +⋯ ∈ S_{18}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{6} +3x^{4} +4x^{3} +6x^{2} +7x+2 = 0. We have ρ_{f} ⊗χ^{9} ∼ ρ_{g}, where g is the weight-6 newform g = q +z^{1}067q^{2} +z^{7}52q^{3} +z^{6}96q^{4} +z^{8}39q^{5} +z^{4}89q^{6} +z^{3}61q^{7} +z^{3}18q^{8} +z^{5}6q^{9} +z^{5}76q^{10} +z^{5}17q^{11} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{8}13)ω^{6} & 0 \\ 0 & u(z^{5}17)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(131446899915469254352+ O(11^{2}0)) ∗ a^{2} − (16668102967084971608 + O(11^{2}0)) ∗ a − 143028618626631968157 + O(11^{2}0)q^{3} + a^{2} − 131072q^{4}+(248431489477194627658+O(11^{2}0))∗a^{2}−(62985048310559850397+O(11^{2}0))∗a−107124721210528445277+ O(11^{2}0)q^{5} + −(112777766827640977646 + O(11^{2}0)) ∗ a^{2} + (115594248386970322716 + O(11^{2}0)) ∗ a − 65634966030451128723+O(11^{2}0)q^{6}+−(74876497582956721159+O(11^{2}0))∗a^{2}+(128308652523796125897+ O(11^{2}0)) ∗ a − 216030589636627076766 + O(11^{2}0)q^{7} + (179872375456932504195 + O(11^{2}0)) ∗ a^{2} + (41167414242319362385+O(11^{2}0))∗a−210841057517064670475+O(11^{2}0)q^{8}+−(196227769989181940335+ O(11^{2}0)) ∗ a^{2} − (104526655010805199407 + O(11^{2}0)) ∗ a + 197438649616604575264 + O(11^{2}0)q^{9} + −(71391365500713487726+O(11^{2}0))∗a^{2}+(237140037334375000868+O(11^{2}0))∗a+136642671148158180752+ O(11^{2}0)q^{10} +−(6471662588803686460∗11+O(11^{2}0))∗a^{2} +(21245108018379929106∗11+O(11^{2}0))∗a− 20686351859699962235∗11+O(11^{2}0)q^{11}+−(267173984533929290358+O(11^{2}0))∗a^{2}+(99754482923420751911+ O(11^{2}0)) ∗ a + 141597952696203725694 + O(11^{2}0)q^{12} + −815730721q^{13} + −(51096450356173876496 + O(11^{2}0)) ∗ a^{2} + (257257083675995671626 + O(11^{2}0)) ∗ a + 237408682042918616888 + O(11^{2}0)q^{14} + −(289934825349939680807+O(11^{2}0))∗a^{2}−(11000262763865428178∗11+O(11^{2}0))∗a+224147636419749071020+ O(11^{2}0)q^{15} + (295283955689600901776 + O(11^{2}0)) ∗ a^{2} − (20206118722874328192 + O(11^{2}0)) ∗ a + 300346790331365632049+O(11^{2}0)q^{16}+(127107690642767188535+O(11^{2}0))∗a^{2}−(290403863786268544102+ O(11^{2}0)) ∗ a − 206063594537721036723 + O(11^{2}0)q^{17} + (35828085643784992027 + O(11^{2}0)) ∗ a^{2} − (217264071391869846631+O(11^{2}0))∗a−126469075039791961352+O(11^{2}0)q^{18}+(21240962067465993280+ O(11^{2}0)) ∗ a^{2} − (14742874762373252012 + O(11^{2}0)) ∗ a + 113231826398361026205 + O(11^{2}0)q^{19} + −(65135236818980544173+O(11^{2}0))∗a^{2}−(278827988407146949560+O(11^{2}0))∗a−7831435695833978885+ O(11^{2}0)q^{20} + (97651087496701653768 + O(11^{2}0)) ∗ a^{2} − (5896933888367694243 ∗ 11 + O(11^{2}0)) ∗ a − 171489876588728548432+O(11^{2}0)q^{21} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3}+−179872375456932504195+O(11^{2}0)x^{2}+−41167414242319624529+ O(11^{2}0)x + 210841057517064670475 + O(11^{2}0). The slope of f_{0} is 1.

21. Consider f = q+3∗z^{5}+7∗z^{4}+3∗z^{3}+6∗z^{2}+9∗z+6q^{2}+5∗z^{5}+8∗z^{4}+5∗z^{3}+10∗z^{2}+4∗zq^{3}+8∗z^{5}+4∗ z^{4}+8∗z^{3}+5∗z^{2}+2∗z+1q^{4}+8∗z^{5}+4∗z^{4}+8∗z^{3}+5∗z^{2}+2∗z+4q^{5}+9∗z^{5}+10∗z^{4}+9∗z^{3}+7∗z^{2}+5∗z+ 1q^{6}+2∗z^{5}+z^{4}+2∗z^{3}+4∗z^{2}+6∗z+2q^{7}+3∗z^{5}+7∗z^{4}+3∗z^{3}+6∗z^{2}+9∗z+3q^{8}+9∗z^{5}+10∗z^{4}+9∗z^{3}+7∗ z^{2}+5∗zq^{9}+8q^{10}+4∗z^{5}+2∗z^{4}+4∗z^{3}+8∗z^{2}+z+10q^{12}+8q^{13}+3∗z^{5}+7∗z^{4}+3∗z^{3}+6∗z^{2}+9∗z+8q^{14}+ 8∗z^{5}+4∗z^{4}+8∗z^{3}+5∗z^{2}+2∗z+10q^{15}+9∗z^{5}+10∗z^{4}+9∗z^{3}+7∗z^{2}+5∗z+5q^{16}+3∗z^{5}+7∗z^{4}+3∗z^{3}+ 6∗z^{2} +9∗z+4q^{17} +3∗z^{5} +7∗z^{4} +3∗z^{3} +6∗z^{2} +9∗z+4q^{18} +6∗z^{5} +3∗z^{4} +6∗z^{3} +z^{2} +7∗z+7q^{19} +z^{5} + 6∗z^{4}+z^{3}+2∗z^{2}+3∗z+9q^{20}+7∗z^{5}+9∗z^{4}+7∗z^{3}+3∗z^{2}+10∗z+8q^{21}+⋯ ∈ S_{18}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{6} + 3x^{4} + 4x^{3} + 6x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{3}1q^{2} + z^{2}5q^{3} + z^{2}2q^{4} + z^{5}q^{5} + z^{5}6q^{6} + z^{3}8q^{7} + z^{5}0q^{8} + z^{7}3q^{9} + 8q^{10} + z^{6}5q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{5}5)ω^{1} & 0 \\ 0 & u(z^{6}5)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(251157685445580017414+ O(11^{2}0))∗a−211522979086522098313+O(11^{2}0)q^{3}+(278917505612855520135+O(11^{2}0))∗a+265801768396367255290+ O(11^{2}0)q^{4}+(95897020197850878544+O(11^{2}0))∗a−168834143089696130454+O(11^{2}0)q^{5}+(241304523491755823630+ O(11^{2}0))∗a−8989424310386254763+O(11^{2}0)q^{6}+(229375404370051590890+O(11^{2}0))∗a−162731959496640124197+ O(11^{2}0)q^{7}+−(239147309550726811805+O(11^{2}0))∗a−84360658941818716108+O(11^{2}0)q^{8}+(320725115062636248902+ O(11^{2}0)) ∗ a − 101278434917921209384 + O(11^{2}0)q^{9} + (722302050528306190 ∗ 11^{2} + O(11^{2}0)) ∗ a − 271905961476176858878+O(11^{2}0)q^{10}+(27995605320306607628∗11+O(11^{2}0))∗a−661298934354984443∗ 11^{2} + O(11^{2}0)q^{11} + (274239627763664940261 + O(11^{2}0)) ∗ a + 38301770559711586666 + O(11^{2}0)q^{12} + −815730721q^{13} + (234084917845784294543 + O(11^{2}0)) ∗ a + 266349477935623914141 + O(11^{2}0)q^{14} + −(103657139201945751489+O(11^{2}0))∗a+278945581592450353146+O(11^{2}0)q^{15}+(71504967044455391505+ O(11^{2}0))∗a−72254448793848104632+O(11^{2}0)q^{16}+(95827848895805211868+O(11^{2}0))∗a+84196480384593878959+ O(11^{2}0)q^{17}+−(51759889030508192918+O(11^{2}0))∗a−329528320418896970043+O(11^{2}0)q^{18}+(205191804344267328854+ O(11^{2}0)) ∗ a − 290907798149866244748 + O(11^{2}0)q^{19} + −(28299446300072876936 + O(11^{2}0)) ∗ a − 167803151868169340494+O(11^{2}0)q^{20}+−(34110964855482709133+O(11^{2}0))∗a−90759072901225433777+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −278917505612855520135 + O(11^{2}0)x + −265801768396367386362 + O(11^{2}0). The slope of f_{0} is 1.

22. Consider f = q+8∗z^{5}+4∗z^{4}+8∗z^{3}+5∗z^{2}+2∗z+4q^{2}+6∗z^{5}+3∗z^{4}+6∗z^{3}+z^{2}+7∗z+4q^{3}+3∗z^{5}+7∗ z^{4}+3∗z^{3}+6∗z^{2}+9∗z+3q^{4}+3∗z^{5}+7∗z^{4}+3∗z^{3}+6∗z^{2}+9∗z+6q^{5}+2∗z^{5}+z^{4}+2∗z^{3}+4∗z^{2}+6∗z+6q^{6}+ 9∗z^{5}+10∗z^{4}+9∗z^{3}+7∗z^{2}+5∗z+8q^{7}+8∗z^{5}+4∗z^{4}+8∗z^{3}+5∗z^{2}+2∗z+1q^{8}+2∗z^{5}+z^{4}+2∗z^{3}+4∗z^{2}+ 6∗z+5q^{9}+8q^{10}+7∗z^{5}+9∗z^{4}+7∗z^{3}+3∗z^{2}+10∗zq^{12}+8q^{13}+8∗z^{5}+4∗z^{4}+8∗z^{3}+5∗z^{2}+2∗z+6q^{14}+3∗ z^{5}+7∗z^{4}+3∗z^{3}+6∗z^{2}+9∗z+1q^{15}+2∗z^{5}+z^{4}+2∗z^{3}+4∗z^{2}+6∗z+10q^{16}+8∗z^{5}+4∗z^{4}+8∗z^{3}+5∗z^{2}+ 2∗z+2q^{17}+8∗z^{5}+4∗z^{4}+8∗z^{3}+5∗z^{2}+2∗z+2q^{18}+5∗z^{5}+8∗z^{4}+5∗z^{3}+10∗z^{2}+4∗z+3q^{19}+10∗z^{5}+ 5∗z^{4}+10∗z^{3}+9∗z^{2}+8∗z+1q^{20}+4∗z^{5}+2∗z^{4}+4∗z^{3}+8∗z^{2}+z+7q^{21}+⋯ ∈ S_{18}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{6} + 3x^{4} + 4x^{3} + 6x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{1}01q^{2} + z^{3}5q^{3} + z^{2}q^{4} + z^{5}5q^{5} + z^{1}6q^{6} + z^{5}8q^{7} + z^{7}0q^{8} + z^{8}3q^{9} + 8q^{10} + z^{1}15q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{5})ω^{1} & 0 \\ 0 & u(z^{1}15)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(251157685445580017414+ O(11^{2}0))∗a−211522979086522098313+O(11^{2}0)q^{3}+(278917505612855520135+O(11^{2}0))∗a+265801768396367255290+ O(11^{2}0)q^{4}+(95897020197850878544+O(11^{2}0))∗a−168834143089696130454+O(11^{2}0)q^{5}+(241304523491755823630+ O(11^{2}0))∗a−8989424310386254763+O(11^{2}0)q^{6}+(229375404370051590890+O(11^{2}0))∗a−162731959496640124197+ O(11^{2}0)q^{7}+−(239147309550726811805+O(11^{2}0))∗a−84360658941818716108+O(11^{2}0)q^{8}+(320725115062636248902+ O(11^{2}0)) ∗ a − 101278434917921209384 + O(11^{2}0)q^{9} + (722302050528306190 ∗ 11^{2} + O(11^{2}0)) ∗ a − 271905961476176858878+O(11^{2}0)q^{10}+(27995605320306607628∗11+O(11^{2}0))∗a−661298934354984443∗ 11^{2} + O(11^{2}0)q^{11} + (274239627763664940261 + O(11^{2}0)) ∗ a + 38301770559711586666 + O(11^{2}0)q^{12} + −815730721q^{13} + (234084917845784294543 + O(11^{2}0)) ∗ a + 266349477935623914141 + O(11^{2}0)q^{14} + −(103657139201945751489+O(11^{2}0))∗a+278945581592450353146+O(11^{2}0)q^{15}+(71504967044455391505+ O(11^{2}0))∗a−72254448793848104632+O(11^{2}0)q^{16}+(95827848895805211868+O(11^{2}0))∗a+84196480384593878959+ O(11^{2}0)q^{17}+−(51759889030508192918+O(11^{2}0))∗a−329528320418896970043+O(11^{2}0)q^{18}+(205191804344267328854+ O(11^{2}0)) ∗ a − 290907798149866244748 + O(11^{2}0)q^{19} + −(28299446300072876936 + O(11^{2}0)) ∗ a − 167803151868169340494+O(11^{2}0)q^{20}+−(34110964855482709133+O(11^{2}0))∗a−90759072901225433777+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −278917505612855520135 + O(11^{2}0)x + −265801768396367386362 + O(11^{2}0). The slope of f_{0} is 1.

23. Consider f = q+6∗z^{5}+8∗z^{4}+8∗z^{3}+6∗z^{2}+10∗z+4q^{2}+8∗z^{5}+8∗z^{4}+8∗z^{3}+5∗z+6q^{3}+8∗z^{5}+9∗z^{4}+ 9∗z^{3}+3∗z^{2}+4∗z+4q^{4}+8∗z^{5}+5∗z^{4}+5∗z^{3}+2∗z^{2}+8∗zq^{5}+2∗z^{5}+z^{4}+z^{3}+8∗z^{2}+5∗z+1q^{6}+10∗z^{5}+ 10∗z^{4}+10∗z^{3}+9∗zq^{7}+5∗z^{5}+8∗z^{4}+8∗z^{3}+9∗z^{2}+7∗zq^{8}+z^{5}+2∗z^{4}+2∗z^{3}+3∗z^{2}+z+2q^{9}+4∗z^{5}+2∗ z^{4}+2∗z^{3}+5∗z^{2}+10∗z+4q^{10}+7∗z^{4}+7∗z^{3}+10∗z^{2}+4∗z+1q^{12}+3q^{13}+7∗z^{5}+10∗z^{4}+10∗z^{3}+9∗z^{2}+ 7q^{14}+7∗z^{5}+z^{4}+z^{3}+4∗z^{2}+9∗z+2q^{15}+3∗z^{5}+3∗z^{4}+3∗z^{3}+6∗z+9q^{16}+10∗z^{5}+7∗z^{4}+7∗z^{3}+2∗z^{2}+ z+6q^{17} +7∗z^{5} +8∗z^{4} +8∗z^{3} +3∗z^{2} +2∗z+3q^{18} +2∗z^{5} +z^{4} +z^{3} +8∗z^{2} +5∗z+4q^{19} +2∗z^{5} +8∗z^{4} +8∗ z^{3} +7∗z^{2} +9∗z+6q^{20} +10∗z^{5} +3∗z^{4} +3∗z^{3} +z^{2} +5∗z+10q^{21} +⋯ ∈ S_{18}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{6} +3x^{4} +4x^{3} +6x^{2} +7x+2 = 0. We have ρ_{f} ⊗χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q +z^{1}067q^{2} +z^{7}52q^{3} +z^{6}96q^{4} +z^{8}39q^{5} +z^{4}89q^{6} +z^{3}61q^{7} +z^{3}18q^{8} +z^{5}6q^{9} +z^{5}76q^{10} +z^{5}17q^{11} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{8}13)ω^{1} & 0 \\ 0 & u(z^{5}17)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(149569575741345744067+ O(11^{2}0)) ∗ a^{2} + (55312490144383855402 + O(11^{2}0)) ∗ a − 311262222554677784262 + O(11^{2}0)q^{3} + a^{2} − 131072q^{4}+(18735205580472032079+O(11^{2}0))∗a^{2}+(74535500139276212860+O(11^{2}0))∗a−112747231876766915259+ O(11^{2}0)q^{5} + (301599169477698661086 + O(11^{2}0)) ∗ a^{2} − (250247632654785028979 + O(11^{2}0)) ∗ a − 169236403449841771533+O(11^{2}0)q^{6}+−(14335411719530183775+O(11^{2}0))∗a^{2}−(71105348579061835897+ O(11^{2}0))∗a−272972490743352732108+O(11^{2}0)q^{7}+(70877947609790528435+O(11^{2}0))∗a^{2}+(177822203473271662231+ O(11^{2}0)) ∗ a − 308645731076688667003 + O(11^{2}0)q^{8} + −(322659750603526840027 + O(11^{2}0)) ∗ a^{2} + (8969640084107277018+O(11^{2}0))∗a−259213134836902462758+O(11^{2}0)q^{9}+(190796495195215443151+ O(11^{2}0)) ∗ a^{2} − (198911597147031627410 + O(11^{2}0)) ∗ a − 56043492265978822175 + O(11^{2}0)q^{10} + −(26650494788326434517∗11+O(11^{2}0))∗a^{2}−(25072698844281164140∗11+O(11^{2}0))∗a+17376098921086446040∗ 11 + O(11^{2}0)q^{11} + −(244554450967432551860 + O(11^{2}0)) ∗ a^{2} + (311914471829335852487 + O(11^{2}0)) ∗ a − 72058598017010758280 + O(11^{2}0)q^{12} + 815730721q^{13} + (330330107196127219636 + O(11^{2}0)) ∗ a^{2} + (332321082741740077111+O(11^{2}0))∗a+20980894756417820882+O(11^{2}0)q^{14}+(218956033555645700275+ O(11^{2}0)) ∗ a^{2} − (2211839037262125497 ∗ 11 + O(11^{2}0)) ∗ a − 59683733851883073711 + O(11^{2}0)q^{15} + (195343373811952570539+O(11^{2}0))∗a^{2}−(81941124735669415598+O(11^{2}0))∗a+66923247509356939137+ O(11^{2}0)q^{16} + −(71777889533206699969 + O(11^{2}0)) ∗ a^{2} − (66884438390569181799 + O(11^{2}0)) ∗ a − 148847017660804663957+O(11^{2}0)q^{17}+−(127958027037448893128+O(11^{2}0))∗a^{2}−(116326071986922203044+ O(11^{2}0))∗a+94634718590851472095+O(11^{2}0)q^{18}+(11592447775930661168+O(11^{2}0))∗a^{2}+(262026825942027521220+ O(11^{2}0)) ∗ a − 284814153681123821615 + O(11^{2}0)q^{19} + −(111452875871509955905 + O(11^{2}0)) ∗ a^{2} − (88876602765122545672+O(11^{2}0))∗a+279884756033401996634+O(11^{2}0)q^{20}+(2398996655199154311+ O(11^{2}0))∗a^{2} +(17729132352949607553∗11+O(11^{2}0))∗a−261594364831706829605+O(11^{2}0)q^{21} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + −70877947609790528435+O(11^{2}0)x^{2}+−177822203473271924375+O(11^{2}0)x+308645731076688667003+ O(11^{2}0). The slope of f_{0} is 1.

24. Consider f = q+z^{5}+3∗z^{4}+3∗z^{3}+6∗z^{2}+8q^{2}+4∗z^{5}+7∗z^{4}+7∗z^{3}+9∗z^{2}+5∗z+3q^{3}+3∗z^{5}+10∗z^{4}+10∗ z^{3}+10∗z^{2}+10∗zq^{4}+7∗z^{5}+9∗z^{4}+9∗z^{3}+6∗z^{2}+zq^{5}+2∗z^{5}+7∗z^{4}+7∗z^{3}+4∗z^{2}+10∗z+4q^{6}+5∗z^{5}+6∗ z^{4}+6∗z^{3}+3∗z^{2}+9∗z+10q^{7}+5∗z^{5}+z^{4}+z^{3}+10∗z^{2}+3∗z+2q^{8}+5∗z^{5}+8∗z^{4}+8∗z^{3}+9∗z^{2}+7∗z+2q^{9}+ 4∗z^{5}+3∗z^{4}+3∗z^{3}+8∗z^{2}+9∗z+10q^{10}+4∗z^{5}+10∗z^{4}+10∗z^{3}+7∗z^{2}+2∗z+5q^{12}+3q^{13}+6∗z^{5}+4∗z^{2}+ 7∗zq^{14}+4∗z^{5}+6∗z^{4}+6∗z^{3}+6∗z^{2}+6∗z+4q^{15}+7∗z^{5}+4∗z^{4}+4∗z^{3}+2∗z^{2}+6∗z+1q^{16}+8∗z^{5}+8∗z^{4}+ 8∗z^{3}+5∗z+8q^{17}+8∗z^{5}+5∗z^{4}+5∗z^{3}+2∗z^{2}+8∗z+9q^{18}+2∗z^{5}+7∗z^{4}+7∗z^{3}+4∗z^{2}+10∗z+7q^{19}+6∗ z^{5}+6∗z^{4}+6∗z^{3}+z+2q^{20}+z^{5}+7∗z^{4}+7∗z^{3}+7∗z^{2}+7∗z+5q^{21}+⋯ ∈ S_{18}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{6} +3x^{4} +4x^{3} +6x^{2} +7x+2 = 0. We have ρ_{f} ⊗χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q+z^{1}097q^{2}+z^{2}92q^{3}+z^{1}006q^{4}+z^{1}249q^{5}+z^{5}9q^{6}+z^{1}311q^{7}+z^{8}38q^{8}+z^{6}16q^{9}+z^{1}016q^{10}+z^{3}67q^{11}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{9}63)ω^{1} & 0 \\ 0 & u(z^{3}67)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(149569575741345744067+ O(11^{2}0)) ∗ a^{2} + (55312490144383855402 + O(11^{2}0)) ∗ a − 311262222554677784262 + O(11^{2}0)q^{3} + a^{2} − 131072q^{4}+(18735205580472032079+O(11^{2}0))∗a^{2}+(74535500139276212860+O(11^{2}0))∗a−112747231876766915259+ O(11^{2}0)q^{5} + (301599169477698661086 + O(11^{2}0)) ∗ a^{2} − (250247632654785028979 + O(11^{2}0)) ∗ a − 169236403449841771533+O(11^{2}0)q^{6}+−(14335411719530183775+O(11^{2}0))∗a^{2}−(71105348579061835897+ O(11^{2}0))∗a−272972490743352732108+O(11^{2}0)q^{7}+(70877947609790528435+O(11^{2}0))∗a^{2}+(177822203473271662231+ O(11^{2}0)) ∗ a − 308645731076688667003 + O(11^{2}0)q^{8} + −(322659750603526840027 + O(11^{2}0)) ∗ a^{2} + (8969640084107277018+O(11^{2}0))∗a−259213134836902462758+O(11^{2}0)q^{9}+(190796495195215443151+ O(11^{2}0)) ∗ a^{2} − (198911597147031627410 + O(11^{2}0)) ∗ a − 56043492265978822175 + O(11^{2}0)q^{10} + −(26650494788326434517∗11+O(11^{2}0))∗a^{2}−(25072698844281164140∗11+O(11^{2}0))∗a+17376098921086446040∗ 11 + O(11^{2}0)q^{11} + −(244554450967432551860 + O(11^{2}0)) ∗ a^{2} + (311914471829335852487 + O(11^{2}0)) ∗ a − 72058598017010758280 + O(11^{2}0)q^{12} + 815730721q^{13} + (330330107196127219636 + O(11^{2}0)) ∗ a^{2} + (332321082741740077111+O(11^{2}0))∗a+20980894756417820882+O(11^{2}0)q^{14}+(218956033555645700275+ O(11^{2}0)) ∗ a^{2} − (2211839037262125497 ∗ 11 + O(11^{2}0)) ∗ a − 59683733851883073711 + O(11^{2}0)q^{15} + (195343373811952570539+O(11^{2}0))∗a^{2}−(81941124735669415598+O(11^{2}0))∗a+66923247509356939137+ O(11^{2}0)q^{16} + −(71777889533206699969 + O(11^{2}0)) ∗ a^{2} − (66884438390569181799 + O(11^{2}0)) ∗ a − 148847017660804663957+O(11^{2}0)q^{17}+−(127958027037448893128+O(11^{2}0))∗a^{2}−(116326071986922203044+ O(11^{2}0))∗a+94634718590851472095+O(11^{2}0)q^{18}+(11592447775930661168+O(11^{2}0))∗a^{2}+(262026825942027521220+ O(11^{2}0)) ∗ a − 284814153681123821615 + O(11^{2}0)q^{19} + −(111452875871509955905 + O(11^{2}0)) ∗ a^{2} − (88876602765122545672+O(11^{2}0))∗a+279884756033401996634+O(11^{2}0)q^{20}+(2398996655199154311+ O(11^{2}0))∗a^{2} +(17729132352949607553∗11+O(11^{2}0))∗a−261594364831706829605+O(11^{2}0)q^{21} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + −70877947609790528435+O(11^{2}0)x^{2}+−177822203473271924375+O(11^{2}0)x+308645731076688667003+ O(11^{2}0). The slope of f_{0} is 1.

25. Consider f = q+4∗z^{5}+10∗z^{2}+z+7q^{2}+10∗z^{5}+7∗z^{4}+7∗z^{3}+2∗z^{2}+z+4q^{3}+3∗z^{4}+3∗z^{3}+9∗z^{2}+8∗ z+7q^{4}+7∗z^{5}+8∗z^{4}+8∗z^{3}+3∗z^{2}+2∗z+5q^{5}+7∗z^{5}+3∗z^{4}+3∗z^{3}+10∗z^{2}+7∗z+1q^{6}+7∗z^{5}+6∗z^{4}+ 6∗z^{3} +8∗z^{2} +4∗z +3q^{7} +z^{5} +2∗z^{4} +2∗z^{3} +3∗z^{2} +zq^{8} +5∗z^{5} +z^{4} +z^{3} +10∗z^{2} +3∗z +4q^{9} +3∗z^{5} +6∗ z^{4} +6∗z^{3} +9∗z^{2} +3∗z +4q^{10} +7∗z^{5} +5∗z^{4} +5∗z^{3} +5∗z^{2} +5∗z +3q^{12} +3q^{13} +9∗z^{5} +z^{4} +z^{3} +9∗z^{2} + 4∗z +1q^{14} +4∗z^{4} +4∗z^{3} +z^{2} +7∗z +6q^{15} +z^{5} +4∗z^{4} +4∗z^{3} +9∗z^{2} +10∗zq^{16} +4∗z^{5} +7∗z^{4} +7∗z^{3} + 9∗z^{2} +5∗z +5q^{17} +7∗z^{5} +9∗z^{4} +9∗z^{3} +6∗z^{2} +z +9q^{18} +7∗z^{5} +3∗z^{4} +3∗z^{3} +10∗z^{2} +7∗z +4q^{19} +3∗ z^{5} +8∗z^{4} +8∗z^{3} +4∗z^{2} +z +8q^{20} +z^{4} +z^{3} +3∗z^{2} +10∗zq^{21} +⋯ ∈ S_{18}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{6} +3x^{4} +4x^{3} +6x^{2} +7x+2 = 0. We have ρ_{f} ⊗χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q +z^{9}7q^{2} +z^{5}52q^{3} +z^{4}26q^{4} +z^{4}39q^{5} +z^{6}49q^{6} +z^{1}121q^{7} +z^{1}238q^{8} +z^{1}26q^{9} +z^{5}36q^{10} +z^{4}7q^{11} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}283)ω^{1} & 0 \\ 0 & u(z^{4}7)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(149569575741345744067+ O(11^{2}0)) ∗ a^{2} + (55312490144383855402 + O(11^{2}0)) ∗ a − 311262222554677784262 + O(11^{2}0)q^{3} + a^{2} − 131072q^{4}+(18735205580472032079+O(11^{2}0))∗a^{2}+(74535500139276212860+O(11^{2}0))∗a−112747231876766915259+ O(11^{2}0)q^{5} + (301599169477698661086 + O(11^{2}0)) ∗ a^{2} − (250247632654785028979 + O(11^{2}0)) ∗ a − 169236403449841771533+O(11^{2}0)q^{6}+−(14335411719530183775+O(11^{2}0))∗a^{2}−(71105348579061835897+ O(11^{2}0))∗a−272972490743352732108+O(11^{2}0)q^{7}+(70877947609790528435+O(11^{2}0))∗a^{2}+(177822203473271662231+ O(11^{2}0)) ∗ a − 308645731076688667003 + O(11^{2}0)q^{8} + −(322659750603526840027 + O(11^{2}0)) ∗ a^{2} + (8969640084107277018+O(11^{2}0))∗a−259213134836902462758+O(11^{2}0)q^{9}+(190796495195215443151+ O(11^{2}0)) ∗ a^{2} − (198911597147031627410 + O(11^{2}0)) ∗ a − 56043492265978822175 + O(11^{2}0)q^{10} + −(26650494788326434517∗11+O(11^{2}0))∗a^{2}−(25072698844281164140∗11+O(11^{2}0))∗a+17376098921086446040∗ 11 + O(11^{2}0)q^{11} + −(244554450967432551860 + O(11^{2}0)) ∗ a^{2} + (311914471829335852487 + O(11^{2}0)) ∗ a − 72058598017010758280 + O(11^{2}0)q^{12} + 815730721q^{13} + (330330107196127219636 + O(11^{2}0)) ∗ a^{2} + (332321082741740077111+O(11^{2}0))∗a+20980894756417820882+O(11^{2}0)q^{14}+(218956033555645700275+ O(11^{2}0)) ∗ a^{2} − (2211839037262125497 ∗ 11 + O(11^{2}0)) ∗ a − 59683733851883073711 + O(11^{2}0)q^{15} + (195343373811952570539+O(11^{2}0))∗a^{2}−(81941124735669415598+O(11^{2}0))∗a+66923247509356939137+ O(11^{2}0)q^{16} + −(71777889533206699969 + O(11^{2}0)) ∗ a^{2} − (66884438390569181799 + O(11^{2}0)) ∗ a − 148847017660804663957+O(11^{2}0)q^{17}+−(127958027037448893128+O(11^{2}0))∗a^{2}−(116326071986922203044+ O(11^{2}0))∗a+94634718590851472095+O(11^{2}0)q^{18}+(11592447775930661168+O(11^{2}0))∗a^{2}+(262026825942027521220+ O(11^{2}0)) ∗ a − 284814153681123821615 + O(11^{2}0)q^{19} + −(111452875871509955905 + O(11^{2}0)) ∗ a^{2} − (88876602765122545672+O(11^{2}0))∗a+279884756033401996634+O(11^{2}0)q^{20}+(2398996655199154311+ O(11^{2}0))∗a^{2} +(17729132352949607553∗11+O(11^{2}0))∗a−261594364831706829605+O(11^{2}0)q^{21} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + −70877947609790528435+O(11^{2}0)x^{2}+−177822203473271924375+O(11^{2}0)x+308645731076688667003+ O(11^{2}0). The slope of f_{0} is 1.

26. Consider f = q+3∗z^{5}+7∗z^{4}+3∗z^{3}+6∗z^{2}+9∗z+7q^{2}+6∗z^{5}+3∗z^{4}+6∗z^{3}+z^{2}+7∗z+4q^{3}+3∗z^{5}+7∗ z^{4}+3∗z^{3}+6∗z^{2}+9∗z+3q^{4}+3∗z^{5}+7∗z^{4}+3∗z^{3}+6∗z^{2}+9∗z+6q^{5}+9∗z^{5}+10∗z^{4}+9∗z^{3}+7∗z^{2}+5∗z+ 5q^{6}+2∗z^{5}+z^{4}+2∗z^{3}+4∗z^{2}+6∗z+3q^{7}+3∗z^{5}+7∗z^{4}+3∗z^{3}+6∗z^{2}+9∗z+10q^{8}+2∗z^{5}+z^{4}+2∗z^{3}+4∗ z^{2}+6∗z+5q^{9}+3q^{10}+7∗z^{5}+9∗z^{4}+7∗z^{3}+3∗z^{2}+10∗zq^{12}+3q^{13}+8∗z^{5}+4∗z^{4}+8∗z^{3}+5∗z^{2}+2∗z+6q^{14}+ 3∗z^{5}+7∗z^{4}+3∗z^{3}+6∗z^{2}+9∗z+1q^{15}+2∗z^{5}+z^{4}+2∗z^{3}+4∗z^{2}+6∗z+10q^{16}+3∗z^{5}+7∗z^{4}+3∗z^{3}+6∗ z^{2}+9∗z+9q^{17}+3∗z^{5}+7∗z^{4}+3∗z^{3}+6∗z^{2}+9∗z+9q^{18}+6∗z^{5}+3∗z^{4}+6∗z^{3}+z^{2}+7∗z+8q^{19}+10∗z^{5}+5∗ z^{4}+10∗z^{3}+9∗z^{2}+8∗z+1q^{20}+7∗z^{5}+9∗z^{4}+7∗z^{3}+3∗z^{2}+10∗z+4q^{21}+⋯ ∈ S_{18}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{6} + 3x^{4} + 4x^{3} + 6x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{1}01q^{2} + z^{3}5q^{3} + z^{2}q^{4} + z^{5}5q^{5} + z^{1}6q^{6} + z^{5}8q^{7} + z^{7}0q^{8} + z^{8}3q^{9} + 8q^{10} + z^{1}15q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{5})ω^{6} & 0 \\ 0 & u(z^{1}15)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(218301364107339573073+ O(11^{2}0))∗a−317426206022229757661+O(11^{2}0)q^{3}+(197909813640634484302+O(11^{2}0))∗a−39775139136064597346+ O(11^{2}0)q^{4}+−(55623529301721924641+O(11^{2}0))∗a−18544470360882193960+O(11^{2}0)q^{5}+−(82871056430802840285+ O(11^{2}0))∗a−311362745582668031367+O(11^{2}0)q^{6}+(330509496135486050391+O(11^{2}0))∗a−5158415913546529731+ O(11^{2}0)q^{7}+−(234999919310435269341+O(11^{2}0))∗a−29602841799772490653+O(11^{2}0)q^{8}+−(35778824788734492864+ O(11^{2}0)) ∗ a − 196825649541021105678 + O(11^{2}0)q^{9} + −(15800652036949667864 ∗ 11 + O(11^{2}0)) ∗ a − 287147621133651957940+O(11^{2}0)q^{10}+(22783843855514819323∗11+O(11^{2}0))∗a−12062036429396396128∗ 11 + O(11^{2}0)q^{11} + (130606226486869935027 + O(11^{2}0)) ∗ a − 265516657881613764415 + O(11^{2}0)q^{12} + 815730721q^{13} + −(51685024526844878038 + O(11^{2}0)) ∗ a + 326796038033106269762 + O(11^{2}0)q^{14} + (172902938990627226437+O(11^{2}0))∗a+221788832920340810184+O(11^{2}0)q^{15}+−(74775165204249792274+ O(11^{2}0))∗a+42567000684170454534+O(11^{2}0)q^{16}+(234940862872644808364+O(11^{2}0))∗a+41759455222102692560+ O(11^{2}0)q^{17} + −(310305400614851292650 + O(11^{2}0)) ∗ a + 293919710981770294357 + O(11^{2}0)q^{18} + (25287435991802393279+O(11^{2}0))∗a−321281036701597434380+O(11^{2}0)q^{19}+(121186396666003681371+ O(11^{2}0))∗a+56889422011153215084+O(11^{2}0)q^{20}+(1800328397102481032+O(11^{2}0))∗a−199963198877026977364+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −197909813640634484302 + O(11^{2}0)x + 39775139136064466274 + O(11^{2}0). The slope of f_{0} is 1.

27. Consider f = q+8∗z^{5} +4∗z^{4} +8∗z^{3} +5∗z^{2} +2∗z +5q^{2} +5∗z^{5} +8∗z^{4} +5∗z^{3} +10∗z^{2} +4∗zq^{3} +8∗z^{5} + 4∗z^{4}+8∗z^{3}+5∗z^{2}+2∗z+1q^{4}+8∗z^{5}+4∗z^{4}+8∗z^{3}+5∗z^{2}+2∗z+4q^{5}+2∗z^{5}+z^{4}+2∗z^{3}+4∗z^{2}+6∗z+ 10q^{6}+9∗z^{5}+10∗z^{4}+9∗z^{3}+7∗z^{2}+5∗z+9q^{7}+8∗z^{5}+4∗z^{4}+8∗z^{3}+5∗z^{2}+2∗z+8q^{8}+9∗z^{5}+10∗z^{4}+9∗ z^{3}+7∗z^{2}+5∗zq^{9}+3q^{10}+4∗z^{5}+2∗z^{4}+4∗z^{3}+8∗z^{2}+z+10q^{12}+3q^{13}+3∗z^{5}+7∗z^{4}+3∗z^{3}+6∗z^{2}+9∗z+ 8q^{14}+8∗z^{5}+4∗z^{4}+8∗z^{3}+5∗z^{2}+2∗z+10q^{15}+9∗z^{5}+10∗z^{4}+9∗z^{3}+7∗z^{2}+5∗z+5q^{16}+8∗z^{5}+4∗z^{4}+8∗ z^{3}+5∗z^{2}+2∗z+7q^{17}+8∗z^{5}+4∗z^{4}+8∗z^{3}+5∗z^{2}+2∗z+7q^{18}+5∗z^{5}+8∗z^{4}+5∗z^{3}+10∗z^{2}+4∗z+4q^{19}+ z^{5}+6∗z^{4}+z^{3}+2∗z^{2}+3∗z+9q^{20}+4∗z^{5}+2∗z^{4}+4∗z^{3}+8∗z^{2}+z+3q^{21}+⋯ ∈ S_{18}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{6} + 3x^{4} + 4x^{3} + 6x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{3}1q^{2} + z^{2}5q^{3} + z^{2}2q^{4} + z^{5}q^{5} + z^{5}6q^{6} + z^{3}8q^{7} + z^{5}0q^{8} + z^{7}3q^{9} + 8q^{10} + z^{6}5q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{5}5)ω^{6} & 0 \\ 0 & u(z^{6}5)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(218301364107339573073+ O(11^{2}0))∗a−317426206022229757661+O(11^{2}0)q^{3}+(197909813640634484302+O(11^{2}0))∗a−39775139136064597346+ O(11^{2}0)q^{4}+−(55623529301721924641+O(11^{2}0))∗a−18544470360882193960+O(11^{2}0)q^{5}+−(82871056430802840285+ O(11^{2}0))∗a−311362745582668031367+O(11^{2}0)q^{6}+(330509496135486050391+O(11^{2}0))∗a−5158415913546529731+ O(11^{2}0)q^{7}+−(234999919310435269341+O(11^{2}0))∗a−29602841799772490653+O(11^{2}0)q^{8}+−(35778824788734492864+ O(11^{2}0)) ∗ a − 196825649541021105678 + O(11^{2}0)q^{9} + −(15800652036949667864 ∗ 11 + O(11^{2}0)) ∗ a − 287147621133651957940+O(11^{2}0)q^{10}+(22783843855514819323∗11+O(11^{2}0))∗a−12062036429396396128∗ 11 + O(11^{2}0)q^{11} + (130606226486869935027 + O(11^{2}0)) ∗ a − 265516657881613764415 + O(11^{2}0)q^{12} + 815730721q^{13} + −(51685024526844878038 + O(11^{2}0)) ∗ a + 326796038033106269762 + O(11^{2}0)q^{14} + (172902938990627226437+O(11^{2}0))∗a+221788832920340810184+O(11^{2}0)q^{15}+−(74775165204249792274+ O(11^{2}0))∗a+42567000684170454534+O(11^{2}0)q^{16}+(234940862872644808364+O(11^{2}0))∗a+41759455222102692560+ O(11^{2}0)q^{17} + −(310305400614851292650 + O(11^{2}0)) ∗ a + 293919710981770294357 + O(11^{2}0)q^{18} + (25287435991802393279+O(11^{2}0))∗a−321281036701597434380+O(11^{2}0)q^{19}+(121186396666003681371+ O(11^{2}0))∗a+56889422011153215084+O(11^{2}0)q^{20}+(1800328397102481032+O(11^{2}0))∗a−199963198877026977364+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −197909813640634484302 + O(11^{2}0)x + 39775139136064466274 + O(11^{2}0). The slope of f_{0} is 1.

28. Consider f = q +7q^{2} +9q^{3} +10q^{4} +5q^{5} +8q^{6} +4q^{7} +6q^{8} +2q^{10} +2q^{12} +6q^{13} +6q^{14} +q^{15} +4q^{16} +8q^{19} + 6q^{20}+3q^{21}+⋯ ∈ S_{20}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{3}+2x+9 = 0. We have ρ_{f} ⊗χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 6q^{2} + 4q^{3} + 6q^{4} + 4q^{5} + 2q^{6} + 9q^{7} + 10q^{8} + 2q^{10} + 7q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{1} & 0 \\ 0 & u(7)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+116423195320276415688+ O(11^{2}0)q^{2}+−224675068198352242176+O(11^{2}0)q^{3}+−150898831612306126702+O(11^{2}0)q^{4}+91653315089999987021+ O(11^{2}0)q^{5}+−255680284324054468739+O(11^{2}0)q^{6}+−180001910866333393284+O(11^{2}0)q^{7}+26354215159703407466+ O(11^{2}0)q^{8} + −23049344351147302029 ∗ 11 + O(11^{2}0)q^{9} + −333779947711910610557 + O(11^{2}0)q^{10} + 7930710172591148955 ∗ 11 + O(11^{2}0)q^{11} + −96869971134591424966 + O(11^{2}0)q^{12} + 10604499373q^{13} + 77611858350341397154+O(11^{2}0)q^{14}+117263288228950941034+O(11^{2}0)q^{15}+−288003090393573503109+ O(11^{2}0)q^{16} + 1563387279478799428 ∗ 11^{2} + O(11^{2}0)q^{17} + 2874233011214049542 ∗ 11 + O(11^{2}0)q^{18} + 183201729836378404170+O(11^{2}0)q^{19}+−306889067529710389069+O(11^{2}0)q^{20}+−212987487555839193865+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

29. Consider f = q +z^{1}35q^{2} +z^{9}72q^{3} +z^{1}048q^{4} +z^{5}87q^{5} +z^{1}107q^{6} +z^{1}47q^{7} +z^{5}75q^{8} +z^{1}227q^{9} +z^{7}22q^{10} + z^{6}90q^{12} + 6q^{13} + z^{2}82q^{14} + z^{2}29q^{15} + z^{1}086q^{16} + z^{1}89q^{17} + z^{3}2q^{18} + z^{9}74q^{19} + z^{3}05q^{20} + z^{1}119q^{21} + ⋯ ∈ S_{20}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{3} + 2x + 9 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{2}q^{2} + z^{1}238q^{3} + z^{7}82q^{4} + z^{5}5q^{5} + z^{1}240q^{6} + z^{5}46q^{7} + z^{1}76q^{8} + z^{4}29q^{9} + z^{5}7q^{10} + z^{6}6q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}264)ω^{8} & 0 \\ 0 & u(z^{6}6)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(261944427659166551128+ O(11^{2}0)) ∗ a^{2} − (1972874748507731608 ∗ 11^{2} + O(11^{2}0)) ∗ a + 49991356319423805930 + O(11^{2}0)q^{3} + a^{2} − 524288q^{4} + −(15629132323290452509 + O(11^{2}0)) ∗ a^{2} + (223651836423998699113 + O(11^{2}0)) ∗ a − 332974324924084149294+O(11^{2}0)q^{5}+−(8951521684482309093+O(11^{2}0))∗a^{2}+(167099212046762761076+ O(11^{2}0)) ∗ a − 30176124299100187435 + O(11^{2}0)q^{6} + −(39771220601587384255 + O(11^{2}0)) ∗ a^{2} + (152277785942337391989+O(11^{2}0))∗a−9136556182243961791+O(11^{2}0)q^{7}+(211246717601930227913+ O(11^{2}0))∗a^{2}−(24698822317382780936+O(11^{2}0))∗a+122246286436849476560+O(11^{2}0)q^{8}+(146836213696897186347+ O(11^{2}0))∗a^{2}−(248191317888897727249+O(11^{2}0))∗a+32711340319554954509+O(11^{2}0)q^{9}+−(30126170303609906331∗ 11 + O(11^{2}0)) ∗ a^{2} − (334779423914629599808 + O(11^{2}0)) ∗ a + 231218683686685030648 + O(11^{2}0)q^{10} + (27724926688272700048∗11+O(11^{2}0))∗a^{2}−(22313158266492118952∗11+O(11^{2}0))∗a−2085444958414109814∗ 11^{2} +O(11^{2}0)q^{11} +(158513676505459650987+O(11^{2}0))∗a^{2} +(265122853337554474545+O(11^{2}0))∗a+ 262840387354599694039+O(11^{2}0)q^{12} +10604499373q^{13} +(21892042919591779545∗11+O(11^{2}0))∗a^{2} + (294884136818829750381+O(11^{2}0))∗a+220609876938828292459+O(11^{2}0)q^{14}+(128470773818748458112+ O(11^{2}0)) ∗ a^{2} + (165105518390895303192 + O(11^{2}0)) ∗ a + 166545980152304693617 + O(11^{2}0)q^{15} + (73314554574014209497+O(11^{2}0))∗a^{2}+(307209712803010740916+O(11^{2}0))∗a+2487159702599744143∗ 11^{2} +O(11^{2}0)q^{16} +(125628264243904985558+O(11^{2}0))∗a^{2} +(578602854718024654∗11+O(11^{2}0))∗a+ 103547999653940705064+O(11^{2}0)q^{17}+−(80571504130870246645+O(11^{2}0))∗a^{2}+(18159551125799218623+ O(11^{2}0)) ∗ a + 46099678710407259426 + O(11^{2}0)q^{18} + (273130915323254570617 + O(11^{2}0)) ∗ a^{2} + (220892462696867813243+O(11^{2}0))∗a+97628645360743973743+O(11^{2}0)q^{19}+(42125126896567443114+ O(11^{2}0)) ∗ a^{2} + (228488996609381764969 + O(11^{2}0)) ∗ a + 237550397517689306781 + O(11^{2}0)q^{20} + (224538403870890159547+O(11^{2}0))∗a^{2}−(153925696639809943195+O(11^{2}0))∗a−303652499311060775277+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + −211246717601930227913 + O(11^{2}0)x^{2} + 24698822317381732360 + O(11^{2}0)x + −122246286436849476560 + O(11^{2}0). The slope of f_{0} is 1.

30. Consider f = q + z^{1}55q^{2} + z^{5}2q^{3} + z^{8}88q^{4} + z^{1}137q^{5} + z^{2}07q^{6} + z^{2}87q^{7} + z^{1}005q^{8} + z^{1}97q^{9} + z^{1}292q^{10} + z^{9}40q^{12} + 6q^{13} + z^{4}42q^{14} + z^{1}189q^{15} + z^{1}306q^{16} + z^{7}49q^{17} + z^{3}52q^{18} + z^{7}4q^{19} + z^{6}95q^{20} + z^{3}39q^{21} + ⋯ ∈ S_{20}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{3} + 2x + 9 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{2}2q^{2} + z^{3}18q^{3} + z^{6}22q^{4} + z^{6}05q^{5} + z^{3}40q^{6} + z^{6}86q^{7} + z^{6}06q^{8} + z^{7}29q^{9} + z^{6}27q^{10} + z^{7}26q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{6}04)ω^{8} & 0 \\ 0 & u(z^{7}26)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(261944427659166551128+ O(11^{2}0)) ∗ a^{2} − (1972874748507731608 ∗ 11^{2} + O(11^{2}0)) ∗ a + 49991356319423805930 + O(11^{2}0)q^{3} + a^{2} − 524288q^{4} + −(15629132323290452509 + O(11^{2}0)) ∗ a^{2} + (223651836423998699113 + O(11^{2}0)) ∗ a − 332974324924084149294+O(11^{2}0)q^{5}+−(8951521684482309093+O(11^{2}0))∗a^{2}+(167099212046762761076+ O(11^{2}0)) ∗ a − 30176124299100187435 + O(11^{2}0)q^{6} + −(39771220601587384255 + O(11^{2}0)) ∗ a^{2} + (152277785942337391989+O(11^{2}0))∗a−9136556182243961791+O(11^{2}0)q^{7}+(211246717601930227913+ O(11^{2}0))∗a^{2}−(24698822317382780936+O(11^{2}0))∗a+122246286436849476560+O(11^{2}0)q^{8}+(146836213696897186347+ O(11^{2}0))∗a^{2}−(248191317888897727249+O(11^{2}0))∗a+32711340319554954509+O(11^{2}0)q^{9}+−(30126170303609906331∗ 11 + O(11^{2}0)) ∗ a^{2} − (334779423914629599808 + O(11^{2}0)) ∗ a + 231218683686685030648 + O(11^{2}0)q^{10} + (27724926688272700048∗11+O(11^{2}0))∗a^{2}−(22313158266492118952∗11+O(11^{2}0))∗a−2085444958414109814∗ 11^{2} +O(11^{2}0)q^{11} +(158513676505459650987+O(11^{2}0))∗a^{2} +(265122853337554474545+O(11^{2}0))∗a+ 262840387354599694039+O(11^{2}0)q^{12} +10604499373q^{13} +(21892042919591779545∗11+O(11^{2}0))∗a^{2} + (294884136818829750381+O(11^{2}0))∗a+220609876938828292459+O(11^{2}0)q^{14}+(128470773818748458112+ O(11^{2}0)) ∗ a^{2} + (165105518390895303192 + O(11^{2}0)) ∗ a + 166545980152304693617 + O(11^{2}0)q^{15} + (73314554574014209497+O(11^{2}0))∗a^{2}+(307209712803010740916+O(11^{2}0))∗a+2487159702599744143∗ 11^{2} +O(11^{2}0)q^{16} +(125628264243904985558+O(11^{2}0))∗a^{2} +(578602854718024654∗11+O(11^{2}0))∗a+ 103547999653940705064+O(11^{2}0)q^{17}+−(80571504130870246645+O(11^{2}0))∗a^{2}+(18159551125799218623+ O(11^{2}0)) ∗ a + 46099678710407259426 + O(11^{2}0)q^{18} + (273130915323254570617 + O(11^{2}0)) ∗ a^{2} + (220892462696867813243+O(11^{2}0))∗a+97628645360743973743+O(11^{2}0)q^{19}+(42125126896567443114+ O(11^{2}0)) ∗ a^{2} + (228488996609381764969 + O(11^{2}0)) ∗ a + 237550397517689306781 + O(11^{2}0)q^{20} + (224538403870890159547+O(11^{2}0))∗a^{2}−(153925696639809943195+O(11^{2}0))∗a−303652499311060775277+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + −211246717601930227913 + O(11^{2}0)x^{2} + 24698822317381732360 + O(11^{2}0)x + −122246286436849476560 + O(11^{2}0). The slope of f_{0} is 1.

31. Consider f = q + z^{3}75q^{2} + z^{5}72q^{3} + z^{4}58q^{4} + z^{5}37q^{5} + z^{9}47q^{6} + z^{4}97q^{7} + z^{4}15q^{8} + z^{8}37q^{9} + z^{9}12q^{10} + z^{1}030q^{12}+6q^{13}+z^{8}72q^{14}+z^{1}109q^{15}+z^{1}066q^{16}+z^{2}59q^{17}+z^{1}212q^{18}+z^{8}14q^{19}+z^{9}95q^{20}+z^{1}069q^{21}+⋯ ∈ S_{20}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{3} + 2x + 9 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{2}42q^{2} + z^{8}38q^{3} + z^{1}92q^{4} + z^{5}q^{5} + z^{1}080q^{6} + z^{8}96q^{7} + z^{1}6q^{8} + z^{3}9q^{9} + z^{2}47q^{10} + z^{6}q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}324)ω^{8} & 0 \\ 0 & u(z^{6})ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(261944427659166551128+ O(11^{2}0)) ∗ a^{2} − (1972874748507731608 ∗ 11^{2} + O(11^{2}0)) ∗ a + 49991356319423805930 + O(11^{2}0)q^{3} + a^{2} − 524288q^{4} + −(15629132323290452509 + O(11^{2}0)) ∗ a^{2} + (223651836423998699113 + O(11^{2}0)) ∗ a − 332974324924084149294+O(11^{2}0)q^{5}+−(8951521684482309093+O(11^{2}0))∗a^{2}+(167099212046762761076+ O(11^{2}0)) ∗ a − 30176124299100187435 + O(11^{2}0)q^{6} + −(39771220601587384255 + O(11^{2}0)) ∗ a^{2} + (152277785942337391989+O(11^{2}0))∗a−9136556182243961791+O(11^{2}0)q^{7}+(211246717601930227913+ O(11^{2}0))∗a^{2}−(24698822317382780936+O(11^{2}0))∗a+122246286436849476560+O(11^{2}0)q^{8}+(146836213696897186347+ O(11^{2}0))∗a^{2}−(248191317888897727249+O(11^{2}0))∗a+32711340319554954509+O(11^{2}0)q^{9}+−(30126170303609906331∗ 11 + O(11^{2}0)) ∗ a^{2} − (334779423914629599808 + O(11^{2}0)) ∗ a + 231218683686685030648 + O(11^{2}0)q^{10} + (27724926688272700048∗11+O(11^{2}0))∗a^{2}−(22313158266492118952∗11+O(11^{2}0))∗a−2085444958414109814∗ 11^{2} +O(11^{2}0)q^{11} +(158513676505459650987+O(11^{2}0))∗a^{2} +(265122853337554474545+O(11^{2}0))∗a+ 262840387354599694039+O(11^{2}0)q^{12} +10604499373q^{13} +(21892042919591779545∗11+O(11^{2}0))∗a^{2} + (294884136818829750381+O(11^{2}0))∗a+220609876938828292459+O(11^{2}0)q^{14}+(128470773818748458112+ O(11^{2}0)) ∗ a^{2} + (165105518390895303192 + O(11^{2}0)) ∗ a + 166545980152304693617 + O(11^{2}0)q^{15} + (73314554574014209497+O(11^{2}0))∗a^{2}+(307209712803010740916+O(11^{2}0))∗a+2487159702599744143∗ 11^{2} +O(11^{2}0)q^{16} +(125628264243904985558+O(11^{2}0))∗a^{2} +(578602854718024654∗11+O(11^{2}0))∗a+ 103547999653940705064+O(11^{2}0)q^{17}+−(80571504130870246645+O(11^{2}0))∗a^{2}+(18159551125799218623+ O(11^{2}0)) ∗ a + 46099678710407259426 + O(11^{2}0)q^{18} + (273130915323254570617 + O(11^{2}0)) ∗ a^{2} + (220892462696867813243+O(11^{2}0))∗a+97628645360743973743+O(11^{2}0)q^{19}+(42125126896567443114+ O(11^{2}0)) ∗ a^{2} + (228488996609381764969 + O(11^{2}0)) ∗ a + 237550397517689306781 + O(11^{2}0)q^{20} + (224538403870890159547+O(11^{2}0))∗a^{2}−(153925696639809943195+O(11^{2}0))∗a−303652499311060775277+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + −211246717601930227913 + O(11^{2}0)x^{2} + 24698822317381732360 + O(11^{2}0)x + −122246286436849476560 + O(11^{2}0). The slope of f_{0} is 1.

32. Consider f = q + 5q^{2} + 7q^{3} + 8q^{4} + 4q^{5} + 2q^{6} + 5q^{7} + 10q^{8} + q^{9} + 9q^{10} + q^{12} + 6q^{13} + 3q^{14} + 6q^{15} + 2q^{16} + 8q^{17} + 5q^{18} + 9q^{19} + 10q^{20} + 2q^{21} + ⋯ ∈ S_{20}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{3} + 2x + 9 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q + 8q^{2} + 6q^{3} + 2q^{4} + 3q^{5} + 4q^{6} + 7q^{7} + 4q^{8} + 5q^{9} + 2q^{10} + 3q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{8} & 0 \\ 0 & u(3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−147619578742164675999+ O(11^{2}0)q^{2}+144268645987620008312+O(11^{2}0)q^{3}+119694842743539212751+O(11^{2}0)q^{4}+80305151870029437135+ O(11^{2}0)q^{5}+−281438782306002185852+O(11^{2}0)q^{6}+115296854666829851108+O(11^{2}0)q^{7}+−84901567145413118+ O(11^{2}0)q^{8}+154682306545564341985+O(11^{2}0)q^{9}+−277264805125848683794+O(11^{2}0)q^{10}+25363522588800060763∗ 11 + O(11^{2}0)q^{11} + −79813011207949546262 + O(11^{2}0)q^{12} + 10604499373q^{13} + 96362691120407490958 + O(11^{2}0)q^{14}+80050292556712691923+O(11^{2}0)q^{15}+204861993258385865272+O(11^{2}0)q^{16}+295848783646984984259+ O(11^{2}0)q^{17}+301646133348570110688+O(11^{2}0)q^{18}+−62837385514577433795+O(11^{2}0)q^{19}+85842204075083025646+ O(11^{2}0)q^{20} + 16393959567367867531 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

33. Consider f = q + 9q^{2} + q^{3} + 9q^{4} + 10q^{5} + 9q^{6} + 8q^{7} + 5q^{8} + 8q^{9} + 2q^{10} + 9q^{12} + 5q^{13} + 6q^{14} + 10q^{15} + 2q^{16} + 6q^{18} + 10q^{19} + 2q^{20} + 8q^{21} + ⋯ ∈ S_{20}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{10} + 7x^{5} + 8x^{4} + 10x^{3} + 6x^{2} + 6x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q + 10q^{2} + 4q^{3} + 5q^{4} + 2q^{5} + 7q^{6} + 9q^{7} + 2q^{8} + 7q^{9} + 9q^{10} + 10q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(10)ω^{8} & 0 \\ 0 & u(10)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−234404086842689588464+ O(11^{2}0)q^{2}+581446077887876323+O(11^{2}0)q^{3}+−135839871186645041242+O(11^{2}0)q^{4}+−72527316704641511802+ O(11^{2}0)q^{5}+235833727071564421545+O(11^{2}0)q^{6}+166676482220483040363+O(11^{2}0)q^{7}+17531890579277680886+ O(11^{2}0)q^{8}+290016715091590569761+O(11^{2}0)q^{9}+−110356065082175692580+O(11^{2}0)q^{10}+18596051557893655597∗ 11+O(11^{2}0)q^{11}+−226321428247023473542+O(11^{2}0)q^{12}+−10604499373q^{13}+−323210050963168389454+ O(11^{2}0)q^{14}+−67502827752870582337+O(11^{2}0)q^{15}+−280767717922816025867+O(11^{2}0)q^{16}+−3468969837238988993∗ 11 + O(11^{2}0)q^{17} + −273725586650761777504 + O(11^{2}0)q^{18} + 283271459612099869415 + O(11^{2}0)q^{19} + 69562629007199173067 + O(11^{2}0)q^{20} + −222249496473888872761 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

34. Consider f = q+6∗z^{9} +7∗z^{7} +7∗z^{6} +3∗z^{5} +z^{2} +z+2q^{2} +6∗z^{9} +7∗z^{7} +7∗z^{6} +3∗z^{5} +z^{2} +z+6q^{3} +3∗ z^{9} +9∗z^{7} +9∗z^{6} +7∗z^{5} +6∗z^{2} +6∗z +4q^{4} +3q^{5} +5∗z^{9} +4∗z^{7} +4∗z^{6} +8∗z^{5} +10∗z^{2} +10∗z +7q^{6} +9∗ z^{9}+5∗z^{7}+5∗z^{6}+10∗z^{5}+7∗z^{2}+7∗z+7q^{7}+10q^{8}+7∗z^{9}+10∗z^{7}+10∗z^{6}+9∗z^{5}+3∗z^{2}+3∗z+5q^{9}+7∗ z^{9}+10∗z^{7}+10∗z^{6}+9∗z^{5}+3∗z^{2}+3∗z+6q^{10}+4∗z^{9}+z^{7}+z^{6}+2∗z^{5}+8∗z^{2}+8∗z+5q^{12}+5q^{13}+z^{9}+3∗ z^{7}+3∗z^{6}+6∗z^{5}+2∗z^{2}+2∗z+1q^{14}+7∗z^{9}+10∗z^{7}+10∗z^{6}+9∗z^{5}+3∗z^{2}+3∗z+7q^{15}+9∗z^{9}+5∗z^{7}+5∗ z^{6}+10∗z^{5}+7∗z^{2}+7∗z+7q^{16}+7∗z^{9}+10∗z^{7}+10∗z^{6}+9∗z^{5}+3∗z^{2}+3∗z+1q^{17}+3∗z^{9}+9∗z^{7}+9∗z^{6}+ 7∗z^{5}+6∗z^{2}+6∗z+6q^{18}+10∗z^{9}+8∗z^{7}+8∗z^{6}+5∗z^{5}+9∗z^{2}+9∗z+8q^{19}+9∗z^{9}+5∗z^{7}+5∗z^{6}+10∗z^{5}+ 7∗z^{2} +7∗z +1q^{20} +4∗z^{9} +z^{7} +z^{6} +2∗z^{5} +8∗z^{2} +8∗z +7q^{21} +⋯ ∈ S_{20}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{10} + 7x^{5} + 8x^{4} + 10x^{3} + 6x^{2} + 6x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{8}9q^{2} + z^{2}0q^{3} + z^{1}17q^{4} + 5q^{5} + z^{1}09q^{6} + z^{6}3q^{7} + 4q^{8} + z^{1}18q^{9} + z^{1}7q^{10} + z^{7}0q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{5}0)ω^{8} & 0 \\ 0 & u(z^{7}0)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(272303827025241883805+ O(11^{2}0))∗a−215322758263783466853+O(11^{2}0)q^{3}+(263073086349674588224+O(11^{2}0))∗a+119157449566346766688+ O(11^{2}0)q^{4} + −(11792744442634703783 ∗ 11 + O(11^{2}0)) ∗ a + 2132467115334317787 + O(11^{2}0)q^{5} + (307370063729186535967+O(11^{2}0))∗a−11527209846235776497+O(11^{2}0)q^{6}+−(311112583582943152740+ O(11^{2}0)) ∗ a − 297917013299758162449 + O(11^{2}0)q^{7} + (6192675389814108818 ∗ 11 + O(11^{2}0)) ∗ a − 27963725221396248802+O(11^{2}0)q^{8}+−(99936423499898059302+O(11^{2}0))∗a−312458379130186459655+ O(11^{2}0)q^{9} + −(296923324283038982515 + O(11^{2}0)) ∗ a + 5803528070715235176 ∗ 11 + O(11^{2}0)q^{10} + (11981597508268817670∗11+O(11^{2}0))∗a−14854164576330428026∗11+O(11^{2}0)q^{11}+−(252721356561247014070+ O(11^{2}0))∗a−18072158248157614059∗11+O(11^{2}0)q^{12} +−10604499373q^{13} +(63439165617033570498+ O(11^{2}0)) ∗ a + 195799181015717743593 + O(11^{2}0)q^{14} + −(129923307831437396636 + O(11^{2}0)) ∗ a + 108587408843261519964+O(11^{2}0)q^{15}+(24214110984069740785+O(11^{2}0))∗a+146559214190227232717+ O(11^{2}0)q^{16}+(173923144566889851136+O(11^{2}0))∗a−3175073011623829013+O(11^{2}0)q^{17}+(234154222990670536593+ O(11^{2}0))∗a+995513828048219595+O(11^{2}0)q^{18}+(295851502337360175853+O(11^{2}0))∗a+316794284444217877848+ O(11^{2}0)q^{19}+(327803949062047361867+O(11^{2}0))∗a−230261960691287358726+O(11^{2}0)q^{20}+(111073886043987669657+ O(11^{2}0)) ∗ a − 15225312207865725271 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −263073086349674588224 + O(11^{2}0)x + −119157449566347290976 + O(11^{2}0). The slope of f_{0} is 1.

35. Consider f = q +4∗z^{9} +z^{7} +z^{6} +2∗z^{5} +8∗z^{2} +8∗z +10q^{2} +2∗z^{9} +6∗z^{7} +6∗z^{6} +z^{5} +4∗z^{2} +4∗z + 3q^{3} +z^{9} +3∗z^{7} +3∗z^{6} +6∗z^{5} +2∗z^{2} +2∗z +5q^{4} +9∗z^{9} +5∗z^{7} +5∗z^{6} +10∗z^{5} +7∗z^{2} +7∗z +9q^{5} +9∗ z^{9} +5∗z^{7} +5∗z^{6} +10∗z^{5} +7∗z^{2} +7∗z +2q^{6} +9q^{7} +9q^{8} +6∗z^{9} +7∗z^{7} +7∗z^{6} +3∗z^{5} +z^{2} +z +2q^{9} +6∗ z^{9} +7∗z^{7} +7∗z^{6} +3∗z^{5} +z^{2} +z +8q^{10} +10∗z^{9} +8∗z^{7} +8∗z^{6} +5∗z^{5} +9∗z^{2} +9∗z +8q^{12} +5q^{13} +3∗z^{9} + 9∗z^{7} +9∗z^{6} +7∗z^{5} +6∗z^{2} +6∗z +2q^{14} +7∗z^{9} +10∗z^{7} +10∗z^{6} +9∗z^{5} +3∗z^{2} +3∗z +8q^{15} +8∗z^{9} +2∗ z^{7} +2∗z^{6} +4∗z^{5} +5∗z^{2} +5∗z +5q^{16} +z^{9} +3∗z^{7} +3∗z^{6} +6∗z^{5} +2∗z^{2} +2∗z +3q^{17} +10∗z^{9} +8∗z^{7} +8∗ z^{6}+5∗z^{5}+9∗z^{2}+9∗z+2q^{18}+3∗z^{9}+9∗z^{7}+9∗z^{6}+7∗z^{5}+6∗z^{2}+6∗z+4q^{19}+2∗z^{9}+6∗z^{7}+6∗z^{6}+z^{5}+ 4∗z^{2} +4∗z +8q^{20} +7∗z^{9} +10∗z^{7} +10∗z^{6} +9∗z^{5} +3∗z^{2} +3∗z +5q^{21} +⋯ ∈ S_{20}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{10} + 7x^{5} + 8x^{4} + 10x^{3} + 6x^{2} + 6x + 2 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q +z^{9}7q^{2} +z^{7}0q^{3} +z^{9}3q^{4} +z^{5}6q^{5} +z^{4}7q^{6} +q^{7} +4q^{8} +z^{5}q^{9} +z^{3}3q^{10} +z^{7}6q^{11} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{4}4)ω^{1} & 0 \\ 0 & u(z^{7}6)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(209867710879757042690+ O(11^{2}0))∗a+202154414094505092526+O(11^{2}0)q^{3}+(22774454677973748929+O(11^{2}0))∗a−59336859581788100381+ O(11^{2}0)q^{4}+(287336633575499315780+O(11^{2}0))∗a−83613365379999442545+O(11^{2}0)q^{5}+−(195058502989790474194+ O(11^{2}0)) ∗ a − 107160137793774659224 + O(11^{2}0)q^{6} + (2563543920251190144 ∗ 11 + O(11^{2}0)) ∗ a − 208680264182933080760+O(11^{2}0)q^{7}+(4021886618287982123∗11+O(11^{2}0))∗a−207115167455718804339+ O(11^{2}0)q^{8}+(4319860259984061734+O(11^{2}0))∗a−193493551960292943636+O(11^{2}0)q^{9}+−(324398354493518668378+ O(11^{2}0)) ∗ a − 137005028109894811940 + O(11^{2}0)q^{10} + −(23393544352390912576 ∗ 11 + O(11^{2}0)) ∗ a + 24475679125356274941∗11+O(11^{2}0)q^{11}+(204319014151714204465+O(11^{2}0))∗a−236580570750573332245+ O(11^{2}0)q^{12} + −10604499373q^{13} + (285691909951492828837 + O(11^{2}0)) ∗ a + 833866046998830099 ∗ 11 + O(11^{2}0)q^{14} + −(5995847152348734660 + O(11^{2}0)) ∗ a − 193241371615301142445 + O(11^{2}0)q^{15} + −(71555344044507954827+O(11^{2}0))∗a+156353267723444211625+O(11^{2}0)q^{16}+−(74223820962819400540+ O(11^{2}0)) ∗ a + 335559460822518978073 + O(11^{2}0)q^{17} + (157295347233511199974 + O(11^{2}0)) ∗ a + 7886004069475144073+O(11^{2}0)q^{18}+−(159895703726444612181+O(11^{2}0))∗a+156054462680065417426+ O(11^{2}0)q^{19}+−(258533341790640675313+O(11^{2}0))∗a+73464025562232043786+O(11^{2}0)q^{20}+−(42408452093388325539+ O(11^{2}0)) ∗ a + 57105879574712163540 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −22774454677973748929 + O(11^{2}0)x + 59336859581787576093 + O(11^{2}0). The slope of f_{0} is 1.

36. Consider f = q +5∗z^{9} +4∗z^{7} +4∗z^{6} +8∗z^{5} +10∗z^{2} +10∗z +4q^{2} +5∗z^{9} +4∗z^{7} +4∗z^{6} +8∗z^{5} +10∗ z^{2} +10∗z +8q^{3} +8∗z^{9} +2∗z^{7} +2∗z^{6} +4∗z^{5} +5∗z^{2} +5∗z +5q^{4} +3q^{5} +6∗z^{9} +7∗z^{7} +7∗z^{6} +3∗z^{5} +z^{2} + z +5q^{6} +2∗z^{9} +6∗z^{7} +6∗z^{6} +z^{5} +4∗z^{2} +4∗z +10q^{7} +10q^{8} +4∗z^{9} +z^{7} +z^{6} +2∗z^{5} +8∗z^{2} +8∗zq^{9} +4∗ z^{9} +z^{7} +z^{6} +2∗z^{5} +8∗z^{2} +8∗z +1q^{10} +7∗z^{9} +10∗z^{7} +10∗z^{6} +9∗z^{5} +3∗z^{2} +3∗z +10q^{12} +5q^{13} +10∗ z^{9} +8∗z^{7} +8∗z^{6} +5∗z^{5} +9∗z^{2} +9∗z +5q^{14} +4∗z^{9} +z^{7} +z^{6} +2∗z^{5} +8∗z^{2} +8∗z +2q^{15} +2∗z^{9} +6∗z^{7} + 6∗z^{6} +z^{5} +4∗z^{2} +4∗z +10q^{16} +4∗z^{9} +z^{7} +z^{6} +2∗z^{5} +8∗z^{2} +8∗z +7q^{17} +8∗z^{9} +2∗z^{7} +2∗z^{6} +4∗ z^{5} +5∗z^{2} +5∗z +7q^{18} +z^{9} +3∗z^{7} +3∗z^{6} +6∗z^{5} +2∗z^{2} +2∗z +4q^{19} +2∗z^{9} +6∗z^{7} +6∗z^{6} +z^{5} +4∗z^{2} + 4∗z +4q^{20} +7∗z^{9} +10∗z^{7} +10∗z^{6} +9∗z^{5} +3∗z^{2} +3∗z +1q^{21} +⋯ ∈ S_{20}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{10} + 7x^{5} + 8x^{4} + 10x^{3} + 6x^{2} + 6x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{1}9q^{2} + z^{1}00q^{3} + z^{8}7q^{4} + 5q^{5} + z^{1}19q^{6} + z^{9}3q^{7} + 4q^{8} + z^{9}8q^{9} + z^{6}7q^{10} + z^{5}0q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{7}0)ω^{8} & 0 \\ 0 & u(z^{5}0)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(272303827025241883805+ O(11^{2}0))∗a−215322758263783466853+O(11^{2}0)q^{3}+(263073086349674588224+O(11^{2}0))∗a+119157449566346766688+ O(11^{2}0)q^{4} + −(11792744442634703783 ∗ 11 + O(11^{2}0)) ∗ a + 2132467115334317787 + O(11^{2}0)q^{5} + (307370063729186535967+O(11^{2}0))∗a−11527209846235776497+O(11^{2}0)q^{6}+−(311112583582943152740+ O(11^{2}0)) ∗ a − 297917013299758162449 + O(11^{2}0)q^{7} + (6192675389814108818 ∗ 11 + O(11^{2}0)) ∗ a − 27963725221396248802+O(11^{2}0)q^{8}+−(99936423499898059302+O(11^{2}0))∗a−312458379130186459655+ O(11^{2}0)q^{9} + −(296923324283038982515 + O(11^{2}0)) ∗ a + 5803528070715235176 ∗ 11 + O(11^{2}0)q^{10} + (11981597508268817670∗11+O(11^{2}0))∗a−14854164576330428026∗11+O(11^{2}0)q^{11}+−(252721356561247014070+ O(11^{2}0))∗a−18072158248157614059∗11+O(11^{2}0)q^{12} +−10604499373q^{13} +(63439165617033570498+ O(11^{2}0)) ∗ a + 195799181015717743593 + O(11^{2}0)q^{14} + −(129923307831437396636 + O(11^{2}0)) ∗ a + 108587408843261519964+O(11^{2}0)q^{15}+(24214110984069740785+O(11^{2}0))∗a+146559214190227232717+ O(11^{2}0)q^{16}+(173923144566889851136+O(11^{2}0))∗a−3175073011623829013+O(11^{2}0)q^{17}+(234154222990670536593+ O(11^{2}0))∗a+995513828048219595+O(11^{2}0)q^{18}+(295851502337360175853+O(11^{2}0))∗a+316794284444217877848+ O(11^{2}0)q^{19}+(327803949062047361867+O(11^{2}0))∗a−230261960691287358726+O(11^{2}0)q^{20}+(111073886043987669657+ O(11^{2}0)) ∗ a − 15225312207865725271 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −263073086349674588224 + O(11^{2}0)x + −119157449566347290976 + O(11^{2}0). The slope of f_{0} is 1.

37. Consider f = q+7∗z^{9} +10∗z^{7} +10∗z^{6} +9∗z^{5} +3∗z^{2} +3∗z+4q^{2} +9∗z^{9} +5∗z^{7} +5∗z^{6} +10∗z^{5} +7∗z^{2} + 7∗zq^{3} +10∗z^{9} +8∗z^{7} +8∗z^{6} +5∗z^{5} +9∗z^{2} +9∗z+9q^{4} +2∗z^{9} +6∗z^{7} +6∗z^{6} +z^{5} +4∗z^{2} +4∗z+1q^{5} +2∗ z^{9} +6∗z^{7} +6∗z^{6} +z^{5} +4∗z^{2} +4∗z +5q^{6} +9q^{7} +9q^{8} +5∗z^{9} +4∗z^{7} +4∗z^{6} +8∗z^{5} +10∗z^{2} +10∗z +4q^{9} + 5∗z^{9} +4∗z^{7} +4∗z^{6} +8∗z^{5} +10∗z^{2} +10∗z+10q^{10} +z^{9} +3∗z^{7} +3∗z^{6} +6∗z^{5} +2∗z^{2} +2∗z+4q^{12} +5q^{13} + 8∗z^{9} +2∗z^{7} +2∗z^{6} +4∗z^{5} +5∗z^{2} +5∗z +3q^{14} +4∗z^{9} +z^{7} +z^{6} +2∗z^{5} +8∗z^{2} +8∗z +3q^{15} +3∗z^{9} +9∗ z^{7} +9∗z^{6} +7∗z^{5} +6∗z^{2} +6∗z +4q^{16} +10∗z^{9} +8∗z^{7} +8∗z^{6} +5∗z^{5} +9∗z^{2} +9∗z +7q^{17} +z^{9} +3∗z^{7} +3∗ z^{6}+6∗z^{5}+2∗z^{2}+2∗z+9q^{18}+8∗z^{9}+2∗z^{7}+2∗z^{6}+4∗z^{5}+5∗z^{2}+5∗z+5q^{19}+9∗z^{9}+5∗z^{7}+5∗z^{6}+10∗ z^{5} +7∗z^{2} +7∗z +5q^{20} +4∗z^{9} +z^{7} +z^{6} +2∗z^{5} +8∗z^{2} +8∗zq^{21} +⋯ ∈ S_{20}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{10} + 7x^{5} + 8x^{4} + 10x^{3} + 6x^{2} + 6x + 2 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + z^{1}07q^{2} + z^{5}0q^{3} + z^{6}3q^{4} + z^{1}6q^{5} + z^{3}7q^{6} + q^{7} + 4q^{8} + z^{5}5q^{9} + z^{3}q^{10} + z^{1}16q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{4})ω^{1} & 0 \\ 0 & u(z^{1}16)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(209867710879757042690+ O(11^{2}0))∗a+202154414094505092526+O(11^{2}0)q^{3}+(22774454677973748929+O(11^{2}0))∗a−59336859581788100381+ O(11^{2}0)q^{4}+(287336633575499315780+O(11^{2}0))∗a−83613365379999442545+O(11^{2}0)q^{5}+−(195058502989790474194+ O(11^{2}0)) ∗ a − 107160137793774659224 + O(11^{2}0)q^{6} + (2563543920251190144 ∗ 11 + O(11^{2}0)) ∗ a − 208680264182933080760+O(11^{2}0)q^{7}+(4021886618287982123∗11+O(11^{2}0))∗a−207115167455718804339+ O(11^{2}0)q^{8}+(4319860259984061734+O(11^{2}0))∗a−193493551960292943636+O(11^{2}0)q^{9}+−(324398354493518668378+ O(11^{2}0)) ∗ a − 137005028109894811940 + O(11^{2}0)q^{10} + −(23393544352390912576 ∗ 11 + O(11^{2}0)) ∗ a + 24475679125356274941∗11+O(11^{2}0)q^{11}+(204319014151714204465+O(11^{2}0))∗a−236580570750573332245+ O(11^{2}0)q^{12} + −10604499373q^{13} + (285691909951492828837 + O(11^{2}0)) ∗ a + 833866046998830099 ∗ 11 + O(11^{2}0)q^{14} + −(5995847152348734660 + O(11^{2}0)) ∗ a − 193241371615301142445 + O(11^{2}0)q^{15} + −(71555344044507954827+O(11^{2}0))∗a+156353267723444211625+O(11^{2}0)q^{16}+−(74223820962819400540+ O(11^{2}0)) ∗ a + 335559460822518978073 + O(11^{2}0)q^{17} + (157295347233511199974 + O(11^{2}0)) ∗ a + 7886004069475144073+O(11^{2}0)q^{18}+−(159895703726444612181+O(11^{2}0))∗a+156054462680065417426+ O(11^{2}0)q^{19}+−(258533341790640675313+O(11^{2}0))∗a+73464025562232043786+O(11^{2}0)q^{20}+−(42408452093388325539+ O(11^{2}0)) ∗ a + 57105879574712163540 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −22774454677973748929 + O(11^{2}0)x + 59336859581787576093 + O(11^{2}0). The slope of f_{0} is 1.

38. Consider f = q+4∗z^{1}9+9∗z^{1}8+8∗z^{1}6+6∗z^{1}5+10∗z^{1}3+10∗z^{1}2+z^{9}+2∗z^{8}+10∗z^{7}+7∗z^{6}+8∗z^{5}+9∗z^{4}+ 6∗z^{2}+10∗z+3q^{2}+4∗z^{1}9+7∗z^{1}8+6∗z^{1}7+10∗z^{1}6+9∗z^{1}5+3∗z^{1}3+9∗z^{1}2+10∗z^{1}1+4∗z^{1}0+9∗z^{9}+3∗z^{8}+7∗ z^{7}+3∗z^{6}+4∗z^{5}+8∗z^{4}+10∗z^{3}+7∗z^{2}+3∗z+9q^{3}+6∗z^{1}9+5∗z^{1}8+5∗z^{1}7+5∗z^{1}6+10∗z^{1}5+9∗z^{1}4+5∗z^{1}3+ 5∗z^{1}2+9∗z^{1}1+10∗z^{1}0+4∗z^{8}+6∗z^{7}+9∗z^{6}+6∗z^{5}+2∗z^{4}+3∗z^{3}+6∗z^{2}+5∗z+6q^{4}+8∗z^{1}9+6∗z^{1}8+10∗z^{1}7+ 10∗z^{1}6+2∗z^{1}4+4∗z^{1}3+9∗z^{1}2+10∗z^{1}1+2∗z^{1}0+8∗z^{9}+3∗z^{8}+3∗z^{7}+2∗z^{6}+6∗z^{5}+z^{4}+9∗z^{3}+6∗z^{2}+8∗z+ 1q^{5}+4∗z^{1}9+2∗z^{1}8+10∗z^{1}7+3∗z^{1}6+7∗z^{1}5+6∗z^{1}4+10∗z^{1}3+9∗z^{1}2+2∗z^{1}1+8∗z^{1}0+4∗z^{8}+5∗z^{7}+7∗z^{6}+ 4∗z^{5}+4∗z^{4}+4∗z^{3}+z^{2}+5∗z+9q^{6}+5∗z^{1}9+4∗z^{1}8+8∗z^{1}7+4∗z^{1}6+8∗z^{1}5+8∗z^{1}4+z^{1}3+9∗z^{1}2+6∗z^{1}1+ z^{1}0+9∗z^{9}+3∗z^{7}+z^{6}+7∗z^{5}+7∗z^{4}+5∗z^{3}+3∗z^{2}+9∗z+5q^{7}+4∗z^{1}9+4∗z^{1}8+8∗z^{1}7+7∗z^{1}6+8∗z^{1}5+10∗ z^{1}4+10∗z^{1}3+z^{1}2+9∗z^{1}1+9∗z^{1}0+5∗z^{9}+3∗z^{8}+7∗z^{7}+2∗z^{6}+4∗z^{5}+9∗z^{4}+3∗z^{3}+9∗z^{2}+7∗z+5q^{8}+5∗z^{1}8+ 5∗z^{1}7+5∗z^{1}6+4∗z^{1}5+8∗z^{1}4+5∗z^{1}3+2∗z^{1}2+5∗z^{1}1+5∗z^{1}0+4∗z^{9}+6∗z^{8}+8∗z^{7}+3∗z^{6}+3∗z^{5}+9∗z^{4}+5∗ z^{3}+10∗z^{2}+3∗z+4q^{9}+10∗z^{1}9+4∗z^{1}8+z^{1}7+10∗z^{1}5+5∗z^{1}4+8∗z^{1}3+8∗z^{1}1+7∗z^{1}0+9∗z^{8}+4∗z^{7}+5∗z^{6}+ z^{5}+3∗z^{4}+3∗z^{3}+8∗z^{2}+10∗z+7q^{10}+6∗z^{1}9+4∗z^{1}8+z^{1}7+7∗z^{1}6+z^{1}5+10∗z^{1}4+7∗z^{1}3+8∗z^{1}2+9∗z^{1}1+9∗ z^{1}0+4∗z^{9}+8∗z^{8}+9∗z^{7}+2∗z^{6}+6∗z^{5}+9∗z^{4}+5∗z^{3}+6∗z^{2}+6∗z+9q^{12}+10q^{13}+10∗z^{1}9+8∗z^{1}8+3∗z^{1}7+8∗ z^{1}6+4∗z^{1}5+7∗z^{1}4+9∗z^{1}3+4∗z^{1}2+9∗z^{1}1+z^{1}0+5∗z^{9}+10∗z^{8}+z^{7}+3∗z^{6}+8∗z^{5}+5∗z^{3}+5∗z^{2}+6∗z+3q^{14}+ 4∗z^{1}9+5∗z^{1}8+7∗z^{1}7+2∗z^{1}6+8∗z^{1}5+10∗z^{1}4+9∗z^{1}2+7∗z^{1}1+6∗z^{1}0+8∗z^{9}+3∗z^{8}+8∗z^{7}+4∗z^{6}+8∗z^{5}+ z^{4}+3∗z^{2}+7∗z+2q^{15}+z^{1}9+10∗z^{1}8+8∗z^{1}6+7∗z^{1}5+7∗z^{1}4+6∗z^{1}3+3∗z^{1}1+3∗z^{1}0+9∗z^{9}+9∗z^{7}+7∗z^{6}+2∗ z^{5}+8∗z^{4}+7∗z^{3}+z^{2}q^{16}+8∗z^{1}9+z^{1}8+4∗z^{1}7+3∗z^{1}5+3∗z^{1}4+4∗z^{1}3+7∗z^{1}2+9∗z^{1}1+5∗z^{1}0+z^{9}+9∗z^{7}+9∗ z^{6}+6∗z^{5}+5∗z^{4}+6∗z^{3}+8∗z^{2}+3∗z+9q^{17}+8∗z^{1}9+3∗z^{1}8+10∗z^{1}6+5∗z^{1}5+6∗z^{1}4+7∗z^{1}3+3∗z^{1}2+2∗z^{1}1+ 8∗z^{1}0+2∗z^{9}+7∗z^{8}+6∗z^{7}+9∗z^{6}+6∗z^{5}+5∗z^{4}+10∗z^{3}+8∗z+4q^{18}+7∗z^{1}9+8∗z^{1}8+8∗z^{1}7+z^{1}6+3∗z^{1}5+5∗ z^{1}4+9∗z^{1}3+9∗z^{1}1+3∗z^{1}0+7∗z^{9}+3∗z^{8}+9∗z^{7}+2∗z^{6}+9∗z^{5}+6∗z^{4}+5∗z^{3}+5∗z^{2}+z+3q^{19}+8∗z^{1}9+6∗z^{1}8+ 3∗z^{1}7+7∗z^{1}6+z^{1}5+5∗z^{1}4+3∗z^{1}3+6∗z^{1}2+5∗z^{1}1+8∗z^{1}0+2∗z^{9}+6∗z^{8}+2∗z^{7}+9∗z^{6}+4∗z^{5}+8∗z^{4}+3∗ z^{3}+10∗z^{2}+8∗z+8q^{20}+z^{1}9+8∗z^{1}8+5∗z^{1}6+9∗z^{1}5+10∗z^{1}4+5∗z^{1}3+8∗z^{1}2+4∗z^{1}1+7∗z^{1}0+9∗z^{9}+7∗z^{8}+ 2∗z^{7}+10∗z^{6}+8∗z^{5}+7∗z^{4}+z^{3}+6∗z^{2}+5∗z+5q^{21}+⋯ ∈ S_{22}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{20}+6x^{19}+5x^{18}+9x^{17}+9x^{16}+2x^{15}+x^{13}+5x^{12}+7x^{11}+10x^{10}+7x^{9}+10x^{8}+9x^{6}+x^{5}+6x^{4}+7x^{3}+x+4 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{1}42133q^{2} + z^{4}0440q^{3} + z^{1}41494q^{4} + z^{1}03104q^{5} + z^{2}1523q^{6} + z^{1}53127q^{7} + z^{9}5697q^{8} + z^{9}9297q^{9} + z^{8}4187q^{10} + z^{6}6312q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{9}4738)ω^{0} & 0 \\ 0 & u(z^{6}6312)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(126620356400921773329+ O(11^{2}0)) ∗ a^{4} + (116124275029971191609 + O(11^{2}0)) ∗ a^{3} + (54200904031282999190 + O(11^{2}0)) ∗ a^{2} + (27192497000010582578 + O(11^{2}0)) ∗ a − 48718083868206747624 + O(11^{2}0)q^{3} + a^{2} − 2097152q^{4} + −(141869525449007678538+O(11^{2}0))∗a^{4}+(151605097979372508+O(11^{2}0))∗a^{3}−(194258345799019839578+ O(11^{2}0))∗a^{2}−(13567311038568361865+O(11^{2}0))∗a−192798036755544856348+O(11^{2}0)q^{5}+−(166098107277332652573+ O(11^{2}0)) ∗ a^{4} + (262077544250279532283 + O(11^{2}0)) ∗ a^{3} − (261813859077022280337 + O(11^{2}0)) ∗ a^{2} + (39649627902202339891+O(11^{2}0))∗a+28297258544846119286+O(11^{2}0)q^{6}+−(27523841579649731365+ O(11^{2}0)) ∗ a^{4} + (20989109331104192790 ∗ 11 + O(11^{2}0)) ∗ a^{3} − (116279107353648712844 + O(11^{2}0)) ∗ a^{2} −(232160258876642163398+O(11^{2}0))∗a+8449694136397374108+O(11^{2}0)q^{7} +a^{3} −4194304∗aq^{8} + −(206728757606538797473+O(11^{2}0))∗a^{4}+(151039960356708325344+O(11^{2}0))∗a^{3}−(143140254869584811747+ O(11^{2}0)) ∗ a^{2} + (158708028880781775585 + O(11^{2}0)) ∗ a + 258252730417370027917 + O(11^{2}0)q^{9} + (2347130664133950252+O(11^{2}0))∗a^{4}−(273666526199816968253+O(11^{2}0))∗a^{3}−(146133167218784058890+ O(11^{2}0)) ∗ a^{2} − (168630994190763393409 + O(11^{2}0)) ∗ a − 77518454593766032615 + O(11^{2}0)q^{10} + (16934259358485004225∗11+O(11^{2}0))∗a^{4}+(21643924749675957590∗11+O(11^{2}0))∗a^{3}−(18870650781680015859∗ 11+O(11^{2}0))∗a^{2}+(843567987186155850∗11^{2}+O(11^{2}0))∗a−11614714628517476181∗11+O(11^{2}0)q^{11}+ −(39441557776730614930+O(11^{2}0))∗a^{4}−(179456845973125848329+O(11^{2}0))∗a^{3}−(66925150743570368834+ O(11^{2}0)) ∗ a^{2} − (245765224208019750989 + O(11^{2}0)) ∗ a + 186082390215892588565 + O(11^{2}0)q^{12} + −137858491849q^{13}+−(8392291303721118361+O(11^{2}0))∗a^{4}+(94080764106716336602+O(11^{2}0))∗a^{3}+ (52533650851688319072+O(11^{2}0))∗a^{2}+(118488488431490924845+O(11^{2}0))∗a+17266215589407957762+ O(11^{2}0)q^{14} + (130120802818579978503 + O(11^{2}0)) ∗ a^{4} + (284867432704604956532 + O(11^{2}0)) ∗ a^{3} + (132137272138135842803+O(11^{2}0))∗a^{2}+(287599407903525171023+O(11^{2}0))∗a−66108796264582541456+ O(11^{2}0)q^{15} + a^{4} − 6291456 ∗ a^{2} + 4398046511104q^{16} + (247263884779259668634 + O(11^{2}0)) ∗ a^{4} − (199320934044999770198+O(11^{2}0))∗a^{3}+(335640184363866694800+O(11^{2}0))∗a^{2}+(111413043909931950062+ O(11^{2}0)) ∗ a − 125895449116754295527 + O(11^{2}0)q^{17} + −(129685198089365853755 + O(11^{2}0)) ∗ a^{4} + (61366251647099302067+O(11^{2}0))∗a^{3}−(145396434073872810176+O(11^{2}0))∗a^{2}+(289716777941210368284+ O(11^{2}0)) ∗ a − 51076396781348042337 + O(11^{2}0)q^{18} + −(102223264689557361929 + O(11^{2}0)) ∗ a^{4} + (332889877538916840034+O(11^{2}0))∗a^{3}+(155178524409514007361+O(11^{2}0))∗a^{2}+(236385904455101109948+ O(11^{2}0)) ∗ a + 267406116352202105485 + O(11^{2}0)q^{19} + −(106802364945428389007 + O(11^{2}0)) ∗ a^{4} − (73484333971793824548+O(11^{2}0))∗a^{3}+(248452910258062260177+O(11^{2}0))∗a^{2}−(237359369857924297976+ O(11^{2}0)) ∗ a + 83016675446571761948 + O(11^{2}0)q^{20} + (226558599902903013364 + O(11^{2}0)) ∗ a^{4} + (306754828431946001816+O(11^{2}0))∗a^{3}+(71714185102327232890+O(11^{2}0))∗a^{2}+(169717460484611971154+ O(11^{2}0)) ∗ a + 26374001414417019471 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{5} + 235939635302487581501 + O(11^{2}0)x^{4} + 19832230278235438204+O(11^{2}0)x^{3}+−75305933702655388704+O(11^{2}0)x^{2}+−286001669361950811530+

O(11^{2}0)x + −253175070989686225911 + O(11^{2}0). The slope of f_{0} is 1.

39. Consider f = q+5∗z^{1}9+8∗z^{1}8+2∗z^{1}7+5∗z^{1}6+3∗z^{1}5+5∗z^{1}4+9∗z^{1}3+2∗z^{1}2+6∗z^{1}1+4∗z^{1}0+z^{9}+3∗z^{8}+ 2∗z^{7}+2∗z^{6}+5∗z^{5}+9∗z^{4}+z^{3}+8∗z^{2}+6q^{2}+4∗z^{1}9+8∗z^{1}8+3∗z^{1}7+5∗z^{1}6+8∗z^{1}5+2∗z^{1}4+3∗z^{1}2+5∗z^{1}1+ 10∗z^{9}+2∗z^{8}+3∗z^{7}+6∗z^{6}+2∗z^{5}+8∗z^{4}+2∗z^{3}+2∗z+2q^{3}+9∗z^{1}8+5∗z^{1}7+2∗z^{1}4+7∗z^{1}3+8∗z^{1}2+3∗z^{1}1+ 8∗z^{1}0+4∗z^{9}+3∗z^{8}+8∗z^{6}+2∗z^{5}+6∗z^{3}+4∗z+6q^{4}+9∗z^{1}9+6∗z^{1}8+5∗z^{1}6+9∗z^{1}5+2∗z^{1}3+7∗z^{1}2+3∗z^{1}1+ 10∗z^{1}0+10∗z^{9}+3∗z^{8}+7∗z^{7}+z^{6}+10∗z^{5}+3∗z^{4}+z^{3}+9q^{5}+7∗z^{1}9+3∗z^{1}8+9∗z^{1}7+6∗z^{1}6+9∗z^{1}5+8∗z^{1}4+ 2∗z^{1}3+4∗z^{1}2+2∗z^{1}1+6∗z^{1}0+2∗z^{9}+4∗z^{8}+5∗z^{7}+10∗z^{6}+z^{5}+10∗z^{4}+6∗z^{3}+6∗z^{2}+4∗z+2q^{6}+4∗z^{1}9+7∗ z^{1}8+3∗z^{1}7+3∗z^{1}6+7∗z^{1}5+z^{1}4+9∗z^{1}3+10∗z^{1}0+z^{9}+2∗z^{8}+5∗z^{7}+9∗z^{6}+10∗z^{5}+4∗z^{4}+z^{2}+6∗z+4q^{7}+3∗ z^{1}9+8∗z^{1}8+5∗z^{1}7+z^{1}6+3∗z^{1}5+3∗z^{1}3+10∗z^{1}2+3∗z^{9}+z^{8}+z^{7}+5∗z^{6}+9∗z^{5}+8∗z^{4}+8∗z^{3}+8∗z^{2}+4q^{8}+5∗ z^{1}9+2∗z^{1}8+3∗z^{1}7+6∗z^{1}6+8∗z^{1}4+5∗z^{1}3+8∗z^{1}2+5∗z^{1}1+5∗z^{1}0+6∗z^{9}+z^{8}+8∗z^{6}+3∗z^{5}+6∗z^{4}+4∗z^{3}+ 4∗z^{2}+2∗z+3q^{9}+4∗z^{1}9+10∗z^{1}8+3∗z^{1}6+z^{1}5+10∗z^{1}4+2∗z^{1}3+3∗z^{1}2+5∗z^{1}1+3∗z^{1}0+5∗z^{9}+8∗z^{8}+8∗ z^{7}+3∗z^{6}+4∗z^{5}+8∗z^{4}+5∗z^{3}+6∗z^{2}+5∗z+9q^{10}+9∗z^{1}7+10∗z^{1}5+2∗z^{1}4+7∗z^{1}3+8∗z^{1}2+8∗z^{1}1+10∗z^{1}0+ 9∗z^{9}+10∗z^{8}+6∗z^{7}+z^{6}+5∗z^{5}+8∗z^{4}+6∗z^{3}+10∗z^{2}+10∗z+4q^{12}+10q^{13}+5∗z^{1}9+10∗z^{1}8+8∗z^{1}7+5∗z^{1}5+ 9∗z^{1}3+z^{1}2+3∗z^{1}1+10∗z^{1}0+6∗z^{9}+9∗z^{8}+2∗z^{7}+9∗z^{6}+9∗z^{5}+7∗z^{4}+5∗z^{3}+4∗z^{2}+5q^{14}+z^{1}9+3∗z^{1}8+2∗ z^{1}5+8∗z^{1}4+10∗z^{1}3+8∗z^{1}2+z^{1}1+10∗z^{1}0+10∗z^{9}+9∗z^{8}+z^{7}+10∗z^{6}+9∗z^{5}+3∗z^{3}+4∗z^{2}+10∗z+3q^{15}+7∗ z^{1}9+10∗z^{1}8+5∗z^{1}7+5∗z^{1}6+9∗z^{1}5+10∗z^{1}4+7∗z^{1}3+7∗z^{1}2+9∗z^{1}1+9∗z^{1}0+7∗z^{8}+3∗z^{7}+3∗z^{6}+6∗z^{5}+ 4∗z^{4}+7∗z^{3}+7∗z^{2}+8∗z+4q^{16}+8∗z^{1}9+10∗z^{1}8+10∗z^{1}7+z^{1}6+2∗z^{1}5+9∗z^{1}4+5∗z^{1}3+3∗z^{1}2+8∗z^{1}1+3∗ z^{1}0+2∗z^{9}+5∗z^{8}+6∗z^{7}+8∗z^{6}+z^{5}+8∗z^{4}+7∗z^{3}+6∗z^{2}+10∗z+2q^{17}+z^{1}9+8∗z^{1}8+9∗z^{1}7+6∗z^{1}6+z^{1}5+6∗ z^{1}4+4∗z^{1}3+8∗z^{1}2+z^{1}1+z^{1}0+3∗z^{9}+2∗z^{8}+8∗z^{7}+8∗z^{6}+3∗z^{5}+5∗z^{3}+8∗z^{2}+zq^{18}+4∗z^{1}9+9∗z^{1}8+10∗ z^{1}7+3∗z^{1}6+2∗z^{1}5+9∗z^{1}4+4∗z^{1}3+10∗z^{1}2+4∗z^{1}1+8∗z^{1}0+9∗z^{9}+5∗z^{8}+2∗z^{7}+6∗z^{6}+2∗z^{3}+7∗z^{2}+8∗ z+10q^{19}+4∗z^{1}9+10∗z^{1}8+z^{1}7+9∗z^{1}6+9∗z^{1}5+6∗z^{1}4+6∗z^{1}2+z^{1}1+z^{1}0+4∗z^{9}+z^{8}+5∗z^{7}+z^{6}+2∗z^{5}+ 2∗z^{4}+z^{3}+8∗z^{2}+4∗z+6q^{20}+8∗z^{1}9+6∗z^{1}8+2∗z^{1}7+3∗z^{1}6+z^{1}5+8∗z^{1}4+9∗z^{1}2+9∗z^{1}1+6∗z^{9}+5∗z^{8}+5∗ z^{7}+5∗z^{6}+8∗z^{5}+6∗z^{4}+3∗z^{3}+3∗z^{2}+3∗z+4q^{21}+⋯ ∈ S_{22}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{20}+6x^{19}+5x^{18}+9x^{17}+9x^{16}+2x^{15}+x^{13}+5x^{12}+7x^{11}+10x^{10}+7x^{9}+10x^{8}+9x^{6}+x^{5}+6x^{4}+7x^{3}+x+4 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{1}06323q^{2} + z^{3}4940q^{3} + z^{6}1064q^{4} + z^{1}6824q^{5} + z^{1}41263q^{6} + z^{8}3787q^{7} + z^{1}43207q^{8} + z^{1}03307q^{9} + z^{1}23147q^{10} + z^{5}872q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}55178)ω^{0} & 0 \\ 0 & u(z^{5}872)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(126620356400921773329+ O(11^{2}0)) ∗ a^{4} + (116124275029971191609 + O(11^{2}0)) ∗ a^{3} + (54200904031282999190 + O(11^{2}0)) ∗ a^{2} + (27192497000010582578 + O(11^{2}0)) ∗ a − 48718083868206747624 + O(11^{2}0)q^{3} + a^{2} − 2097152q^{4} + −(141869525449007678538+O(11^{2}0))∗a^{4}+(151605097979372508+O(11^{2}0))∗a^{3}−(194258345799019839578+ O(11^{2}0))∗a^{2}−(13567311038568361865+O(11^{2}0))∗a−192798036755544856348+O(11^{2}0)q^{5}+−(166098107277332652573+ O(11^{2}0)) ∗ a^{4} + (262077544250279532283 + O(11^{2}0)) ∗ a^{3} − (261813859077022280337 + O(11^{2}0)) ∗ a^{2} + (39649627902202339891+O(11^{2}0))∗a+28297258544846119286+O(11^{2}0)q^{6}+−(27523841579649731365+ O(11^{2}0)) ∗ a^{4} + (20989109331104192790 ∗ 11 + O(11^{2}0)) ∗ a^{3} − (116279107353648712844 + O(11^{2}0)) ∗ a^{2} −(232160258876642163398+O(11^{2}0))∗a+8449694136397374108+O(11^{2}0)q^{7} +a^{3} −4194304∗aq^{8} + −(206728757606538797473+O(11^{2}0))∗a^{4}+(151039960356708325344+O(11^{2}0))∗a^{3}−(143140254869584811747+ O(11^{2}0)) ∗ a^{2} + (158708028880781775585 + O(11^{2}0)) ∗ a + 258252730417370027917 + O(11^{2}0)q^{9} + (2347130664133950252+O(11^{2}0))∗a^{4}−(273666526199816968253+O(11^{2}0))∗a^{3}−(146133167218784058890+ O(11^{2}0)) ∗ a^{2} − (168630994190763393409 + O(11^{2}0)) ∗ a − 77518454593766032615 + O(11^{2}0)q^{10} + (16934259358485004225∗11+O(11^{2}0))∗a^{4}+(21643924749675957590∗11+O(11^{2}0))∗a^{3}−(18870650781680015859∗ 11+O(11^{2}0))∗a^{2}+(843567987186155850∗11^{2}+O(11^{2}0))∗a−11614714628517476181∗11+O(11^{2}0)q^{11}+ −(39441557776730614930+O(11^{2}0))∗a^{4}−(179456845973125848329+O(11^{2}0))∗a^{3}−(66925150743570368834+ O(11^{2}0)) ∗ a^{2} − (245765224208019750989 + O(11^{2}0)) ∗ a + 186082390215892588565 + O(11^{2}0)q^{12} + −137858491849q^{13}+−(8392291303721118361+O(11^{2}0))∗a^{4}+(94080764106716336602+O(11^{2}0))∗a^{3}+ (52533650851688319072+O(11^{2}0))∗a^{2}+(118488488431490924845+O(11^{2}0))∗a+17266215589407957762+ O(11^{2}0)q^{14} + (130120802818579978503 + O(11^{2}0)) ∗ a^{4} + (284867432704604956532 + O(11^{2}0)) ∗ a^{3} + (132137272138135842803+O(11^{2}0))∗a^{2}+(287599407903525171023+O(11^{2}0))∗a−66108796264582541456+ O(11^{2}0)q^{15} + a^{4} − 6291456 ∗ a^{2} + 4398046511104q^{16} + (247263884779259668634 + O(11^{2}0)) ∗ a^{4} − (199320934044999770198+O(11^{2}0))∗a^{3}+(335640184363866694800+O(11^{2}0))∗a^{2}+(111413043909931950062+ O(11^{2}0)) ∗ a − 125895449116754295527 + O(11^{2}0)q^{17} + −(129685198089365853755 + O(11^{2}0)) ∗ a^{4} + (61366251647099302067+O(11^{2}0))∗a^{3}−(145396434073872810176+O(11^{2}0))∗a^{2}+(289716777941210368284+ O(11^{2}0)) ∗ a − 51076396781348042337 + O(11^{2}0)q^{18} + −(102223264689557361929 + O(11^{2}0)) ∗ a^{4} + (332889877538916840034+O(11^{2}0))∗a^{3}+(155178524409514007361+O(11^{2}0))∗a^{2}+(236385904455101109948+ O(11^{2}0)) ∗ a + 267406116352202105485 + O(11^{2}0)q^{19} + −(106802364945428389007 + O(11^{2}0)) ∗ a^{4} − (73484333971793824548+O(11^{2}0))∗a^{3}+(248452910258062260177+O(11^{2}0))∗a^{2}−(237359369857924297976+ O(11^{2}0)) ∗ a + 83016675446571761948 + O(11^{2}0)q^{20} + (226558599902903013364 + O(11^{2}0)) ∗ a^{4} + (306754828431946001816+O(11^{2}0))∗a^{3}+(71714185102327232890+O(11^{2}0))∗a^{2}+(169717460484611971154+ O(11^{2}0)) ∗ a + 26374001414417019471 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{5} + 235939635302487581501 + O(11^{2}0)x^{4} + 19832230278235438204+O(11^{2}0)x^{3}+−75305933702655388704+O(11^{2}0)x^{2}+−286001669361950811530+ O(11^{2}0)x + −253175070989686225911 + O(11^{2}0). The slope of f_{0} is 1.

40. Consider f = q+6∗z^{1}9+3∗z^{1}8+10∗z^{1}7+6∗z^{1}6+3∗z^{1}5+5∗z^{1}4+4∗z^{1}3+5∗z^{1}2+z^{1}1+2∗z^{1}0+3∗z^{8}+6∗ z^{7}+3∗z^{6}+9∗z^{5}+9∗z^{4}+7∗z^{3}+9∗z^{2}+6∗z+3q^{2}+5∗z^{1}9+9∗z^{1}8+10∗z^{1}7+6∗z^{1}6+4∗z^{1}5+4∗z^{1}4+2∗z^{1}3+ 3∗z^{1}2+z^{1}1+3∗z^{1}0+9∗z^{7}+9∗z^{6}+9∗z^{5}+7∗z^{4}+3∗z^{3}+8∗z^{2}+3∗z+9q^{3}+6∗z^{1}9+8∗z^{1}7+6∗z^{1}6+z^{1}5+z^{1}4+ z^{1}3+5∗z^{1}1+z^{1}0+2∗z^{9}+7∗z^{7}+6∗z^{6}+3∗z^{4}+6∗z^{3}+7∗z^{2}+10∗z+10q^{4}+8∗z^{1}9+9∗z^{1}8+10∗z^{1}7+3∗z^{1}6+ 6∗z^{1}5+6∗z^{1}4+4∗z^{1}3+z^{1}2+3∗z^{1}1+4∗z^{1}0+10∗z^{9}+8∗z^{7}+10∗z^{6}+2∗z^{5}+10∗z^{4}+8∗z^{3}+8∗z^{2}+2∗z+1q^{5}+ 5∗z^{1}9+3∗z^{1}8+8∗z^{1}6+5∗z^{1}5+z^{1}4+z^{1}3+z^{1}2+10∗z^{1}0+z^{9}+5∗z^{8}+7∗z^{7}+z^{6}+8∗z^{5}+4∗z^{3}+7∗z^{2}+2∗z+ 3q^{6}+4∗z^{1}9+3∗z^{1}8+z^{1}7+z^{1}6+4∗z^{1}5+7∗z^{1}4+4∗z^{1}3+3∗z^{1}2+10∗z^{1}1+8∗z^{1}0+5∗z^{9}+10∗z^{8}+8∗z^{7}+4∗ z^{6}+9∗z^{5}+5∗z^{4}+8∗z^{3}+3∗z+8q^{7}+z^{1}9+9∗z^{1}8+8∗z^{1}6+3∗z^{1}5+5∗z^{1}4+10∗z^{1}3+3∗z^{1}2+9∗z^{1}1+3∗z^{1}0+ 3∗z^{9}+3∗z^{8}+4∗z^{6}+z^{5}+7∗z^{4}+z^{3}+8∗z^{2}+9∗z+4q^{8}+10∗z^{1}8+9∗z^{1}7+7∗z^{1}6+z^{1}5+2∗z^{1}4+10∗z^{1}3+10∗ z^{1}2+3∗z^{1}1+8∗z^{1}0+8∗z^{9}+7∗z^{8}+8∗z^{7}+10∗z^{6}+z^{4}+8∗z^{3}+5∗z^{2}+9∗z+8q^{9}+7∗z^{1}9+2∗z^{1}8+z^{1}7+10∗ z^{1}5+6∗z^{1}4+5∗z^{1}3+10∗z^{1}2+7∗z^{1}1+10∗z^{1}0+z^{9}+5∗z^{8}+9∗z^{7}+7∗z^{6}+3∗z^{5}+10∗z^{4}+3∗z^{3}+2∗z^{2}+5∗z+ 10q^{10}+2∗z^{1}9+9∗z^{1}8+2∗z^{1}7+5∗z^{1}6+4∗z^{1}5+z^{1}4+5∗z^{1}3+3∗z^{1}2+9∗z^{1}1+7∗z^{1}0+9∗z^{9}+z^{8}+z^{7}+2∗z^{6}+ 10∗z^{5}+8∗z^{4}+8∗z^{3}+3∗z^{2}+z+10q^{12}+10q^{13}+3∗z^{1}9+8∗z^{1}8+10∗z^{1}7+7∗z^{1}6+2∗z^{1}5+3∗z^{1}4+9∗z^{1}3+8∗ z^{1}2+2∗z^{1}1+6∗z^{9}+7∗z^{8}+8∗z^{7}+z^{6}+8∗z^{5}+8∗z^{4}+3∗z^{3}+z^{2}+z+8q^{14}+8∗z^{1}8+7∗z^{1}7+8∗z^{1}5+9∗z^{1}4+4∗ z^{1}3+3∗z^{1}2+z^{1}1+9∗z^{1}0+7∗z^{8}+7∗z^{7}+4∗z^{6}+7∗z^{5}+2∗z^{4}+5∗z^{3}+8∗z^{2}+3∗zq^{15}+3∗z^{1}9+z^{1}8+3∗z^{1}7+ 8∗z^{1}6+2∗z^{1}5+z^{1}4+4∗z^{1}3+5∗z^{1}2+6∗z^{1}1+8∗z^{1}0+8∗z^{9}+4∗z^{8}+10∗z^{7}+8∗z^{6}+3∗z^{5}+3∗z^{4}+2∗z^{3}+8∗ z+2q^{16}+7∗z^{1}9+5∗z^{1}8+10∗z^{1}7+3∗z^{1}6+7∗z^{1}4+9∗z^{1}3+7∗z^{1}2+8∗z^{1}1+5∗z^{1}0+5∗z^{9}+5∗z^{8}+9∗z^{7}+8∗ z^{6}+3∗z^{5}+9∗z^{4}+10∗z^{3}+z+2q^{17}+5∗z^{1}9+7∗z^{1}8+z^{1}7+7∗z^{1}6+z^{1}5+3∗z^{1}4+6∗z^{1}3+4∗z^{1}2+5∗z^{1}1+10∗ z^{1}0+8∗z^{9}+3∗z^{8}+7∗z^{7}+10∗z^{6}+3∗z^{5}+4∗z^{4}+5∗z^{3}+4∗z^{2}+8∗z+4q^{18}+8∗z^{1}9+4∗z^{1}8+3∗z^{1}7+9∗z^{1}6+ z^{1}5+6∗z^{1}3+7∗z^{1}2+6∗z^{1}1+9∗z^{1}0+4∗z^{9}+4∗z^{8}+6∗z^{7}+8∗z^{6}+4∗z^{5}+9∗z^{4}+9∗z^{3}+8∗z^{2}+9∗z+7q^{19}+9∗ z^{1}9+8∗z^{1}8+8∗z^{1}7+10∗z^{1}6+8∗z^{1}5+10∗z^{1}4+3∗z^{1}3+6∗z^{1}2+3∗z^{1}1+10∗z^{9}+3∗z^{8}+z^{7}+10∗z^{6}+8∗z^{5}+ 8∗z^{4}+z^{3}+5∗z^{2}+4∗z+3q^{20}+5∗z^{1}9+7∗z^{1}8+10∗z^{1}7+5∗z^{1}5+3∗z^{1}4+3∗z^{1}3+2∗z^{1}2+2∗z^{1}1+z^{9}+8∗z^{8}+ 2∗z^{7}+10∗z^{6}+z^{5}+7∗z^{4}+3∗z^{3}+4∗z^{2}+6∗z+7q^{21}+⋯ ∈ S_{22}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{20}+6x^{19}+5x^{18}+9x^{17}+9x^{16}+2x^{15}+x^{13}+5x^{12}+7x^{11}+10x^{10}+7x^{9}+10x^{8}+9x^{6}+x^{5}+6x^{4}+7x^{3}+x+4 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{4}2203q^{2} + z^{6}2240q^{3} + z^{2}7504q^{4} + z^{2}4014q^{5} + z^{1}04443q^{6} + z^{1}16407q^{7} + z^{1}25827q^{8} + z^{9}027q^{9} + z^{6}6217q^{10} + z^{6}4592q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{9}6458)ω^{0} & 0 \\ 0 & u(z^{6}4592)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(126620356400921773329+ O(11^{2}0)) ∗ a^{4} + (116124275029971191609 + O(11^{2}0)) ∗ a^{3} + (54200904031282999190 + O(11^{2}0)) ∗ a^{2} + (27192497000010582578 + O(11^{2}0)) ∗ a − 48718083868206747624 + O(11^{2}0)q^{3} + a^{2} − 2097152q^{4} + −(141869525449007678538+O(11^{2}0))∗a^{4}+(151605097979372508+O(11^{2}0))∗a^{3}−(194258345799019839578+ O(11^{2}0))∗a^{2}−(13567311038568361865+O(11^{2}0))∗a−192798036755544856348+O(11^{2}0)q^{5}+−(166098107277332652573+ O(11^{2}0)) ∗ a^{4} + (262077544250279532283 + O(11^{2}0)) ∗ a^{3} − (261813859077022280337 + O(11^{2}0)) ∗ a^{2} + (39649627902202339891+O(11^{2}0))∗a+28297258544846119286+O(11^{2}0)q^{6}+−(27523841579649731365+ O(11^{2}0)) ∗ a^{4} + (20989109331104192790 ∗ 11 + O(11^{2}0)) ∗ a^{3} − (116279107353648712844 + O(11^{2}0)) ∗ a^{2} −(232160258876642163398+O(11^{2}0))∗a+8449694136397374108+O(11^{2}0)q^{7} +a^{3} −4194304∗aq^{8} + −(206728757606538797473+O(11^{2}0))∗a^{4}+(151039960356708325344+O(11^{2}0))∗a^{3}−(143140254869584811747+ O(11^{2}0)) ∗ a^{2} + (158708028880781775585 + O(11^{2}0)) ∗ a + 258252730417370027917 + O(11^{2}0)q^{9} + (2347130664133950252+O(11^{2}0))∗a^{4}−(273666526199816968253+O(11^{2}0))∗a^{3}−(146133167218784058890+ O(11^{2}0)) ∗ a^{2} − (168630994190763393409 + O(11^{2}0)) ∗ a − 77518454593766032615 + O(11^{2}0)q^{10} + (16934259358485004225∗11+O(11^{2}0))∗a^{4}+(21643924749675957590∗11+O(11^{2}0))∗a^{3}−(18870650781680015859∗ 11+O(11^{2}0))∗a^{2}+(843567987186155850∗11^{2}+O(11^{2}0))∗a−11614714628517476181∗11+O(11^{2}0)q^{11}+ −(39441557776730614930+O(11^{2}0))∗a^{4}−(179456845973125848329+O(11^{2}0))∗a^{3}−(66925150743570368834+ O(11^{2}0)) ∗ a^{2} − (245765224208019750989 + O(11^{2}0)) ∗ a + 186082390215892588565 + O(11^{2}0)q^{12} + −137858491849q^{13}+−(8392291303721118361+O(11^{2}0))∗a^{4}+(94080764106716336602+O(11^{2}0))∗a^{3}+ (52533650851688319072+O(11^{2}0))∗a^{2}+(118488488431490924845+O(11^{2}0))∗a+17266215589407957762+ O(11^{2}0)q^{14} + (130120802818579978503 + O(11^{2}0)) ∗ a^{4} + (284867432704604956532 + O(11^{2}0)) ∗ a^{3} + (132137272138135842803+O(11^{2}0))∗a^{2}+(287599407903525171023+O(11^{2}0))∗a−66108796264582541456+ O(11^{2}0)q^{15} + a^{4} − 6291456 ∗ a^{2} + 4398046511104q^{16} + (247263884779259668634 + O(11^{2}0)) ∗ a^{4} − (199320934044999770198+O(11^{2}0))∗a^{3}+(335640184363866694800+O(11^{2}0))∗a^{2}+(111413043909931950062+ O(11^{2}0)) ∗ a − 125895449116754295527 + O(11^{2}0)q^{17} + −(129685198089365853755 + O(11^{2}0)) ∗ a^{4} + (61366251647099302067+O(11^{2}0))∗a^{3}−(145396434073872810176+O(11^{2}0))∗a^{2}+(289716777941210368284+ O(11^{2}0)) ∗ a − 51076396781348042337 + O(11^{2}0)q^{18} + −(102223264689557361929 + O(11^{2}0)) ∗ a^{4} + (332889877538916840034+O(11^{2}0))∗a^{3}+(155178524409514007361+O(11^{2}0))∗a^{2}+(236385904455101109948+ O(11^{2}0)) ∗ a + 267406116352202105485 + O(11^{2}0)q^{19} + −(106802364945428389007 + O(11^{2}0)) ∗ a^{4} − (73484333971793824548+O(11^{2}0))∗a^{3}+(248452910258062260177+O(11^{2}0))∗a^{2}−(237359369857924297976+ O(11^{2}0)) ∗ a + 83016675446571761948 + O(11^{2}0)q^{20} + (226558599902903013364 + O(11^{2}0)) ∗ a^{4} + (306754828431946001816+O(11^{2}0))∗a^{3}+(71714185102327232890+O(11^{2}0))∗a^{2}+(169717460484611971154+ O(11^{2}0)) ∗ a + 26374001414417019471 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{5} + 235939635302487581501 + O(11^{2}0)x^{4} + 19832230278235438204+O(11^{2}0)x^{3}+−75305933702655388704+O(11^{2}0)x^{2}+−286001669361950811530+ O(11^{2}0)x + −253175070989686225911 + O(11^{2}0). The slope of f_{0} is 1.

41. Consider f = q+3∗z^{1}9+2∗z^{1}8+5∗z^{1}7+9∗z^{1}6+6∗z^{1}4+7∗z^{1}3+8∗z^{1}2+3∗z^{1}1+4∗z^{1}0+z^{9}+9∗z^{8}+2∗z^{7}+ 7∗z^{6}+3∗z^{4}+8∗z^{2}+8∗z+9q^{2}+10∗z^{1}9+10∗z^{1}8+3∗z^{1}7+3∗z^{1}6+3∗z^{1}5+8∗z^{1}4+8∗z^{1}3+5∗z^{1}2+8∗z^{1}1+4∗ z^{1}0+4∗z^{9}+2∗z^{8}+8∗z^{7}+2∗z^{6}+5∗z^{5}+5∗z^{3}+9∗z^{2}+3∗zq^{3}+7∗z^{1}9+9∗z^{1}8+6∗z^{1}7+z^{1}6+6∗z^{1}5+9∗z^{1}4+ z^{1}3+5∗z^{1}2+2∗z^{1}0+10∗z^{8}+2∗z^{7}+6∗z^{6}+4∗z^{5}+4∗z^{4}+6∗z^{3}+8∗z^{2}+2∗z+10q^{4}+2∗z^{1}9+4∗z^{1}8+9∗z^{1}7+ 3∗z^{1}6+7∗z^{1}5+7∗z^{1}4+6∗z^{1}2+4∗z^{1}0+5∗z^{9}+z^{7}+6∗z^{6}+2∗z^{5}+5∗z^{4}+5∗z^{3}+8∗z^{2}+6∗z+7q^{5}+9∗z^{1}9+4∗ z^{1}7+3∗z^{1}4+8∗z^{1}3+5∗z^{1}2+z^{1}1+4∗z^{1}0+9∗z^{9}+10∗z^{8}+7∗z^{7}+7∗z^{6}+z^{4}+7∗z^{3}+7∗z^{2}+5∗z+10q^{6}+4∗ z^{1}9+8∗z^{1}8+7∗z^{1}7+z^{1}5+6∗z^{1}4+4∗z^{1}3+5∗z^{1}2+5∗z^{1}0+z^{9}+2∗z^{8}+2∗z^{7}+8∗z^{6}+9∗z^{5}+z^{4}+9∗z^{2}+8∗z+ 10q^{7}+2∗z^{1}9+10∗z^{1}8+z^{1}7+8∗z^{1}6+3∗z^{1}5+8∗z^{1}4+8∗z^{1}3+2∗z^{1}2+7∗z^{1}1+8∗z^{1}0+2∗z^{9}+2∗z^{8}+2∗z^{7}+ 2∗z^{6}+z^{4}+10∗z^{2}+10∗z+1q^{8}+3∗z^{1}9+9∗z^{1}8+z^{1}7+z^{1}6+5∗z^{1}5+6∗z^{1}4+2∗z^{1}3+z^{1}2+10∗z^{1}1+9∗z^{1}0+5∗ z^{9}+7∗z^{8}+10∗z^{6}+9∗z^{5}+3∗z^{4}+4∗z^{3}+10∗z^{2}+6q^{9}+2∗z^{1}9+z^{1}8+2∗z^{1}7+z^{1}6+9∗z^{1}4+3∗z^{1}3+4∗z^{1}2+2∗ z^{1}1+5∗z^{1}0+10∗z^{9}+8∗z^{8}+6∗z^{7}+z^{6}+9∗z^{5}+5∗z^{4}+z^{3}+3∗z^{2}+8∗z+3q^{10}+6∗z^{1}9+2∗z^{1}8+8∗z^{1}7+5∗z^{1}6+ 5∗z^{1}5+z^{1}3+2∗z^{1}2+4∗z^{1}1+6∗z^{1}0+7∗z^{9}+9∗z^{8}+3∗z^{7}+4∗z^{6}+z^{5}+9∗z^{4}+9∗z^{3}+z^{2}+6∗z+3q^{12}+10q^{13}+3∗ z^{1}9+9∗z^{1}8+4∗z^{1}7+7∗z^{1}6+3∗z^{1}5+4∗z^{1}3+7∗z^{1}2+9∗z^{1}1+8∗z^{1}0+6∗z^{9}+z^{8}+2∗z^{7}+7∗z^{6}+2∗z^{5}+5∗z^{3}+ 2∗z^{2}+7q^{14}+z^{1}9+3∗z^{1}8+3∗z^{1}7+8∗z^{1}6+8∗z^{1}5+z^{1}4+7∗z^{1}3+9∗z^{1}2+10∗z^{1}0+3∗z^{9}+6∗z^{8}+3∗z^{7}+z^{6}+ 4∗z^{5}+2∗z^{2}+4∗z+9q^{15}+7∗z^{1}9+7∗z^{1}7+6∗z^{1}6+7∗z^{1}5+4∗z^{1}4+10∗z^{1}3+10∗z^{1}2+6∗z^{1}1+5∗z^{1}0+4∗z^{9}+ z^{8}+6∗z^{7}+2∗z^{6}+3∗z^{5}+4∗z^{4}+5∗z^{3}+4∗z^{2}+8∗z+7q^{16}+3∗z^{1}9+z^{1}8+5∗z^{1}6+5∗z^{1}5+8∗z^{1}2+7∗z^{1}1+5∗ z^{1}0+6∗z^{9}+9∗z^{8}+8∗z^{7}+2∗z^{6}+2∗z^{5}+5∗z^{4}+6∗z^{3}+6∗z^{2}+10∗zq^{17}+3∗z^{1}9+8∗z^{1}8+6∗z^{1}7+3∗z^{1}6+6∗z^{1}5+ 9∗z^{1}4+z^{1}2+7∗z^{1}1+7∗z^{1}0+7∗z^{9}+10∗z^{8}+9∗z^{7}+4∗z^{6}+3∗z^{5}+7∗z^{4}+z^{3}+9∗z^{2}+7q^{18}+5∗z^{1}9+9∗z^{1}8+8∗ z^{1}7+2∗z^{1}6+9∗z^{1}5+8∗z^{1}4+8∗z^{1}3+10∗z^{1}2+9∗z^{1}1+3∗z^{9}+4∗z^{8}+4∗z^{7}+10∗z^{5}+9∗z^{4}+6∗z^{3}+6∗z^{2}+3∗z+ 7q^{19}+6∗z^{1}9+5∗z^{1}8+3∗z^{1}7+3∗z^{1}6+z^{1}5+z^{1}4+6∗z^{1}3+7∗z^{1}2+6∗z^{1}1+8∗z^{1}0+6∗z^{9}+z^{7}+5∗z^{6}+9∗z^{5}+ 7∗z^{4}+2∗z^{3}+10∗z^{2}+10∗z+9q^{20}+2∗z^{1}8+5∗z^{1}7+4∗z^{1}6+10∗z^{1}5+8∗z^{1}4+3∗z^{1}3+8∗z^{1}2+z^{1}1+10∗z^{1}0+ z^{9}+8∗z^{8}+3∗z^{7}+8∗z^{6}+10∗z^{5}+10∗z^{4}+6∗z^{2}+2q^{21}+⋯ ∈ S_{22}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{20}+6x^{19}+5x^{18}+9x^{17}+9x^{16}+2x^{15}+x^{13}+5x^{12}+7x^{11}+10x^{10}+7x^{9}+10x^{8}+9x^{6}+x^{5}+6x^{4}+7x^{3}+x+4 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{1}26793q^{2} + z^{6}1740q^{3} + z^{4}9474q^{4} + z^{7}4734q^{5} + z^{2}7483q^{6} + z^{7}617q^{7} + z^{1}44787q^{8} + z^{9}7237q^{9} + z^{4}0477q^{10} + z^{1}32302q^{11} + · · · . We deduce

(\begin{matrix}u(z^{2}8748)ω^{0} & 0 \\ 0 & u(z^{1}32302)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(126620356400921773329+ O(11^{2}0)) ∗ a^{4} + (116124275029971191609 + O(11^{2}0)) ∗ a^{3} + (54200904031282999190 + O(11^{2}0)) ∗ a^{2} + (27192497000010582578 + O(11^{2}0)) ∗ a − 48718083868206747624 + O(11^{2}0)q^{3} + a^{2} − 2097152q^{4} + −(141869525449007678538+O(11^{2}0))∗a^{4}+(151605097979372508+O(11^{2}0))∗a^{3}−(194258345799019839578+ O(11^{2}0))∗a^{2}−(13567311038568361865+O(11^{2}0))∗a−192798036755544856348+O(11^{2}0)q^{5}+−(166098107277332652573+ O(11^{2}0)) ∗ a^{4} + (262077544250279532283 + O(11^{2}0)) ∗ a^{3} − (261813859077022280337 + O(11^{2}0)) ∗ a^{2} + (39649627902202339891+O(11^{2}0))∗a+28297258544846119286+O(11^{2}0)q^{6}+−(27523841579649731365+ O(11^{2}0)) ∗ a^{4} + (20989109331104192790 ∗ 11 + O(11^{2}0)) ∗ a^{3} − (116279107353648712844 + O(11^{2}0)) ∗ a^{2} −(232160258876642163398+O(11^{2}0))∗a+8449694136397374108+O(11^{2}0)q^{7} +a^{3} −4194304∗aq^{8} + −(206728757606538797473+O(11^{2}0))∗a^{4}+(151039960356708325344+O(11^{2}0))∗a^{3}−(143140254869584811747+ O(11^{2}0)) ∗ a^{2} + (158708028880781775585 + O(11^{2}0)) ∗ a + 258252730417370027917 + O(11^{2}0)q^{9} + (2347130664133950252+O(11^{2}0))∗a^{4}−(273666526199816968253+O(11^{2}0))∗a^{3}−(146133167218784058890+ O(11^{2}0)) ∗ a^{2} − (168630994190763393409 + O(11^{2}0)) ∗ a − 77518454593766032615 + O(11^{2}0)q^{10} + (16934259358485004225∗11+O(11^{2}0))∗a^{4}+(21643924749675957590∗11+O(11^{2}0))∗a^{3}−(18870650781680015859∗ 11+O(11^{2}0))∗a^{2}+(843567987186155850∗11^{2}+O(11^{2}0))∗a−11614714628517476181∗11+O(11^{2}0)q^{11}+ −(39441557776730614930+O(11^{2}0))∗a^{4}−(179456845973125848329+O(11^{2}0))∗a^{3}−(66925150743570368834+ O(11^{2}0)) ∗ a^{2} − (245765224208019750989 + O(11^{2}0)) ∗ a + 186082390215892588565 + O(11^{2}0)q^{12} + −137858491849q^{13}+−(8392291303721118361+O(11^{2}0))∗a^{4}+(94080764106716336602+O(11^{2}0))∗a^{3}+ (52533650851688319072+O(11^{2}0))∗a^{2}+(118488488431490924845+O(11^{2}0))∗a+17266215589407957762+ O(11^{2}0)q^{14} + (130120802818579978503 + O(11^{2}0)) ∗ a^{4} + (284867432704604956532 + O(11^{2}0)) ∗ a^{3} + (132137272138135842803+O(11^{2}0))∗a^{2}+(287599407903525171023+O(11^{2}0))∗a−66108796264582541456+ O(11^{2}0)q^{15} + a^{4} − 6291456 ∗ a^{2} + 4398046511104q^{16} + (247263884779259668634 + O(11^{2}0)) ∗ a^{4} − (199320934044999770198+O(11^{2}0))∗a^{3}+(335640184363866694800+O(11^{2}0))∗a^{2}+(111413043909931950062+ O(11^{2}0)) ∗ a − 125895449116754295527 + O(11^{2}0)q^{17} + −(129685198089365853755 + O(11^{2}0)) ∗ a^{4} + (61366251647099302067+O(11^{2}0))∗a^{3}−(145396434073872810176+O(11^{2}0))∗a^{2}+(289716777941210368284+ O(11^{2}0)) ∗ a − 51076396781348042337 + O(11^{2}0)q^{18} + −(102223264689557361929 + O(11^{2}0)) ∗ a^{4} + (332889877538916840034+O(11^{2}0))∗a^{3}+(155178524409514007361+O(11^{2}0))∗a^{2}+(236385904455101109948+ O(11^{2}0)) ∗ a + 267406116352202105485 + O(11^{2}0)q^{19} + −(106802364945428389007 + O(11^{2}0)) ∗ a^{4} − (73484333971793824548+O(11^{2}0))∗a^{3}+(248452910258062260177+O(11^{2}0))∗a^{2}−(237359369857924297976+ O(11^{2}0)) ∗ a + 83016675446571761948 + O(11^{2}0)q^{20} + (226558599902903013364 + O(11^{2}0)) ∗ a^{4} + (306754828431946001816+O(11^{2}0))∗a^{3}+(71714185102327232890+O(11^{2}0))∗a^{2}+(169717460484611971154+ O(11^{2}0)) ∗ a + 26374001414417019471 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{5} + 235939635302487581501 + O(11^{2}0)x^{4} + 19832230278235438204+O(11^{2}0)x^{3}+−75305933702655388704+O(11^{2}0)x^{2}+−286001669361950811530+ O(11^{2}0)x + −253175070989686225911 + O(11^{2}0). The slope of f_{0} is 1.

42. Consider f = q+4∗z^{1}9+5∗z^{1}7+5∗z^{1}6+10∗z^{1}5+6∗z^{1}4+3∗z^{1}3+8∗z^{1}2+z^{1}1+z^{1}0+8∗z^{9}+5∗z^{8}+2∗z^{7}+ 3∗z^{6}+3∗z^{4}+3∗z^{3}+2∗z^{2}+9∗z+9q^{2}+10∗z^{1}9+10∗z^{1}8+9∗z^{1}6+9∗z^{1}5+8∗z^{1}4+9∗z^{1}3+2∗z^{1}2+9∗z^{1}1+10∗ z^{9}+4∗z^{8}+6∗z^{7}+2∗z^{6}+2∗z^{5}+10∗z^{4}+2∗z^{3}+9∗z^{2}+1q^{3}+3∗z^{1}9+10∗z^{1}8+9∗z^{1}7+10∗z^{1}6+5∗z^{1}5+z^{1}4+ 8∗z^{1}3+4∗z^{1}2+5∗z^{1}1+z^{1}0+5∗z^{9}+5∗z^{8}+7∗z^{7}+4∗z^{6}+10∗z^{5}+2∗z^{4}+z^{3}+z^{2}+z+4q^{4}+6∗z^{1}9+8∗z^{1}8+4∗ z^{1}7+z^{1}6+7∗z^{1}4+z^{1}3+10∗z^{1}2+6∗z^{1}1+2∗z^{1}0+5∗z^{8}+3∗z^{7}+3∗z^{6}+2∗z^{5}+3∗z^{4}+10∗z^{3}+6∗z+10q^{5}+8∗ z^{1}9+3∗z^{1}8+10∗z^{1}7+5∗z^{1}6+z^{1}5+4∗z^{1}4+z^{1}3+3∗z^{1}2+6∗z^{1}1+5∗z^{1}0+10∗z^{9}+10∗z^{8}+9∗z^{7}+8∗z^{6}+9∗ z^{5}+7∗z^{4}+z^{3}+z^{2}+6∗z+1q^{6}+5∗z^{1}9+3∗z^{1}7+3∗z^{1}6+2∗z^{1}5+4∗z^{1}3+5∗z^{1}2+6∗z^{1}1+9∗z^{1}0+6∗z^{9}+8∗z^{8}+ 4∗z^{7}+9∗z^{5}+5∗z^{4}+9∗z^{3}+9∗z^{2}+7∗z+2q^{7}+z^{1}9+2∗z^{1}8+8∗z^{1}7+9∗z^{1}6+5∗z^{1}5+10∗z^{1}4+2∗z^{1}3+6∗z^{1}2+ 8∗z^{1}1+2∗z^{1}0+9∗z^{9}+2∗z^{8}+z^{7}+9∗z^{6}+8∗z^{5}+8∗z^{4}+10∗z^{3}+9∗z^{2}+7∗z+9q^{8}+3∗z^{1}9+7∗z^{1}8+4∗z^{1}7+3∗ z^{1}6+z^{1}5+9∗z^{1}4+z^{1}2+10∗z^{1}1+6∗z^{1}0+10∗z^{9}+z^{8}+6∗z^{7}+2∗z^{6}+7∗z^{5}+3∗z^{4}+z^{3}+4∗z^{2}+8∗z+3q^{9}+10∗ z^{1}9+5∗z^{1}8+7∗z^{1}7+7∗z^{1}6+z^{1}5+3∗z^{1}4+4∗z^{1}3+5∗z^{1}2+8∗z^{1}0+6∗z^{9}+3∗z^{8}+6∗z^{7}+6∗z^{6}+5∗z^{5}+7∗z^{4}+ 10∗z^{3}+3∗z^{2}+5∗z+1q^{10}+8∗z^{1}9+7∗z^{1}8+2∗z^{1}7+5∗z^{1}6+2∗z^{1}5+9∗z^{1}4+2∗z^{1}3+z^{1}2+3∗z^{1}1+z^{1}0+4∗z^{9}+ 5∗z^{8}+3∗z^{7}+2∗z^{6}+10∗z^{4}+5∗z^{3}+2∗z^{2}+10∗z+8q^{12}+10q^{13}+z^{1}9+9∗z^{1}8+8∗z^{1}7+8∗z^{1}5+z^{1}4+2∗z^{1}3+2∗ z^{1}2+10∗z^{1}1+3∗z^{1}0+10∗z^{9}+6∗z^{8}+9∗z^{7}+2∗z^{6}+6∗z^{5}+7∗z^{4}+4∗z^{3}+10∗z^{2}+4∗z+3q^{14}+5∗z^{1}9+3∗z^{1}8+ 5∗z^{1}7+z^{1}6+7∗z^{1}5+5∗z^{1}4+z^{1}3+4∗z^{1}2+2∗z^{1}1+9∗z^{1}0+z^{9}+8∗z^{8}+3∗z^{7}+3∗z^{6}+5∗z^{5}+8∗z^{4}+3∗z^{3}+5∗ z^{2}+9∗z+4q^{15}+4∗z^{1}9+z^{1}8+7∗z^{1}7+6∗z^{1}6+8∗z^{1}5+6∗z^{1}3+9∗z^{1}1+8∗z^{1}0+z^{9}+10∗z^{8}+5∗z^{7}+2∗z^{6}+8∗ z^{5}+3∗z^{4}+z^{3}+10∗z^{2}+9∗z+8q^{16}+7∗z^{1}9+5∗z^{1}8+9∗z^{1}7+2∗z^{1}6+z^{1}5+3∗z^{1}4+4∗z^{1}3+8∗z^{1}2+z^{1}1+4∗ z^{1}0+8∗z^{9}+3∗z^{8}+z^{7}+6∗z^{6}+10∗z^{5}+6∗z^{4}+4∗z^{3}+2∗z^{2}+9∗z+10q^{17}+5∗z^{1}9+7∗z^{1}8+6∗z^{1}7+7∗z^{1}6+9∗ z^{1}5+9∗z^{1}4+5∗z^{1}3+6∗z^{1}2+7∗z^{1}1+7∗z^{1}0+2∗z^{9}+3∗z^{7}+2∗z^{6}+7∗z^{5}+6∗z^{4}+z^{3}+z^{2}+5∗zq^{18}+9∗z^{1}9+3∗ z^{1}8+4∗z^{1}7+7∗z^{1}6+7∗z^{1}5+6∗z^{1}3+6∗z^{1}2+5∗z^{1}1+2∗z^{1}0+10∗z^{9}+6∗z^{8}+z^{7}+6∗z^{6}+10∗z^{5}+9∗z^{4}+7∗z^{2}+ z+8q^{19}+6∗z^{1}9+4∗z^{1}8+7∗z^{1}7+4∗z^{1}6+3∗z^{1}5+10∗z^{1}3+8∗z^{1}2+7∗z^{1}1+5∗z^{1}0+z^{8}+2∗z^{7}+8∗z^{6}+10∗z^{5}+ 8∗z^{4}+4∗z^{3}+7∗zq^{20}+8∗z^{1}9+10∗z^{1}8+5∗z^{1}7+10∗z^{1}6+8∗z^{1}5+4∗z^{1}4+6∗z^{1}2+6∗z^{1}1+5∗z^{1}0+5∗z^{9}+5∗ z^{8}+10∗z^{7}+6∗z^{5}+3∗z^{4}+4∗z^{3}+3∗z^{2}+8∗z+5q^{21}+⋯ ∈ S_{22}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{20}+6x^{19}+5x^{18}+9x^{17}+9x^{16}+2x^{15}+x^{13}+5x^{12}+7x^{11}+10x^{10}+7x^{9}+10x^{8}+9x^{6}+x^{5}+6x^{4}+7x^{3}+x+4 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{1}14013q^{2} + z^{1}22740q^{3} + z^{1}06984q^{4} + z^{6}794q^{5} + z^{7}5703q^{6} + z^{7}3897q^{7} + z^{8}6367q^{8} + z^{1}25967q^{9} + z^{1}20807q^{10} + z^{8}5232q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{7}5818)ω^{0} & 0 \\ 0 & u(z^{8}5232)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(126620356400921773329+ O(11^{2}0)) ∗ a^{4} + (116124275029971191609 + O(11^{2}0)) ∗ a^{3} + (54200904031282999190 + O(11^{2}0)) ∗ a^{2} + (27192497000010582578 + O(11^{2}0)) ∗ a − 48718083868206747624 + O(11^{2}0)q^{3} + a^{2} − 2097152q^{4} + −(141869525449007678538+O(11^{2}0))∗a^{4}+(151605097979372508+O(11^{2}0))∗a^{3}−(194258345799019839578+ O(11^{2}0))∗a^{2}−(13567311038568361865+O(11^{2}0))∗a−192798036755544856348+O(11^{2}0)q^{5}+−(166098107277332652573+ O(11^{2}0)) ∗ a^{4} + (262077544250279532283 + O(11^{2}0)) ∗ a^{3} − (261813859077022280337 + O(11^{2}0)) ∗ a^{2} + (39649627902202339891+O(11^{2}0))∗a+28297258544846119286+O(11^{2}0)q^{6}+−(27523841579649731365+ O(11^{2}0)) ∗ a^{4} + (20989109331104192790 ∗ 11 + O(11^{2}0)) ∗ a^{3} − (116279107353648712844 + O(11^{2}0)) ∗ a^{2} −(232160258876642163398+O(11^{2}0))∗a+8449694136397374108+O(11^{2}0)q^{7} +a^{3} −4194304∗aq^{8} + −(206728757606538797473+O(11^{2}0))∗a^{4}+(151039960356708325344+O(11^{2}0))∗a^{3}−(143140254869584811747+ O(11^{2}0)) ∗ a^{2} + (158708028880781775585 + O(11^{2}0)) ∗ a + 258252730417370027917 + O(11^{2}0)q^{9} + (2347130664133950252+O(11^{2}0))∗a^{4}−(273666526199816968253+O(11^{2}0))∗a^{3}−(146133167218784058890+ O(11^{2}0)) ∗ a^{2} − (168630994190763393409 + O(11^{2}0)) ∗ a − 77518454593766032615 + O(11^{2}0)q^{10} + (16934259358485004225∗11+O(11^{2}0))∗a^{4}+(21643924749675957590∗11+O(11^{2}0))∗a^{3}−(18870650781680015859∗ 11+O(11^{2}0))∗a^{2}+(843567987186155850∗11^{2}+O(11^{2}0))∗a−11614714628517476181∗11+O(11^{2}0)q^{11}+ −(39441557776730614930+O(11^{2}0))∗a^{4}−(179456845973125848329+O(11^{2}0))∗a^{3}−(66925150743570368834+ O(11^{2}0)) ∗ a^{2} − (245765224208019750989 + O(11^{2}0)) ∗ a + 186082390215892588565 + O(11^{2}0)q^{12} + −137858491849q^{13}+−(8392291303721118361+O(11^{2}0))∗a^{4}+(94080764106716336602+O(11^{2}0))∗a^{3}+ (52533650851688319072+O(11^{2}0))∗a^{2}+(118488488431490924845+O(11^{2}0))∗a+17266215589407957762+ O(11^{2}0)q^{14} + (130120802818579978503 + O(11^{2}0)) ∗ a^{4} + (284867432704604956532 + O(11^{2}0)) ∗ a^{3} + (132137272138135842803+O(11^{2}0))∗a^{2}+(287599407903525171023+O(11^{2}0))∗a−66108796264582541456+ O(11^{2}0)q^{15} + a^{4} − 6291456 ∗ a^{2} + 4398046511104q^{16} + (247263884779259668634 + O(11^{2}0)) ∗ a^{4} − (199320934044999770198+O(11^{2}0))∗a^{3}+(335640184363866694800+O(11^{2}0))∗a^{2}+(111413043909931950062+ O(11^{2}0)) ∗ a − 125895449116754295527 + O(11^{2}0)q^{17} + −(129685198089365853755 + O(11^{2}0)) ∗ a^{4} + (61366251647099302067+O(11^{2}0))∗a^{3}−(145396434073872810176+O(11^{2}0))∗a^{2}+(289716777941210368284+ O(11^{2}0)) ∗ a − 51076396781348042337 + O(11^{2}0)q^{18} + −(102223264689557361929 + O(11^{2}0)) ∗ a^{4} + (332889877538916840034+O(11^{2}0))∗a^{3}+(155178524409514007361+O(11^{2}0))∗a^{2}+(236385904455101109948+ O(11^{2}0)) ∗ a + 267406116352202105485 + O(11^{2}0)q^{19} + −(106802364945428389007 + O(11^{2}0)) ∗ a^{4} − (73484333971793824548+O(11^{2}0))∗a^{3}+(248452910258062260177+O(11^{2}0))∗a^{2}−(237359369857924297976+ O(11^{2}0)) ∗ a + 83016675446571761948 + O(11^{2}0)q^{20} + (226558599902903013364 + O(11^{2}0)) ∗ a^{4} + (306754828431946001816+O(11^{2}0))∗a^{3}+(71714185102327232890+O(11^{2}0))∗a^{2}+(169717460484611971154+ O(11^{2}0)) ∗ a + 26374001414417019471 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{5} + 235939635302487581501 + O(11^{2}0)x^{4} + 19832230278235438204+O(11^{2}0)x^{3}+−75305933702655388704+O(11^{2}0)x^{2}+−286001669361950811530+ O(11^{2}0)x + −253175070989686225911 + O(11^{2}0). The slope of f_{0} is 1.

43. Consider f = q+7q^{2}+6q^{3}+3q^{4}+3q^{5}+9q^{6}+7q^{8}+10q^{10}+7q^{12}+q^{13}+7q^{15}+10q^{16}+4q^{17}+4q^{19}+9q^{20}+⋯ ∈ S_{22}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{12} + x^{8} + x^{7} + 4x^{6} + 2x^{5} + 5x^{4} + 5x^{3} + 6x^{2} + 5x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + 9q^{2} + 2q^{3} + 9q^{4} + 5q^{5} + 7q^{6} + 5q^{8} + q^{10} + 3q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{0} & 0 \\ 0 & u(3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+262181893417714305427+ O(11^{2}0)q^{2}+−174822783418317580342+O(11^{2}0)q^{3}+6771797187940663024+O(11^{2}0)q^{4}+155278044844411215334+ O(11^{2}0)q^{5}+244135640518448845406+O(11^{2}0)q^{6}+−21212439213639365510∗11+O(11^{2}0)q^{7}+−278283433385866024208+ O(11^{2}0)q^{8} + −14411991429224620334 ∗ 11 + O(11^{2}0)q^{9} + −184213244744869145565 + O(11^{2}0)q^{10} + 30072391562676185467∗11+O(11^{2}0)q^{11}+−251630010589184376791+O(11^{2}0)q^{12}+137858491849q^{13}+ 8402159660865650986∗11+O(11^{2}0)q^{14}+−230434857609411452573+O(11^{2}0)q^{15}+265255014858109617505+ O(11^{2}0)q^{16} + 155374830379721516425 + O(11^{2}0)q^{17} + −29212630374336327562 ∗ 11 + O(11^{2}0)q^{18} + −146929748077572279574+O(11^{2}0)q^{19}+−317229535125808514497+O(11^{2}0)q^{20}+−13446932166260753333∗ 11 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

44. Consider f = q+6∗z^{1}1+5∗z^{1}0+5∗z^{9}+7∗z^{8}+9∗z^{7}+9∗z^{6}+10∗z^{5}+8∗z^{4}+5∗z^{3}+3∗z^{2}+7∗z+1q^{2}+3∗ z^{1}1+3∗z^{1}0+3∗z^{9}+z^{8}+5∗z^{7}+2∗z^{6}+2∗z^{5}+3∗z^{4}+9∗z^{3}+8∗z^{2}+6∗z+10q^{3}+9∗z^{1}1+z^{1}0+z^{9}+10∗z^{8}+ 7∗z^{7}+2∗z^{6}+10∗z^{5}+3∗z^{4}+8∗z^{2}+8q^{4}+10∗z^{1}1+4∗z^{1}0+4∗z^{9}+4∗z^{8}+7∗z^{7}+6∗z^{5}+7∗z^{3}+z+6q^{5}+5∗ z^{1}1+4∗z^{1}0+4∗z^{9}+3∗z^{8}+z^{6}+2∗z^{5}+7∗z^{4}+2∗z^{3}+4∗z^{2}+5∗z+3q^{6}+z^{1}1+z^{1}0+z^{9}+4∗z^{8}+9∗z^{7}+8∗ z^{6}+8∗z^{5}+z^{4}+3∗z^{3}+10∗z^{2}+2∗z+2q^{7}+5∗z^{1}1+2∗z^{1}0+2∗z^{9}+2∗z^{8}+9∗z^{7}+3∗z^{5}+9∗z^{3}+6∗z+4q^{8}+ 4∗z^{1}1+z^{1}0+z^{9}+9∗z^{8}+3∗z^{6}+6∗z^{5}+10∗z^{4}+6∗z^{3}+z^{2}+4∗zq^{9}+10∗z^{1}1+5∗z^{1}0+5∗z^{9}+10∗z^{8}+8∗z^{7}+ 6∗z^{6}+9∗z^{4}+9∗z^{3}+2∗z^{2}+6∗z+7q^{10}+4∗z^{1}1+2∗z^{1}0+2∗z^{9}+4∗z^{8}+z^{7}+9∗z^{6}+8∗z^{4}+8∗z^{3}+3∗z^{2}+9∗ z+9q^{12}+q^{13}+z^{1}1+2∗z^{1}0+2∗z^{9}+10∗z^{8}+10∗z^{7}+3∗z^{6}+2∗z^{5}+10∗z^{4}+5∗z^{3}+z^{2}+7∗z+7q^{14}+6∗z^{1}1+ 4∗z^{1}0+4∗z^{9}+z^{8}+8∗z^{7}+3∗z^{6}+5∗z^{5}+10∗z^{4}+3∗z^{3}+z^{2}+2∗z+8q^{15}+4∗z^{1}1+3∗z^{8}+10∗z^{7}+8∗z^{6}+z^{5}+ z^{4}+4∗z^{3}+10∗z^{2}+10∗z+5q^{16}+z^{1}1+5∗z^{1}0+5∗z^{9}+6∗z^{8}+2∗z^{7}+10∗z^{6}+6∗z^{5}+4∗z^{4}+7∗z^{2}+1q^{17}+ 9∗z^{1}1+z^{1}0+z^{9}+10∗z^{8}+7∗z^{7}+2∗z^{6}+10∗z^{5}+3∗z^{4}+8∗z^{2}+1q^{18}+4∗z^{1}1+7∗z^{1}0+7∗z^{9}+z^{8}+6∗z^{7}+ 6∗z^{6}+3∗z^{5}+9∗z^{4}+7∗z^{3}+2∗z^{2}+z+3q^{19}+4∗z^{1}1+2∗z^{1}0+2∗z^{9}+4∗z^{8}+z^{7}+9∗z^{6}+8∗z^{4}+8∗z^{3}+3∗ z^{2}+9∗z+6q^{20}+z^{1}1+9∗z^{8}+8∗z^{7}+2∗z^{6}+3∗z^{5}+3∗z^{4}+z^{3}+8∗z^{2}+8∗z+6q^{21}+⋯ ∈ S_{22}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{12} + x^{8} + x^{7} + 4x^{6} + 2x^{5} + 5x^{4} + 5x^{3} + 6x^{2} + 5x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{8}69q^{2} + z^{8}25q^{3} + z^{1}223q^{4} + z^{8}19q^{5} + z^{3}64q^{6} + z^{1}257q^{7} + z^{1}122q^{8} + z^{2}77q^{9} + z^{3}58q^{10} + z^{6}84q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{6}46)ω^{0} & 0 \\ 0 & u(z^{6}84)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(228494176128019091054+ O(11^{2}0)) ∗ a^{2} + (126232126130457204851 + O(11^{2}0)) ∗ a + 1007663414535709320 ∗ 11 + O(11^{2}0)q^{3} + a^{2} − 2097152q^{4} + −(95366962654606477607 + O(11^{2}0)) ∗ a^{2} + (286975885151558282622 + O(11^{2}0)) ∗ a − 311941616959375706649+O(11^{2}0)q^{5}+−(283882396171998047147+O(11^{2}0))∗a^{2}−(213752326028574652862+ O(11^{2}0)) ∗ a + 196019062370748611157 + O(11^{2}0)q^{6} + (149635348179479808402 + O(11^{2}0)) ∗ a^{2} − (329976420198962930333+O(11^{2}0))∗a−223322087805285745160+O(11^{2}0)q^{7}+−(18829220815627542624+ O(11^{2}0))∗a^{2}−(207662230975403199057+O(11^{2}0))∗a−68567498352877310860+O(11^{2}0)q^{8}+−(72550889623189489209+ O(11^{2}0))∗a^{2}−(9542890798164020166+O(11^{2}0))∗a−263903860454195927970+O(11^{2}0)q^{9}+−(257941820253429481272+ O(11^{2}0)) ∗ a^{2} + (71089598886485630521 + O(11^{2}0)) ∗ a + 268333037649251759590 + O(11^{2}0)q^{10} + (30167737553010492191∗11+O(11^{2}0))∗a^{2}+(17609105808763440729∗11+O(11^{2}0))∗a−8809998171776527779∗ 11 + O(11^{2}0)q^{11} + −(302101307293963381738 + O(11^{2}0)) ∗ a^{2} − (138292184510539719423 + O(11^{2}0)) ∗ a−287404695168779539376+O(11^{2}0)q^{12}+137858491849q^{13}+−(15435291878799267170+O(11^{2}0))∗a^{2}− (29332476439212035770+O(11^{2}0))∗a+14271130288616791857+O(11^{2}0)q^{14}+−(209594106586688373846+ O(11^{2}0)) ∗ a^{2} − (216960595009583517615 + O(11^{2}0)) ∗ a − 333513083638291995489 + O(11^{2}0)q^{15} + (79696005886396998655+O(11^{2}0))∗a^{2}+(160632323172559380816+O(11^{2}0))∗a+138839897361701815992+ O(11^{2}0)q^{16} + −(190492124401294766397 + O(11^{2}0)) ∗ a^{2} − (5148513524370172444 ∗ 11 + O(11^{2}0)) ∗ a + 18894816919005700998+O(11^{2}0)q^{17}+(264441398572581411358+O(11^{2}0))∗a^{2}−(21067508477419604608∗ 11+O(11^{2}0))∗a+296698086672838798004+O(11^{2}0)q^{18}+−(23941721301908660132∗11+O(11^{2}0))∗a^{2}− (72314759768821085937+O(11^{2}0))∗a+193621124714059036077+O(11^{2}0)q^{19}+−(203072463007584879990+ O(11^{2}0)) ∗ a^{2} − (138593765599293151966 + O(11^{2}0)) ∗ a − 221831975660220712232 + O(11^{2}0)q^{20} + −(161604655615340483558+O(11^{2}0))∗a^{2}−(287500465515811730987+O(11^{2}0))∗a+156087460811032487425+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3}+18829220815627542624+O(11^{2}0)x^{2}+207662230975399004753+O(11^{2}0)x+68567498352877310860+ O(11^{2}0). The slope of f_{0} is 1.

45. Consider f = q+5∗z^{1}1+7∗z^{1}0+7∗z^{9}+10∗z^{8}+3∗z^{7}+8∗z^{6}+6∗z^{5}+z^{4}+8∗z^{3}+10∗z^{2}+9∗zq^{2}+5∗z^{1}1+ z^{1}0+z^{9}+7∗z^{8}+8∗z^{7}+5∗z^{6}+9∗z^{5}+2∗z^{4}+7∗z^{3}+9∗z^{2}+z+5q^{3}+8∗z^{1}1+10∗z^{1}0+10∗z^{9}+8∗z^{7}+10∗z^{6}+ 8∗z^{5}+4∗z^{4}+6∗z^{3}+7∗z^{2}+4∗z+10q^{4}+5∗z^{1}1+4∗z^{1}0+4∗z^{9}+3∗z^{8}+z^{6}+2∗z^{5}+7∗z^{4}+2∗z^{3}+4∗z^{2}+5∗z+ 3q^{5}+7∗z^{1}1+3∗z^{1}0+3∗z^{9}+4∗z^{8}+4∗z^{7}+10∗z^{6}+3∗z^{5}+4∗z^{4}+2∗z^{3}+7∗z^{2}+5∗zq^{6}+9∗z^{1}1+4∗z^{1}0+4∗z^{9}+ 6∗z^{8}+10∗z^{7}+9∗z^{6}+3∗z^{5}+8∗z^{4}+6∗z^{3}+3∗z^{2}+4∗z+4q^{7}+8∗z^{1}1+2∗z^{1}0+2∗z^{9}+7∗z^{8}+6∗z^{6}+z^{5}+9∗z^{4}+ z^{3}+2∗z^{2}+8∗z+8q^{8}+10∗z^{1}1+9∗z^{1}0+9∗z^{9}+z^{8}+z^{7}+8∗z^{6}+9∗z^{5}+z^{4}+6∗z^{3}+10∗z^{2}+4∗z+2q^{9}+10∗z^{1}1+ 10∗z^{1}0+10∗z^{9}+7∗z^{8}+2∗z^{7}+3∗z^{6}+3∗z^{5}+10∗z^{4}+8∗z^{3}+z^{2}+9∗z+6q^{10}+4∗z^{1}1+4∗z^{1}0+4∗z^{9}+5∗z^{8}+ 3∗z^{7}+10∗z^{6}+10∗z^{5}+4∗z^{4}+z^{3}+7∗z^{2}+8∗z+2q^{12}+q^{13}+3∗z^{1}1+10∗z^{1}0+10∗z^{9}+10∗z^{8}+z^{7}+4∗z^{5}+z^{3}+ 8∗zq^{14}+z^{1}0+z^{9}+6∗z^{8}+z^{7}+6∗z^{6}+5∗z^{5}+9∗z^{4}+2∗z^{3}+2∗z^{2}+5∗z+5q^{15}+5∗z^{1}1+3∗z^{1}0+3∗z^{9}+8∗z^{8}+ 10∗z^{7}+6∗z^{6}+8∗z^{5}+9∗z^{4}+2∗z^{2}+5q^{16}+7∗z^{1}1+6∗z^{1}0+6∗z^{9}+7∗z^{7}+6∗z^{6}+7∗z^{5}+9∗z^{4}+8∗z^{3}+2∗z^{2}+9∗ zq^{17}+8∗z^{1}1+10∗z^{1}0+10∗z^{9}+8∗z^{7}+10∗z^{6}+8∗z^{5}+4∗z^{4}+6∗z^{3}+7∗z^{2}+4∗z+3q^{18}+7∗z^{1}1+z^{1}0+z^{9}+3∗ z^{8}+2∗z^{7}+9∗z^{6}+4∗z^{5}+8∗z^{4}+9∗z^{3}+3∗z^{2}+6∗z+6q^{19}+4∗z^{1}1+4∗z^{1}0+4∗z^{9}+5∗z^{8}+3∗z^{7}+10∗z^{6}+10∗z^{5}+ 4∗z^{4}+z^{3}+7∗z^{2}+8∗z+10q^{20}+4∗z^{1}1+9∗z^{1}0+9∗z^{9}+2∗z^{8}+8∗z^{7}+7∗z^{6}+2∗z^{5}+5∗z^{4}+6∗z^{2}+6q^{21}+⋯ ∈ S_{22}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{12} + x^{8} + x^{7} + 4x^{6} + 2x^{5} + 5x^{4} + 5x^{3} + 6x^{2} + 5x + 2 = 0. We have ρ_{f} ⊗χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q +z^{2}49q^{2} +z^{1}095q^{3} +z^{1}53q^{4} +z^{1}029q^{5} + z^{1}4q^{6} + z^{5}27q^{7} + z^{3}72q^{8} + z^{3}87q^{9} + z^{1}278q^{10} + z^{8}74q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{4}56)ω^{0} & 0 \\ 0 & u(z^{8}74)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(228494176128019091054+ O(11^{2}0)) ∗ a^{2} + (126232126130457204851 + O(11^{2}0)) ∗ a + 1007663414535709320 ∗ 11 + O(11^{2}0)q^{3} + a^{2} − 2097152q^{4} + −(95366962654606477607 + O(11^{2}0)) ∗ a^{2} + (286975885151558282622 + O(11^{2}0)) ∗ a − 311941616959375706649+O(11^{2}0)q^{5}+−(283882396171998047147+O(11^{2}0))∗a^{2}−(213752326028574652862+ O(11^{2}0)) ∗ a + 196019062370748611157 + O(11^{2}0)q^{6} + (149635348179479808402 + O(11^{2}0)) ∗ a^{2} − (329976420198962930333+O(11^{2}0))∗a−223322087805285745160+O(11^{2}0)q^{7}+−(18829220815627542624+ O(11^{2}0))∗a^{2}−(207662230975403199057+O(11^{2}0))∗a−68567498352877310860+O(11^{2}0)q^{8}+−(72550889623189489209+ O(11^{2}0))∗a^{2}−(9542890798164020166+O(11^{2}0))∗a−263903860454195927970+O(11^{2}0)q^{9}+−(257941820253429481272+ O(11^{2}0)) ∗ a^{2} + (71089598886485630521 + O(11^{2}0)) ∗ a + 268333037649251759590 + O(11^{2}0)q^{10} + (30167737553010492191∗11+O(11^{2}0))∗a^{2}+(17609105808763440729∗11+O(11^{2}0))∗a−8809998171776527779∗ 11 + O(11^{2}0)q^{11} + −(302101307293963381738 + O(11^{2}0)) ∗ a^{2} − (138292184510539719423 + O(11^{2}0)) ∗ a−287404695168779539376+O(11^{2}0)q^{12}+137858491849q^{13}+−(15435291878799267170+O(11^{2}0))∗a^{2}− (29332476439212035770+O(11^{2}0))∗a+14271130288616791857+O(11^{2}0)q^{14}+−(209594106586688373846+ O(11^{2}0)) ∗ a^{2} − (216960595009583517615 + O(11^{2}0)) ∗ a − 333513083638291995489 + O(11^{2}0)q^{15} + (79696005886396998655+O(11^{2}0))∗a^{2}+(160632323172559380816+O(11^{2}0))∗a+138839897361701815992+ O(11^{2}0)q^{16} + −(190492124401294766397 + O(11^{2}0)) ∗ a^{2} − (5148513524370172444 ∗ 11 + O(11^{2}0)) ∗ a + 18894816919005700998+O(11^{2}0)q^{17}+(264441398572581411358+O(11^{2}0))∗a^{2}−(21067508477419604608∗ 11+O(11^{2}0))∗a+296698086672838798004+O(11^{2}0)q^{18}+−(23941721301908660132∗11+O(11^{2}0))∗a^{2}− (72314759768821085937+O(11^{2}0))∗a+193621124714059036077+O(11^{2}0)q^{19}+−(203072463007584879990+ O(11^{2}0)) ∗ a^{2} − (138593765599293151966 + O(11^{2}0)) ∗ a − 221831975660220712232 + O(11^{2}0)q^{20} + −(161604655615340483558+O(11^{2}0))∗a^{2}−(287500465515811730987+O(11^{2}0))∗a+156087460811032487425+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3}+18829220815627542624+O(11^{2}0)x^{2}+207662230975399004753+O(11^{2}0)x+68567498352877310860+ O(11^{2}0). The slope of f_{0} is 1.

46. Consider f = q+10∗z^{1}0+10∗z^{9}+5∗z^{8}+10∗z^{7}+5∗z^{6}+6∗z^{5}+2∗z^{4}+9∗z^{3}+9∗z^{2}+6∗z+3q^{2}+3∗z^{1}1+ 7∗z^{1}0+7∗z^{9}+3∗z^{8}+9∗z^{7}+4∗z^{6}+6∗z^{4}+6∗z^{3}+5∗z^{2}+4∗z+7q^{3}+5∗z^{1}1+z^{8}+7∗z^{7}+10∗z^{6}+4∗z^{5}+4∗ z^{4}+5∗z^{3}+7∗z^{2}+7∗z+8q^{4}+7∗z^{1}1+3∗z^{1}0+3∗z^{9}+4∗z^{8}+4∗z^{7}+10∗z^{6}+3∗z^{5}+4∗z^{4}+2∗z^{3}+7∗z^{2}+5∗ zq^{5}+10∗z^{1}1+4∗z^{1}0+4∗z^{9}+4∗z^{8}+7∗z^{7}+6∗z^{5}+7∗z^{3}+z+6q^{6}+z^{1}1+6∗z^{1}0+6∗z^{9}+z^{8}+3∗z^{7}+5∗z^{6}+ 2∗z^{4}+2∗z^{3}+9∗z^{2}+5∗z+1q^{7}+9∗z^{1}1+7∗z^{1}0+7∗z^{9}+2∗z^{8}+2∗z^{7}+5∗z^{6}+7∗z^{5}+2∗z^{4}+z^{3}+9∗z^{2}+8∗z+ 1q^{8}+8∗z^{1}1+z^{1}0+z^{9}+z^{8}+10∗z^{7}+7∗z^{5}+10∗z^{3}+3∗z+9q^{9}+2∗z^{1}1+7∗z^{1}0+7∗z^{9}+5∗z^{8}+z^{7}+2∗z^{6}+8∗ z^{5}+3∗z^{4}+5∗z^{3}+8∗z^{2}+7∗z+4q^{10}+3∗z^{1}1+5∗z^{1}0+5∗z^{9}+2∗z^{8}+7∗z^{7}+3∗z^{6}+z^{5}+10∗z^{4}+2∗z^{3}+z^{2}+ 5∗z+10q^{12}+q^{13}+7∗z^{1}1+10∗z^{1}0+10∗z^{9}+2∗z^{8}+8∗z^{6}+5∗z^{5}+z^{4}+5∗z^{3}+10∗z^{2}+7∗z+9q^{14}+5∗z^{1}1+ 6∗z^{1}0+6∗z^{9}+4∗z^{8}+2∗z^{7}+2∗z^{6}+z^{5}+3∗z^{4}+6∗z^{3}+8∗z^{2}+4∗z+7q^{15}+2∗z^{1}1+8∗z^{1}0+8∗z^{9}+2∗z^{7}+8∗ z^{6}+2∗z^{5}+z^{4}+7∗z^{3}+10∗z^{2}+zq^{16}+3∗z^{1}1+5∗z^{8}+2∗z^{7}+6∗z^{6}+9∗z^{5}+9∗z^{4}+3∗z^{3}+2∗z^{2}+2∗z+1q^{17}+5∗ z^{1}1+z^{8}+7∗z^{7}+10∗z^{6}+4∗z^{5}+4∗z^{4}+5∗z^{3}+7∗z^{2}+7∗z+1q^{18}+3∗z^{1}0+3∗z^{9}+7∗z^{8}+3∗z^{7}+7∗z^{6}+4∗z^{5}+ 5∗z^{4}+6∗z^{3}+6∗z^{2}+4∗z+8q^{19}+3∗z^{1}1+5∗z^{1}0+5∗z^{9}+2∗z^{8}+7∗z^{7}+3∗z^{6}+z^{5}+10∗z^{4}+2∗z^{3}+z^{2}+5∗z+ 7q^{20}+6∗z^{1}1+2∗z^{1}0+2∗z^{9}+6∗z^{7}+2∗z^{6}+6∗z^{5}+3∗z^{4}+10∗z^{3}+8∗z^{2}+3∗z+2q^{21}+⋯ ∈ S_{22}(Γ_{0}(13); \overline{F}_{11}). Also, z satisfies the equation x^{12} + x^{8} + x^{7} + 4x^{6} + 2x^{5} + 5x^{4} + 5x^{3} + 6x^{2} + 5x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{7}9q^{2} + z^{7}5q^{3} + z^{3}53q^{4} + z^{6}79q^{5} + z^{1}54q^{6} + z^{4}77q^{7} + z^{1}02q^{8} + z^{2}67q^{9} + z^{7}58q^{10} + z^{3}04q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}026)ω^{0} & 0 \\ 0 & u(z^{3}04)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(228494176128019091054+ O(11^{2}0)) ∗ a^{2} + (126232126130457204851 + O(11^{2}0)) ∗ a + 1007663414535709320 ∗ 11 + O(11^{2}0)q^{3} + a^{2} − 2097152q^{4} + −(95366962654606477607 + O(11^{2}0)) ∗ a^{2} + (286975885151558282622 + O(11^{2}0)) ∗ a − 311941616959375706649+O(11^{2}0)q^{5}+−(283882396171998047147+O(11^{2}0))∗a^{2}−(213752326028574652862+ O(11^{2}0)) ∗ a + 196019062370748611157 + O(11^{2}0)q^{6} + (149635348179479808402 + O(11^{2}0)) ∗ a^{2} − (329976420198962930333+O(11^{2}0))∗a−223322087805285745160+O(11^{2}0)q^{7}+−(18829220815627542624+ O(11^{2}0))∗a^{2}−(207662230975403199057+O(11^{2}0))∗a−68567498352877310860+O(11^{2}0)q^{8}+−(72550889623189489209+ O(11^{2}0))∗a^{2}−(9542890798164020166+O(11^{2}0))∗a−263903860454195927970+O(11^{2}0)q^{9}+−(257941820253429481272+ O(11^{2}0)) ∗ a^{2} + (71089598886485630521 + O(11^{2}0)) ∗ a + 268333037649251759590 + O(11^{2}0)q^{10} + (30167737553010492191∗11+O(11^{2}0))∗a^{2}+(17609105808763440729∗11+O(11^{2}0))∗a−8809998171776527779∗ 11 + O(11^{2}0)q^{11} + −(302101307293963381738 + O(11^{2}0)) ∗ a^{2} − (138292184510539719423 + O(11^{2}0)) ∗ a−287404695168779539376+O(11^{2}0)q^{12}+137858491849q^{13}+−(15435291878799267170+O(11^{2}0))∗a^{2}− (29332476439212035770+O(11^{2}0))∗a+14271130288616791857+O(11^{2}0)q^{14}+−(209594106586688373846+ O(11^{2}0)) ∗ a^{2} − (216960595009583517615 + O(11^{2}0)) ∗ a − 333513083638291995489 + O(11^{2}0)q^{15} + (79696005886396998655+O(11^{2}0))∗a^{2}+(160632323172559380816+O(11^{2}0))∗a+138839897361701815992+ O(11^{2}0)q^{16} + −(190492124401294766397 + O(11^{2}0)) ∗ a^{2} − (5148513524370172444 ∗ 11 + O(11^{2}0)) ∗ a + 18894816919005700998+O(11^{2}0)q^{17}+(264441398572581411358+O(11^{2}0))∗a^{2}−(21067508477419604608∗ 11+O(11^{2}0))∗a+296698086672838798004+O(11^{2}0)q^{18}+−(23941721301908660132∗11+O(11^{2}0))∗a^{2}− (72314759768821085937+O(11^{2}0))∗a+193621124714059036077+O(11^{2}0)q^{19}+−(203072463007584879990+ O(11^{2}0)) ∗ a^{2} − (138593765599293151966 + O(11^{2}0)) ∗ a − 221831975660220712232 + O(11^{2}0)q^{20} + −(161604655615340483558+O(11^{2}0))∗a^{2}−(287500465515811730987+O(11^{2}0))∗a+156087460811032487425+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3}+18829220815627542624+O(11^{2}0)x^{2}+207662230975399004753+O(11^{2}0)x+68567498352877310860+ O(11^{2}0). The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

47. Consider f = q+6q2 +4q3 +6q4 +4q5 +2q6 +9q7 +10q8 +2q10 +7q11 +2q12 +2q13 +10q14 +5q15 +q16 +6q19 +2q20 +3q21 +⋯ ∈ S14(Γ0(13); \overline{F}11). Since f is ordinary, we do not continue our analysis.

48. Consider f = q + 6q2 + q3 + 6q4 + 8q5 + 6q6 + q7 + 10q8 + 7q9 + 4q10 + 6q12 + 2q13 + 6q14 + 8q15 + q16 + 9q17 + 9q18 + 6q19 + 4q20 + q21 + ⋯ ∈ S14(Γ0(13); \overline{F}11). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

49. Consider f = q + 6 ∗ z9 + 7 ∗ z7 + 7 ∗ z6 + 3 ∗ z5 + z2 + z + 5q2 + 4 ∗ z9 + z7 + z6 + 2 ∗ z5 + 8 ∗ z2 + 8 ∗ zq3 + 6 ∗ z9 + 7 ∗ z7 + 7 ∗ z6 + 3 ∗ z5 + z2 + z + 1q4 + 6 ∗ z9 + 7 ∗ z7 + 7 ∗ z6 + 3 ∗ z5 + z2 + z + 3q5 + 6 ∗ z9 + 7 ∗ z7 + 7 ∗ z6 + 3 ∗ z5 + z2 + z + 4q6 + q7 + 4q8 + 9 ∗ z9 + 5 ∗ z7 + 5 ∗ z6 + 10 ∗ z5 + 7 ∗ z2 + 7 ∗ z + 5q9 + 5 ∗ z9 + 4 ∗ z7 + 4 ∗ z6 + 8 ∗ z5 + 10 ∗ z2 + 10 ∗ z + 10q10 + 6 ∗ z9 + 7 ∗ z7 + 7 ∗ z6 + 3 ∗ z5 + z2 + z + 6q11 + z9 + 3 ∗ z7 + 3 ∗ z6 + 6 ∗ z5 + 2 ∗ z2 + 2 ∗ z + 4q12 + 9q13 + 6 ∗ z9 + 7 ∗ z7 + 7 ∗ z6 + 3 ∗ z5 + z2 + z + 5q14 + 9 ∗ z9 + 5 ∗ z7 + 5 ∗ z6 + 10 ∗ z5 + 7 ∗ z2 + 7 ∗ z + 4q15 + 9 ∗ z9 + 5 ∗ z7 + 5 ∗ z6 + 10 ∗ z5 + 7 ∗ z2 + 7 ∗ z + 1q16 + 8 ∗ z9 + 2 ∗ z7 + 2 ∗ z6 + 4 ∗ z5 + 5 ∗ z2 + 5 ∗ z + 10q17 + 5 ∗ z9 + 4 ∗ z7 + 4 ∗ z6 + 8 ∗ z5 + 10 ∗ z2 + 10 ∗ z + 1q18 + 6 ∗ z9 + 7 ∗ z7 + 7 ∗ z6 + 3 ∗ z5 + z2 + z + 1q19 + 3 ∗ z9 + 9 ∗ z7 + 9 ∗ z6 + 7 ∗ z5 + 6 ∗ z2 + 6 ∗ z + 9q20 + 4 ∗ z9 + z7 + z6 + 2 ∗ z5 + 8 ∗ z2 + 8 ∗ zq21 + ⋯ ∈ S14(Γ0(13); \overline{F}11). Also, z satisfies the equation x10 + 7x5 + 8x4 + 10x3 + 6x2 + 6x + 2 = 0. Since f is ordinary, we do not continue our analysis.

50. Consider f = q +5∗z9 +4∗z7 +4∗z6 +8∗z5 +10∗z2 +10∗z +7q2 +7∗z9 +10∗z7 +10∗z6 +9∗z5 +3∗z2 +3∗z +5q3 +5∗z9 +4∗z7 +4∗z6 +8∗z5 + 10∗z2 +10∗z +3q4 +5∗z9 +4∗z7 +4∗z6 +8∗z5 +10∗z2 +10∗z +5q5 +5∗z9 +4∗z7 +4∗z6 +8∗z5 +10∗z2 +10∗z +6q6 +q7 +4q8 +2∗z9 +6∗z7 +6∗ z6 +z5 +4∗z2 +4∗z +8q9 +6∗z9 +7∗z7 +7∗z6 +3∗z5 +z2 +z +8q10 +5∗z9 +4∗z7 +4∗z6 +8∗z5 +10∗z2 +10∗z +8q11 +10∗z9 +8∗z7 +8∗z6 +5∗ z5 +9∗z2 +9∗z+8q12 +9q13 +5∗z9 +4∗z7 +4∗z6 +8∗z5 +10∗z2 +10∗z+7q14 +2∗z9 +6∗z7 +6∗z6 +z5 +4∗z2 +4∗z+7q15 +2∗z9 +6∗z7 +6∗z6 + z5 +4∗z2 +4∗z +4q16 +3∗z9 +9∗z7 +9∗z6 +7∗z5 +6∗z2 +6∗z +9q17 +6∗z9 +7∗z7 +7∗z6 +3∗z5 +z2 +z +10q18 +5∗z9 +4∗z7 +4∗z6 +8∗z5 + 10∗z2 +10∗z +3q19 +8∗z9 +2∗z7 +2∗z6 +4∗z5 +5∗z2 +5∗z +10q20 +7∗z9 +10∗z7 +10∗z6 +9∗z5 +3∗z2 +3∗z +5q21 +⋯ ∈ S14(Γ0(13); \overline{F}11). Also, z satisfies the equation x10 + 7x5 + 8x4 + 10x3 + 6x2 + 6x + 2 = 0. Since f is ordinary, we do not continue our analysis.

51. Consider f = q + z101q2 + z35q3 + z2q4 + z55q5 + z16q6 + z58q7 + z70q8 + z83q9 + 8q10 + z115q11 + z37q12 + 7q13 + z39q14 + z90q15 + z116q16 + z40q17 + z64q18 + z104q19 + z57q20 + z93q21 + ⋯ ∈ S16(Γ0(13); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

52. Consider f = q + z31q2 + z25q3 + z22q4 + z5q5 + z56q6 + z38q7 + z50q8 + z73q9 + 8q10 + z65q11 + z47q12 + 7q13 + z69q14 + z30q15 + z76q16 + z80q17 + z104q18 + z64q19 + z27q20 + z63q21 + ⋯ ∈ S16(Γ0(13); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

53. Consider f = q + z1067q2 + z752q3 + z696q4 + z839q5 + z489q6 + z361q7 + z318q8 + z56q9 + z576q10 + z517q11 + z118q12 + 4q13 + z98q14 + z261q15 + z122q16 + z902q17 + z1123q18 + z794q19 + z205q20 + z1113q21 + ⋯ ∈ S16(Γ0(13); \overline{F}11). Also, z satisfies the equation x3 + 2x + 9 = 0. Since f is ordinary, we do not continue our analysis.

54. Consider f = q + z1097q2 + z292q3 + z1006q4 + z1249q5 + z59q6 + z1311q7 + z838q8 + z616q9 + z1016q10 + z367q11 + z1298q12 + 4q13 + z1078q14 + z211q15 + z12q16 + z612q17 + z383q18 + z754q19 + z925q20 + z273q21 + ⋯ ∈ S16(Γ0(13); \overline{F}11). Also, z satisfies the equation x3 + 2x + 9 = 0. Since f is ordinary, we do not continue our analysis.

55. Consider f = q + z97q2 + z552q3 + z426q4 + z439q5 + z649q6 + z1121q7 + z1238q8 + z126q9 + z536q10 + z47q11 + z978q12 + 4q13 + z1218q14 + z991q15 + z132q16 + z82q17 + z223q18 + z314q19 + z865q20 + z343q21 + ⋯ ∈ S16(Γ0(13); \overline{F}11). Also, z satisfies the equation x3 + 2x + 9 = 0. Since f is ordinary, we do not continue our analysis.

56. Consider f = q + 10q2 + 4q3 + 5q4 + 2q5 + 7q6 + 9q7 + 2q8 + 7q9 + 9q10 + 10q11 + 9q12 + 8q13 + 2q14 + 8q15 + 7q16 + 4q18 + 4q19 + 10q20 + 3q21 + ⋯ ∈ S18(Γ0(13); \overline{F}11). Also, z satisfies the equation x6 + 3x4 + 4x3 + 6x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

57. Consider f = q+2∗z5 +z4 +2∗z3 +4∗z2 +6∗z+4q2 +5∗z5 +8∗z4 +5∗z3 +10∗z2 +4∗z+4q3 +6∗z5 +3∗z4 +6∗z3 +z2 +7∗z+10q4 +5q5 +3∗z5 +7∗ z4 +3∗z3 +6∗z2 +9∗z+2q6 +4∗z5 +2∗z4 +4∗z3 +8∗z2 +z+7q7 +4q8 +5∗z5 +8∗z4 +5∗z3 +10∗z2 +4∗z+5q9 +10∗z5 +5∗z4 +10∗z3 +9∗z2 +8∗ z+9q10 +z5 +6∗z4 +z3 +2∗z2 +3∗z+1q11 +10∗z5 +5∗z4 +10∗z3 +9∗z2 +8∗z+9q12 +8q13 +10∗z5 +5∗z4 +10∗z3 +9∗z2 +8∗z+8q14 +3∗z5 +7∗ z4 +3∗z3 +6∗z2 +9∗z+9q15 +10∗z5 +5∗z4 +10∗z3 +9∗z2 +8∗z+1q16 +2∗z5 +z4 +2∗z3 +4∗z2 +6∗z+5q17 +5∗z5 +8∗z4 +5∗z3 +10∗z2 +4∗z+ 6q18 +10∗z5 +5∗z4 +10∗z3 +9∗z2 +8∗z+5q19 +8∗z5 +4∗z4 +8∗z3 +5∗z2 +2∗z+6q20 +z5 +6∗z4 +z3 +2∗z2 +3∗zq21 +⋯ ∈ S18(Γ0(13); \overline{F}11). Also, z satisfies the equation x6 + 3x4 + 4x3 + 6x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

58. Consider f = q +9∗z5 +10∗z4 +9∗z3 +7∗z2 +5∗z +10q2 +6∗z5 +3∗z4 +6∗z3 +z2 +7∗z +8q3 +5∗z5 +8∗z4 +5∗z3 +10∗z2 +4∗z +6q4 +5q5 + 8∗z5 +4∗z4 +8∗z3 +5∗z2 +2∗zq6 +7∗z5 +9∗z4 +7∗z3 +3∗z2 +10∗z +8q7 +4q8 +6∗z5 +3∗z4 +6∗z3 +z2 +7∗z +9q9 +z5 +6∗z4 +z3 +2∗z2 + 3∗z +6q10 +10∗z5 +5∗z4 +10∗z3 +9∗z2 +8∗z +4q11 +z5 +6∗z4 +z3 +2∗z2 +3∗z +6q12 +8q13 +z5 +6∗z4 +z3 +2∗z2 +3∗z +5q14 +8∗z5 +4∗ z4 +8∗z3 +5∗z2 +2∗z +7q15 +z5 +6∗z4 +z3 +2∗z2 +3∗z +9q16 +9∗z5 +10∗z4 +9∗z3 +7∗z2 +5∗zq17 +6∗z5 +3∗z4 +6∗z3 +z2 +7∗z +10q18 + z5 +6∗z4 +z3 +2∗z2 +3∗z +2q19 +3∗z5 +7∗z4 +3∗z3 +6∗z2 +9∗z +8q20 +10∗z5 +5∗z4 +10∗z3 +9∗z2 +8∗z +3q21 +⋯ ∈ S18(Γ0(13); \overline{F}11). Also, z satisfies the equation x6 + 3x4 + 4x3 + 6x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

59. Consider f = q+5∗z5 +5∗z4 +5∗z3 +10∗z+2q2 +9∗z5 +10∗z4 +10∗z3 +3∗z2 +6∗zq3 +3∗z5 +7∗z4 +7∗z3 +z2 +2∗z+7q4 +10∗z5 +6∗z4 +6∗z3 +10∗ z2 +2∗z+9q5 +2∗z5 +9∗z4 +9∗z3 +10∗z2 +8∗zq6 +5∗z5 +3∗z4 +3∗z3 +5∗z2 +z+3q7 +8∗z5 +3∗z4 +3∗z3 +7∗z2 +10∗zq8 +6∗z5 +2∗z4 +2∗z3 +10∗ z2 +5∗z+10q9 +10∗z5 +10∗z4 +10∗z3 +9∗zq10 +z4 +z3 +3∗z2 +10∗z+1q11 +10∗z5 +z4 +z3 +6∗z2 +7∗z+10q12 +3q13 +3∗z5 +3∗z4 +3∗z3 +6∗ z+8q14 +5∗z4 +5∗z3 +4∗z2 +6∗z+1q15 +7∗z5 +4∗z4 +4∗z3 +2∗z2 +6∗z+2q16 +z5 +6∗z4 +6∗z3 +4∗z2 +8∗z+7q17 +3∗z5 +2∗z4 +2∗z3 +8∗z2 +7∗ z+5q18 +9∗z5 +2∗z4 +2∗z3 +z2 +3∗zq19 +4∗z5 +7∗z4 +7∗z3 +9∗z2 +5∗z+1q20 +3∗z5 +4∗z4 +4∗z3 +3∗z2 +5∗z+6q21 +⋯ ∈ S18(Γ0(13); \overline{F}11). Also, z satisfies the equation x6 + 3x4 + 4x3 + 6x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

60. Consider f = q+8∗z5 +3∗z4 +3∗z3 +7∗z2 +10∗z+7q2 +9∗z5 +4∗z4 +4∗z3 +7∗z2 +z+8q3 +3∗z5 +5∗z4 +5∗z3 +6∗z2 +4∗z+6q4 +9∗z5 +5∗z4 +5∗ z3+10∗z2+1q5+5∗z5+9∗z4+9∗z3+z2+6∗z+6q6+10∗z5+8∗z4+8∗z3+5∗z2+10q7+9∗z5+3∗z4+3∗z3+4∗z2+2∗z+2q8+7∗z5+7∗z4+7∗z3+3∗ z+9q9 +5∗z5 +6∗z4 +6∗z3 +3∗z2 +9∗z+10q10 +10∗z5 +3∗z4 +3∗z3 +z2 +5∗zq11 +3∗z5 +z4 +z3 +5∗z2 +8∗z+7q12 +3q13 +7∗z5 +4∗z4 +4∗z3 +2∗ z2 +6∗zq14 +6∗z5 +4∗z4 +4∗z3 +5∗z2 +3∗z+7q15 +z5 +4∗z4 +4∗z3 +9∗z2 +10∗z+1q16 +z5 +9∗z4 +9∗z3 +2∗z2 +5∗z+3q17 +8∗z5 +z4 +z3 +z2 + z+9q18 +6∗z5 +2∗z4 +2∗z3 +10∗z2 +5∗z+5q19 +10∗z5 +7∗z4 +7∗z3 +2∗z2 +z+2q20 +6∗z5 +7∗z4 +7∗z3 +3∗z2 +8q21 +⋯ ∈ S18(Γ0(13); \overline{F}11). Also, z satisfies the equation x6 + 3x4 + 4x3 + 6x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

61. Consider f = q +9∗z5 +3∗z4 +3∗z3 +4∗z2 +2∗z +9q2 +4∗z5 +8∗z4 +8∗z3 +z2 +4∗zq3 +5∗z5 +10∗z4 +10∗z3 +4∗z2 +5∗z +7q4 +3∗z5 +2∗ z2 +9∗z +3q5 +4∗z5 +4∗z4 +4∗z3 +8∗z +7q6 +7∗z5 +z2 +10∗zq7 +5∗z5 +5∗z4 +5∗z3 +10∗z +6q8 +9∗z5 +2∗z4 +2∗z3 +z2 +3∗z +5q9 +7∗ z5 +6∗z4 +6∗z3 +8∗z2 +4∗z +3q10 +z5 +7∗z4 +7∗z3 +7∗z2 +7∗z +6q11 +9∗z5 +9∗z4 +9∗z3 +7∗z +1q12 +3q13 +z5 +4∗z4 +4∗z3 +9∗z2 + 10 ∗ z + 10q14 + 5 ∗ z5 + 2 ∗ z4 + 2 ∗ z3 + 2 ∗ z2 + 2 ∗ z + 4q15 + 3 ∗ z5 + 3 ∗ z4 + 3 ∗ z3 + 6 ∗ z + 10q16 + 9 ∗ z5 + 7 ∗ z4 + 7 ∗ z3 + 5 ∗ z2 + 9 ∗ z + 7q17 + 8 ∗ z4 + 8 ∗ z3 + 2 ∗ z2 + 3 ∗ z + 2q18 + 7 ∗ z5 + 7 ∗ z4 + 7 ∗ z3 + 3 ∗ z + 4q19 + 8 ∗ z5 + 8 ∗ z4 + 8 ∗ z3 + 5 ∗ z + 4q20 + 2 ∗ z5 + 5 ∗ z2 + 6 ∗ z + 2q21 + ⋯ ∈ S18(Γ0(13); \overline{F}11). Also, z satisfies the equation x6 + 3x4 + 4x3 + 6x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

62. Consider f = q +8q2 +6q3 +2q4 +3q5 +4q6 +7q7 +4q8 +5q9 +2q10 +3q11 +q12 +3q13 +q14 +7q15 +7q16 +5q17 +7q18 +8q19 +6q20 +9q21 +⋯ ∈ S18(Γ0(13); \overline{F}11). Also, z satisfies the equation x6 + 3x4 + 4x3 + 6x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

63. Consider f = q + 9q2 + 2q3 + 9q4 + 5q5 + 7q6 + 5q8 + q10 + 3q11 + 7q12 + 6q13 + 10q15 + 2q16 + 8q17 + 6q19 + q20 + ⋯ ∈ S20(Γ0(13); \overline{F}11). Also, z satisfies the equation x3 + 2x + 9 = 0. Since f is ordinary, we do not continue our analysis.

64. Consider f = q + z869q2 + z825q3 + z1223q4 + z819q5 + z364q6 + z1257q7 + z1122q8 + z277q9 + z358q10 + z684q11 + z718q12 + 6q13 + z796q14 + z314q15 + z554q16 + z907q17 + z1146q18 + z1099q19 + z712q20 + z752q21 + ⋯ ∈ S20(Γ0(13); \overline{F}11). Also, z satisfies the equation x3 + 2x + 9 = 0. Since f is ordinary, we do not continue our analysis.

65. Consider f = q + z79q2 + z75q3 + z353q4 + z679q5 + z154q6 + z477q7 + z102q8 + z267q9 + z758q10 + z304q11 + z428q12 + 6q13 + z556q14 + z754q15 + z534q16 + z687q17 + z346q18 + z1309q19 + z1032q20 + z552q21 + ⋯ ∈ S20(Γ0(13); \overline{F}11). Also, z satisfies the equation x3 + 2x + 9 = 0. Since f is ordinary, we do not continue our analysis.

66. Consider f = q + z249q2 + z1095q3 + z153q4 + z1029q5 + z14q6 + z527q7 + z372q8 + z387q9 + z1278q10 + z874q11 + z1248q12 + 6q13 + z776q14 + z794q15 + z774q16 + z667q17 + z636q18 + z119q19 + z1182q20 + z292q21 + ⋯ ∈ S20(Γ0(13); \overline{F}11). Also, z satisfies the equation x3 + 2x + 9 = 0. Since f is ordinary, we do not continue our analysis.

67. Consider f = q+z9 +8∗z8 +10∗z7 +10∗z6 +7∗z5 +8∗z4 +2∗z3 +2∗z2 +8∗z+10q2 +9∗z9 +9∗z8 +z7 +z6 +5∗z4 +10∗z3 +3∗z2 +10∗z+6q3 +2∗z8 +4∗ z7+9∗z6+5∗z5+9∗z4+8∗z3+8∗z2+9∗zq4+7∗z9+3∗z8+6∗z7+5∗z5+2∗z4+z3+7∗z2+9∗z+7q5+4∗z9+7∗z8+3∗z7+7∗z6+3∗z5+8∗z4+5∗z3+ 6∗z2+10∗z+6q6+6∗z9+z8+5∗z7+8∗z6+6∗z5+9∗z3+6∗z2+3∗z+2q7+10∗z9+z8+9∗z7+5∗z6+6∗z5+5∗z4+4∗z3+4∗z2+3∗z+1q8+8∗z9+z8+ 7∗z7+6∗z6+z5+10∗z4+2∗z3+10∗z2+7∗z+7q9+9∗z9+8∗z8+10∗z7+10∗z6+6∗z5+7∗z4+z3+7∗z2+2∗zq10+8∗z8+5∗z7+5∗z6+7∗z5+6∗z4+6∗ z3+z2+9∗z+6q11+8∗z9+5∗z8+2∗z7+9∗z6+8∗z5+6∗z4+z3+9∗z2+7∗z+3q12+5q13+3∗z9+10∗z8+3∗z6+9∗z5+10∗z4+7∗z3+6∗z2+8∗zq14+ 10∗z9+2∗z8+3∗z7+8∗z6+4∗z5+z4+10∗z3+4∗z2+z+7q15+z8+10∗z7+10∗z6+3∗z5+8∗z4+5∗z3+z2+z+5q16+10∗z9+2∗z8+10∗z7+10∗z6+ 10∗z5+5∗z4+2∗z2+6∗z+8q17+7∗z9+2∗z8+z7+7∗z6+8∗z5+10∗z4+8∗z3+6∗z2+8∗z+8q18+2∗z9+3∗z8+5∗z7+4∗z6+5∗z5+2∗z4+10∗z3+8∗ z+4q19+3∗z9+8∗z8+2∗z7+3∗z6+5∗z5+2∗z4+z3+2∗z2q20+5∗z9+4∗z8+5∗z7+6∗z6+4∗z5+7∗z4+6∗z3+5∗z+5q21+⋯ ∈ S20(Γ0(13); \overline{F}11). Also, z satisfies the equation x10 + 7x5 + 8x4 + 10x3 + 6x2 + 6x + 2 = 0. Since f is ordinary, we do not continue our analysis.

68. Consider f = q+8∗z8 +4∗z7 +4∗z6 +10∗z5 +2∗z4 +z3 +10∗z2 +3∗z+7q2 +7∗z8 +9∗z7 +8∗z6 +7∗z5 +3∗z4 +7∗z3 +5∗z2 +z+8q3 +4∗z9 +4∗z8 +2∗ z7+3∗z6+9∗z5+2∗z4+9∗z3+3∗z2+7∗z+9q4+6∗z9+5∗z8+z6+z5+3∗z4+6∗z3+6∗z2+6∗z+4q5+9∗z9+8∗z8+5∗z7+4∗z5+7∗z4+8∗z3+3∗z2+ z+3q6 +4∗z9 +5∗z8 +6∗z7 +8∗z6 +7∗z5 +4∗z4 +4∗z3 +4∗z2 +8∗zq7 +7∗z9 +7∗z8 +10∗z7 +2∗z6 +6∗z5 +8∗z4 +10∗z3 +3∗z2 +4q8 +5∗z9 +2∗z8 + 2∗z7+8∗z6+8∗z5+6∗z4+5∗z3+4∗z2+10∗z+9q9+6∗z9+3∗z8+5∗z7+8∗z6+6∗z5+2∗z4+7∗z3+z2+6∗z+1q10+7∗z9+10∗z8+10∗z7+8∗z5+8∗ z4 +10∗z2 +4∗z+3q11 +2∗z9 +5∗z8 +7∗z7 +7∗z6 +9∗z5 +8∗z4 +5∗z3 +4∗z+5q12 +5q13 +z9 +3∗z8 +6∗z7 +8∗z6 +6∗z5 +z4 +5∗z3 +3∗z2 +10∗z+ 10q14+7∗z9+6∗z8+2∗z7+4∗z6+9∗z5+z4+6∗z3+5∗z2+8∗z+9q15+8∗z9+4∗z8+3∗z7+10∗z6+5∗z5+9∗z4+7∗z3+8∗z2+6∗z+3q16+6∗z9+10∗ z8+3∗z7+7∗z6+8∗z4+10∗z3+8∗z2+2∗z+10q17+5∗z9+9∗z8+7∗z7+5∗z6+z5+10∗z4+6∗z3+5∗z2+10∗z+9q18+10∗z7+3∗z6+10∗z5+2∗z4+9∗ z3 +2∗z2 +6∗z+10q19 +5∗z9 +z8 +9∗z7 +2∗z5 +5∗z4 +5∗z3 +9q20 +3∗z9 +3∗z8 +7∗z7 +z6 +8∗z5 +z4 +z3 +z2 +7∗zq21 +⋯ ∈ S20(Γ0(13); \overline{F}11). Also, z satisfies the equation x10 + 7x5 + 8x4 + 10x3 + 6x2 + 6x + 2 = 0. Since f is ordinary, we do not continue our analysis.

69. Consider f = q+5∗z9 +10∗z8 +8∗z7 +8∗z6 +7∗z5 +7∗z4 +6∗z3 +z2 +7∗zq2 +z9 +10∗z8 +2∗z7 +8∗z6 +3∗z5 +9∗z4 +3∗z3 +9∗z2 +3∗zq3 +4∗z9 + 8∗z8 +5∗z7 +z6 +5∗z5 +5∗z4 +8∗z3 +5∗z2 +5∗z+8q4 +8∗z9 +5∗z8 +9∗z7 +9∗z6 +5∗z5 +7∗z4 +6∗z3 +5∗z2 +6∗z+7q5 +7∗z9 +z8 +5∗z7 +2∗z6 +5∗ z5 +6∗z4 +3∗z3 +2∗z2 +9∗z+8q6 +z9 +4∗z8 +4∗z7 +3∗z6 +4∗z5 +6∗z4 +10∗z2 +7∗z+2q7 +2∗z9 +2∗z7 +5∗z6 +10∗z5 +4∗z4 +z3 +3∗z+6q8 +z9 + 3∗z8 +3∗z7 +9∗z6 +6∗z4 +4∗z3 +z2 +4∗z+9q9 +7∗z9 +z8 +5∗z7 +10∗z6 +8∗z5 +7∗z4 +9∗z3 +10∗z2 +9q10 +7∗z9 +6∗z8 +9∗z7 +6∗z6 +4∗z5 +5∗ z4 +7∗z3 +3∗z2 +2∗z+7q11 +9∗z9 +5∗z8 +5∗z7 +5∗z6 +10∗z5 +6∗z4 +z3 +10∗z2 +6∗z+4q12 +5q13 +5∗z9 +z8 +3∗z7 +7∗z6 +7∗z5 +6∗z4 +4∗z3 + z2 +3∗z+3q14 +2∗z9 +7∗z8 +7∗z6 +z5 +9∗z4 +9∗z3 +10∗z2 +8∗z+1q15 +8∗z9 +4∗z8 +7∗z7 +3∗z6 +4∗z5 +3∗z4 +5∗z3 +5∗z2 +8∗z+6q16 +z9 + 3∗z8 +6∗z7 +5∗z6 +9∗z5 +2∗z4 +zq17 +7∗z9 +6∗z6 +4∗z4 +5∗z3 +9∗z2 +10∗z+7q18 +10∗z9 +4∗z8 +8∗z6 +10∗z5 +z4 +9∗z3 +7∗z2 +6∗z+2q19 + 9∗z9 +8∗z8 +10∗z6 +4∗z5 +4∗z4 +8∗z3 +8∗z2 +5∗z+10q20 +6∗z9 +3∗z7 +2∗z6 +5∗z5 +7∗z4 +8∗z3 +6∗z2 +5∗zq21 +⋯ ∈ S20(Γ0(13); \overline{F}11). Also, z satisfies the equation x10 + 7x5 + 8x4 + 10x3 + 6x2 + 6x + 2 = 0. Since f is ordinary, we do not continue our analysis.

70. Consider f = q+8∗z9+8∗z8+5∗z7+4∗z6+7∗z5+9∗z4+7∗z3+5∗z2+6q2+6∗z9+9∗z8+3∗z7+8∗z6+10∗z5+3∗z4+9∗z3+6∗z2+9∗z+2q3+6∗z9+ 3∗z8+z7+4∗z6+7∗z5+8∗z4+9∗z3+4∗z2+4∗z+8q4+10∗z9+4∗z8+6∗z7+6∗z5+4∗z4+z3+5∗z2+4q5+7∗z9+10∗z8+7∗z7+6∗z6+8∗z5+4∗z4+ 2∗z2 +2∗z+8q6 +10∗z9 +6∗z8 +6∗z7 +7∗z6 +10∗z5 +7∗z3 +7∗z2 +2∗zq7 +8∗z9 +9∗z8 +3∗z7 +2∗z6 +2∗z5 +2∗z4 +7∗z3 +4∗z2 +6∗zq8 +3∗z8 +2∗ z7+2∗z6+5∗z5+7∗z3+2∗z2+8q9+4∗z9+8∗z8+4∗z7+2∗z6+6∗z5+4∗z4+10∗z3+5∗z2+5∗z+7q10+5∗z9+8∗z8+3∗z7+4∗z6+10∗z5+3∗z4+3∗ z3+8∗z2+10∗z+6q11+z9+2∗z8+8∗z7+z6+9∗z5+z3+6∗z2+10∗z+3q12+5q13+9∗z9+4∗z8+3∗z7+10∗z6+9∗z5+2∗z4+10∗z2+8∗z+1q14+8∗ z9+3∗z8+4∗z7+2∗z6+4∗z4+9∗z3+9∗z2+7q15+z9+z8+9∗z7+2∗z6+6∗z5+3∗z4+7∗z3+5∗z2+9∗z+3q16+z9+2∗z8+7∗z7+z6+3∗z4+5∗z3+ 5∗z2 +7∗z+1q17 +3∗z9 +z8 +6∗z6 +6∗z5 +10∗z3 +4∗z2 +9∗z+9q18 +5∗z9 +5∗z8 +7∗z7 +2∗z6 +4∗z5 +7∗z4 +5∗z3 +z2 +6∗z+1q19 +4∗z9 +6∗z8 + 8∗z7 +7∗z6 +4∗z5 +3∗z4 +8∗z3 +8∗z2 +7∗z+4q20 +6∗z9 +2∗z8 +9∗z7 +10∗z6 +7∗z5 +3∗z4 +10∗z3 +4∗z2 +6∗z+9q21 +⋯ ∈ S20(Γ0(13); \overline{F}11). Also, z satisfies the equation x10 + 7x5 + 8x4 + 10x3 + 6x2 + 6x + 2 = 0. Since f is ordinary, we do not continue our analysis.

71. Consider f = q+8∗z9+10∗z8+6∗z7+7∗z6+2∗z5+7∗z4+6∗z3+4∗z2+4∗z+3q2+6∗z9+9∗z8+7∗z7+8∗z6+2∗z5+2∗z4+4∗z3+10∗z2+10∗z+2q3+ 8∗z9+5∗z8+10∗z7+5∗z6+7∗z5+9∗z4+10∗z3+2∗z2+8∗z+6q4+2∗z9+5∗z8+z7+z6+5∗z5+6∗z4+8∗z3+10∗z2+z+10q5+6∗z9+7∗z8+2∗z7+7∗ z6+2∗z5+8∗z4+6∗z3+9∗z2+3q6+z9+6∗z8+z7+7∗z6+6∗z5+z4+2∗z3+6∗z2+2∗z+8q7+6∗z9+5∗z8+9∗z7+8∗z6+9∗z5+3∗z4+10∗z+7q8+8∗ z9+2∗z8+8∗z7+8∗z6+8∗z5+4∗z3+5∗z2+z+10q9+7∗z9+2∗z8+9∗z7+3∗z6+7∗z5+2∗z4+6∗z3+10∗z2+9∗z+8q10+3∗z9+z8+6∗z7+7∗z6+4∗ z5+6∗z3+8∗z+9q11+2∗z9+5∗z8+8∗z5+2∗z4+3∗z3+8∗z2+6∗z+8q12+5q13+4∗z9+4∗z8+10∗z7+5∗z6+2∗z5+3∗z4+6∗z3+2∗z2+4∗z+2q14+6∗ z9+4∗z8+2∗z7+z6+8∗z5+7∗z4+10∗z3+5∗z2+5∗z+8q15+5∗z9+z8+4∗z7+8∗z6+4∗z5+10∗z4+9∗z3+3∗z2+9∗z+7q16+4∗z9+5∗z8+7∗z7+10∗ z6+3∗z5+4∗z4+7∗z3+7∗z2+6∗z+5q17+10∗z8+3∗z7+9∗z6+7∗z5+9∗z4+4∗z3+9∗z2+7∗z+10q18+5∗z9+10∗z8+5∗z6+4∗z5+10∗z4+z2+7∗z+ 8q19+z9+10∗z8+3∗z7+2∗z6+7∗z5+8∗z4+4∗z2+10∗z+8q20+2∗z9+2∗z8+9∗z7+3∗z6+9∗z5+4∗z4+8∗z3+10∗z+7q21+⋯ ∈ S20(Γ0(13); \overline{F}11). Also, z satisfies the equation x10 + 7x5 + 8x4 + 10x3 + 6x2 + 6x + 2 = 0. Since f is ordinary, we do not continue our analysis.

72. Consider f = q + 10q3 + 9q4 + 10q5 + 9q7 + 9q9 + 10q11 + 2q12 + 10q13 + q15 + 4q16 + 7q17 + 2q19 + 2q20 + 2q21 + ⋯ ∈ S22(Γ0(13); \overline{F}11). Also, z satisfies the equation x20 + 6x19 + 5x18 + 9x17 + 9x16 + 2x15 + x13 + 5x12 + 7x11 + 10x10 + 7x9 + 10x8 + 9x6 + x5 + 6x4 + 7x3 + x + 4 = 0. Since f is ordinary, we do not continue our analysis.

73. Consider f = q+4∗z19+5∗z18+3∗z17+7∗z16+7∗z15+6∗z14+6∗z13+7∗z12+z11+4∗z10+4∗z9+9∗z8+8∗z7+7∗z5+5∗z4+z3+6∗z2+9∗z+6q2+8∗ z19+9∗z18+z17+9∗z16+6∗z15+2∗z14+9∗z13+3∗z10+4∗z9+5∗z8+4∗z7+z6+3∗z4+5∗z3+4∗z2+8∗z+4q3+3∗z19+2∗z18+4∗z17+z16+7∗z15+ 5∗z14+6∗z12+6∗z11+3∗z10+8∗z9+8∗z8+2∗z7+10∗z6+5∗z5+5∗z4+10∗z3+3∗z2+3∗z+2q4+2∗z19+6∗z18+9∗z17+z16+2∗z14+z13+2∗z12+ z11+2∗z10+3∗z9+2∗z8+z7+2∗z6+4∗z5+4∗z3+6∗z2+z+8q5+5∗z19+8∗z18+2∗z17+2∗z16+7∗z15+7∗z14+z13+8∗z12+7∗z11+5∗z10+10∗z8+ 3∗z7+z6+9∗z5+5∗z4+3∗z3+9∗z2+4∗z+9q6+4∗z19+8∗z18+6∗z17+9∗z16+2∗z15+6∗z14+6∗z12+8∗z11+9∗z10+7∗z9+8∗z8+z7+8∗z6+4∗z5+ 9∗z4 +6∗z+1q7 +7∗z19+10∗z18+10∗z17+10∗z16+9∗z15+z12+3∗z11+z10+4∗z9 +5∗z8 +3∗z7 +7∗z6 +9∗z5 +3∗z4 +10∗z3 +3∗z2 +9∗z+3q8 +6∗ z19+3∗z18+4∗z17+10∗z16+2∗z15+8∗z14+z13+8∗z12+9∗z11+10∗z9+10∗z8+2∗z7+10∗z6+8∗z5+9∗z4+4∗z3+6∗z2+7∗z+8q9+9∗z19+8∗z18+ 9∗z17+9∗z16+z15+8∗z14+9∗z13+4∗z12+2∗z11+10∗z10+3∗z9+3∗z8+10∗z7+6∗z6+8∗z5+6∗z4+7∗z3+9∗z2+7∗zq10+q11+4∗z19+5∗z18+9∗ z17+8∗z16+5∗z15+10∗z14+8∗z13+z12+6∗z11+9∗z10+3∗z9+7∗z8+2∗z7+2∗z5+8∗z4+8∗z3+10∗z2+9∗z+1q12+10q13+3∗z19+6∗z18+z17+ 8∗z16+10∗z15+4∗z14+8∗z13+8∗z12+4∗z11+2∗z10+4∗z9 +9∗z8 +7∗z7 +6∗z6 +5∗z5 +5∗z4 +6∗z3 +5∗z2 +10∗z+1q14 +2∗z19+6∗z18+8∗z17+ 10∗z16+4∗z15+5∗z14+8∗z13+3∗z12+2∗z11+3∗z10+5∗z9+6∗z8+2∗z7+2∗z6+3∗z5+5∗z4+z3+9∗z2+z+5q15+z19+9∗z17+8∗z16+8∗z15+6∗ z14+9∗z13+8∗z11+5∗z10+5∗z8+4∗z6+7∗z5+8∗z4+9∗z3+3∗z2+9∗z+1q16+z19+2∗z17+5∗z16+3∗z15+5∗z14+3∗z13+7∗z12+4∗z11+z10+3∗ z9+7∗z7+4∗z6+10∗z4+10∗z3+2∗z2+9∗z+6q17+8∗z19+9∗z18+7∗z17+10∗z16+4∗z15+6∗z14+5∗z12+5∗z11+8∗z10+3∗z9+3∗z8+9∗z7+z6+ 6∗z5+6∗z4+z3+8∗z2+8∗z+6q18+4∗z19+z17+6∗z16+7∗z15+8∗z14+6∗z13+2∗z12+z11+4∗z9+6∗z8+2∗z7+5∗z6+4∗z5+9∗z4+8∗z3+7∗z2+3∗ zq19+5∗z19+10∗z18+9∗z17+6∗z16+2∗z15+3∗z14+6∗z13+6∗z12+3∗z11+7∗z10+3∗z9+4∗z8+8∗z7+10∗z6+z5+z4+10∗z3+z2+2∗z+1q20+6∗ z19+2∗z18+9∗z17+3∗z16+9∗z15+z14+5∗z13+6∗z12+8∗z11+7∗z10+8∗z9+z8+2∗z7+4∗z5+7∗z4+4∗z3+z2+8∗zq21+⋯ ∈ S22(Γ0(13); \overline{F}11). Also, z satisfies the equation x20 + 6x19 + 5x18 + 9x17 + 9x16 + 2x15 + x13 + 5x12 + 7x11 + 10x10 + 7x9 + 10x8 + 9x6 + x5 + 6x4 + 7x3 + x + 4 = 0. Since f is ordinary, we do not continue our analysis.

74. Consider f = q+10∗z19+6∗z18+4∗z17+10∗z16+9∗z15+6∗z14+7∗z13+2∗z12+10∗z11+9∗z10+3∗z9 +9∗z8 +z7 +z6 +5∗z5 +9∗z4 +10∗z3 +8∗ z2+7∗z+4q2+2∗z18+3∗z17+7∗z16+8∗z14+2∗z12+6∗z10+10∗z8+9∗z7+9∗z6+10∗z5+5∗z4+6∗z3+4∗z2+9∗z+4q3+4∗z19+9∗z18+5∗z17+2∗ z16+z15+z14+z13+4∗z12+5∗z11+9∗z10+z9 +z8 +8∗z7 +7∗z6 +z5 +2∗z4 +z3 +4∗z2 +5∗z+1q4 +z19+5∗z18+9∗z17+5∗z16+5∗z15+10∗z14+ z13+7∗z12+10∗z11+4∗z9 +5∗z8 +8∗z7 +10∗z6 +8∗z5 +3∗z4 +7∗z3 +8∗z2 +4∗z+6q5 +5∗z19+3∗z18+3∗z17+7∗z16+6∗z15+2∗z13+4∗z11+9∗ z10+5∗z9 +6∗z8 +5∗z7 +6∗z6 +9∗z5 +5∗z4 +8∗z3 +z2 +9∗z+6q6 +z19+3∗z18+2∗z17+z16+10∗z15+3∗z14+7∗z13+9∗z12+3∗z11+9∗z10+z9 + 3∗z7 +z6 +5∗z5 +7∗z4 +5∗z2 +6∗z+7q7 +5∗z19+z18+7∗z17+3∗z16+4∗z14+8∗z13+2∗z12+8∗z11+7∗z10+2∗z9 +z8 +8∗z6 +6∗z5 +9∗z4 +z3 + 9∗z2 +8q8 +2∗z19+8∗z18+6∗z16+4∗z15+2∗z14+8∗z13+5∗z12+2∗z11+9∗z10+5∗z9 +5∗z8 +2∗z5 +10∗z4 +7∗z3 +2∗z2 +10∗z+1q9 +2∗z19+3∗ z18+2∗z17+2∗z16+10∗z15+3∗z14+2∗z13+7∗z12+9∗z11+z10+8∗z9 +8∗z8 +z7 +5∗z6 +3∗z5 +5∗z4 +4∗z3 +2∗z2 +4∗z+7q10 +q11 +6∗z19+6∗ z18+7∗z17+z16+8∗z15+8∗z14+6∗z13+7∗z12+5∗z11+5∗z10+2∗z9+9∗z8+6∗z7+7∗z6+5∗z5+2∗z4+3∗z3+4∗zq12+10q13+4∗z19+5∗z18+8∗ z17+6∗z16+9∗z15+2∗z14+4∗z13+2∗z12+7∗z11+10∗z10+5∗z9+3∗z7+z5+2∗z4+5∗z3+2∗z2+9∗z+4q14+9∗z19+5∗z18+3∗z17+z16+7∗z15+ 6∗z14+3∗z13+8∗z12+9∗z11+8∗z10+6∗z9+5∗z8+9∗z7+9∗z6+8∗z5+6∗z4+10∗z3+2∗z2+10∗z+1q15+10∗z19+2∗z17+3∗z16+3∗z15+5∗z14+ 2∗z13+3∗z11+6∗z10+6∗z8+7∗z6+4∗z5+3∗z4+2∗z3+8∗z2+2∗z+7q16+4∗z17+8∗z16+6∗z15+10∗z14+5∗z13+7∗z12+7∗z11+7∗z10+3∗z9+6∗ z8+7∗z7+4∗z5+2∗z4+z3+10∗z2+9q17+7∗z19+2∗z18+6∗z17+9∗z16+10∗z15+10∗z14+10∗z13+7∗z12+6∗z11+2∗z10+10∗z9+10∗z8+3∗z7+ 4∗z6 +10∗z5 +9∗z4 +10∗z3 +7∗z2 +6∗z+7q18 +z19+7∗z17+4∗z16+5∗z15+z14+z13+2∗z12+10∗z11+7∗z10+4∗z9 +2∗z8 +2∗z7 +4∗z6 +5∗z5 + 7∗z4+3∗z3+9∗z2+9∗z+9q19+3∗z19+z18+6∗z17+10∗z16+4∗z15+7∗z14+3∗z13+7∗z12+8∗z11+2∗z10+z9+5∗z7+9∗z5+7∗z4+z3+7∗z2+4∗ z+6q20 +z19+9∗z18+10∗z15+5∗z14+7∗z13+4∗z12+3∗z11+5∗z10+z9 +8∗z8 +8∗z7 +6∗z6 +2∗z5 +7∗z3 +6∗z2 +6q21 +⋯ ∈ S22(Γ0(13); \overline{F}11). Also, z satisfies the equation x20 + 6x19 + 5x18 + 9x17 + 9x16 + 2x15 + x13 + 5x12 + 7x11 + 10x10 + 7x9 + 10x8 + 9x6 + x5 + 6x4 + 7x3 + x + 4 = 0. Since f is ordinary, we do not continue our analysis.

75. Consider f = q+z19+6∗z18+9∗z17+2∗z14+6∗z13+4∗z12+6∗z11+10∗z10+7∗z9 +5∗z8 +3∗z7 +9∗z6 +6∗z5 +2∗z4 +9∗z2 +3∗z+8q2 +6∗z19+3∗ z18+6∗z17+3∗z16+5∗z15+2∗z14+9∗z13+6∗z12+7∗z11+9∗z10+6∗z9 +2∗z8 +10∗z6 +10∗z5 +10∗z4 +10∗z3 +7∗z+3q3 +3∗z19+z18+z17+2∗ z15+5∗z13+z12+2∗z11+7∗z10+9∗z8+10∗z7+4∗z5+10∗z4+8∗z3+4∗z+2q4+7∗z19+10∗z18+5∗z17+7∗z15+4∗z14+2∗z13+6∗z12+8∗z11+6∗ z10+3∗z9+3∗z8+8∗z7+7∗z6+4∗z5+6∗z4+6∗z3+7∗z2+9∗z+1q5+10∗z19+6∗z17+9∗z15+4∗z14+7∗z13+7∗z12+10∗z11+2∗z10+3∗z9+z8+7∗ z7+7∗z6+8∗z5+5∗z4+3∗z3+7∗z2+2∗z+2q6+7∗z19+8∗z18+4∗z17+8∗z16+10∗z15+z14+10∗z13+2∗z12+6∗z11+9∗z10+10∗z9+7∗z8+8∗z7+ 9∗z6+7∗z5+2∗z4+6∗z3+8∗z2+5q7+10∗z19+9∗z18+5∗z17+8∗z16+z15+z14+4∗z13+3∗z12+8∗z11+5∗z10+10∗z9+5∗z8+7∗z7+9∗z6+z4+3∗ z3+8∗z2+4∗z+7q8+3∗z19+7∗z18+9∗z17+8∗z16+6∗z15+5∗z14+z13+8∗z12+3∗z11+4∗z10+2∗z9+10∗z8+5∗z7+5∗z6+8∗z4+z3+4∗z2+9∗z+ 5q9+4∗z19+4∗z18+6∗z16+2∗z15+z14+2∗z12+8∗z11+8∗z10+7∗z9+10∗z8+5∗z7+z6+6∗z5+10∗z4+10∗z3+3∗z2+10∗z+6q10+q11+3∗z19+2∗ z18+2∗z17+8∗z16+4∗z15+3∗z13+8∗z12+8∗z11+5∗z10+z9+5∗z8+4∗z7+10∗z6+3∗z5+4∗z4+4∗z3+9∗z2+3∗z+4q12+10q13+5∗z19+z18+8∗ z17+5∗z16+7∗z15+z14+z13+z12+7∗z11+6∗z10+8∗z8+10∗z7+8∗z6+7∗z5+4∗z4+4∗z3+6∗z2+8q14+3∗z19+7∗z18+8∗z17+6∗z16+10∗z15+ 8∗z14+8∗z13+9∗z12+4∗z11+5∗z10+4∗z9+3∗z8+6∗z7+5∗z6+10∗z5+2∗z4+9∗z3+7∗z2+9∗z+7q15+9∗z18+7∗z17+2∗z16+4∗z15+6∗z14+7∗ z13+10∗z12+z11+6∗z10+2∗z9 +5∗z8 +3∗z7 +2∗z6 +2∗z3 +10∗z2 +2∗z+5q16 +5∗z19+10∗z18+6∗z17+3∗z15+z14+5∗z13+9∗z12+6∗z11+10∗ z10+9∗z9+5∗z8+10∗z6+9∗z5+5∗z4+z3+10∗z2+2∗z+7q17+8∗z19+10∗z18+10∗z17+9∗z15+6∗z13+10∗z12+9∗z11+4∗z10+2∗z8+z7+7∗z5+ z4+3∗z3+7∗z+6q18+3∗z19+8∗z18+z17+9∗z16+10∗z14+7∗z13+2∗z12+7∗z11+10∗z9+9∗z8+8∗z7+4∗z6+z5+3∗z4+3∗z3+7∗z2+8∗z+1q19+ z19+9∗z18+6∗z17+z16+8∗z15+9∗z14+9∗z13+9∗z12+8∗z11+10∗z10+6∗z8+2∗z7+6∗z6+8∗z5+3∗z4+3∗z3+10∗z2+9q20+3∗z19+6∗z18+6∗ z16+3∗z15+7∗z14+4∗z13+9∗z12+5∗z11+3∗z10+6∗z9 +2∗z8 +8∗z7 +6∗z6 +4∗z5 +10∗z4 +3∗z3 +8∗z2 +10∗z+1q21 +⋯ ∈ S22(Γ0(13); \overline{F}11). Also, z satisfies the equation x20 + 6x19 + 5x18 + 9x17 + 9x16 + 2x15 + x13 + 5x12 + 7x11 + 10x10 + 7x9 + 10x8 + 9x6 + x5 + 6x4 + 7x3 + x + 4 = 0. Since f is ordinary, we do not continue our analysis.

76. Consider f = q+7∗z19+5∗z18+6∗z17+5∗z16+6∗z15+8∗z14+3∗z13+9∗z12+5∗z11+10∗z10+8∗z9 +10∗z8 +10∗z7 +z6 +4∗z5 +6∗z4 +10∗z2 +3∗ z+7q2+8∗z19+8∗z18+z17+3∗z16+10∗z14+4∗z13+3∗z12+4∗z11+4∗z10+z9+5∗z8+9∗z7+2∗z6+2∗z5+4∗z4+z3+3∗z2+9∗zq3+z19+10∗z18+ z17+8∗z16+z15+5∗z14+5∗z13+9∗z11+3∗z10+2∗z9+4∗z8+2∗z7+5∗z6+z5+5∗z4+3∗z3+4∗z2+10∗z+9q4+z19+z18+10∗z17+5∗z16+10∗z15+ 6∗z14+7∗z13+7∗z12+3∗z11+3∗z10+z9+z8+5∗z7+3∗z6+6∗z5+2∗z4+5∗z3+z2+8∗z+7q5+2∗z19+2∗z16+z13+7∗z12+z11+6∗z10+3∗z9+5∗ z8+7∗z7+8∗z6+7∗z5+7∗z4+8∗z3+5∗z2+7∗z+4q6+10∗z19+3∗z18+10∗z17+4∗z16+z14+5∗z13+5∗z12+5∗z11+6∗z10+4∗z9+7∗z8+10∗z7+4∗ z6+6∗z5+4∗z4+5∗z3+9∗z2+10∗z+4q7+2∗z18+z16+z15+6∗z14+10∗z13+5∗z12+3∗z11+9∗z10+6∗z9+z7+9∗z6+7∗z5+9∗z4+8∗z3+2∗z2+9∗ z+2q8+4∗z18+9∗z17+9∗z16+10∗z15+7∗z14+z13+z12+8∗z11+9∗z10+5∗z9+8∗z8+4∗z7+7∗z6+z5+6∗z4+10∗z3+10∗z2+7∗z+10q9+7∗z19+ 7∗z18+5∗z16+9∗z15+10∗z14+9∗z12+3∗z11+3∗z10+4∗z9+z8+6∗z7+10∗z6+5∗z5+z4+z3+8∗z2+z+1q10+q11+9∗z19+9∗z18+4∗z17+5∗z16+ 5∗z15+4∗z14+5∗z13+6∗z12+3∗z11+3∗z10+5∗z9 +z8 +10∗z7 +5∗z6 +z5 +8∗z4 +7∗z3 +3∗z2 +6∗z+10q12 +10q13 +10∗z19+10∗z18+5∗z17+3∗ z16+7∗z15+4∗z14+9∗z13+4∗z11+4∗z10+2∗z9+5∗z8+2∗z7+8∗z6+9∗z5+7∗z3+9∗z2+3∗z+5q14+8∗z19+4∗z18+3∗z17+5∗z16+z15+3∗z14+3∗ z13+2∗z12+7∗z11+6∗z10+7∗z9+8∗z8+5∗z7+6∗z6+z5+9∗z4+2∗z3+4∗z2+2∗z+10q15+2∗z18+4∗z17+9∗z16+7∗z15+5∗z14+4∗z13+z12+10∗ z11+5∗z10+9∗z9+6∗z8+8∗z7+9∗z6+9∗z3+z2+9∗z+3q16+5∗z19+z18+10∗z17+9∗z16+10∗z15+6∗z14+9∗z13+10∗z12+5∗z11+4∗z10+7∗z9+ 8∗z7+8∗z6+9∗z5+5∗z4+10∗z3+6q17+10∗z19+z18+10∗z17+3∗z16+10∗z15+6∗z14+6∗z13+2∗z11+8∗z10+9∗z9+7∗z8+9∗z7+6∗z6+10∗z5+6∗ z4+8∗z3+7∗z2+z+10q18+3∗z19+3∗z18+2∗z17+3∗z16+10∗z15+3∗z14+8∗z13+5∗z12+4∗z11+4∗z10+4∗z9+5∗z8+10∗z7+9∗z6+z5+3∗z4+8∗ z3+10∗z2+2∗z+9q19+2∗z19+2∗z18+z17+5∗z16+8∗z15+3∗z14+4∗z13+3∗z11+3∗z10+7∗z9+z8+7∗z7+6∗z6+4∗z5+8∗z3+4∗z2+5∗z+4q20+ z19+5∗z18+2∗z17+2∗z16+9∗z14+6∗z13+3∗z12+6∗z11+7∗z10+7∗z9+4∗z7+10∗z6+z5+5∗z4+8∗z3+7∗z2+4∗z+2q21+⋯ ∈ S22(Γ0(13); \overline{F}11). Also, z satisfies the equation x20 + 6x19 + 5x18 + 9x17 + 9x16 + 2x15 + x13 + 5x12 + 7x11 + 10x10 + 7x9 + 10x8 + 9x6 + x5 + 6x4 + 7x3 + x + 4 = 0. Since f is ordinary, we do not continue our analysis.

77. Consider f = q+z11+6∗z10+5∗z9 +5∗z8 +2∗z7 +3∗z6 +3∗z5 +2∗z4 +6∗z3 +z2 +7q2 +7∗z10+7∗z9 +7∗z8 +10∗z7 +7∗z6 +5∗z4 +4∗z3 +7∗z2 +3∗ z+1q3+6∗z11+4∗z10+6∗z9+z8+10∗z6+8∗z5+4∗z4+2∗z2+6∗z+8q4+8∗z11+10∗z10+5∗z9+10∗z8+5∗z7+6∗z6+z5+7∗z4+8∗z2+8∗z+2q5+ 8∗z11+8∗z10+6∗z7 +6∗z6 +2∗z5 +z2 +8∗z+2q6 +4∗z11+10∗z10+7∗z9 +5∗z8 +4∗z7 +z6 +8∗z5 +3∗z4 +2∗z3 +10∗z2 +3q7 +z11+5∗z10+6∗z9 + 2∗z8+9∗z7+8∗z6+6∗z5+5∗z4+z3+7∗z2+10∗zq8+10∗z11+2∗z10+10∗z9+7∗z8+2∗z7+4∗z6+2∗z5+z4+9∗z3+4∗z2+3∗z+9q9+8∗z10+8∗z9+ 8∗z8+2∗z7+8∗z6+z4+3∗z3+8∗z2+5∗zq10+10q11+9∗z11+4∗z8+2∗z5+4∗z4+4∗z3+3∗z2+z+2q12+q13+7∗z11+7∗z10+7∗z9+4∗z8+10∗z7+ 7∗z6 +4∗z5 +2∗z4 +z3 +2∗z2 +5∗z+3q14 +3∗z11+8∗z10+3∗z9 +7∗z8 +10∗z7 +4∗z6 +6∗z5 +3∗z4 +z3 +8∗z2 +z+4q15 +8∗z11+8∗z10+8∗z9 +3∗ z8 +2∗z7 +8∗z6 +3∗z5 +7∗z4 +9∗z3 +7∗z2 +z+10q16 +9∗z11+10∗z10+z9 +z8 +7∗z7 +5∗z6 +5∗z5 +7∗z4 +10∗z3 +9∗z2 +8q17 +7∗z11+3∗z10+3∗ z8 +5∗z7 +5∗z6 +5∗z5 +3∗z4 +3∗z3 +4∗z2 +z+7q18 +3∗z11+8∗z10+2∗z9 +8∗z8 +5∗z7 +z6 +10∗z5 +9∗z4 +4∗z3 +10∗z2 +6∗z+8q19 +3∗z11+9∗ z10+z8 +4∗z7 +4∗z6 +z4 +z3 +6∗z2 +z+6q20 +4∗z11+3∗z10+8∗z9 +z8 +z7 +7∗z6 +9∗z5 +5∗z4 +7∗z3 +9∗z2 +z+8q21 +⋯ ∈ S22(Γ0(13); \overline{F}11). Also, z satisfies the equation x12 + x8 + x7 + 4x6 + 2x5 + 5x4 + 5x3 + 6x2 + 5x + 2 = 0. Since f is ordinary, we do not continue our analysis.

78. Consider f = q +4∗z11+9∗z10+3∗z9 +7∗z8 +8∗z7 +2∗z6 +9∗z5 +3∗z4 +z3 +4∗z2 +2∗z +3q2 +10∗z11+2∗z10+6∗z9 +4∗z7 +3∗z6 +7∗z5 + 3∗z4 +10∗z3 +z2 +z +2q3 +3∗z11+7∗z9 +9∗z8 +2∗z7 +10∗z6 +2∗z5 +7∗z4 +6∗z3 +9∗z2 +2∗z +7q4 +8∗z11+7∗z10+8∗z9 +z8 +4∗z7 +10∗ z6 +10∗z5 +5∗z4 +7∗z3 +4∗z2 +5∗z +3q5 +4∗z10+7∗z9 +z8 +5∗z7 +2∗z6 +10∗z5 +10∗z4 +9∗z3 +9∗z2 +4∗z +2q6 +4∗z11+4∗z10+5∗z9 + 6∗z8 +6∗z7 +7∗z6 +3∗z5 +3∗z4 +7∗z3 +7∗z2 +z +2q7 +6∗z11+3∗z10+9∗z9 +7∗z8 +6∗z7 +10∗z6 +3∗z5 +6∗z4 +4∗z2 +6∗zq8 +10∗z11+9∗ z9 +4∗z8 +z7 +5∗z6 +9∗z5 +3∗z4 +8∗z3 +5∗z +9q9 +2∗z11+7∗z10+10∗z9 +3∗z7 +5∗z6 +8∗z5 +5∗z4 +2∗z3 +9∗z2 +9∗z +9q10 +10q11 + z11+10∗z10+10∗z9 +8∗z8 +8∗z7 +10∗z6 +10∗z5 +2∗z4 +10∗z3 +3∗z2 +3∗z+6q12 +q13 +z11+4∗z9 +5∗z8 +9∗z7 +z6 +5∗z5 +7∗z4 +8∗z3 + 7∗z2 +7∗z +1q14 +10∗z11+3∗z9 +10∗z8 +4∗z7 +9∗z6 +6∗z4 +4∗z3 +z2 +2∗z +10q15 +9∗z11+3∗z9 +z8 +4∗z7 +9∗z6 +z5 +8∗z4 +6∗z3 +8∗ z2 +8∗z +3q16 +3∗z11+4∗z10+5∗z9 +8∗z8 +6∗z7 +7∗z6 +4∗z5 +5∗z4 +9∗z3 +3∗z2 +7∗z +5q17 +9∗z11+9∗z10+6∗z9 +5∗z8 +z7 +3∗z5 + 8∗z4 +4∗z3 +7∗z2 +z+10q18 +2∗z11+6∗z10+9∗z9 +10∗z8 +4∗z6 +8∗z5 +9∗z4 +3∗z3 +8∗z2 +7∗z+8q19 +3∗z11+7∗z10+9∗z9 +10∗z8 +9∗ z7 +2∗z6 +9∗z4 +3∗z3 +4∗z2 +8∗z+7q20 +3∗z11+9∗z10+6∗z9 +6∗z8 +z7 +9∗z5 +9∗z4 +5∗z3 +5∗z2 +4∗z+10q21 +⋯ ∈ S22(Γ0(13); \overline{F}11). Also, z satisfies the equation x12 + x8 + x7 + 4x6 + 2x5 + 5x4 + 5x3 + 6x2 + 5x + 2 = 0. Since f is ordinary, we do not continue our analysis.

79. Consider f = q + 3q2 + 4q3 + 7q4 + 6q5 + q6 + 8q7 + 4q8 + 2q9 + 7q10 + 6q12 + q13 + 2q14 + 2q15 + 9q16 + 7q17 + 6q18 + 9q19 + 9q20 + 10q21 + ⋯ ∈ S22(Γ0(13); \overline{F}11). Also, z satisfies the equation x12 + x8 + x7 + 4x6 + 2x5 + 5x4 + 5x3 + 6x2 + 5x + 2 = 0. I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

80. Consider f = q +7∗z11+z10+6∗z9 +4∗z8 +6∗z7 +5∗z6 +6∗z5 +7∗z4 +3∗z3 +8∗z2 +z +5q2 +z11+9∗z9 +z7 +5∗z6 +7∗z5 +10∗z4 +4∗z3 + 8∗z2 +4∗z +4q3 +8∗z11+9∗z10+2∗z8 +4∗z7 +4∗z6 +6∗z5 +2∗z4 +2∗z3 +4∗z2 +4∗z +6q4 +3∗z11+5∗z10+5∗z9 +10∗z8 +4∗z7 +5∗z6 +8∗ z5 +7∗z4 +6∗z2 +3∗z +1q5 +8∗z10+6∗z9 +10∗z8 +3∗z7 +2∗z6 +8∗z5 +2∗z4 +9∗z3 +z2 +4∗z +5q6 +7∗z11+3∗z10+9∗z9 +5∗z8 +6∗z7 + 10∗z6 +2∗z5 +4∗z4 +9∗z3 +8∗z2 +1q7 +5∗z11+10∗z10+6∗z9 +4∗z8 +10∗z7 +9∗z6 +7∗z4 +3∗z3 +5∗z2 +10∗z+9q8 +z11+3∗z10+8∗z9 +7∗ z8 +z7 +7∗z6 +z5 +2∗z3 +8∗z2 +8∗z +1q9 +9∗z11+4∗z9 +9∗z7 +z6 +8∗z5 +2∗z4 +3∗z3 +6∗z2 +3∗z +5q10 +10q11 +10∗z11+10∗z9 +3∗z8 + 6∗z7 +8∗z6 +5∗z5 +8∗z4 +5∗z3 +9∗z2 +6q12 +q13 +10∗z11+7∗z9 +6∗z8 +2∗z7 +10∗z6 +6∗z5 +4∗z4 +3∗z3 +4∗z2 +4∗z +3q14 +z11+5∗ z10+7∗z9 +z8 +3∗z7 +2∗z5 +10∗z4 +9∗z3 +5∗z2 +5∗z+9q15 +2∗z11+8∗z9 +10∗z8 +7∗z7 +2∗z6 +10∗z5 +3∗z4 +5∗z3 +3∗z2 +3∗z+10q16 + 8∗z11+9∗z10+10∗z9 +3∗z8 +10∗z7 +z6 +10∗z5 +8∗z4 +5∗z3 +6∗z2 +9∗z +1q17 +2∗z11+3∗z10+7∗z9 +6∗z8 +7∗z7 +4∗z6 +4∗z5 +4∗z4 + 3∗z3 +3∗z2 +7∗z +7q18 +9∗z11+6∗z10+4∗z9 +z8 +8∗z7 +10∗z5 +3∗z4 +4∗z3 +2∗z2 +z +1q19 +8∗z11+9∗z10+z9 +z8 +9∗z7 +7∗z6 +2∗ z5 +7∗z4 +10∗z3 +2∗z2 +10∗z +9q20 +8∗z11+6∗z10+2∗z9 +5∗z8 +9∗z7 +5∗z6 +8∗z5 +6∗z4 +z3 +2∗z2 +6∗zq21 +⋯ ∈ S22(Γ0(13); \overline{F}11). Also, z satisfies the equation x12 + x8 + x7 + 4x6 + 2x5 + 5x4 + 5x3 + 6x2 + 5x + 2 = 0. Since f is ordinary, we do not continue our analysis.

81. Consider f = q+10∗z11+6∗z10+8∗z9 +6∗z8 +6∗z7 +z6 +4∗z5 +10∗z4 +z3 +9∗z2 +8∗z+9q2 +2∗z10+4∗z8 +7∗z7 +7∗z6 +8∗z5 +4∗z4 +4∗z3 + 6∗z2 +3∗zq3 +5∗z11+9∗z10+9∗z9 +10∗z8 +5∗z7 +9∗z6 +6∗z5 +9∗z4 +3∗z3 +7∗z2 +10∗z+10q4 +3∗z11+4∗z9 +z8 +9∗z7 +z6 +3∗z5 +3∗z4 + 4∗z3 +4∗z2 +6∗z+5q5 +3∗z11+2∗z10+9∗z9 +8∗z7 +z6 +2∗z5 +10∗z4 +4∗z3 +6∗z+8q6 +7∗z11+5∗z10+z9 +6∗z8 +6∗z7 +4∗z6 +9∗z5 +z4 +4∗ z3 +8∗z2 +10∗z+10q7 +10∗z11+4∗z10+z9 +9∗z8 +8∗z7 +6∗z6 +2∗z5 +4∗z4 +7∗z3 +6∗z2 +7∗z+4q8 +z11+6∗z10+6∗z9 +4∗z8 +7∗z7 +6∗z6 + 10∗z5 +7∗z4 +3∗z3 +10∗z2 +6∗z+10q9 +7∗z10+3∗z8 +8∗z7 +8∗z6 +6∗z5 +3∗z4 +3∗z3 +10∗z2 +5∗z+2q10 +10q11 +2∗z11+z10+2∗z9 +7∗z8 + 8∗z7 +4∗z6 +5∗z5 +8∗z4 +3∗z3 +7∗z2 +7∗z+7q12 +q13 +4∗z11+4∗z10+4∗z9 +7∗z8 +z7 +4∗z6 +7∗z5 +9∗z4 +10∗z3 +9∗z2 +6∗z+1q14 +8∗ z11+9∗z10+9∗z9 +4∗z8 +5∗z7 +9∗z6 +3∗z5 +3∗z4 +8∗z3 +8∗z2 +3∗z+8q15 +3∗z11+3∗z10+3∗z9 +8∗z8 +9∗z7 +3∗z6 +8∗z5 +4∗z4 +2∗z3 + 4∗z2 +10∗z+3q16 +2∗z11+10∗z10+6∗z9 +10∗z8 +10∗z7 +9∗z6 +3∗z5 +2∗z4 +9∗z3 +4∗z2 +6∗z+4q17 +4∗z11+7∗z10+9∗z9 +8∗z8 +9∗z7 +2∗ z6 +10∗z5 +7∗z4 +z3 +8∗z2 +2∗z+2q18 +8∗z11+2∗z10+7∗z9 +3∗z8 +9∗z7 +6∗z6 +5∗z5 +z4 +2∗z2 +8∗z+4q19 +8∗z11+8∗z10+z9 +10∗z8 + 9∗z6 +9∗z5 +5∗z4 +8∗z3 +10∗z2 +3∗z+3q20 +7∗z11+4∗z10+6∗z9 +10∗z8 +10∗z6 +7∗z5 +2∗z4 +9∗z3 +6∗z2 +3q21 +⋯ ∈ S22(Γ0(13); \overline{F}11). Also, z satisfies the equation x12 + x8 + x7 + 4x6 + 2x5 + 5x4 + 5x3 + 6x2 + 5x + 2 = 0. Since f is ordinary, we do not continue our analysis.

82. Consider f = q+3∗z10+6∗z9 +2∗z7 +z6 +10∗z5 +3∗z4 +10∗z3 +8∗z2 +2∗zq2 +5∗z10+10∗z9 +7∗z7 +9∗z6 +2∗z5 +5∗z4 +2∗z3 +6∗z2 +7∗z+7q3 + 5∗z10+10∗z9 +7∗z7 +9∗z6 +2∗z5 +5∗z4 +2∗z3 +6∗z2 +7∗z+3q4 +6∗z10+z9 +4∗z7 +2∗z6 +9∗z5 +6∗z4 +9∗z3 +5∗z2 +4∗z+8q5 +q6 +3∗z10+6∗ z9 +2∗z7 +z6 +10∗z5 +3∗z4 +10∗z3 +8∗z2 +2∗z+6q7 +4∗z10+8∗z9 +10∗z7 +5∗z6 +6∗z5 +4∗z4 +6∗z3 +7∗z2 +10∗z+1q8 +2∗z10+4∗z9 +5∗z7 + 8∗z6 +3∗z5 +2∗z4 +3∗z3 +9∗z2 +5∗zq9 +z10+2∗z9 +8∗z7 +4∗z6 +7∗z5 +z4 +7∗z3 +10∗z2 +8∗z+10q10 +10q11 +4∗z10+8∗z9 +10∗z7 +5∗z6 +6∗ z5 +4∗z4 +6∗z3 +7∗z2 +10∗z+8q12 +q13 +z10+2∗z9 +8∗z7 +4∗z6 +7∗z5 +z4 +7∗z3 +10∗z2 +8∗z+5q14 +7∗z10+3∗z9 +z7 +6∗z6 +5∗z5 +7∗z4 + 5∗z3 +4∗z2 +z+3q15 +7∗z10+3∗z9 +z7 +6∗z6 +5∗z5 +7∗z4 +5∗z3 +4∗z2 +z+8q16 +5∗z10+10∗z9 +7∗z7 +9∗z6 +2∗z5 +5∗z4 +2∗z3 +6∗z2 +7∗ zq17 +7∗z10+3∗z9 +z7 +6∗z6 +5∗z5 +7∗z4 +5∗z3 +4∗z2 +z+7q18 +10∗z10+9∗z9 +3∗z7 +7∗z6 +4∗z5 +10∗z4 +4∗z3 +z2 +3∗z+8q19 +5∗z10+10∗ z9 +7∗z7 +9∗z6 +2∗z5 +5∗z4 +2∗z3 +6∗z2 +7∗z+4q20 +8∗z10+5∗z9 +9∗z7 +10∗z6 +z5 +8∗z4 +z3 +3∗z2 +9∗z+10q21 +⋯ ∈ S22(Γ0(13); \overline{F}11). Also, z satisfies the equation x12 + x8 + x7 + 4x6 + 2x5 + 5x4 + 5x3 + 6x2 + 5x + 2 = 0. Since f is ordinary, we do not continue our analysis.

83. Consider f = q+8∗z10+5∗z9 +9∗z7 +10∗z6 +z5 +8∗z4 +z3 +3∗z2 +9∗z+9q2 +6∗z10+z9 +4∗z7 +2∗z6 +9∗z5 +6∗z4 +9∗z3 +5∗z2 +4∗zq3 +6∗z10+ z9+4∗z7+2∗z6+9∗z5+6∗z4+9∗z3+5∗z2+4∗z+7q4+5∗z10+10∗z9+7∗z7+9∗z6+2∗z5+5∗z4+2∗z3+6∗z2+7∗z+4q5+q6+8∗z10+5∗z9+9∗z7+ 10∗z6 +z5 +8∗z4 +z3 +3∗z2 +9∗z+4q7 +7∗z10+3∗z9 +z7 +6∗z6 +5∗z5 +7∗z4 +5∗z3 +4∗z2 +z+2q8 +9∗z10+7∗z9 +6∗z7 +3∗z6 +8∗z5 +9∗z4 +8∗ z3 +2∗z2 +6∗z+6q9 +10∗z10+9∗z9 +3∗z7 +7∗z6 +4∗z5 +10∗z4 +4∗z3 +z2 +3∗z+2q10 +10q11 +7∗z10+3∗z9 +z7 +6∗z6 +5∗z5 +7∗z4 +5∗z3 +4∗ z2 +z+9q12 +q13 +10∗z10+9∗z9 +3∗z7 +7∗z6 +4∗z5 +10∗z4 +4∗z3 +z2 +3∗z+8q14 +4∗z10+8∗z9 +10∗z7 +5∗z6 +6∗z5 +4∗z4 +6∗z3 +7∗z2 +10∗ z+2q15 +4∗z10+8∗z9 +10∗z7 +5∗z6 +6∗z5 +4∗z4 +6∗z3 +7∗z2 +10∗z+7q16 +6∗z10+z9 +4∗z7 +2∗z6 +9∗z5 +6∗z4 +9∗z3 +5∗z2 +4∗z+4q17 + 4∗z10+8∗z9 +10∗z7 +5∗z6 +6∗z5 +4∗z4 +6∗z3 +7∗z2 +10∗z+6q18 +z10+2∗z9 +8∗z7 +4∗z6 +7∗z5 +z4 +7∗z3 +10∗z2 +8∗z+5q19 +6∗z10+z9 + 4∗z7 +2∗z6 +9∗z5 +6∗z4 +9∗z3 +5∗z2 +4∗z+8q20 +3∗z10+6∗z9 +2∗z7 +z6 +10∗z5 +3∗z4 +10∗z3 +8∗z2 +2∗z+1q21 +⋯ ∈ S22(Γ0(13); \overline{F}11). Also, z satisfies the equation x12 + x8 + x7 + 4x6 + 2x5 + 5x4 + 5x3 + 6x2 + 5x + 2 = 0. Since f is ordinary, we do not continue our analysis.

11.12 N = 14, ℓ = 11 (38 forms)

1. Consider f = q + 2q^{2} + z^{8}8q^{3} + 4q^{4} + z^{2}5q^{5} + z^{1}00q^{6} + 4q^{7} + 8q^{8} + z^{7}4q^{9} + z^{3}7q^{10} + z^{1}12q^{12} + z^{5}5q^{13} + 8q^{14} + z^{1}13q^{15} + 5q^{16} + z^{7}8q^{17} + z^{8}6q^{18} + z^{7}q^{19} + z^{4}9q^{20} + z^{1}12q^{21} + ⋯ ∈ S_{14}(Γ_{0}(14); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 6q^{2} + z^{1}6q^{3} + 3q^{4} + z^{4}9q^{5} + z^{4}q^{6} + 3q^{7} + 7q^{8} + z^{5}0q^{9} + z^{3}7q^{10} + z^{9}1q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}9)ω^{1} & 0 \\ 0 & u(z^{9}1)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −64q^{2} + aq^{3} + 4096q^{4} + 63 ∗ a − 13986q^{5} + −64 ∗ aq^{6} + 117649q^{7} + −262144q^{8} + 952 ∗ a − 1420383q^{9} + −4032 ∗ a + 895104q^{10} + 1008 ∗ 11 ∗ a − 541296 ∗ 11q^{11} + 4096 ∗ aq^{12} + −50895 ∗ a + 25981214q^{13} + −7529536q^{14} + 45990∗a+10958220q^{15}+16777216q^{16}+128286∗a+47791842q^{17}+−60928∗a+90904512q^{18}+−114129∗ a + 349993280q^{19} + 258048 ∗ a − 57286656q^{20} + 117649 ∗ aq^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −952x + −173940. The slope of f_{0} is 1.

2. Consider f = q + 2q^{2} + z^{8}q^{3} + 4q^{4} + z^{3}5q^{5} + z^{2}0q^{6} + 4q^{7} + 8q^{8} + z^{9}4q^{9} + z^{4}7q^{10} + z^{3}2q^{12} + z^{5}q^{13} + 8q^{14} + z^{4}3q^{15} + 5q^{16} + z^{1}8q^{17} + z^{1}06q^{18} + z^{7}7q^{19} + z^{5}9q^{20} + z^{3}2q^{21} + ⋯ ∈ S_{14}(Γ_{0}(14); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 6q^{2} + z^{5}6q^{3} + 3q^{4} + z^{5}9q^{5} + z^{4}4q^{6} + 3q^{7} + 7q^{8} + z^{7}0q^{9} + z^{4}7q^{10} + z^{4}1q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{7}9)ω^{1} & 0 \\ 0 & u(z^{4}1)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −64q^{2} + aq^{3} + 4096q^{4} + 63 ∗ a − 13986q^{5} + −64 ∗ aq^{6} + 117649q^{7} + −262144q^{8} + 952 ∗ a − 1420383q^{9} + −4032 ∗ a + 895104q^{10} + 1008 ∗ 11 ∗ a − 541296 ∗ 11q^{11} + 4096 ∗ aq^{12} + −50895 ∗ a + 25981214q^{13} + −7529536q^{14} + 45990∗a+10958220q^{15}+16777216q^{16}+128286∗a+47791842q^{17}+−60928∗a+90904512q^{18}+−114129∗ a + 349993280q^{19} + 258048 ∗ a − 57286656q^{20} + 117649 ∗ aq^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −952x + −173940. The slope of f_{0} is 1.

3. Consider f = q + 9q^{2} + 5q^{3} + 4q^{4} + q^{6} + 7q^{7} + 3q^{8} + 9q^{9} + 9q^{12} + 3q^{13} + 8q^{14} + 5q^{16} + 3q^{17} + 4q^{18} + 5q^{19} + 2q^{21} + ⋯ ∈ S_{14}(Γ_{0}(14); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-2 newform g = q + 10q^{2} + 9q^{3} + q^{4} + 2q^{6} + q^{7} + 10q^{8} + q^{9} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{13}. The form f_{0} = q +64q^{2} +−71455084965079460354+O(11^{2}0)q^{3} + 4096q^{4}+−13972696888007613799∗11+O(11^{2}0)q^{5}+136124526762834601751+O(11^{2}0)q^{6}+−117649q^{7}+ 262144q^{8}+−317574564268362000514+O(11^{2}0)q^{9}+23133755893730911229∗11+O(11^{2}0)q^{10}+−5231466589223644857∗ 11^{2} + O(11^{2}1)q^{11} + −33780221301865607549 + O(11^{2}0)q^{12} + 92043363160735822289 + O(11^{2}0)q^{13} + −7529536q^{14} + −30042712725034765960 ∗ 11 + O(11^{2}0)q^{15} + 16777216q^{16} + 28266995949799157411 + O(11^{2}0)q^{17}+−142272265198367756866+O(11^{2}0)q^{18}+39644719207806030323+O(11^{2}0)q^{19}+12742206436829207672∗ 11 + O(11^{2}0)q^{20} + −64645620636443787950 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 2.

4. Consider f = q + 9q^{2} + q^{3} + 4q^{4} + 4q^{5} + 9q^{6} + 7q^{7} + 3q^{8} + 7q^{9} + 3q^{10} + 4q^{12} + 10q^{13} + 8q^{14} + 4q^{15} + 5q^{16} + 8q^{17} + 8q^{18} + 4q^{19} + 5q^{20} + 7q^{21} + ⋯ ∈ S_{14}(Γ_{0}(14); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 5q^{2} + 5q^{3} + 3q^{4} + 5q^{5} + 3q^{6} + 8q^{7} + 4q^{8} + 10q^{9} + 3q^{10} + 6q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(6)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+64q^{2}+71455084965079461460+ O(11^{2}0)q^{3} + 4096q^{4} + 153699665768083827319 + O(11^{2}0)q^{5} + −136124526762834530967 + O(11^{2}0)q^{6} + −117649q^{7}+262144q^{8}+317574564268363372736+O(11^{2}0)q^{9}+−254471314831035189599+O(11^{2}0)q^{10}+ 57546132481460396687∗11+O(11^{2}1)q^{11}+33780221301870137725+O(11^{2}0)q^{12}+−92043363160727824187+ O(11^{2}0)q^{13}+−7529536q^{14}+330469839975454997800+O(11^{2}0)q^{15}+16777216q^{16}+−28266995949760132579+ O(11^{2}0)q^{17}+142272265198455579074+O(11^{2}0)q^{18}+−39644719207681937389+O(11^{2}0)q^{19}+−140164270804811913512+ O(11^{2}0)q^{20} + 64645620636313668156 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

5. Consider f = q + 4q^{2} + 5q^{3} + 5q^{4} + 6q^{5} + 9q^{6} + 5q^{7} + 9q^{8} + 2q^{9} + 2q^{10} + 3q^{12} + 2q^{13} + 9q^{14} + 8q^{15} + 3q^{16} + 9q^{17} + 8q^{18} + 5q^{19} + 8q^{20} + 3q^{21} + ⋯ ∈ S_{16}(Γ_{0}(14); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-4 newform g = q + 2q^{2} + 9q^{3} + 4q^{4} + 10q^{5} + 7q^{6} + 7q^{7} + 8q^{8} + 10q^{9} + 9q^{10} + 4q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(3)ω^{4} & 0 \\ 0 & u(4)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −128q^{2} + 7807203393452746695+O(11^{2}0)q^{3}+16384q^{4}+−1989857992595405055502+O(11^{2}1)q^{5}+−326572039429391567759+ O(11^{2}0)q^{6}+−823543q^{7}+−2097152q^{8}+−287284188935161181619+O(11^{2}0)q^{9}+3093324947434403663082+ O(11^{2}1)q^{10}+206047481755422047368∗11+O(11^{2}1)q^{11}+90721361143400102690+O(11^{2}0)q^{12}+2486283182446847126+ O(11^{2}0)q^{13}+105413504q^{14}+211963906453239825904+O(11^{2}0)q^{15}+268435456q^{16}+21209118725945449649+ O(11^{2}0)q^{17}+−228873537590169258823+O(11^{2}0)q^{18}+102562633712366418640+O(11^{2}0)q^{19}+3667903718336976590898+ O(11^{2}1)q^{20} + −96002683779363506428 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

6. Consider f = q + 4q^{2} + 2q^{3} + 5q^{4} + 6q^{5} + 8q^{6} + 5q^{7} + 9q^{8} + 3q^{9} + 2q^{10} + 10q^{12} + 6q^{13} + 9q^{14} + q^{15} + 3q^{16} + q^{18} + 2q^{19} + 8q^{20} + 10q^{21} + ⋯ ∈ S_{16}(Γ_{0}(14); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + 3q^{2} + 6q^{3} + 9q^{4} + 8q^{5} + 7q^{6} + 9q^{7} + 5q^{8} + 5q^{9} + 2q^{10} + 10q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(10)ω^{1} & 0 \\ 0 & u(10)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −128q^{2} + −7807203393452751385+O(11^{2}0)q^{3}+16384q^{4}+1989857992595404977480+O(11^{2}1)q^{5}+326572039429392168079+ O(11^{2}0)q^{6}+−823543q^{7}+−2097152q^{8}+287284188935143819873+O(11^{2}0)q^{9}+−3093324947434393676266+ O(11^{2}1)q^{10} + −206047481755431118388 ∗ 11 + O(11^{2}1)q^{11} + −90721361143476943650 + O(11^{2}0)q^{12} + −2486283182720724128+O(11^{2}0)q^{13}+105413504q^{14}+−211963906452822617840+O(11^{2}0)q^{15}+268435456q^{16}+ −1928101702300032707∗11+O(11^{2}0)q^{17}+228873537592391562311+O(11^{2}0)q^{18}+−102562633712056193718+ O(11^{2}0)q^{19} + −3667903718338254903346 + O(11^{2}1)q^{20} + 96002683783225923098 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

7. Consider f = q +7q^{2} +5q^{4} +8q^{5} +5q^{7} +2q^{8} +10q^{9} +q^{10} +4q^{13} +2q^{14} +3q^{16} +3q^{17} +4q^{18} +7q^{20} +⋯ ∈ S_{16}(Γ_{0}(14); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + 8q^{2} + 9q^{4} + 7q^{5} + 9q^{7} + 6q^{8} + 2q^{9} + q^{10} + 7q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{1} & 0 \\ 0 & u(7)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+128q^{2}+−14819577472334149740∗ 11 + O(11^{2}0)q^{3} + 16384q^{4} + 282160195088452661692 + O(11^{2}0)q^{5} + −974112557920231699 ∗ 11 + O(11^{2}0)q^{6}+−823543q^{7}+2097152q^{8}+37216725750716179755+O(11^{2}0)q^{9}+−211994755036299800278+ O(11^{2}0)q^{10} + −30396379129595511327 ∗ 11 + O(11^{2}0)q^{11} + −2368226516960564890 ∗ 11 + O(11^{2}0)q^{12} + −65621189036914420312 + O(11^{2}0)q^{13} + −105413504q^{14} + 27373917109294621881 ∗ 11 + O(11^{2}0)q^{15} + 268435456q^{16}+−325802019173778417817+O(11^{2}0)q^{17}+54490931563750944233+O(11^{2}0)q^{18}+5287040495333281188∗ 11+O(11^{2}0)q^{19}+−225328847343974067544+O(11^{2}0)q^{20}+18154955566308774606∗11+O(11^{2}0)q^{21}+⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

8. Consider f = q + 7q^{2} + 8q^{3} + 5q^{4} + 6q^{5} + q^{6} + 6q^{7} + 2q^{8} + 8q^{9} + 9q^{10} + 7q^{12} + 10q^{13} + 9q^{14} + 4q^{15} + 3q^{16} + 5q^{17} + q^{18} + 8q^{19} + 8q^{20} + 4q^{21} + ⋯ ∈ S_{16}(Γ_{0}(14); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + 8q^{2} + 2q^{3} + 9q^{4} + 8q^{5} + 5q^{6} + 2q^{7} + 6q^{8} + 6q^{9} + 9q^{10} + 6q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(6)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 128q^{2} + aq^{3} + 16384q^{4} + −(301527548504564531876 + O(11^{2}0)) ∗ a − 50786220545473251220 + O(11^{2}0)q^{5} + 128 ∗ aq^{6} + 823543q^{7} + 2097152q^{8} + −(119543894017838527485 + O(11^{2}0)) ∗ a + 242071684065802915050 + O(11^{2}0)q^{9}+−(248776497428339555671+O(11^{2}0))∗a+226863719505023935850+O(11^{2}0)q^{10}+(3316803033550209851311∗ 11+O(11^{2}2))∗a+558648426000206911838∗11+O(11^{2}2)q^{11}+16384∗aq^{12}+−(333495731664369472685+ O(11^{2}0)) ∗ a + 104725640414720122469 + O(11^{2}0)q^{13} + 105413504q^{14} + −(93379820821668550600 + O(11^{2}0))∗a−76988548011903891365+O(11^{2}0)q^{15}+268435456q^{16}+(31122373792035376489+O(11^{2}0))∗ a+31197315424436046300+O(11^{2}0)q^{17}+(171631449165548693543+O(11^{2}0))∗a+38675793525012703154+ O(11^{2}0)q^{18} + −(2950194440308259991241 + O(11^{2}1)) ∗ a − 241792668037098997593 ∗ 11 + O(11^{2}1)q^{19} + −(224141908997142693441 + O(11^{2}0)) ∗ a + 110306314542983393157 + O(11^{2}0)q^{20} + 823543 ∗ aq^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 119543894017838527485 + O(11^{2}0)x + −242071684065817263957 + O(11^{2}0). The slope of f_{0} is 1.

9. Consider f = q + 7q^{2} + 2q^{3} + 5q^{4} + 7q^{5} + 3q^{6} + 6q^{7} + 2q^{8} + 3q^{9} + 5q^{10} + 10q^{12} + 3q^{13} + 9q^{14} + 3q^{15} + 3q^{16} + 4q^{17} + 10q^{18} + 2q^{19} + 2q^{20} + q^{21} + ⋯ ∈ S_{16}(Γ_{0}(14); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-4 newform g = q + 9q^{2} + 8q^{3} + 4q^{4} + 8q^{5} + 6q^{6} + 4q^{7} + 3q^{8} + 4q^{9} + 6q^{10} + 5q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(9)ω^{4} & 0 \\ 0 & u(5)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+128q^{2}+119543894017838530297+ O(11^{2}0)q^{3} +16384q^{4} +144166041367141754726+O(11^{2}0)q^{5} +−171631449165548333607+O(11^{2}0)q^{6} + 823543q^{7} +2097152q^{8} +213110822748698693085+O(11^{2}0)q^{9} +289003431815024356501+O(11^{2}0)q^{10} + −278197395039631288811∗11+O(11^{2}1)q^{11}+231924339584293601937+O(11^{2}0)q^{12}+−223966876872820512450+ O(11^{2}0)q^{13}+105413504q^{14}+199850018668791469630+O(11^{2}0)q^{15}+268435456q^{16}+48867907097327702751+ O(11^{2}0)q^{17}+−304564480401527662361+O(11^{2}0)q^{18}+312680128874906060644+O(11^{2}0)q^{19}+−8810448967682873927+ O(11^{2}0)q^{20} + −24397303102430082868 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

10. Consider f = q + 3q^{2} + 8q^{3} + 9q^{4} + 2q^{5} + 2q^{6} + 2q^{7} + 5q^{8} + 6q^{10} + 6q^{12} + 2q^{13} + 6q^{14} + 5q^{15} + 4q^{16} + 8q^{17} + 9q^{19} + 7q^{20} + 5q^{21} + ⋯ ∈ S_{18}(Γ_{0}(14); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-6 newform g = q + 7q^{2} + 10q^{3} + 5q^{4} + 7q^{5} + 4q^{6} + 5q^{7} + 2q^{8} + 5q^{10} + 5q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(9)ω^{6} & 0 \\ 0 & u(5)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+256q^{2}+31812061040660681393+ O(11^{2}0)q^{3} + 65536q^{4} + 95436183121981591573 + O(11^{2}0)q^{5} + 70887687218414326196 + O(11^{2}0)q^{6} + −5764801q^{7} + 16777216q^{8} + −12781841416893272764 ∗ 11 + O(11^{2}0)q^{9} + 212663061655127111452 + O(11^{2}0)q^{10} + 24603590073936218861 ∗ 11 + O(11^{2}1)q^{11} + −17001935265052742251 + O(11^{2}0)q^{12} + 295100372327335945152 + O(11^{2}0)q^{13} + −1475789056q^{14} + 116641405556423788122 + O(11^{2}0)q^{15} + 4294967296q^{16} + −198563495704474277864 + O(11^{2}0)q^{17} + 30439481489707672130 ∗ 11 + O(11^{2}0)q^{18} + 16194745014267417222+O(11^{2}0)q^{19}+−51005805824820213569+O(11^{2}0)q^{20}+101819364256633126405+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

11. Consider f = q + 3q^{2} + 2q^{3} + 9q^{4} + 6q^{5} + 6q^{6} + 2q^{7} + 5q^{8} + 6q^{9} + 7q^{10} + 7q^{12} + 5q^{13} + 6q^{14} + q^{15} + 4q^{16} + 3q^{17} + 7q^{18} + 3q^{19} + 10q^{20} + 4q^{21} + ⋯ ∈ S_{18}(Γ_{0}(14); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 4q^{2} + 8q^{3} + 5q^{4} + 10q^{5} + 10q^{6} + 6q^{7} + 9q^{8} + 8q^{9} + 7q^{10} + q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+256q^{2}+−31812061040660675751+ O(11^{2}0)q^{3} + 65536q^{4} + −95436183121982479859 + O(11^{2}0)q^{5} + −70887687218412881844 + O(11^{2}0)q^{6} + −5764801q^{7}+16777216q^{8}+140600255586121906482+O(11^{2}0)q^{9}+−212663061655354512668+O(11^{2}0)q^{10}+ −24603590073914263377∗11+O(11^{2}1)q^{11}+17001935265422496363+O(11^{2}0)q^{12}+−295100372321456095554+ O(11^{2}0)q^{13}+−1475789056q^{14}+−116641405557314831962+O(11^{2}0)q^{15}+4294967296q^{16}+198563495694349299000+ O(11^{2}0)q^{17}+−334834296311032437462+O(11^{2}0)q^{18}+−16194744799470256280+O(11^{2}0)q^{19}+51005805766605502273+ O(11^{2}0)q^{20} + −101819364289158133647 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

12. Consider f = q + 8q^{2} + 8q^{3} + 9q^{4} + 2q^{5} + 9q^{6} + 9q^{7} + 6q^{8} + 5q^{10} + 6q^{12} + 9q^{13} + 6q^{14} + 5q^{15} + 4q^{16} + 3q^{17} + 2q^{19} + 7q^{20} + 6q^{21} + ⋯ ∈ S_{18}(Γ_{0}(14); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 7q^{2} + 10q^{3} + 5q^{4} + 7q^{5} + 4q^{6} + 5q^{7} + 2q^{8} + 5q^{10} + 5q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(9)ω^{1} & 0 \\ 0 & u(5)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −256q^{2} + 253815811816238495084+O(11^{2}0)q^{3}+65536q^{4}+−229649504323810436292+O(11^{2}0)q^{5}+279901683501266150993+ O(11^{2}0)q^{6}+5764801q^{7}+−16777216q^{8}+−19467233249146164759∗11+O(11^{2}0)q^{9}+261023547762750890265+ O(11^{2}0)q^{10}+23472007902006542545∗11+O(11^{2}0)q^{11}+329418481459786330299+O(11^{2}0)q^{12}+170733113515581954214+ O(11^{2}0)q^{13}+−1475789056q^{14}+−82110775435645210161+O(11^{2}0)q^{15}+4294967296q^{16}+−13371441927349415247+ O(11^{2}0)q^{17}+29725385459839928733∗11+O(11^{2}0)q^{18}+68847267374141406020+O(11^{2}0)q^{19}+−219778728940786996941+ O(11^{2}0)q^{20} + 44295492100687023334 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

13. Consider f = q + 8q^{2} + 2q^{3} + 9q^{4} + 6q^{5} + 5q^{6} + 9q^{7} + 6q^{8} + 6q^{9} + 4q^{10} + 7q^{12} + 6q^{13} + 6q^{14} + q^{15} + 4q^{16} + 8q^{17} + 4q^{18} + 8q^{19} + 10q^{20} + 7q^{21} + ⋯ ∈ S_{18}(Γ_{0}(14); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-6 newform g = q + 4q^{2} + 8q^{3} + 5q^{4} + 10q^{5} + 10q^{6} + 6q^{7} + 9q^{8} + 8q^{9} + 7q^{10} + q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{6} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −256q^{2} + 41930144096046101557+O(11^{2}0)q^{3}+65536q^{4}+331880770779098262739+O(11^{2}0)q^{5}+29883030333158148624+ O(11^{2}0)q^{6}+5764801q^{7}+−16777216q^{8}+31906908150322110923+O(11^{2}0)q^{9}+−194977957946594101858+ O(11^{2}0)q^{10} + 6936549970480912135 ∗ 11 + O(11^{2}0)q^{11} + −249805821030325946533 + O(11^{2}0)q^{12} + 298835465118840732055 + O(11^{2}0)q^{13} + −1475789056q^{14} + −218455948637300923350 + O(11^{2}0)q^{15} + 4294967296q^{16}+80359725564331409721+O(11^{2}0)q^{17}+−95168547291740285876+O(11^{2}0)q^{18}+94848546895699174047+ O(11^{2}0)q^{19} + 130857609318649394774 + O(11^{2}0)q^{20} + −136564238149004024143 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

14. Consider f = q +5q^{2} +3q^{4} +2q^{5} +8q^{7} +4q^{8} +7q^{9} +10q^{10} +6q^{13} +7q^{14} +9q^{16} +2q^{17} +2q^{18} +6q^{20} +⋯ ∈ S_{20}(Γ_{0}(14); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q + 8q^{2} + 9q^{4} + 7q^{5} + 9q^{7} + 6q^{8} + 2q^{9} + q^{10} + 7q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{8} & 0 \\ 0 & u(7)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −512q^{2} + −28733405673548700037∗11+O(11^{2}0)q^{3}+262144q^{4}+−317102298187116756587+O(11^{2}0)q^{5}+−27837093210971237187∗ 11+O(11^{2}0)q^{6}+40353607q^{7}+−134217728q^{8}+69049025165309437981+O(11^{2}0)q^{9}+223627893056817155103+ O(11^{2}0)q^{10} + 20969592249212352982 ∗ 11 + O(11^{2}0)q^{11} + 2523649536684153941 ∗ 11 + O(11^{2}0)q^{12} + −164713839026173572008+O(11^{2}0)q^{13}+−20661046784q^{14}+−18116445064347617715∗11+O(11^{2}0)q^{15}+ 68719476736q^{16} + 217349377416645842535 + O(11^{2}0)q^{17} + 302648846787248241381 + O(11^{2}0)q^{18} + 122717808346203518∗11^{2}+O(11^{2}0)q^{19}+−129982106555181848566+O(11^{2}0)q^{20}+−18033049919959533214∗ 11 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

15. Consider f = q + 5q^{2} + 7q^{3} + 3q^{4} + 10q^{5} + 2q^{6} + 3q^{7} + 4q^{8} + q^{9} + 6q^{10} + 10q^{12} + 10q^{13} + 4q^{14} + 4q^{15} + 9q^{16} + 10q^{17} + 5q^{18} + 4q^{19} + 8q^{20} + 10q^{21} + ⋯ ∈ S_{20}(Γ_{0}(14); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 9q^{2} + 8q^{3} + 4q^{4} + 8q^{5} + 6q^{6} + 4q^{7} + 3q^{8} + 4q^{9} + 6q^{10} + 5q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(9)ω^{1} & 0 \\ 0 & u(5)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −512q^{2} + −88543766322218070626+O(11^{2}0)q^{3}+262144q^{4}+−289792308788191874776+O(11^{2}0)q^{5}+260158696494131544045+ O(11^{2}0)q^{6}+−40353607q^{7}+−134217728q^{8}+317791722004941507137+O(11^{2}0)q^{9}+−304086780541522148109+ O(11^{2}0)q^{10}+220225206379366684713∗11+O(11^{2}1)q^{11}+3246391651531270758+O(11^{2}0)q^{12}+−38112882514750438759+ O(11^{2}0)q^{13}+20661046784q^{14}+226042674673868636973+O(11^{2}0)q^{15}+68719476736q^{16}+−71712117644626233798+ O(11^{2}0)q^{17}+96137107149470572498+O(11^{2}0)q^{18}+−52026293102723072624+O(11^{2}0)q^{19}+287182807837977706377+ O(11^{2}0)q^{20} + −313869559253647831545 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

16. Consider f = q + 5q^{2} + 6q^{3} + 3q^{4} + 7q^{5} + 8q^{6} + 3q^{7} + 4q^{8} + 10q^{9} + 2q^{10} + 7q^{12} + 4q^{13} + 4q^{14} + 9q^{15} + 9q^{16} + 7q^{17} + 6q^{18} + 5q^{19} + 10q^{20} + 7q^{21} + ⋯ ∈ S_{20}(Γ_{0}(14); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q + 8q^{2} + 2q^{3} + 9q^{4} + 8q^{5} + 5q^{6} + 2q^{7} + 6q^{8} + 6q^{9} + 9q^{10} + 6q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{8} & 0 \\ 0 & u(6)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −512q^{2} + aq^{3} + 262144q^{4} + −(322513104493254508046 + O(11^{2}0)) ∗ a + 244564215959920200276 + O(11^{2}0)q^{5} + −512 ∗ aq^{6} +−40353607q^{7} +−134217728q^{8} +(88543766322218101918+O(11^{2}0))∗a+27901650363128691780+ O(11^{2}0)q^{9}+(302960742069105865307+O(11^{2}0))∗a−85379514022980829926+O(11^{2}0)q^{10}+−(182922628804035616394∗ 11+O(11^{2}1))∗a−255239115454051740229∗11+O(11^{2}1)q^{11}+262144∗aq^{12}+(131754868830797288158+ O(11^{2}0)) ∗ a − 260625834194036769220 + O(11^{2}0)q^{13} + 20661046784q^{14} + (45228092828269722046 + O(11^{2}0)) ∗ a − 218578168036968076556 + O(11^{2}0)q^{15} + 68719476736q^{16} + −(39640398502677007787 + O(11^{2}0)) ∗ a + 24148215320092318013 + O(11^{2}0)q^{17} + −(260158696494147565549 + O(11^{2}0)) ∗ a − 157895092338129998139+O(11^{2}0)q^{18}+(2878242378016569516996+O(11^{2}1))∗a+212462003907491390601∗ 11 + O(11^{2}1)q^{19} + (289348890039159088247 + O(11^{2}0)) ∗ a − 14438490850215675953 + O(11^{2}0)q^{20} + −40353607 ∗ aq^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −88543766322218101918 + O(11^{2}0)x + −27901650364290953247 + O(11^{2}0). The slope of f_{0} is 1.

17. Consider f = q + 6q^{2} + 7q^{3} + 3q^{4} + 7q^{5} + 9q^{6} + 8q^{7} + 7q^{8} + q^{9} + 9q^{10} + 10q^{12} + 9q^{13} + 4q^{14} + 5q^{15} + 9q^{16} + 6q^{18} + 4q^{19} + 10q^{20} + q^{21} + ⋯ ∈ S_{20}(Γ_{0}(14); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q + 3q^{2} + 6q^{3} + 9q^{4} + 8q^{5} + 7q^{6} + 9q^{7} + 5q^{8} + 5q^{9} + 2q^{10} + 10q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(10)ω^{8} & 0 \\ 0 & u(10)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+512q^{2}+−212435844131104615608+ O(11^{2}0)q^{3}+262144q^{4}+−166350572082550052839+O(11^{2}0)q^{5}+218346983949158299266+O(11^{2}0)q^{6}+ 40353607q^{7}+134217728q^{8}+−151640659715407733573+O(11^{2}0)q^{9}+267756450169494114959+O(11^{2}0)q^{10}+ −29016309934628874733∗11+O(11^{2}0)q^{11}+117156623164087696826+O(11^{2}0)q^{12}+−30163309034502243360+ O(11^{2}0)q^{13}+20661046784q^{14}+49241021454514169087+O(11^{2}0)q^{15}+68719476736q^{16}+20458140033638085737∗ 11 + O(11^{2}0)q^{17} + −273768357044358531261 + O(11^{2}0)q^{18} + −183025940344440506826 + O(11^{2}0)q^{19} + −149696479461255017996 + O(11^{2}0)q^{20} + −100601974791207067903 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

18. Consider f = q + 6q^{2} + q^{3} + 3q^{4} + 7q^{5} + 6q^{6} + 8q^{7} + 7q^{8} + 8q^{9} + 9q^{10} + 3q^{12} + 3q^{13} + 4q^{14} + 7q^{15} + 9q^{16} + 6q^{17} + 4q^{18} + 10q^{19} + 10q^{20} + 8q^{21} + ⋯ ∈ S_{20}(Γ_{0}(14); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 2q^{2} + 9q^{3} + 4q^{4} + 10q^{5} + 7q^{6} + 7q^{7} + 8q^{8} + 10q^{9} + 9q^{10} + 4q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(3)ω^{1} & 0 \\ 0 & u(4)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+512q^{2}+98397500895388515283+ O(11^{2}0)q^{3} + 262144q^{4} + 80466606255616733555 + O(11^{2}0)q^{5} + −76729161503080865179 + O(11^{2}0)q^{6} + 40353607q^{7}+134217728q^{8}+164242079669970465545+O(11^{2}0)q^{9}+161152711989607018899+O(11^{2}0)q^{10}+ 24560838001429294949∗11+O(11^{2}0)q^{11}+−265830983488922437990+O(11^{2}0)q^{12}+−83754095754752859587+ O(11^{2}0)q^{13}+20661046784q^{14}+287074315542583532861+O(11^{2}0)q^{15}+68719476736q^{16}+313109924524578582550+ O(11^{2}0)q^{17}+−1804575545122791085+O(11^{2}0)q^{18}+−94370087850337774586+O(11^{2}0)q^{19}+−238060838026087455435+ O(11^{2}0)q^{20} + −175176380507302319203 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

19. Consider f = q + q^{2} + z^{1}12q^{3} + q^{4} + z^{9}7q^{5} + z^{1}12q^{6} + 10q^{7} + q^{8} + z^{2}q^{9} + z^{9}7q^{10} + z^{1}12q^{12} + z^{4}3q^{13} + 10q^{14} + z^{8}9q^{15} + q^{16} + z^{9}0q^{17} + z^{2}q^{18} + z^{9}1q^{19} + z^{9}7q^{20} + z^{5}2q^{21} + ⋯ ∈ S_{22}(Γ_{0}(14); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + 6q^{2} + z^{1}6q^{3} + 3q^{4} + z^{4}9q^{5} + z^{4}q^{6} + 3q^{7} + 7q^{8} + z^{5}0q^{9} + z^{3}7q^{10} + z^{9}1q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}9)ω^{0} & 0 \\ 0 & u(z^{9}1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 1024q^{2} + aq^{3} + 1048576q^{4} + −(178039985429450202359 + O(11^{2}0)) ∗ a − 248156152481185400376 + O(11^{2}0)q^{5} + 1024 ∗ aq^{6}+−282475249q^{7}+1073741824q^{8}+(306002974327305080828+O(11^{2}0))∗a+3115271135234090870+ O(11^{2}0)q^{9}+(2303546966755277855+O(11^{2}0))∗a+187597943773833492954+O(11^{2}0)q^{10}+−(1771554646678636890∗ 11+O(11^{2}0))∗a+21613591510010758474∗11+O(11^{2}0)q^{11}+1048576∗aq^{12}+(270889201092463293959+ O(11^{2}0))∗a+183042683592197536407+O(11^{2}0)q^{13}+−289254654976q^{14}+−(181670430045426612893+ O(11^{2}0))∗a+122100611589142006749+O(11^{2}0)q^{15}+1099511627776q^{16}+−(239133139984143310563+ O(11^{2}0)) ∗ a + 122187333694336029953 + O(11^{2}0)q^{17} + −(154451927412561519794 + O(11^{2}0)) ∗ a − 173712332183090995125+O(11^{2}0)q^{18}+−(36171358559449276390+O(11^{2}0))∗a+306516068180385297832+ O(11^{2}0)q^{19} + −(332167885772835513284 + O(11^{2}0)) ∗ a − 306204126306665846590 + O(11^{2}0)q^{20} + −282475249 ∗ aq^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −306002974327305080828 + O(11^{2}0)x + −3115271145694444073 + O(11^{2}0). The slope of f_{0} is 1.

20. Consider f = q + q^{2} + z^{3}2q^{3} + q^{4} + z^{1}07q^{5} + z^{3}2q^{6} + 10q^{7} + q^{8} + z^{2}2q^{9} + z^{1}07q^{10} + z^{3}2q^{12} + z^{1}13q^{13} + 10q^{14} + z^{1}9q^{15} + q^{16} + z^{3}0q^{17} + z^{2}2q^{18} + z^{4}1q^{19} + z^{1}07q^{20} + z^{9}2q^{21} + ⋯ ∈ S_{22}(Γ_{0}(14); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + 6q^{2} + z^{5}6q^{3} + 3q^{4} + z^{5}9q^{5} + z^{4}4q^{6} + 3q^{7} + 7q^{8} + z^{7}0q^{9} + z^{4}7q^{10} + z^{4}1q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{7}9)ω^{0} & 0 \\ 0 & u(z^{4}1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 1024q^{2} + aq^{3} + 1048576q^{4} + −(178039985429450202359 + O(11^{2}0)) ∗ a − 248156152481185400376 + O(11^{2}0)q^{5} + 1024 ∗ aq^{6}+−282475249q^{7}+1073741824q^{8}+(306002974327305080828+O(11^{2}0))∗a+3115271135234090870+ O(11^{2}0)q^{9}+(2303546966755277855+O(11^{2}0))∗a+187597943773833492954+O(11^{2}0)q^{10}+−(1771554646678636890∗ 11+O(11^{2}0))∗a+21613591510010758474∗11+O(11^{2}0)q^{11}+1048576∗aq^{12}+(270889201092463293959+ O(11^{2}0))∗a+183042683592197536407+O(11^{2}0)q^{13}+−289254654976q^{14}+−(181670430045426612893+ O(11^{2}0))∗a+122100611589142006749+O(11^{2}0)q^{15}+1099511627776q^{16}+−(239133139984143310563+ O(11^{2}0)) ∗ a + 122187333694336029953 + O(11^{2}0)q^{17} + −(154451927412561519794 + O(11^{2}0)) ∗ a − 173712332183090995125+O(11^{2}0)q^{18}+−(36171358559449276390+O(11^{2}0))∗a+306516068180385297832+ O(11^{2}0)q^{19} + −(332167885772835513284 + O(11^{2}0)) ∗ a − 306204126306665846590 + O(11^{2}0)q^{20} + −282475249 ∗ aq^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −306002974327305080828 + O(11^{2}0)x + −3115271145694444073 + O(11^{2}0). The slope of f_{0} is 1.

21. Consider f = q + 10q^{2} + 4q^{3} + q^{4} + 3q^{5} + 7q^{6} + q^{7} + 10q^{8} + 2q^{9} + 8q^{10} + 4q^{12} + 5q^{13} + 10q^{14} + q^{15} + q^{16} + 5q^{17} + 9q^{18} + 6q^{19} + 3q^{20} + 4q^{21} + ⋯ ∈ S_{22}(Γ_{0}(14); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + 5q^{2} + 5q^{3} + 3q^{4} + 5q^{5} + 3q^{6} + 8q^{7} + 4q^{8} + 10q^{9} + 3q^{10} + 6q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{0} & 0 \\ 0 & u(6)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −1024q^{2} + −243127244563705540012+O(11^{2}0)q^{3}+1048576q^{4}+−59655181185898158231+O(11^{2}0)q^{5}+44800308187269567918+ O(11^{2}0)q^{6}+282475249q^{7}+−1073741824q^{8}+231597047784313100639+O(11^{2}0)q^{9}+−133344004503246808747+ O(11^{2}0)q^{10} + −339997677635748457 ∗ 11 + O(11^{1}9)q^{11} + −128515928349956922364 + O(11^{2}0)q^{12} + 134558513680610573955+O(11^{2}0)q^{13} +−289254654976q^{14} +−144129109102847490434+O(11^{2}0)q^{15} + 1099511627776q^{16} + 144359684245610315575 + O(11^{2}0)q^{17} + 325371280057068193617 + O(11^{2}0)q^{18} + 31196334320133697908+O(11^{2}0)q^{19}+−23988359984949710875+O(11^{2}0)q^{20}+−13624314745968555966+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

22. Consider f = q + 10q^{2} + 9q^{3} + q^{4} + 2q^{6} + q^{7} + 10q^{8} + q^{9} + 9q^{12} + 7q^{13} + 10q^{14} + q^{16} + 6q^{17} + 10q^{18} + 2q^{19} + 9q^{21} + ⋯ ∈ S_{22}(Γ_{0}(14); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{10} ∼ ρ_{g}, where g is the weight-2 newform g = q + 10q^{2} + 9q^{3} + q^{4} + 2q^{6} + q^{7} + 10q^{8} + q^{9} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{1}. The form f_{0} = q + −1024q^{2} + aq^{3} + 1048576q^{4}+(82522415730337645153+O(11^{2}0))∗a+58771318789010497346+O(11^{2}0)q^{5}+−1024∗aq^{6}+ 282475249q^{7} + −1073741824q^{8} + (243127244563705387376 + O(11^{2}0)) ∗ a + 176033703029438919702 + O(11^{2}0)q^{9}+(263545653636812522654+O(11^{2}0))∗a−307080890948908463415+O(11^{2}0)q^{10}+−(236775406310654129615∗ 11^{−}1+O(11^{1}9))∗a−244195287253577509858∗11^{−}1+O(11^{1}9)q^{11}+1048576∗aq^{12}+−(20234923239646134248+ O(11^{2}0)) ∗ a − 13555505478385604938 ∗ 11 + O(11^{2}0)q^{13} + −289254654976q^{14} + (883862396903721559 + O(11^{2}0)) ∗ a + 43495215476657951280 + O(11^{2}0)q^{15} + 1099511627776q^{16} + (156656221204214568343 + O(11^{2}0)) ∗ a + 280995984318022920240 + O(11^{2}0)q^{17} + −(44800308187113268654 + O(11^{2}0)) ∗ a + 38486739780628691020+O(11^{2}0)q^{18} +(323304623022660072+O(11^{2}0))∗a−240349055759379841301+ O(11^{2}0)q^{19}+−(98001356139459508095+O(11^{2}0))∗a+276584698176742240093+O(11^{2}0)q^{20}+282475249∗ aq^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −243127244563705387376 + O(11^{2}0)x + −176033703039899272905 + O(11^{2}0). The slope of f_{0} is 1/2.

—————————————————————————————————- (Uninteresting cases below.)

23. Consider f = q+9q2 +8q3 +4q4 +8q5 +6q6 +4q7 +3q8 +4q9 +6q10 +5q11 +10q12 +7q13 +3q14 +9q15 +5q16 +8q17 +3q18 +3q19 +10q20 +10q21 +⋯ ∈ S14(Γ0(14); \overline{F}11). Since f is ordinary, we do not continue our analysis.

24. Consider f = q+2q2 +9q3 +4q4 +10q5 +7q6 +7q7 +8q8 +10q9 +9q10 +4q11 +3q12 +q13 +3q14 +2q15 +5q16 +7q17 +9q18 +2q19 +7q20 +8q21 +⋯ ∈ S14(Γ0(14); \overline{F}11). Since f is ordinary, we do not continue our analysis.

25. Consider f = q+4q2 +8q3 +5q4 +10q5 +10q6 +6q7 +9q8 +8q9 +7q10 +q11 +7q12 +3q13 +2q14 +3q15 +3q16 +5q17 +10q18 +q19 +6q20 +4q21 +⋯ ∈ S16(Γ0(14); \overline{F}11). Since f is ordinary, we do not continue our analysis.

26. Consider f = q + 7q2 + 10q3 + 5q4 + 7q5 + 4q6 + 5q7 + 2q8 + 5q10 + 5q11 + 6q12 + q13 + 2q14 + 4q15 + 3q16 + 5q17 + 8q19 + 2q20 + 6q21 + ⋯ ∈ S16(Γ0(14); \overline{F}11). Since f is ordinary, we do not continue our analysis.

27. Consider f = q +3q2 +6q3 +9q4 +8q5 +7q6 +9q7 +5q8 +5q9 +2q10 +10q11 +10q12 +10q13 +5q14 +4q15 +4q16 +4q18 +6q19 +6q20 +10q21 +⋯ ∈ S18(Γ0(14); \overline{F}11). Since f is ordinary, we do not continue our analysis.

28. Consider f = q+8q2 +2q3 +9q4 +8q5 +5q6 +2q7 +6q8 +6q9 +9q10 +6q11 +7q12 +2q13 +5q14 +5q15 +4q16 +3q17 +4q18 +2q19 +6q20 +4q21 +⋯ ∈ S18(Γ0(14); \overline{F}11). Since f is ordinary, we do not continue our analysis.

29. Consider f = q + 8q2 + 9q4 + 7q5 + 9q7 + 6q8 + 2q9 + q10 + 7q11 + 3q13 + 6q14 + 4q16 + 4q17 + 5q18 + 8q20 + ⋯ ∈ S18(Γ0(14); \overline{F}11). Since f is ordinary, we do not continue our analysis.

30. Consider f = q+5q2 +5q3 +3q4 +5q5 +3q6 +8q7 +4q8 +10q9 +3q10 +6q11 +4q12 +8q13 +7q14 +3q15 +9q16 +10q17 +6q18 +9q19 +4q20 +7q21 +⋯ ∈ S20(Γ0(14); \overline{F}11). Since f is ordinary, we do not continue our analysis.

31. Consider f = q + 6q2 + z16q3 + 3q4 + z49q5 + z4q6 + 3q7 + 7q8 + z50q9 + z37q10 + z91q11 + z112q12 + z31q13 + 7q14 + z65q15 + 9q16 + z102q17 + z38q18 + z55q19 + z25q20 + z112q21 + ⋯ ∈ S20(Γ0(14); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

32. Consider f = q + 6q2 + z56q3 + 3q4 + z59q5 + z44q6 + 3q7 + 7q8 + z70q9 + z47q10 + z41q11 + z32q12 + z101q13 + 7q14 + z115q15 + 9q16 + z42q17 + z58q18 + z5q19 + z35q20 + z32q21 + ⋯ ∈ S20(Γ0(14); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

33. Consider f = q+6q2 +5q3 +3q4 +10q5 +8q6 +8q7 +7q8 +10q9 +5q10 +q11 +4q12 +7q13 +4q14 +6q15 +9q16 +3q17 +5q18 +8q19 +8q20 +7q21 +⋯ ∈ S20(Γ0(14); \overline{F}11). Since f is ordinary, we do not continue our analysis.

34. Consider f = q + 10q2 + q4 + 7q5 + 10q7 + 10q8 + 8q9 + 4q10 + 10q11 + 2q13 + q14 + q16 + 7q17 + 3q18 + 5q19 + 7q20 + ⋯ ∈ S22(Γ0(14); \overline{F}11). Since f is ordinary, we do not continue our analysis.

35. Consider f = q + 10q2 + 2q3 + q4 + 2q5 + 9q6 + 10q7 + 10q8 + q9 + 9q10 + q11 + 2q12 + 7q13 + q14 + 4q15 + q16 + 10q18 + 4q19 + 2q20 + 9q21 + ⋯ ∈ S22(Γ0(14); \overline{F}11). Since f is ordinary, we do not continue our analysis.

36. Consider f = q +q2 +6q3 +q4 +5q5 +6q6 +q7 +q8 +5q10 +q11 +6q12 +3q13 +q14 +8q15 +q16 +q17 +10q19 +5q20 +6q21 +⋯ ∈ S22(Γ0(14); \overline{F}11). Since f is ordinary, we do not continue our analysis.

37. Consider f = q + q2 + 3q3 + q4 + 8q5 + 3q6 + q7 + q8 + 6q9 + 8q10 + q11 + 3q12 + 6q13 + q14 + 2q15 + q16 + 6q17 + 6q18 + 2q19 + 8q20 + 3q21 + ⋯ ∈ S22(Γ0(14); \overline{F}11). Since f is ordinary, we do not continue our analysis.

38. Consider f = q + q2 + q4 + 2q5 + 10q7 + q8 + 8q9 + 2q10 + 10q11 + 2q13 + 10q14 + q16 + 2q17 + 8q18 + 2q20 + ⋯ ∈ S22(Γ0(14); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

11.13 N = 15, ℓ = 11 (55 forms)

1. Consider f = q + 8q^{3} + 3q^{4} + 6q^{5} + 2q^{7} + 9q^{9} + 2q^{12} + 4q^{15} + 9q^{16} + 6q^{17} + q^{19} + 7q^{20} + 5q^{21} + ⋯ ∈ S_{14}(Γ_{0}(15); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 7q^{3} + 5q^{4} + 2q^{5} + 7q^{7} + 5q^{9} + 4q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(3)ω^{1} & 0 \\ 0 & u(4)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−9751629919976711636∗ 11+O(11^{2}0)q^{2}+−729q^{3}+−156251017811293567302+O(11^{2}0)q^{4}+−15625q^{5}+14483719646935412888∗ 11 + O(11^{2}0)q^{6} + −91184784065611439971 + O(11^{2}0)q^{7} + −683186076967310296 ∗ 11 + O(11^{2}0)q^{8} + 531441q^{9} + 21923192635484501619 ∗ 11 + O(11^{2}0)q^{10} + −16230108369330856444 ∗ 11 + O(11^{2}0)q^{11} + 212242840830369008189+O(11^{2}0)q^{12}+22275811034683964292∗11+O(11^{2}0)q^{13}+22127209722227597018∗ 11 + O(11^{2}0)q^{14} + 11390625q^{15} + −296819141045490644759 + O(11^{2}0)q^{16} + 189957442124460364030 + O(11^{2}0)q^{17} + 21891024959800512991 ∗ 11 + O(11^{2}0)q^{18} + −139074851511684630068 + O(11^{2}0)q^{19} + 12421691201715703321 + O(11^{2}0)q^{20} + −128541914492701172040 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + z^{3}7q^{2} + 3q^{3} + z^{9}3q^{4} + 6q^{5} + z^{1}3q^{6} + z^{7}1q^{7} + 7q^{8} + 9q^{9} + z^{2}5q^{10} + z^{6}9q^{12} + z^{4}9q^{13} + 6q^{14} + 7q^{15} + z^{8}7q^{16} + 3q^{17} + z^{1}09q^{18} + 3q^{19} + z^{8}1q^{20} + z^{4}7q^{21} + ⋯ ∈ S_{14}(Γ_{0}(15); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{1}3q^{2} + 4q^{3} + z^{4}5q^{4} + 2q^{5} + z^{3}7q^{6} + z^{2}3q^{7} + 2q^{8} + 5q^{9} + z^{2}5q^{10} + z^{5}5q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{6}5)ω^{1} & 0 \\ 0 & u(z^{5}5)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+729q^{3}+131∗a− 5460q^{4} +−15625q^{5} +729∗aq^{6} +1624∗a+141764q^{7} +29∗11^{2} ∗a+357892q^{8} +531441q^{9} +−15625∗aq^{10} + 3232∗11∗a+72208∗11q^{11}+95499∗a−3980340q^{12}+3016∗a+62094∗11q^{13}+32228∗11∗a+4436768q^{14}+ −11390625q^{15} + −255581 ∗ a + 54314908q^{16} + −55400 ∗ 11 ∗ a + 63991502q^{17} + 531441 ∗ aq^{18} + −378808 ∗ 11 ∗ a + 292168836q^{19} + −2046875 ∗ a + 85312500q^{20} + 1183896 ∗ a + 103345956q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} +−131x+−2732. The slope of f_{0} is 1.

3. Consider f = q + z^{4}7q^{2} + 3q^{3} + z^{6}3q^{4} + 6q^{5} + z^{2}3q^{6} + z^{6}1q^{7} + 7q^{8} + 9q^{9} + z^{3}5q^{10} + z^{3}9q^{12} + z^{5}9q^{13} + 6q^{14} + 7q^{15} + z^{1}17q^{16} + 3q^{17} + z^{1}19q^{18} + 3q^{19} + z^{5}1q^{20} + z^{3}7q^{21} + ⋯ ∈ S_{14}(Γ_{0}(15); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{2}3q^{2} + 4q^{3} + z^{1}5q^{4} + 2q^{5} + z^{4}7q^{6} + z^{1}3q^{7} + 2q^{8} + 5q^{9} + z^{3}5q^{10} + z^{5}q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}15)ω^{1} & 0 \\ 0 & u(z^{5})ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+729q^{3}+131∗a− 5460q^{4} +−15625q^{5} +729∗aq^{6} +1624∗a+141764q^{7} +29∗11^{2} ∗a+357892q^{8} +531441q^{9} +−15625∗aq^{10} + 3232∗11∗a+72208∗11q^{11}+95499∗a−3980340q^{12}+3016∗a+62094∗11q^{13}+32228∗11∗a+4436768q^{14}+ −11390625q^{15} + −255581 ∗ a + 54314908q^{16} + −55400 ∗ 11 ∗ a + 63991502q^{17} + 531441 ∗ aq^{18} + −378808 ∗ 11 ∗ a + 292168836q^{19} + −2046875 ∗ a + 85312500q^{20} + 1183896 ∗ a + 103345956q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} +−131x+−2732. The slope of f_{0} is 1.

4. Consider f = q+9q^{2}+8q^{3}+7q^{4}+5q^{5}+6q^{6}+2q^{8}+9q^{9}+q^{10}+q^{12}+7q^{13}+7q^{15}+6q^{16}+q^{17}+4q^{18}+10q^{19}+ 2q^{20} + ⋯ ∈ S_{14}(Γ_{0}(15); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-2 newform g = q + 10q^{2} + 10q^{3} + 10q^{4} + q^{5} + q^{6} + 3q^{8} + q^{9} + 10q^{10} + 7q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{2} & 0 \\ 0 & u(7)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−16435229468117994987+ O(11^{2}0)q^{2} + −729q^{3} + 321336111263610465447 + O(11^{2}0)q^{4} + 15625q^{5} + −128217626528061820095 + O(11^{2}0)q^{6}+6527467962427583892∗11+O(11^{2}0)q^{7}+−305467132190083741634+O(11^{2}0)q^{8}+531441q^{9}+ 190037624894251842907+O(11^{2}0)q^{10}+−176088697983489315312∗11+O(11^{2}1)q^{11}+−137026874641146108915+ O(11^{2}0)q^{12} + 302802986877339314021 + O(11^{2}0)q^{13} + −30539046920140088276 ∗ 11 + O(11^{2}0)q^{14} + −11390625q^{15} + 293540100060264551599 + O(11^{2}0)q^{16} + −175657480920550390926 + O(11^{2}0)q^{17} + −41599556668774429684+O(11^{2}0)q^{18}+−98819649541076636833+O(11^{2}0)q^{19}+143526312218173942312+ O(11^{2}0)q^{20} + 11884910366625953430 ∗ 11 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

5. Consider f = q + z^{7}3q^{2} + 8q^{3} + z^{1}03q^{4} + 5q^{5} + z^{1}09q^{6} + z^{2}7q^{7} + z^{1}01q^{8} + 9q^{9} + zq^{10} + z^{1}9q^{12} + z^{8}5q^{13} + z^{1}00q^{14} + 7q^{15} + z^{3}8q^{16} + z^{5}9q^{17} + z^{2}5q^{18} + z^{3}q^{19} + z^{3}1q^{20} + z^{6}3q^{21} + ⋯ ∈ S_{14}(Γ_{0}(15); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{4}9q^{2} + 7q^{3} + z^{5}5q^{4} + 9q^{5} + z^{1}3q^{6} + z^{9}9q^{7} + z^{2}9q^{8} + 5q^{9} + zq^{10} + z^{4}4q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{7}6)ω^{1} & 0 \\ 0 & u(z^{4}4)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + −729q^{3} + (16435229468117994922 + O(11^{2}0)) ∗ a + 74203809231499201161 + O(11^{2}0)q^{4} + 15625q^{5} + −729 ∗ aq^{6} + (179065013231154110692+O(11^{2}0))∗a−146869377507410534228+O(11^{2}0)q^{7}+(277210074437450350126+ O(11^{2}0))∗a−113997635519528443410+O(11^{2}0)q^{8}+531441q^{9}+15625∗aq^{10}+(146822651952784174193∗ 11+O(11^{2}1))∗a+316478084322265966920∗11+O(11^{2}1)q^{11}+(128217626528061867480+O(11^{2}0))∗a− 274577335158116910289+O(11^{2}0)q^{12}+(602555301763172103865+O(11^{2}1))∗a−38857563297046352189∗ 11 + O(11^{2}1)q^{13} + (75067229920706614024 + O(11^{2}0)) ∗ a − 201912429175003893001 + O(11^{2}0)q^{14} + −11390625q^{15} + (332063951398997197529 + O(11^{2}0)) ∗ a − 82162044654808286690 + O(11^{2}0)q^{16} + (190979793642920608565 + O(11^{2}0)) ∗ a − 76308082936306932757 + O(11^{2}0)q^{17} + 531441 ∗ aq^{18} + −(85611631633033831717+O(11^{2}0))∗a−220973960899392707388+O(11^{2}0)q^{19}+−(190037624894252858532+ O(11^{2}0)) ∗ a + 286277973374122287302 + O(11^{2}0)q^{20} + −(24895628594704909474 + O(11^{2}0)) ∗ a + 100527008625237989253+O(11^{2}0)q^{21} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2}+−16435229468117994922+O(11^{2}0)x+−74203809231499209353+ O(11^{2}0). The slope of f_{0} is 1.

6. Consider f = q + z^{8}3q^{2} + 8q^{3} + z^{5}3q^{4} + 5q^{5} + z^{1}19q^{6} + z^{5}7q^{7} + z^{3}1q^{8} + 9q^{9} + z^{1}1q^{10} + z^{8}9q^{12} + z^{9}5q^{13} + z^{2}0q^{14} + 7q^{15} + z^{5}8q^{16} + z^{4}9q^{17} + z^{3}5q^{18} + z^{3}3q^{19} + z^{1}01q^{20} + z^{9}3q^{21} + ⋯ ∈ S_{14}(Γ_{0}(15); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{5}9q^{2} + 7q^{3} + z^{5}q^{4} + 9q^{5} + z^{2}3q^{6} + z^{9}q^{7} + z^{7}9q^{8} + 5q^{9} + z^{1}1q^{10} + z^{4}q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}16)ω^{1} & 0 \\ 0 & u(z^{4})ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + −729q^{3} + (16435229468117994922 + O(11^{2}0)) ∗ a + 74203809231499201161 + O(11^{2}0)q^{4} + 15625q^{5} + −729 ∗ aq^{6} + (179065013231154110692+O(11^{2}0))∗a−146869377507410534228+O(11^{2}0)q^{7}+(277210074437450350126+ O(11^{2}0))∗a−113997635519528443410+O(11^{2}0)q^{8}+531441q^{9}+15625∗aq^{10}+(146822651952784174193∗ 11+O(11^{2}1))∗a+316478084322265966920∗11+O(11^{2}1)q^{11}+(128217626528061867480+O(11^{2}0))∗a− 274577335158116910289+O(11^{2}0)q^{12}+(602555301763172103865+O(11^{2}1))∗a−38857563297046352189∗ 11 + O(11^{2}1)q^{13} + (75067229920706614024 + O(11^{2}0)) ∗ a − 201912429175003893001 + O(11^{2}0)q^{14} + −11390625q^{15} + (332063951398997197529 + O(11^{2}0)) ∗ a − 82162044654808286690 + O(11^{2}0)q^{16} + (190979793642920608565 + O(11^{2}0)) ∗ a − 76308082936306932757 + O(11^{2}0)q^{17} + 531441 ∗ aq^{18} + −(85611631633033831717+O(11^{2}0))∗a−220973960899392707388+O(11^{2}0)q^{19}+−(190037624894252858532+ O(11^{2}0)) ∗ a + 286277973374122287302 + O(11^{2}0)q^{20} + −(24895628594704909474 + O(11^{2}0)) ∗ a + 100527008625237989253+O(11^{2}0)q^{21} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2}+−16435229468117994922+O(11^{2}0)x+−74203809231499209353+ O(11^{2}0). The slope of f_{0} is 1.

7. Consider f = q + 9q^{3} + q^{4} + 8q^{5} + 5q^{7} + 4q^{9} + 9q^{12} + 3q^{13} + 6q^{15} + q^{16} + 4q^{17} + 8q^{19} + 8q^{20} + q^{21} + ⋯ ∈ S_{16}(Γ_{0}(15); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + 5q^{3} + 4q^{4} + 7q^{5} + 9q^{7} + 3q^{9} + 4q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(3)ω^{1} & 0 \\ 0 & u(4)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+260488084415886969∗ 11^{2} +O(11^{2}0)q^{2} +2187q^{3} +−270761233335006997175+O(11^{2}0)q^{4} +−78125q^{5} +2575874641337190141∗ 11^{2} + O(11^{2}0)q^{6} + −231162659124525621202 + O(11^{2}0)q^{7} + 1906158953185967235 ∗ 11^{2} + O(11^{2}0)q^{8} + 4782969q^{9} + −1334227609602232665 ∗ 11^{2} + O(11^{2}0)q^{10} + 29835902138823472635 ∗ 11 + O(11^{2}0)q^{11} + −134821763007494724845+O(11^{2}0)q^{12}+−178454116099437923511+O(11^{2}0)q^{13}+−708594907426405441∗ 11^{2}+O(11^{2}0)q^{14}+−170859375q^{15}+−139825144528303425047+O(11^{2}0)q^{16}+−2261985481387611453129+ O(11^{2}1)q^{17} + 1241602036804348114 ∗ 11^{2} + O(11^{2}0)q^{18} + −315186207215027237290 + O(11^{2}0)q^{19} + −56736367062715010168 + O(11^{2}0)q^{20} + −317489310984966658823 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

8. Consider f = q + 6q^{2} + 2q^{3} + 4q^{4} + 8q^{5} + q^{6} + 8q^{7} + 8q^{8} + 4q^{9} + 4q^{10} + 8q^{12} + 5q^{13} + 4q^{14} + 5q^{15} + 8q^{16} + 5q^{17} + 2q^{18} + 9q^{19} + 10q^{20} + 5q^{21} + ⋯ ∈ S_{16}(Γ_{0}(15); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-4 newform g = q + 3q^{2} + 8q^{3} + q^{4} + 6q^{5} + 2q^{6} + 9q^{7} + q^{8} + 9q^{9} + 7q^{10} + 9q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{4} & 0 \\ 0 & u(9)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−133846224224865215552+ O(11^{2}0)q^{2} + −2187q^{3} + −45817125007865263246 + O(11^{2}0)q^{4} + −78125q^{5} + 75444584116622409789 + O(11^{2}0)q^{6}+−312623324206051918513+O(11^{2}0)q^{7}+−210129644913620890175+O(11^{2}0)q^{8}+4782969q^{9}+ 183096330814741988857+O(11^{2}0)q^{10}+−29007296767092437525∗11+O(11^{2}0)q^{11}+−37696852750110651947+ O(11^{2}0)q^{12}+290743075577355649704+O(11^{2}0)q^{13}+119691740227359379347+O(11^{2}0)q^{14}+170859375q^{15}+ −81173452332401553215+O(11^{2}0)q^{16}+−318966492302716197792+O(11^{2}0)q^{17}+−173556704576007954298+ O(11^{2}0)q^{18}+−177831421681845354469+O(11^{2}0)q^{19}+−239831796678117864771+O(11^{2}0)q^{20}+193215187154576439715+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

9. Consider f = q + q^{2} + 2q^{3} + 2q^{4} + 8q^{5} + 2q^{6} + 4q^{7} + 3q^{8} + 4q^{9} + 8q^{10} + 4q^{12} + 4q^{13} + 4q^{14} + 5q^{15} + 5q^{16} + q^{17} + 4q^{18} + 10q^{19} + 5q^{20} + 8q^{21} + ⋯ ∈ S_{16}(Γ_{0}(15); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + 9q^{2} + 6q^{3} + 8q^{4} + 7q^{5} + 10q^{6} + 5q^{7} + 9q^{8} + 3q^{9} + 8q^{10} + 7q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{1} & 0 \\ 0 & u(7)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+140340332360087566829+ O(11^{2}0)q^{2}+−2187q^{3}+−286498081121202048646+O(11^{2}0)q^{4}+−78125q^{5}+−150309182264144459367+ O(11^{2}0)q^{6}+306181955836661377334+O(11^{2}0)q^{7}+−106880417729400798029+O(11^{2}0)q^{8}+4782969q^{9}+ −281798215910688566928+O(11^{2}0)q^{10}+16372805778670364971∗11+O(11^{2}0)q^{11}+241058129855511822671+ O(11^{2}0)q^{12}+305284089591231202478+O(11^{2}0)q^{13}+101758978916901986387+O(11^{2}0)q^{14}+170859375q^{15}+ 72214326856386649324+O(11^{2}0)q^{16}+−48988938352072850966+O(11^{2}0)q^{17}+−248565910337911863660+ O(11^{2}0)q^{18}+331434853145305979387+O(11^{2}0)q^{19}+270256187638544351480+O(11^{2}0)q^{20}+−233692456881223074463+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

10. Consider f = q + z^{1}12q^{2} + 9q^{3} + z^{2}1q^{4} + 3q^{5} + z^{6}4q^{6} + z^{7}1q^{7} + z^{8}2q^{8} + 4q^{9} + z^{8}8q^{10} + z^{9}3q^{12} + z^{1}0q^{13} + z^{6}3q^{14} + 5q^{15} + z^{6}4q^{16} + z^{1}0q^{17} + z^{1}6q^{18} + z^{6}8q^{19} + z^{1}17q^{20} + z^{2}3q^{21} + ⋯ ∈ S_{16}(Γ_{0}(15); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{6}4q^{2} + 5q^{3} + z^{4}5q^{4} + 4q^{5} + z^{1}12q^{6} + z^{9}5q^{7} + z^{5}8q^{8} + 3q^{9} + z^{8}8q^{10} + z^{9}1q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}9)ω^{1} & 0 \\ 0 & u(z^{9}1)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + 2187q^{3} + −(1173233073618776421+O(11^{2}0))∗a+170130902257686544534+O(11^{2}0)q^{4} +78125q^{5} +2187∗aq^{6} + −(14078796883425318632+O(11^{2}0))∗a+23320842294559336013+O(11^{2}0)q^{7}+−(224087517061186304777+ O(11^{2}0))∗a+203002574455249602762+O(11^{2}0)q^{8}+4782969q^{9}+78125∗aq^{10}+(272190073079556157872∗ 11+O(11^{2}1))∗a+221880377237764170119∗11+O(11^{2}1)q^{11}+(125139247725976004077+O(11^{2}0))∗a+ 45536039854787807705+O(11^{2}0)q^{12}+−(126998270124211985643+O(11^{2}0))∗a+230983493093805052022+ O(11^{2}0)q^{13}+−(162591953484007680519+O(11^{2}0))∗a+38203319611966739580+O(11^{2}0)q^{14}+170859375q^{15}+ −(1941196452709988715100+O(11^{2}1))∗a+18877802782488392113∗11+O(11^{2}1)q^{16}+−(246406075522738024543+ O(11^{2}0)) ∗ a + 20770961843675563051 + O(11^{2}0)q^{17} + 4782969 ∗ aq^{18} + −(183763481661406868834 + O(11^{2}0)) ∗ a − 97056046915720777000 + O(11^{2}0)q^{19} + −(164834565638746639289 + O(11^{2}0)) ∗ a − 44911000826810065407+O(11^{2}0)q^{20}+(156170982846588575062+O(11^{2}0))∗a−126317516673292838845+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 1173233073618776421 + O(11^{2}0)x + −170130902257686577302 + O(11^{2}0). The slope of f_{0} is 1.

11. Consider f = q + 2q^{2} + 9q^{3} + 5q^{4} + 3q^{5} + 7q^{6} + 8q^{7} + q^{8} + 4q^{9} + 6q^{10} + q^{12} + 5q^{14} + 5q^{15} + 7q^{16} + 4q^{17} + 8q^{18} + 5q^{19} + 4q^{20} + 6q^{21} + ⋯ ∈ S_{16}(Γ_{0}(15); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-4 newform g = q + q^{2} + 3q^{3} + 4q^{4} + 5q^{5} + 3q^{6} + 9q^{7} + 7q^{8} + 9q^{9} + 5q^{10} + 8q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(7)ω^{4} & 0 \\ 0 & u(8)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+1173233073618776612+ O(11^{2}0)q^{2} + 2187q^{3} + 53956614803499779992 + O(11^{2}0)q^{4} + 78125q^{5} + −125139247725975586360 + O(11^{2}0)q^{6}+139271111189451160554+O(11^{2}0)q^{7}+−182304461358787128305+O(11^{2}0)q^{8}+4782969q^{9}+ 164834565638761561164+O(11^{2}0)q^{10}+−288011801271806825827∗11+O(11^{2}1)q^{11}+271867462056017232329+ O(11^{2}0)q^{12} + −14371612738478646004 ∗ 11 + O(11^{2}0)q^{13} + −96327995460906973374 + O(11^{2}0)q^{14} + 170859375q^{15}+−56075959961317515454+O(11^{2}0)q^{16}+327371976706545664227+O(11^{2}0)q^{17}+129713160843316375487+ O(11^{2}0)q^{18}+12395119427673751554+O(11^{2}0)q^{19}+−90936724000705778466+O(11^{2}0)q^{20}+−169827533119996036455+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

12. Consider f = q + z^{3}2q^{2} + 9q^{3} + z^{1}11q^{4} + 3q^{5} + z^{1}04q^{6} + z^{6}1q^{7} + z^{6}2q^{8} + 4q^{9} + z^{8}q^{10} + z^{6}3q^{12} + z^{1}10q^{13} + z^{9}3q^{14} + 5q^{15} + z^{1}04q^{16} + z^{1}10q^{17} + z^{5}6q^{18} + z^{2}8q^{19} + z^{8}7q^{20} + z^{1}3q^{21} + ⋯ ∈ S_{16}(Γ_{0}(15); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{1}04q^{2} + 5q^{3} + z^{1}5q^{4} + 4q^{5} + z^{3}2q^{6} + z^{8}5q^{7} + z^{3}8q^{8} + 3q^{9} + z^{8}q^{10} + z^{4}1q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{7}9)ω^{1} & 0 \\ 0 & u(z^{4}1)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + 2187q^{3} + −(1173233073618776421+O(11^{2}0))∗a+170130902257686544534+O(11^{2}0)q^{4} +78125q^{5} +2187∗aq^{6} + −(14078796883425318632+O(11^{2}0))∗a+23320842294559336013+O(11^{2}0)q^{7}+−(224087517061186304777+ O(11^{2}0))∗a+203002574455249602762+O(11^{2}0)q^{8}+4782969q^{9}+78125∗aq^{10}+(272190073079556157872∗ 11+O(11^{2}1))∗a+221880377237764170119∗11+O(11^{2}1)q^{11}+(125139247725976004077+O(11^{2}0))∗a+ 45536039854787807705+O(11^{2}0)q^{12}+−(126998270124211985643+O(11^{2}0))∗a+230983493093805052022+ O(11^{2}0)q^{13}+−(162591953484007680519+O(11^{2}0))∗a+38203319611966739580+O(11^{2}0)q^{14}+170859375q^{15}+ −(1941196452709988715100+O(11^{2}1))∗a+18877802782488392113∗11+O(11^{2}1)q^{16}+−(246406075522738024543+ O(11^{2}0)) ∗ a + 20770961843675563051 + O(11^{2}0)q^{17} + 4782969 ∗ aq^{18} + −(183763481661406868834 + O(11^{2}0)) ∗ a − 97056046915720777000 + O(11^{2}0)q^{19} + −(164834565638746639289 + O(11^{2}0)) ∗ a − 44911000826810065407+O(11^{2}0)q^{20}+(156170982846588575062+O(11^{2}0))∗a−126317516673292838845+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 1173233073618776421 + O(11^{2}0)x + −170130902257686577302 + O(11^{2}0). The slope of f_{0} is 1.

13. Consider f = q + 8q^{2} + 5q^{3} + 2q^{4} + 7q^{5} + 7q^{6} + 4q^{7} + 4q^{8} + 3q^{9} + q^{10} + 10q^{12} + q^{13} + 10q^{14} + 2q^{15} + 7q^{16} + 9q^{17} + 2q^{18} + 3q^{19} + 3q^{20} + 9q^{21} + ⋯ ∈ S_{18}(Γ_{0}(15); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 7q^{2} + 9q^{3} + 6q^{4} + 8q^{5} + 8q^{6} + q^{7} + 5q^{8} + 4q^{9} + q^{10} + 2q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{1} & 0 \\ 0 & u(2)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+144091499507995928776+ O(11^{2}0)q^{2} + 6561q^{3} + 109476704870136626651 + O(11^{2}0)q^{4} + −390625q^{5} + 170585391714475771931 + O(11^{2}0)q^{6} +58429780546332325607+O(11^{2}0)q^{7} +250147478686674133613+O(11^{2}0)q^{8} +43046721q^{9} + −113669278276508323335+O(11^{2}0)q^{10}+−8007983966730741025∗11+O(11^{2}0)q^{11}+−220333935007682369457+ O(11^{2}0)q^{12}+−131611622561919349299+O(11^{2}0)q^{13}+−16950338279492379015+O(11^{2}0)q^{14}+−2562890625q^{15}+ −124372889088372345832+O(11^{2}0)q^{16}+−296235042838219826662+O(11^{2}0)q^{17}+−245236529104315671173+ O(11^{2}0)q^{18}+−311591404790892123675+O(11^{2}0)q^{19}+−311662014010240676109+O(11^{2}0)q^{20}+−109706947072816937043+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

14. Consider f = q + 3q^{2} + 5q^{3} + 2q^{4} + 7q^{5} + 4q^{6} + 7q^{7} + 7q^{8} + 3q^{9} + 10q^{10} + 10q^{12} + 10q^{13} + 10q^{14} + 2q^{15} + 7q^{16} + 2q^{17} + 9q^{18} + 8q^{19} + 3q^{20} + 2q^{21} + ⋯ ∈ S_{18}(Γ_{0}(15); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-6 newform g = q + 7q^{2} + 9q^{3} + 6q^{4} + 8q^{5} + 8q^{6} + q^{7} + 5q^{8} + 4q^{9} + q^{10} + 2q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{6} & 0 \\ 0 & u(2)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−83056145758732442722+ O(11^{2}0)q^{2} + 6561q^{3} + 157074771589844060988 + O(11^{2}0)q^{4} + −390625q^{5} + −3876427669949246232 + O(11^{2}0)q^{6}+−186594992576883609294+O(11^{2}0)q^{7}+−320166393848196123529+O(11^{2}0)q^{8}+43046721q^{9}+ −234318612778565446176+O(11^{2}0)q^{10}+−16828793367704097809∗11+O(11^{2}0)q^{11}+−85415835715049953664+ O(11^{2}0)q^{12}+−315405969455155319610+O(11^{2}0)q^{13}+56282909335615634495+O(11^{2}0)q^{14}+−2562890625q^{15}+ 110475912834999708883+O(11^{2}0)q^{16}+132013113992893347403+O(11^{2}0)q^{17}+131257864900275821486+ O(11^{2}0)q^{18}+−173945939567497880896+O(11^{2}0)q^{19}+157885546366755730504+O(11^{2}0)q^{20}+155244480325856167886+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

15. Consider f = q + 7q^{2} + 6q^{3} + 9q^{4} + 7q^{5} + 9q^{6} + 3q^{8} + 3q^{9} + 5q^{10} + 10q^{12} + 3q^{13} + 9q^{15} + 2q^{16} + 10q^{18} + 6q^{19} + 8q^{20} + ⋯ ∈ S_{18}(Γ_{0}(15); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-6 newform g = q+9q^{2}+2q^{3}+5q^{4}+8q^{5}+7q^{6}+10q^{8}+4q^{9}+6q^{10}+10q^{11}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(10)ω^{6} & 0 \\ 0 & u(10)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−238740975694266776999+ O(11^{2}0)q^{2} + −6561q^{3} + 67697048275815856401 + O(11^{2}0)q^{4} + −390625q^{5} + 217553327084622470511 + O(11^{2}0)q^{6} + −24609630446542978132 ∗ 11 + O(11^{2}0)q^{7} + −100959907687354864409 + O(11^{2}0)q^{8} + 43046721q^{9} + 243833031626169773353 + O(11^{2}0)q^{10} + 29656704464430710225 ∗ 11 + O(11^{2}0)q^{11} + −145337082138227774301+O(11^{2}0)q^{12}+64314536475298563688+O(11^{2}0)q^{13}+7167329417656227442∗ 11 + O(11^{2}0)q^{14} + 2562890625q^{15} + 302628621484872093512 + O(11^{2}0)q^{16} + 28620898847104368031 ∗ 11 + O(11^{2}0)q^{17} + 208110244684310501851 + O(11^{2}0)q^{18} + 12300253256379420525 + O(11^{2}0)q^{19} + 297318068499935032283 + O(11^{2}0)q^{20} + 3786575954077315812 ∗ 11 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

16. Consider f = q + 4q^{2} + 6q^{3} + 9q^{4} + 7q^{5} + 2q^{6} + 8q^{8} + 3q^{9} + 6q^{10} + 10q^{12} + 8q^{13} + 9q^{15} + 2q^{16} + q^{18} + 5q^{19} + 8q^{20} + ⋯ ∈ S_{18}(Γ_{0}(15); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q+9q^{2}+2q^{3}+5q^{4}+8q^{5}+7q^{6}+10q^{8}+4q^{9}+6q^{10}+10q^{11}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(10)ω^{1} & 0 \\ 0 & u(10)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−135805765055399654970+ O(11^{2}0)q^{2}+−6561q^{3}+−148335122963298741054+O(11^{2}0)q^{4}+−390625q^{5}+300631237767684076046+ O(11^{2}0)q^{6}+7004482980090190389∗11+O(11^{2}0)q^{7}+−229869742498938163053+O(11^{2}0)q^{8}+43046721q^{9}+ 98874353403257120596+O(11^{2}0)q^{10}+−27269869694657921619∗11+O(11^{2}0)q^{11}+−242500905211293258553+ O(11^{2}0)q^{12} + −316840883562028248690 + O(11^{2}0)q^{13} + −20522955797606941227 ∗ 11 + O(11^{2}0)q^{14} + 2562890625q^{15} + −50591760435654548933 + O(11^{2}0)q^{16} + 355347975234023944 ∗ 11 + O(11^{2}0)q^{17} + 61434148490724039526+O(11^{2}0)q^{18}+295147361900172477151+O(11^{2}0)q^{19}+123093992109691745821+ O(11^{2}0)q^{20} + −25935905612414877688 ∗ 11 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

17. Consider f = q + z^{6}5q^{2} + 6q^{3} + z^{7}3q^{4} + 4q^{5} + z^{5}3q^{6} + z^{3}8q^{7} + z^{8}3q^{8} + 3q^{9} + z^{8}9q^{10} + z^{6}1q^{12} + z^{4}5q^{13} + z^{1}03q^{14} + 2q^{15} + z^{1}01q^{16} + z^{1}10q^{17} + z^{4}1q^{18} + z^{2}9q^{19} + z^{9}7q^{20} + z^{2}6q^{21} + ⋯ ∈ S_{18}(Γ_{0}(15); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{5}3q^{2} + 2q^{3} + z^{4}9q^{4} + 3q^{5} + z^{6}5q^{6} + z^{7}4q^{7} + z^{4}7q^{8} + 4q^{9} + z^{2}9q^{10} + z^{8}8q^{11} + · · · . We deduce

(\begin{matrix}u(z^{3}2)ω^{6} & 0 \\ 0 & u(z^{8}8)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + −6561q^{3} + −(294239901055521235729 + O(11^{2}0)) ∗ a + 114822294975997915421 + O(11^{2}0)q^{4} + 390625q^{5} + −6561 ∗ aq^{6}+−(182860995531218352206+O(11^{2}0))∗a−160315214313877611370+O(11^{2}0)q^{7}+(298581499405063963629+ O(11^{2}0))∗a−175470314453330113483+O(11^{2}0)q^{8}+43046721q^{9}+390625∗aq^{10}+−(30356078969905714178∗ 11+O(11^{2}1))∗a+256005584773313217905∗11+O(11^{2}1)q^{11}+−(284494631172398788901+O(11^{2}0))∗a+ 130916986944887227939+O(11^{2}0)q^{12}+−(244172570624505595298+O(11^{2}0))∗a−11795596608678715293+ O(11^{2}0)q^{13}+−(21534421088057794978+O(11^{2}0))∗a+46665876610504753447+O(11^{2}0)q^{14}+−2562890625q^{15}+ −(150051480140991475401+O(11^{2}0))∗a−19610294317155925594∗11+O(11^{2}0)q^{16}+(33385607141272890121+ O(11^{2}0)) ∗ a + 97740510554191706217 + O(11^{2}0)q^{17} + 43046721 ∗ aq^{18} + −(1322015103737185895811 + O(11^{2}1)) ∗ a + 7661160748071970522 ∗ 11 + O(11^{2}1)q^{19} + −(142965569902814677378 + O(11^{2}0)) ∗ a + 216812845409897897455+O(11^{2}0)q^{20}+(237750715569112418183+O(11^{2}0))∗a+319879033759713817407+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 294239901055521235729 + O(11^{2}0)x + −114822294975998046493 + O(11^{2}0). The slope of f_{0} is 1.

18. Consider f = q + z^{1}15q^{2} + 6q^{3} + z^{8}3q^{4} + 4q^{5} + z^{1}03q^{6} + z^{5}8q^{7} + z^{7}3q^{8} + 3q^{9} + z^{1}9q^{10} + z^{7}1q^{12} + z^{1}5q^{13} + z^{5}3q^{14} + 2q^{15} + z^{3}1q^{16} + z^{1}0q^{17} + z^{9}1q^{18} + z^{7}9q^{19} + z^{1}07q^{20} + z^{4}6q^{21} + ⋯ ∈ S_{18}(Γ_{0}(15); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{1}03q^{2} + 2q^{3} + z^{5}9q^{4} + 3q^{5} + z^{1}15q^{6} + z^{9}4q^{7} + z^{3}7q^{8} + 4q^{9} + z^{7}9q^{10} + z^{8}q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}12)ω^{6} & 0 \\ 0 & u(z^{8})ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + −6561q^{3} + −(294239901055521235729 + O(11^{2}0)) ∗ a + 114822294975997915421 + O(11^{2}0)q^{4} + 390625q^{5} + −6561 ∗ aq^{6}+−(182860995531218352206+O(11^{2}0))∗a−160315214313877611370+O(11^{2}0)q^{7}+(298581499405063963629+ O(11^{2}0))∗a−175470314453330113483+O(11^{2}0)q^{8}+43046721q^{9}+390625∗aq^{10}+−(30356078969905714178∗ 11+O(11^{2}1))∗a+256005584773313217905∗11+O(11^{2}1)q^{11}+−(284494631172398788901+O(11^{2}0))∗a+ 130916986944887227939+O(11^{2}0)q^{12}+−(244172570624505595298+O(11^{2}0))∗a−11795596608678715293+ O(11^{2}0)q^{13}+−(21534421088057794978+O(11^{2}0))∗a+46665876610504753447+O(11^{2}0)q^{14}+−2562890625q^{15}+ −(150051480140991475401+O(11^{2}0))∗a−19610294317155925594∗11+O(11^{2}0)q^{16}+(33385607141272890121+ O(11^{2}0)) ∗ a + 97740510554191706217 + O(11^{2}0)q^{17} + 43046721 ∗ aq^{18} + −(1322015103737185895811 + O(11^{2}1)) ∗ a + 7661160748071970522 ∗ 11 + O(11^{2}1)q^{19} + −(142965569902814677378 + O(11^{2}0)) ∗ a + 216812845409897897455+O(11^{2}0)q^{20}+(237750715569112418183+O(11^{2}0))∗a+319879033759713817407+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 294239901055521235729 + O(11^{2}0)x + −114822294975998046493 + O(11^{2}0). The slope of f_{0} is 1.

19. Consider f = q + z^{5}q^{2} + 6q^{3} + z^{7}3q^{4} + 4q^{5} + z^{1}13q^{6} + z^{9}8q^{7} + z^{2}3q^{8} + 3q^{9} + z^{2}9q^{10} + z^{6}1q^{12} + z^{1}05q^{13} + z^{1}03q^{14} + 2q^{15} + z^{1}01q^{16} + z^{5}0q^{17} + z^{1}01q^{18} + z^{8}9q^{19} + z^{9}7q^{20} + z^{8}6q^{21} + ⋯ ∈ S_{18}(Γ_{0}(15); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{5}3q^{2} + 2q^{3} + z^{4}9q^{4} + 3q^{5} + z^{6}5q^{6} + z^{7}4q^{7} + z^{4}7q^{8} + 4q^{9} + z^{2}9q^{10} + z^{8}8q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{3}2)ω^{1} & 0 \\ 0 & u(z^{8}8)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + −6561q^{3} + (294239901055521235696+O(11^{2}0))∗a−7164693628703225324+O(11^{2}0)q^{4} +390625q^{5} +−6561∗aq^{6} + (43589035088909358432+O(11^{2}0))∗a+33634958679821262189+O(11^{2}0)q^{7}+(266510894180201532688+ O(11^{2}0))∗a−13480555344569036689+O(11^{2}0)q^{8}+43046721q^{9}+390625∗aq^{10}+(38360630785935188218∗ 11+O(11^{2}1))∗a+183347903536338807198∗11+O(11^{2}1)q^{11}+(284494631172399005414+O(11^{2}0))∗a− 84944747357339293306+O(11^{2}0)q^{12}+(165411008537564897871+O(11^{2}0))∗a+286924629454608078229+ O(11^{2}0)q^{13}+(148214676722131727263+O(11^{2}0))∗a+114219855589201553036+O(11^{2}0)q^{14}+−2562890625q^{15}+ −(333747644993601406631+O(11^{2}0))∗a−11422432062816364250∗11+O(11^{2}0)q^{16}+−(208419025718548025544+ O(11^{2}0)) ∗ a + 206972516718361955204 + O(11^{2}0)q^{17} + 43046721 ∗ aq^{18} + −(3526835559388213305149 + O(11^{2}1)) ∗ a + 305778424457468140583 ∗ 11 + O(11^{2}1)q^{19} + (142965569902801786753 + O(11^{2}0)) ∗ a − 68469792747753911340+O(11^{2}0)q^{20}+−(68911371996296761927+O(11^{2}0))∗a−16965560427618204101+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −294239901055521235696 + O(11^{2}0)x + 7164693628703094252 + O(11^{2}0). The slope of f_{0} is 1.

20. Consider f = q + z^{5}5q^{2} + 6q^{3} + z^{8}3q^{4} + 4q^{5} + z^{4}3q^{6} + z^{1}18q^{7} + z^{1}3q^{8} + 3q^{9} + z^{7}9q^{10} + z^{7}1q^{12} + z^{7}5q^{13} + z^{5}3q^{14} + 2q^{15} + z^{3}1q^{16} + z^{7}0q^{17} + z^{3}1q^{18} + z^{1}9q^{19} + z^{1}07q^{20} + z^{1}06q^{21} + ⋯ ∈ S_{18}(Γ_{0}(15); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{1}03q^{2} + 2q^{3} + z^{5}9q^{4} + 3q^{5} + z^{1}15q^{6} + z^{9}4q^{7} + z^{3}7q^{8} + 4q^{9} + z^{7}9q^{10} + z^{8}q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}12)ω^{1} & 0 \\ 0 & u(z^{8})ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + −6561q^{3} + (294239901055521235696+O(11^{2}0))∗a−7164693628703225324+O(11^{2}0)q^{4} +390625q^{5} +−6561∗aq^{6} + (43589035088909358432+O(11^{2}0))∗a+33634958679821262189+O(11^{2}0)q^{7}+(266510894180201532688+ O(11^{2}0))∗a−13480555344569036689+O(11^{2}0)q^{8}+43046721q^{9}+390625∗aq^{10}+(38360630785935188218∗ 11+O(11^{2}1))∗a+183347903536338807198∗11+O(11^{2}1)q^{11}+(284494631172399005414+O(11^{2}0))∗a− 84944747357339293306+O(11^{2}0)q^{12}+(165411008537564897871+O(11^{2}0))∗a+286924629454608078229+ O(11^{2}0)q^{13}+(148214676722131727263+O(11^{2}0))∗a+114219855589201553036+O(11^{2}0)q^{14}+−2562890625q^{15}+ −(333747644993601406631+O(11^{2}0))∗a−11422432062816364250∗11+O(11^{2}0)q^{16}+−(208419025718548025544+ O(11^{2}0)) ∗ a + 206972516718361955204 + O(11^{2}0)q^{17} + 43046721 ∗ aq^{18} + −(3526835559388213305149 + O(11^{2}1)) ∗ a + 305778424457468140583 ∗ 11 + O(11^{2}1)q^{19} + (142965569902801786753 + O(11^{2}0)) ∗ a − 68469792747753911340+O(11^{2}0)q^{20}+−(68911371996296761927+O(11^{2}0))∗a−16965560427618204101+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −294239901055521235696 + O(11^{2}0)x + 7164693628703094252 + O(11^{2}0). The slope of f_{0} is 1.

21. Consider f = q + 9q^{2} + 7q^{3} + 9q^{4} + 2q^{5} + 8q^{6} + 4q^{7} + 5q^{8} + 5q^{9} + 7q^{10} + 8q^{12} + 2q^{13} + 3q^{14} + 3q^{15} + 2q^{16} + 7q^{17} + q^{18} + 7q^{19} + 7q^{20} + 6q^{21} + ⋯ ∈ S_{20}(Γ_{0}(15); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 3q^{2} + 8q^{3} + q^{4} + 6q^{5} + 2q^{6} + 9q^{7} + q^{8} + 9q^{9} + 7q^{10} + 9q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{1} & 0 \\ 0 & u(9)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+136785601983731981234+ O(11^{2}0)q^{2}+−19683q^{3}+300595354336332175646+O(11^{2}0)q^{4}+−1953125q^{5}+−5524125691429806420+ O(11^{2}0)q^{6}+97665506940152702247+O(11^{2}0)q^{7}+−326623126542062680908+O(11^{2}0)q^{8}+387420489q^{9}+ −264636832957793801135+O(11^{2}0)q^{10}+−2925828859094444631∗11+O(11^{2}0)q^{11}+217846029839067682577+ O(11^{2}0)q^{12}+−222451549851631958955+O(11^{2}0)q^{13}+−272487727606148055845+O(11^{2}0)q^{14}+38443359375q^{15}+ 254347926289653450410+O(11^{2}0)q^{16}+183928672015313821845+O(11^{2}0)q^{17}+−254133194661841725702+ O(11^{2}0)q^{18}+316399609497009090466+O(11^{2}0)q^{19}+−126610437783809000663+O(11^{2}0)q^{20}+−303437580701692040444+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

22. Consider f = q + 7q^{2} + 7q^{3} + 10q^{4} + 2q^{5} + 5q^{6} + 2q^{7} + 6q^{8} + 5q^{9} + 3q^{10} + 4q^{12} + 6q^{13} + 3q^{14} + 3q^{15} + 4q^{16} + 8q^{17} + 2q^{18} + 9q^{19} + 9q^{20} + 3q^{21} + ⋯ ∈ S_{20}(Γ_{0}(15); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q + 9q^{2} + 6q^{3} + 8q^{4} + 7q^{5} + 10q^{6} + 5q^{7} + 9q^{8} + 3q^{9} + 8q^{10} + 7q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{8} & 0 \\ 0 & u(7)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−256522122560367353837+ O(11^{2}0)q^{2}+−19683q^{3}+−293809834342100467161+O(11^{2}0)q^{4}+−1953125q^{5}+136226386847756520166+ O(11^{2}0)q^{6}+35320998511839791434+O(11^{2}0)q^{7}+223275966655047534003+O(11^{2}0)q^{8}+387420489q^{9}+ −24100387657929406909+O(11^{2}0)q^{10}+28239351567159364725∗11+O(11^{2}0)q^{11}+100012915277656038167+ O(11^{2}0)q^{12}+−174186288794626801445+O(11^{2}0)q^{13}+141171898553229274162+O(11^{2}0)q^{14}+38443359375q^{15}+ −323366018483208830301+O(11^{2}0)q^{16}+102804194724677631435+O(11^{2}0)q^{17}+237507476792610247808+ O(11^{2}0)q^{18}+140582060582945287733+O(11^{2}0)q^{19}+332771875410355494738+O(11^{2}0)q^{20}+−272468943208125290789+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

23. Consider f = q+4q^{3}+5q^{4}+2q^{5}+8q^{7}+5q^{9}+9q^{12}+10q^{13}+8q^{15}+3q^{16}+10q^{17}+5q^{19}+10q^{20}+10q^{21}+⋯ ∈ S_{20}(Γ_{0}(15); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q+5q^{3} +4q^{4} +7q^{5} +9q^{7} +3q^{9} +4q^{11} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(3)ω^{8} & 0 \\ 0 & u(4)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−13385944252755311599∗ 11+O(11^{2}0)q^{2}+19683q^{3}+2091421336074949739481+O(11^{2}1)q^{4}+−1953125q^{5}+−2179075212932781489∗ 11 + O(11^{2}0)q^{6} + 28281943370054692271 + O(11^{2}0)q^{7} + −8129216555991772275 ∗ 11 + O(11^{2}0)q^{8} + 387420489q^{9} + 12065593570889227613 ∗ 11 + O(11^{2}0)q^{10} + 30556705630075235650 ∗ 11 + O(11^{2}0)q^{11} + −2144281944908920832270+O(11^{2}1)q^{12}+−236753426854890540387+O(11^{2}0)q^{13}+3264900363460145792∗ 11+O(11^{2}0)q^{14}+−38443359375q^{15}+−25046761515325317083+O(11^{2}0)q^{16}+131406425079796706784+ O(11^{2}0)q^{17} + −18215011817341097996 ∗ 11 + O(11^{2}0)q^{18} + −55677164650094837845 + O(11^{2}0)q^{19} + −2532289878480937177923 + O(11^{2}1)q^{20} + 309245543559380360866 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

24. Consider f = q + z^{7}6q^{2} + 4q^{3} + z^{6}9q^{4} + 9q^{5} + z^{1}00q^{6} + z^{5}9q^{7} + z^{9}4q^{8} + 5q^{9} + z^{2}8q^{10} + z^{9}3q^{12} + z^{9}4q^{13} + z^{1}5q^{14} + 3q^{15} + z^{4}0q^{16} + z^{4}6q^{17} + z^{4}q^{18} + z^{8}0q^{19} + z^{2}1q^{20} + z^{8}3q^{21} + ⋯ ∈ S_{20}(Γ_{0}(15); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{6}4q^{2} + 5q^{3} + z^{4}5q^{4} + 4q^{5} + z^{1}12q^{6} + z^{9}5q^{7} + z^{5}8q^{8} + 3q^{9} + z^{8}8q^{10} + z^{9}1q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}9)ω^{8} & 0 \\ 0 & u(z^{9}1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + 19683q^{3} + −(222610656384762127963+O(11^{2}0))∗a+313721011611536908837+O(11^{2}0)q^{4} +1953125q^{5} +19683∗ aq^{6}+(64606395817342351997+O(11^{2}0))∗a−30942266912528557953+O(11^{2}0)q^{7}+(168187788115108491586+ O(11^{2}0))∗a+181945891371709285937+O(11^{2}0)q^{8}+387420489q^{9}+1953125∗aq^{10}+(21927269682871384572∗ 11+O(11^{2}0))∗a−27901945986853852644∗11+O(11^{2}0)q^{11} +−(24832625509624769616+O(11^{2}0))∗a− 201531936087347817308+O(11^{2}0)q^{12}+−(20491099359732284273+O(11^{2}0))∗a+242848904165154107321+ O(11^{2}0)q^{13}+(314725666747629700026+O(11^{2}0))∗a+98018528546516815802+O(11^{2}0)q^{14}+38443359375q^{15}+ −(312996449125848242924+O(11^{2}0))∗a−19626892139790663056∗11+O(11^{2}0)q^{16}+(69199051019447779465+ O(11^{2}0)) ∗ a − 26102353076075774059 + O(11^{2}0)q^{17} + 387420489 ∗ aq^{18} + (69256629551252867002 + O(11^{2}0)) ∗ a + 78999739937773680145 + O(11^{2}0)q^{19} + −(226026483783311293693 + O(11^{2}0)) ∗ a − 308080823035947949969+O(11^{2}0)q^{20}+(150198450211096967061+O(11^{2}0))∗a−197894225332797861994+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 222610656384762127963 + O(11^{2}0)x + −313721011611537433125 + O(11^{2}0). The slope of f_{0} is 1.

25. Consider f = q + 3q^{2} + 4q^{3} + 3q^{4} + 9q^{5} + q^{6} + 4q^{7} + 2q^{8} + 5q^{9} + 5q^{10} + q^{12} + q^{14} + 3q^{15} + 10q^{16} + 10q^{17} + 4q^{18} + 10q^{19} + 5q^{20} + 5q^{21} + ⋯ ∈ S_{20}(Γ_{0}(15); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + q^{2} + 3q^{3} + 4q^{4} + 5q^{5} + 3q^{6} + 9q^{7} + 7q^{8} + 9q^{9} + 5q^{10} + 8q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(7)ω^{1} & 0 \\ 0 & u(8)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+26052737474758099680+ O(11^{2}0)q^{2} +19683q^{3} +−105970766635179373518+O(11^{2}0)q^{4} +1953125q^{5} +160535577052948990278+ O(11^{2}0)q^{6}+221620687952904688353+O(11^{2}0)q^{7}+−17269459981731541934+O(11^{2}0)q^{8}+387420489q^{9}+ 134263667804581573164+O(11^{2}0)q^{10}+−28120477431931020076∗11+O(11^{2}0)q^{11}+−297615389299580431694+ O(11^{2}0)q^{12}+23134896465683320752∗11+O(11^{2}0)q^{13}+212520985605451783118+O(11^{2}0)q^{14}+38443359375q^{15}+ −203747445321801421096+O(11^{2}0)q^{16}+−161063351795525658568+O(11^{2}0)q^{17}+−84963065039387575223+ O(11^{2}0)q^{18}+254786177121018190911+O(11^{2}0)q^{19}+73356647103168380704+O(11^{2}0)q^{20}+49033834303881192815+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

26. Consider f = q + z^{1}16q^{2} + 4q^{3} + z^{3}9q^{4} + 9q^{5} + z^{2}0q^{6} + z^{4}9q^{7} + z^{7}4q^{8} + 5q^{9} + z^{6}8q^{10} + z^{6}3q^{12} + z^{7}4q^{13} + z^{4}5q^{14} + 3q^{15} + z^{8}0q^{16} + z^{2}6q^{17} + z^{4}4q^{18} + z^{4}0q^{19} + z^{1}11q^{20} + z^{7}3q^{21} + ⋯ ∈ S_{20}(Γ_{0}(15); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{1}04q^{2} + 5q^{3} + z^{1}5q^{4} + 4q^{5} + z^{3}2q^{6} + z^{8}5q^{7} + z^{3}8q^{8} + 3q^{9} + z^{8}q^{10} + z^{4}1q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{7}9)ω^{8} & 0 \\ 0 & u(z^{4}1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + 19683q^{3} + −(222610656384762127963+O(11^{2}0))∗a+313721011611536908837+O(11^{2}0)q^{4} +1953125q^{5} +19683∗ aq^{6}+(64606395817342351997+O(11^{2}0))∗a−30942266912528557953+O(11^{2}0)q^{7}+(168187788115108491586+ O(11^{2}0))∗a+181945891371709285937+O(11^{2}0)q^{8}+387420489q^{9}+1953125∗aq^{10}+(21927269682871384572∗ 11+O(11^{2}0))∗a−27901945986853852644∗11+O(11^{2}0)q^{11} +−(24832625509624769616+O(11^{2}0))∗a− 201531936087347817308+O(11^{2}0)q^{12}+−(20491099359732284273+O(11^{2}0))∗a+242848904165154107321+ O(11^{2}0)q^{13}+(314725666747629700026+O(11^{2}0))∗a+98018528546516815802+O(11^{2}0)q^{14}+38443359375q^{15}+ −(312996449125848242924+O(11^{2}0))∗a−19626892139790663056∗11+O(11^{2}0)q^{16}+(69199051019447779465+ O(11^{2}0)) ∗ a − 26102353076075774059 + O(11^{2}0)q^{17} + 387420489 ∗ aq^{18} + (69256629551252867002 + O(11^{2}0)) ∗ a + 78999739937773680145 + O(11^{2}0)q^{19} + −(226026483783311293693 + O(11^{2}0)) ∗ a − 308080823035947949969+O(11^{2}0)q^{20}+(150198450211096967061+O(11^{2}0))∗a−197894225332797861994+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 222610656384762127963 + O(11^{2}0)x + −313721011611537433125 + O(11^{2}0). The slope of f_{0} is 1.

27. Consider f = q + 10q^{3} + 9q^{4} + 10q^{5} + 5q^{7} + q^{9} + 2q^{12} + q^{15} + 4q^{16} + q^{17} + 7q^{19} + 2q^{20} + 6q^{21} + ⋯ ∈ S_{22}(Γ_{0}(15); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q+7q^{3}+5q^{4}+2q^{5}+7q^{7}+5q^{9}+4q^{11}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(3)ω^{0} & 0 \\ 0 & u(4)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−449696466023333965∗ 11^{2}+O(11^{2}0)q^{2}+−59049q^{3}+23947209566261751142756+O(11^{2}2)q^{4}+−9765625q^{5}+−38467027050253971∗ 11^{2} + O(11^{2}0)q^{6} + −25461347783878707718 + O(11^{2}0)q^{7} + 899034572903146433 ∗ 11^{2} + O(11^{2}0)q^{8} + 3486784401q^{9} + −358060480190092497 ∗ 11^{2} + O(11^{2}0)q^{10} + 29672750548738992867 ∗ 11 + O(11^{2}0)q^{11} + −11618079396652929099953+O(11^{2}2)q^{12}+−1867632963716818571∗11^{2}+O(11^{2}0)q^{13}+2079505566176968202∗ 11^{2}+O(11^{2}0)q^{14}+576650390625q^{15}+316899002075757946024+O(11^{2}0)q^{16}+76120333812466968740+ O(11^{2}0)q^{17} + −2566700927875942150 ∗ 11^{2} + O(11^{2}0)q^{18} + 240718489574801163691 + O(11^{2}0)q^{19} + −33192303955833611174551 + O(11^{2}2)q^{20} + −129113384017808524053 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

28. Consider f = q + z^{2}5q^{2} + q^{3} + z^{6}9q^{4} + 10q^{5} + z^{2}5q^{6} + z^{1}07q^{7} + 5q^{8} + q^{9} + z^{8}5q^{10} + z^{6}9q^{12} + z^{3}7q^{13} + 2q^{14} + 10q^{15} + z^{3}9q^{16} + 6q^{17} + z^{2}5q^{18} + 10q^{19} + z^{9}q^{20} + z^{1}07q^{21} + ⋯ ∈ S_{22}(Γ_{0}(15); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{1}3q^{2} + 4q^{3} + z^{4}5q^{4} + 2q^{5} + z^{3}7q^{6} + z^{2}3q^{7} + 2q^{8} + 5q^{9} + z^{2}5q^{10} + z^{5}5q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{6}5)ω^{0} & 0 \\ 0 & u(z^{5}5)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + 59049q^{3} + (142138691934458746246+O(11^{2}0))∗a−282172714833381759872+O(11^{2}0)q^{4} +−9765625q^{5} +59049∗ aq^{6}+(523364949406308688329+O(11^{2}1))∗a−491097273432350022734+O(11^{2}1)q^{7}+(25897689747917138049∗ 11^{2} + O(11^{2}1)) ∗ a + 2823058260794606093957 + O(11^{2}1)q^{8} + 3486784401q^{9} + −9765625 ∗ aq^{10} + (19871364951828585302∗11+O(11^{2}0))∗a−4697816821901356801∗11+O(11^{2}0)q^{11}+−(81316740764167711622+ O(11^{2}0)) ∗ a − 17513701645790800561 + O(11^{2}0)q^{12} + (1583649235420496908864 + O(11^{2}1)) ∗ a − 4586203085723360785∗11^{2}+O(11^{2}1)q^{13}+(2647011600635843899∗11^{2}+O(11^{2}0))∗a+266753617092379466920+ O(11^{2}0)q^{14} +−576650390625q^{15} +−(282143660104994842115+O(11^{2}0))∗a+67855275535540325538+ O(11^{2}0)q^{16} + (1173271191052172353 ∗ 11^{2} + O(11^{2}0)) ∗ a − 114501784419210639896 + O(11^{2}0)q^{17} + 3486784401∗aq^{18}+−(1940809024862119702∗11^{2}+O(11^{2}0))∗a−158868869932209905480+O(11^{2}0)q^{19}+ −(208378006412904336068+O(11^{2}0))∗a+193301043827782584387+O(11^{2}0)q^{20}+(733130271045154481985+ O(11^{2}1)) ∗ a + 2776632640892944225943 + O(11^{2}1)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −142138691934458746246 + O(11^{2}0)x + 282172714833379662720 + O(11^{2}0). The slope of f_{0} is 1.

29. Consider f = q + z^{3}5q^{2} + q^{3} + z^{3}9q^{4} + 10q^{5} + z^{3}5q^{6} + z^{9}7q^{7} + 5q^{8} + q^{9} + z^{9}5q^{10} + z^{3}9q^{12} + z^{4}7q^{13} + 2q^{14} + 10q^{15} + z^{6}9q^{16} + 6q^{17} + z^{3}5q^{18} + 10q^{19} + z^{9}9q^{20} + z^{9}7q^{21} + ⋯ ∈ S_{22}(Γ_{0}(15); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{2}3q^{2} + 4q^{3} + z^{1}5q^{4} + 2q^{5} + z^{4}7q^{6} + z^{1}3q^{7} + 2q^{8} + 5q^{9} + z^{3}5q^{10} + z^{5}q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}15)ω^{0} & 0 \\ 0 & u(z^{5})ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + 59049q^{3} + (142138691934458746246+O(11^{2}0))∗a−282172714833381759872+O(11^{2}0)q^{4} +−9765625q^{5} +59049∗ aq^{6}+(523364949406308688329+O(11^{2}1))∗a−491097273432350022734+O(11^{2}1)q^{7}+(25897689747917138049∗ 11^{2} + O(11^{2}1)) ∗ a + 2823058260794606093957 + O(11^{2}1)q^{8} + 3486784401q^{9} + −9765625 ∗ aq^{10} + (19871364951828585302∗11+O(11^{2}0))∗a−4697816821901356801∗11+O(11^{2}0)q^{11}+−(81316740764167711622+ O(11^{2}0)) ∗ a − 17513701645790800561 + O(11^{2}0)q^{12} + (1583649235420496908864 + O(11^{2}1)) ∗ a − 4586203085723360785∗11^{2}+O(11^{2}1)q^{13}+(2647011600635843899∗11^{2}+O(11^{2}0))∗a+266753617092379466920+ O(11^{2}0)q^{14} +−576650390625q^{15} +−(282143660104994842115+O(11^{2}0))∗a+67855275535540325538+ O(11^{2}0)q^{16} + (1173271191052172353 ∗ 11^{2} + O(11^{2}0)) ∗ a − 114501784419210639896 + O(11^{2}0)q^{17} + 3486784401∗aq^{18}+−(1940809024862119702∗11^{2}+O(11^{2}0))∗a−158868869932209905480+O(11^{2}0)q^{19}+ −(208378006412904336068+O(11^{2}0))∗a+193301043827782584387+O(11^{2}0)q^{20}+(733130271045154481985+ O(11^{2}1)) ∗ a + 2776632640892944225943 + O(11^{2}1)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −142138691934458746246 + O(11^{2}0)x + 282172714833379662720 + O(11^{2}0). The slope of f_{0} is 1.

30. Consider f = q + 10q^{2} + 10q^{3} + 10q^{4} + q^{5} + q^{6} + 3q^{8} + q^{9} + 10q^{10} + q^{12} + 9q^{13} + 10q^{15} + 10q^{16} + 2q^{17} + 10q^{18} + 4q^{19} + 10q^{20} + ⋯ ∈ S_{22}(Γ_{0}(15); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{10} ∼ ρ_{g}, where g is the weight-2 newform g = q + 10q^{2} + 10q^{3} + 10q^{4} + q^{5} + q^{6} + 3q^{8} + q^{9} + 10q^{10} + 7q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{1} & 0 \\ 0 & u(7)ω^{0} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+211952552600926859490+ O(11^{2}0)q^{2} + −59049q^{3} + 71922571603221830886 + O(11^{2}0)q^{4} + 9765625q^{5} + 254627193216285150394 + O(11^{2}0)q^{6}+−26641012234877748969∗11+O(11^{2}0)q^{7}+187250963787395858342+O(11^{2}0)q^{8}+3486784401q^{9}+ −90165586085541676249+O(11^{2}0)q^{10}+1110088387966295818∗11+O(11^{1}9)q^{11}+114787410605446098499+ O(11^{2}0)q^{12} + −223338416819695273032 + O(11^{2}0)q^{13} + 20464147735831640255 ∗ 11 + O(11^{2}0)q^{14} + −576650390625q^{15} +112119154895973352434+O(11^{2}0)q^{16} +−225579095873058280219+O(11^{2}0)q^{17} + −191495480638199982157+O(11^{2}0)q^{18}+−117257756067506383644+O(11^{2}0)q^{19}+−295646742636479998677+ O(11^{2}0)q^{20} + −8993056822760826621 ∗ 11 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

31. Consider f = q + z^{6}1q^{2} + 10q^{3} + z^{7}9q^{4} + q^{5} + zq^{6} + z^{6}3q^{7} + z^{6}5q^{8} + q^{9} + z^{6}1q^{10} + z^{1}9q^{12} + z^{7}3q^{13} + z^{4}q^{14} + 10q^{15} + z^{1}10q^{16} + z^{7}1q^{17} + z^{6}1q^{18} + z^{8}7q^{19} + z^{7}9q^{20} + z^{3}q^{21} + ⋯ ∈ S_{22}(Γ_{0}(15); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{4}9q^{2} + 7q^{3} + z^{5}5q^{4} + 9q^{5} + z^{1}3q^{6} + z^{9}9q^{7} + z^{2}9q^{8} + 5q^{9} + zq^{10} + z^{4}4q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{7}6)ω^{0} & 0 \\ 0 & u(z^{4}4)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + −59049q^{3} + −(116531532108928555337+O(11^{2}0))∗a+180709794610884586494+O(11^{2}0)q^{4}+9765625q^{5}+−59049∗ aq^{6}+−(81977712067328285062+O(11^{2}0))∗a+71042637830402397156+O(11^{2}0)q^{7}+(295550882087195481669+ O(11^{2}0))∗a+329256802437333451480+O(11^{2}0)q^{8}+3486784401q^{9}+9765625∗aq^{10}+−(1983659416617462016∗ 11+O(11^{1}9))∗a−1654147170006205599∗11+O(11^{1}9)q^{11} +(183491329898489986685+O(11^{2}0))∗a− 244992352789641947145+O(11^{2}0)q^{12}+(91123167035900883336+O(11^{2}0))∗a−1441280425060559551∗ 11^{2} + O(11^{2}0)q^{13} + (266796386610660125413 + O(11^{2}0)) ∗ a + 318332889382363925866 + O(11^{2}0)q^{14} + −576650390625q^{15}+−(205943630303816993381+O(11^{2}0))∗a−222020281969157405306+O(11^{2}0)q^{16}+ −(22027375094205645099 + O(11^{2}0)) ∗ a + 267067300110921826567 + O(11^{2}0)q^{17} + 3486784401 ∗ aq^{18} + (68672191816808670520+O(11^{2}0))∗a+161768551049321168270+O(11^{2}0)q^{19}+−(207073179821008764256+ O(11^{2}0)) ∗ a − 243710272974955410430 + O(11^{2}0)q^{20} + (265706323898638424843 + O(11^{2}0)) ∗ a + 272247152013067712792+O(11^{2}0)q^{21} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2}+116531532108928555337+O(11^{2}0)x+−180709794610886683646+ O(11^{2}0). The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

32. Consider f = q + q2 + 3q3 + 4q4 + 5q5 + 3q6 + 9q7 + 7q8 + 9q9 + 5q10 + 8q11 + q12 + 9q14 + 4q15 + 8q16 + 8q17 + 9q18 + 2q19 + 9q20 + 5q21 + ⋯ ∈ S14(Γ0(15); \overline{F}11). Since f is ordinary, we do not continue our analysis.

33. Consider f = q +3q2 +8q3 +q4 +6q5 +2q6 +9q7 +q8 +9q9 +7q10 +9q11 +8q12 +8q13 +5q14 +4q15 +6q16 +10q17 +5q18 +8q19 +6q20 +6q21 +⋯ ∈ S14(Γ0(15); \overline{F}11). Since f is ordinary, we do not continue our analysis.

34. Consider f = q +7q2 +9q3 +6q4 +8q5 +8q6 +q7 +5q8 +4q9 +q10 +2q11 +10q12 +5q13 +7q14 +6q15 +8q16 +4q17 +6q18 +q19 +4q20 +9q21 +⋯ ∈ S16(Γ0(15); \overline{F}11). Since f is ordinary, we do not continue our analysis.

35. Consider f = q + z103q2 + 2q3 + z59q4 + 3q5 + z115q6 + z94q7 + z37q8 + 4q9 + z79q10 + z8q11 + z71q12 + z3q13 + z77q14 + 6q15 + z103q16 + z22q17 + z7q18 + z43q19 + z35q20 + z106q21 + ⋯ ∈ S16(Γ0(15); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

36. Consider f = q + z53q2 + 2q3 + z49q4 + 3q5 + z65q6 + z74q7 + z47q8 + 4q9 + z29q10 + z88q11 + z61q12 + z33q13 + z7q14 + 6q15 + z53q16 + z2q17 + z77q18 + z113q19 + z25q20 + z86q21 + ⋯ ∈ S16(Γ0(15); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

37. Consider f = q + 9q2 + 2q3 + 5q4 + 8q5 + 7q6 + 10q8 + 4q9 + 6q10 + 10q11 + 10q12 + 7q13 + 5q15 + 7q16 + 3q18 + 9q19 + 7q20 + ⋯ ∈ S16(Γ0(15); \overline{F}11). Since f is ordinary, we do not continue our analysis.

38. Consider f = q + z64q2 + 5q3 + z45q4 + 4q5 + z112q6 + z95q7 + z58q8 + 3q9 + z88q10 + z91q11 + z93q12 + z82q13 + z39q14 + 9q15 + z112q16 + z58q17 + z40q18 + z44q19 + z69q20 + z23q21 + ⋯ ∈ S18(Γ0(15); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

39. Consider f = q + z104q2 + 5q3 + z15q4 + 4q5 + z32q6 + z85q7 + z38q8 + 3q9 + z8q10 + z41q11 + z63q12 + z62q13 + z69q14 + 9q15 + z32q16 + z38q17 + z80q18 + z4q19 + z39q20 + z13q21 + ⋯ ∈ S18(Γ0(15); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

40. Consider f = q + 5q3 + 4q4 + 7q5 + 9q7 + 3q9 + 4q11 + 9q12 + 5q13 + 2q15 + 5q16 + 9q17 + 2q19 + 6q20 + q21 + ⋯ ∈ S18(Γ0(15); \overline{F}11). Since f is ordinary, we do not continue our analysis.

41. Consider f = q +9q2 +6q3 +8q4 +7q5 +10q6 +5q7 +9q8 +3q9 +8q10 +7q11 +4q12 +3q13 +q14 +9q15 +3q16 +5q17 +5q18 +8q19 +q20 +8q21 +⋯ ∈ S18(Γ0(15); \overline{F}11). Since f is ordinary, we do not continue our analysis.

42. Consider f = q + z49q2 + 7q3 + z55q4 + 9q5 + z13q6 + z99q7 + z29q8 + 5q9 + zq10 + z44q11 + z19q12 + z61q13 + z28q14 + 8q15 + z62q16 + z83q17 + z97q18 + z51q19 + z7q20 + z63q21 + ⋯ ∈ S20(Γ0(15); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

43. Consider f = q + z59q2 + 7q3 + z5q4 + 9q5 + z23q6 + z9q7 + z79q8 + 5q9 + z11q10 + z4q11 + z89q12 + z71q13 + z68q14 + 8q15 + z82q16 + z73q17 + z107q18 + z81q19 + z77q20 + z93q21 + ⋯ ∈ S20(Γ0(15); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

44. Consider f = q + 7q3 + 5q4 + 2q5 + 7q7 + 5q9 + 4q11 + 2q12 + 3q15 + 3q16 + 2q17 + 5q19 + 10q20 + 5q21 + ⋯ ∈ S20(Γ0(15); \overline{F}11). Since f is ordinary, we do not continue our analysis.

45. Consider f = q + z13q2 + 4q3 + z45q4 + 2q5 + z37q6 + z23q7 + 2q8 + 5q9 + z25q10 + z55q11 + z69q12 + z25q13 + 8q14 + 8q15 + z111q16 + q17 + z61q18 + 4q19 + z57q20 + z47q21 + ⋯ ∈ S20(Γ0(15); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

46. Consider f = q + z23q2 + 4q3 + z15q4 + 2q5 + z47q6 + z13q7 + 2q8 + 5q9 + z35q10 + z5q11 + z39q12 + z35q13 + 8q14 + 8q15 + z21q16 + q17 + z71q18 + 4q19 + z27q20 + z37q21 + ⋯ ∈ S20(Γ0(15); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

47. Consider f = q+7q2 +4q3 +10q4 +9q5 +6q6 +9q7 +6q8 +5q9 +8q10 +q11 +7q12 +7q13 +8q14 +3q15 +4q16 +3q17 +2q18 +8q19 +2q20 +3q21 +⋯ ∈ S20(Γ0(15); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

48. Consider f = q + 5q2 + q3 + q4 + 10q5 + 5q6 + 2q7 + 6q8 + q9 + 6q10 + 10q11 + q12 + 3q13 + 10q14 + 10q15 + 6q16 + 5q18 + 3q19 + 10q20 + 2q21 + ⋯ ∈ S22(Γ0(15); \overline{F}11). Since f is ordinary, we do not continue our analysis.

49. Consider f = q + z65q2 + 10q3 + z32q4 + 10q5 + z5q6 + z7q7 + z117q8 + q9 + z5q10 + 10q11 + z92q12 + z90q13 + 9q14 + q15 + 3q16 + z73q17 + z65q18 + z83q19 + z92q20 + z67q21 + ⋯ ∈ S22(Γ0(15); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

50. Consider f = q + z115q2 + 10q3 + z112q4 + 10q5 + z55q6 + z77q7 + z87q8 + q9 + z55q10 + 10q11 + z52q12 + z30q13 + 9q14 + q15 + 3q16 + z83q17 + z115q18 + z73q19 + z52q20 + z17q21 + ⋯ ∈ S22(Γ0(15); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

51. Consider f = q + 6q2 + q3 + q4 + 10q5 + 6q6 + 2q7 + 5q8 + q9 + 5q10 + 10q11 + q12 + q13 + q14 + 10q15 + 6q16 + 6q18 + q19 + 10q20 + 2q21 + ⋯ ∈ S22(Γ0(15); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

52. Consider f = q + z880q2 + q3 + z294q4 + q5 + z880q6 + z279q7 + z434q8 + q9 + z880q10 + q11 + z294q12 + z1203q13 + z1159q14 + q15 + z340q16 + z403q17 + z880q18 + z194q19 + z294q20 + z279q21 + ⋯ ∈ S22(Γ0(15); \overline{F}11). Also, z satisfies the equation x3 + 2x + 9 = 0. Since f is ordinary, we do not continue our analysis.

53. Consider f = q + z80q2 + q3 + z994q4 + q5 + z80q6 + z509q7 + z644q8 + q9 + z80q10 + q11 + z994q12 + z593q13 + z589q14 + q15 + z1240q16 + z883q17 + z80q18 + z864q19 + z994q20 + z509q21 + ⋯ ∈ S22(Γ0(15); \overline{F}11). Also, z satisfies the equation x3 + 2x + 9 = 0. Since f is ordinary, we do not continue our analysis.

54. Consider f = q + z370q2 + q3 + z574q4 + q5 + z370q6 + z409q7 + z784q8 + q9 + z370q10 + q11 + z574q12 + z1263q13 + z779q14 + q15 + z1080q16 + z443q17 + z370q18 + z804q19 + z574q20 + z409q21 + ⋯ ∈ S22(Γ0(15); \overline{F}11). Also, z satisfies the equation x3 + 2x + 9 = 0. Since f is ordinary, we do not continue our analysis.

55. Consider f = q+10q2+10q3+10q4+q5+q6+3q8+q9+10q10+7q11+q12+9q13+10q15+10q16+2q17+10q18+4q19+10q20+⋯ ∈ S22(Γ0(15); \overline{F}11). Since f is ordinary, we do not continue our analysis.

11.14 N = 16, ℓ = 11 (37 forms)

1. Consider f = q + q^{3} + 4q^{5} + 6q^{7} + 7q^{9} + 4q^{13} + 4q^{15} + 10q^{17} + 8q^{19} + 6q^{21} + ⋯ ∈ S_{14}(Γ_{0}(16); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 5q^{3} + 5q^{5} + 10q^{7} + 10q^{9} + 10q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(10)ω^{1} & 0 \\ 0 & u(10)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 12q^{3} + −4330q^{5} + 139992q^{7}+−1594179q^{9}+589484∗11q^{11}+−22588034q^{13}+−51960q^{15}+−23732270q^{17}+−325344836q^{19}+ 1679904q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 5q^{3} + 4q^{5} + 4q^{7} + 9q^{9} + 8q^{13} + 9q^{15} + 2q^{17} + 8q^{19} + 9q^{21} + ⋯ ∈ S_{14}(Γ_{0}(16); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 3q^{3} + 5q^{5} + 3q^{7} + 5q^{9} + 5q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(9)ω^{1} & 0 \\ 0 & u(5)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −468q^{3} + 56214q^{5} + −333032q^{7}+−1375299q^{9}+581580∗11q^{11}+15199742q^{13}+−26308152q^{15}+43114194q^{17}+365115484q^{19}+ 155858976q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q + 7q^{3} + 3q^{5} + 8q^{7} + 5q^{13} + 10q^{15} + 9q^{17} + 9q^{19} + q^{21} + ⋯ ∈ S_{14}(Γ_{0}(16); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 2q^{3} + q^{5} + 6q^{7} + 4q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(3)ω^{1} & 0 \\ 0 & u(4)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −1236q^{3} + −57450q^{5} + −64232q^{7} + −6057 ∗ 11q^{9} + −224052 ∗ 11q^{11} + 8032766q^{13} + 71008200q^{15} + 71112402q^{17} + −136337060q^{19} + 79390752q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

4. Consider f = q + 4q^{3} + 9q^{5} + 9q^{7} + 3q^{15} + 6q^{17} + 3q^{21} + ⋯ ∈ S_{14}(Γ_{0}(16); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{10} ∼ ρ_{g}, where g is the weight-4 newform g = q + 4q^{3} + 9q^{5} + 9q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{3}. The form f_{0} = q + aq^{3} + −12 ∗ a + 4006q^{5} + −222 ∗ a − 152256q^{7}+−872∗a+1414557q^{9}+531∗a−8005504q^{11}+10452∗a+13929358q^{13}+14470∗a−36106560q^{15}+ −38184∗a−93545038q^{17} +−8529∗11∗a+1678976∗11q^{19} +41328∗a−667971360q^{21} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2}+872x+−3008880. The slope of f_{0} is 1/2.

5. Consider f = q + 6q^{3} + 10q^{5} + 4q^{7} + 2q^{9} + 10q^{13} + 5q^{15} + 3q^{17} + 4q^{19} + 2q^{21} + ⋯ ∈ S_{16}(Γ_{0}(16); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + 7q^{3} + 6q^{5} + 5q^{7} + 7q^{9} + 5q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(9)ω^{1} & 0 \\ 0 & u(5)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −2700q^{3} + −251890q^{5}+−1374072q^{7}+−7058907q^{9}+3935156∗11q^{11}+−323161466q^{13}+680103000q^{15}+−191653646q^{17}+ 6515456644q^{19} + 3709994400q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

6. Consider f = q +q^{3} +q^{5} +8q^{7} +q^{15} +3q^{17} +8q^{21} +⋯ ∈ S_{16}(Γ_{0}(16); \overline{F}_{11}). We have ρ_{f} ⊗χ^{9} ∼ ρ_{g}, where g is the weight-4 newform g = q +4q^{3} +9q^{5} +9q^{7} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{15}. The form f_{0} = q + 3444q^{3} + 313358q^{5} + 2324616q^{7} + −226161 ∗ 11q^{9} + 456604 ∗ 11^{2}q^{11} + −10023598 ∗ 11q^{13} + 1079204952q^{15} + −2601428750q^{17} + −177465844 ∗ 11q^{19} + 8005977504q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 2.

7. Consider f = q + 7q^{3} + 2q^{5} + 10q^{7} + 4q^{9} + 2q^{13} + 3q^{15} + 2q^{17} + 7q^{19} + 4q^{21} + ⋯ ∈ S_{16}(Γ_{0}(16); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + 10q^{3} + 10q^{5} + 7q^{7} + 3q^{9} + 8q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(7)ω^{1} & 0 \\ 0 & u(8)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −6252q^{3} + 90510q^{5} +−56q^{7} +24738597q^{9} +8717268∗11q^{11} +−59782138q^{13} +−565868520q^{15} +−1355814414q^{17} + −3783593180q^{19} + 350112q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

8. Consider f = q + 9q^{5} + 9q^{7} + 10q^{9} + 10q^{13} + 2q^{17} + 3q^{19} + ⋯ ∈ S_{16}(Γ_{0}(16); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q+q^{5}+3q^{7}+2q^{9}+7q^{11}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{1} & 0 \\ 0 & u(7)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−29659132878511889008∗ 11+O(11^{2}0)q^{3}+308266706037187893917+O(11^{2}0)q^{5}+238458227305870040588+O(11^{2}0)q^{7}+189360097501490988236+ O(11^{2}0)q^{9} + −2366594861980184581 ∗ 11 + O(11^{2}0)q^{11} + −145163955559332709737 + O(11^{2}0)q^{13} + −29743906027638592201∗11+O(11^{2}0)q^{15}+−96112956299898508947+O(11^{2}0)q^{17}+−3473901814486932800936+ O(11^{2}1)q^{19} + 6385330492892579432 ∗ 11 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

(\begin{matrix}u(7)ω^{1} & 0 \\ 0 & u(8)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−237853258544855010238+ O(11^{2}0)q^{3}+143445619107540422809+O(11^{2}0)q^{5}+151011389797562871838+O(11^{2}0)q^{7}+280446157570764627678+ O(11^{2}0)q^{9}+6128292804387041005∗11+O(11^{2}0)q^{11}+136748187992773101625+O(11^{2}0)q^{13}+181024828818751938484+ O(11^{2}0)q^{15}+−335098537127383232363+O(11^{2}0)q^{17}+−779378992030693016008+O(11^{2}1)q^{19}+306402192853309311070+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

10. Consider f = q + 3q^{3} + 6q^{5} + 7q^{15} + ⋯ ∈ S_{18}(Γ_{0}(16); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + q^{3} + 10q^{5} + 10q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(10)ω^{1} & 0 \\ 0 & u(10)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{3} + 36 ∗ a + 196182q^{5} + 54 ∗ 11 ∗ a − 1311040 ∗ 11q^{7} + 5880 ∗ a + 207300141q^{9} + 315 ∗ 11^{2} ∗ a − 67443840 ∗ 11q^{11} + 14796 ∗ 11 ∗ a − 9409330 ∗ 11^{2}q^{13} + 407862 ∗ a + 12111850944q^{15} + 15912 ∗ 11 ∗ a − 1296700650 ∗ 11q^{17} + −260145∗11∗a−3832157056∗11q^{19} +−90320∗11^{2} ∗a+18167776416∗11q^{21} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −5880x + −336440304. The slope of f_{0} is 1.

11. Consider f = q + 6q^{3} + 4q^{5} + 3q^{7} + 5q^{9} + 10q^{13} + 2q^{15} + 6q^{17} + 2q^{19} + 7q^{21} + ⋯ ∈ S_{18}(Γ_{0}(16); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-6 newform g = q + 2q^{3} + 3q^{5} + 2q^{7} + 3q^{9} + 8q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(7)ω^{6} & 0 \\ 0 & u(8)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+27120212785074008683+ O(11^{2}0)q^{3}+−174075515412647438993+O(11^{2}0)q^{5}+151823049696759149382+O(11^{2}0)q^{7}+−203258971072395800392+ O(11^{2}0)q^{9}+1912073832274955453∗11+O(11^{2}0)q^{11}+318101605297731572008+O(11^{2}0)q^{13}+−321580757079223955300+ O(11^{2}0)q^{15}+269735114044000647664+O(11^{2}0)q^{17}+−220005465942469156519+O(11^{2}0)q^{19}+−173162390089023029182+ O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

12. Consider f = q + 8q^{3} + 6q^{5} + 5q^{7} + 5q^{9} + 3q^{13} + 4q^{15} + 5q^{17} + 3q^{19} + 7q^{21} + ⋯ ∈ S_{20}(Γ_{0}(16); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q + 10q^{3} + 10q^{5} + 7q^{7} + 3q^{9} + 8q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(7)ω^{8} & 0 \\ 0 & u(8)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 53028q^{3} + −5556930q^{5}+44496424q^{7}+1649707317q^{9}+−574606812∗11q^{11}+−33124973098q^{13}+−294672884040q^{15}+ −722355252174q^{17} +1312620671860q^{19} +2359556371872q^{21} +⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

13. Consider f = q + 5q^{5} + 10q^{7} + 7q^{9} + 4q^{13} + 5q^{17} + 6q^{19} + ⋯ ∈ S_{20}(Γ_{0}(16); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q+q^{5}+3q^{7}+2q^{9}+7q^{11}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{8} & 0 \\ 0 & u(7)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−28762896031311307819∗ 11 + O(11^{2}0)q^{3} + 879258209361647217272 + O(11^{2}1)q^{5} + 2856029319604017868398 + O(11^{2}1)q^{7} + 59689194141760551716+O(11^{2}0)q^{9}+17682337531174718495∗11+O(11^{2}0)q^{11}+−292846419252694870801+ O(11^{2}0)q^{13} + −30133086800404770211 ∗ 11 + O(11^{2}0)q^{15} + 88133675594113824405 + O(11^{2}0)q^{17} + 246354975700501317689 + O(11^{2}0)q^{19} + 8799218661165635436 ∗ 11 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

14. Consider f = q + 10q^{3} + 8q^{5} + 2q^{7} + 8q^{9} + 4q^{13} + 3q^{15} + 2q^{17} + 8q^{19} + 9q^{21} + ⋯ ∈ S_{20}(Γ_{0}(16); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-8 newform g = q + 7q^{3} + 6q^{5} + 5q^{7} + 7q^{9} + 5q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(9)ω^{8} & 0 \\ 0 & u(5)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−3012953843206747342+ O(11^{2}0)q^{3}+237341441956642283587+O(11^{2}0)q^{5}+−272316958307207048365+O(11^{2}0)q^{7}+151616048286617080251+ O(11^{2}0)q^{9} + −13055498409929419156 ∗ 11 + O(11^{2}0)q^{11} + 293985161745674099474 + O(11^{2}0)q^{13} + 6080281819353922040+O(11^{2}0)q^{15}+−106390654215567499567+O(11^{2}0)q^{17}+−133460784114714809888+ O(11^{2}0)q^{19} + 241850928540593178288 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

15. Consider f = q + 9q^{3} + 3q^{5} + 4q^{7} + 5q^{15} + 2q^{17} + 3q^{21} + ⋯ ∈ S_{20}(Γ_{0}(16); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 4q^{3} + 9q^{5} + 9q^{7} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{99}. The form f_{0} = q + aq^{3} + −(71739913100518356081 + O(11^{2}0)) ∗ a − 286771297198932320206 + O(11^{2}0)q^{5} + −(89298272145247099096 + O(11^{2}0)) ∗ a + 319324010414393401097+O(11^{2}0)q^{7}+(3012953843206723610+O(11^{2}0))∗a+40305095844010507117+ O(11^{2}0)q^{9}+−(261868908749327877655+O(11^{2}0))∗a−306917766934413219031+O(11^{2}0)q^{11}+(146960528936824745075+ O(11^{2}0))∗a+283648462680551198426+O(11^{2}0)q^{13}+(49429855242292176837+O(11^{2}1))∗a−2114529052840512092740+ O(11^{2}1)q^{15}+(5222660501278696115+O(11^{2}0))∗a−183128850258256009838+O(11^{2}0)q^{17}+(9252652515355046232∗ 11+O(11^{2}0))∗a−526357107575210889∗11^{2} +O(11^{2}0)q^{19} +−(47007052107242204452+O(11^{2}0))∗a+ 61024949400868555464 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} +−3012953843206723610+O(11^{2}0)x+−40305095845172768584+ O(11^{2}0). The slope of f_{0} is 1/2.

16. Consider f = q + 6q^{3} + 5q^{5} + 9q^{7} + 8q^{13} + 8q^{15} + 7q^{17} + 8q^{19} + 10q^{21} + ⋯ ∈ S_{22}(Γ_{0}(16); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + 2q^{3} + q^{5} + 6q^{7} + 4q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(3)ω^{0} & 0 \\ 0 & u(4)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −71604q^{3} + −28693770q^{5}+853202392q^{7}+−484838217∗11q^{9}+−7884652692∗11q^{11}+−895323442786q^{13}+2054588707080q^{15}+ 3257566804818q^{17} +−23032467644420q^{19} +−61092704076768q^{21} +⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

17. Consider f = q + 9q^{3} + 3q^{5} + 10q^{7} + q^{9} + 4q^{13} + 5q^{15} + 4q^{17} + q^{19} + 2q^{21} + ⋯ ∈ S_{22}(Γ_{0}(16); \overline{F}_{11}). We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + 3q^{3} + 5q^{5} + 3q^{7} + 5q^{9} + 5q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(9)ω^{0} & 0 \\ 0 & u(5)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+197815801794667479828+ O(11^{2}0)q^{3}+10576129841434816530+O(11^{2}0)q^{5}+186431864848526734843+O(11^{2}0)q^{7}+118422391371438141362+ O(11^{2}0)q^{9}+24976012745428922040∗11+O(11^{2}0)q^{11}+298761524669349109867+O(11^{2}0)q^{13}+310158248916494961548+ O(11^{2}0)q^{15}+66120954489554434749+O(11^{2}0)q^{17}+−39625584201932371499+O(11^{2}0)q^{19}+−3351172969707456781248+ O(11^{2}1)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

18. Consider f = q+5q^{3}+4q^{5}+6q^{7}+9q^{15}+q^{17}+8q^{21}+⋯ ∈ S_{22}(Γ_{0}(16); \overline{F}_{11}). We have ρ_{f} ⊗χ^{1} ∼ ρ_{g}, where g is the weight-4 newform g = q +4q^{3} +9q^{5} +9q^{7} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{111}. The form f_{0} = q + −192011347013165958526 + O(11^{2}0)q^{3} + 196273029332040884436 + O(11^{2}0)q^{5} + 8163195851865990623 + O(11^{2}0)q^{7} + 22955383498403253067 ∗ 11+O(11^{2}0)q^{9}+650348885411941718∗11^{2}+O(11^{2}0)q^{11}+−117066435239381991577∗11+O(11^{2}1)q^{13}+ −94228476522828356579+O(11^{2}0)q^{15}+−162433658798202972476+O(11^{2}0)q^{17}+−26058816584734320422∗ 11 + O(11^{2}0)q^{19} + −235635184637186998876 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 2.

19. Consider f = q + 4q^{3} + 3q^{5} + 4q^{7} + 2q^{9} + 2q^{13} + q^{15} + 9q^{17} + q^{19} + 5q^{21} + ⋯ ∈ S_{22}(Γ_{0}(16); \overline{F}_{11}). Also, z satisfies the equation x^{2} + 7x + 2 = 0. We have ρ_{f} ⊗ χ^{9} ∼ ρ_{g}, where g is the weight-10 newform g = q + 5q^{3} + 5q^{5} + 10q^{7} + 10q^{9} + 10q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(10)ω^{0} & 0 \\ 0 & u(10)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+54482294323998791782+ O(11^{2}0)q^{3}+325033101411364126993+O(11^{2}0)q^{5}+36661258535087029210+O(11^{2}0)q^{7}+165475433795445166417+ O(11^{2}0)q^{9} + −16179817711094748369 ∗ 11 + O(11^{2}0)q^{11} + −322845341485774539144 + O(11^{2}0)q^{13} + 329062144376499511957+O(11^{2}0)q^{15}+267123841190985321431+O(11^{2}0)q^{17}+−70862335466729695272+ O(11^{2}0)q^{19} + 180401648425834546243 + O(11^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

20. Consider f = q + 10q3 + 8q5 + 10q7 + 7q9 + 6q13 + 3q15 + 9q17 + 5q19 + q21 + ⋯ ∈ S14(Γ0(16); \overline{F}11). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

21. Consider f = q + q3 + 10q5 + 10q11 + 10q15 + ⋯ ∈ S16(Γ0(16); \overline{F}11). Since f is ordinary, we do not continue our analysis.

22. Consider f = q + 4q3 + 3q5 + q7 + 4q9 + 7q13 + q15 + 4q17 + 10q19 + 4q21 + ⋯ ∈ S16(Γ0(16); \overline{F}11). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

23. Consider f = q + 2q3 + 3q5 + 2q7 + 3q9 + 8q11 + 5q13 + 6q15 + q17 + 3q19 + 4q21 + ⋯ ∈ S16(Γ0(16); \overline{F}11). Since f is ordinary, we do not continue our analysis.

24. Consider f = q + 5q3 + 10q5 + 5q9 + 6q15 + ⋯ ∈ S18(Γ0(16); \overline{F}11). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

25. Consider f = q + 10q3 + 10q5 + 7q7 + 3q9 + 8q11 + 7q13 + q15 + 10q17 + 10q19 + 4q21 + ⋯ ∈ S18(Γ0(16); \overline{F}11). Since f is ordinary, we do not continue our analysis.

26. Consider f = q + q5 + 3q7 + 2q9 + 7q11 + 2q13 + 10q17 + 9q19 + ⋯ ∈ S18(Γ0(16); \overline{F}11). Since f is ordinary, we do not continue our analysis.

27. Consider f = q + 7q3 + 6q5 + 5q7 + 7q9 + 5q11 + 2q13 + 9q15 + 4q17 + q19 + 2q21 + ⋯ ∈ S18(Γ0(16); \overline{F}11). Since f is ordinary, we do not continue our analysis.

28. Consider f = q + 3q3 + 5q5 + 3q7 + 5q9 + 5q11 + 2q13 + 4q15 + 8q17 + 7q19 + 9q21 + ⋯ ∈ S20(Γ0(16); \overline{F}11). Since f is ordinary, we do not continue our analysis.

29. Consider f = q + 2q3 + q5 + 6q7 + 4q11 + 4q13 + 2q15 + 3q17 + q19 + q21 + ⋯ ∈ S20(Γ0(16); \overline{F}11). Since f is ordinary, we do not continue our analysis.

30. Consider f = q + 3q3 + 9q5 + 6q7 + 5q9 + 5q13 + 5q15 + 10q17 + 9q19 + 7q21 + ⋯ ∈ S20(Γ0(16); \overline{F}11). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

31. Consider f = q + 5q3 + 5q5 + 10q7 + 10q9 + 10q11 + q13 + 3q15 + 7q17 + 7q19 + 6q21 + ⋯ ∈ S20(Γ0(16); \overline{F}11). Since f is ordinary, we do not continue our analysis.

32. Consider f = q + 7q3 + 6q5 + 3q7 + 2q9 + 3q13 + 9q15 + 7q17 + 2q19 + 10q21 + ⋯ ∈ S22(Γ0(16); \overline{F}11). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

33. Consider f = q + q3 + q5 + 2q7 + 9q9 + 10q11 + 4q13 + q15 + 9q17 + 2q21 + ⋯ ∈ S22(Γ0(16); \overline{F}11). Since f is ordinary, we do not continue our analysis.

34. Consider f = q + 10q3 + 8q5 + 9q7 + 9q9 + q11 + 7q13 + 3q15 + 6q17 + 3q19 + 2q21 + ⋯ ∈ S22(Γ0(16); \overline{F}11). Since f is ordinary, we do not continue our analysis.

35. Consider f = q + 3q3 + 8q5 + 2q7 + 6q9 + q11 + 2q15 + 5q17 + 7q19 + 6q21 + ⋯ ∈ S22(Γ0(16); \overline{F}11). Since f is ordinary, we do not continue our analysis.

36. Consider f = q + z37q3 + z116q5 + z109q7 + z76q9 + q11 + z59q13 + z33q15 + 2q17 + 4q19 + z26q21 + ⋯ ∈ S22(Γ0(16); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

37. Consider f = q + z47q3 + z76q5 + z119q7 + z116q9 + q11 + z49q13 + z3q15 + 2q17 + 4q19 + z46q21 + ⋯ ∈ S22(Γ0(16); \overline{F}11). Also, z satisfies the equation x2 + 7x + 2 = 0. Since f is ordinary, we do not continue our analysis.

13 2 ≤ N ≤ 10, ℓ = 13

13.1 N = 2, ℓ = 13 (10 forms)

1. Consider f = q + 9q^{2} + 3q^{4} + 12q^{5} + 12q^{7} + q^{8} + 4q^{9} + 4q^{10} + 10q^{11} + 4q^{14} + 9q^{16} + 9q^{17} + 10q^{18} + ⋯ ∈ S_{18}(Γ_{0}(2); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 3q^{2} + 9q^{4} + 12q^{5} + 10q^{7} + q^{8} + 12q^{9} + 10q^{10} + 12q^{11} + 7q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(7)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 256q^{2} + 36 ∗ 13^{2}q^{3} + 65536q^{4} + 1255110q^{5} + 9216 ∗ 13^{2}q^{6} + −22465912q^{7} + 16777216q^{8} + −92125107q^{9} + 321308160q^{10} + 172399692q^{11}+2359296∗13^{2}q^{12}+−167703802∗13q^{13}+−5751273472q^{14}+45183960∗13^{2}q^{15}+4294967296q^{16}+ 30163933458q^{17} + −23584027392q^{18} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 8q^{2} + 12q^{3} + 12q^{4} + 2q^{5} + 5q^{6} + 11q^{7} + 5q^{8} + 11q^{9} + 3q^{10} + q^{12} + 10q^{14} + 11q^{15} + q^{16} + 3q^{17} + 10q^{18} + ⋯ ∈ S_{20}(Γ_{0}(2); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + 5q^{2} + 12q^{3} + 12q^{4} + 11q^{5} + 8q^{6} + 2q^{7} + 8q^{8} + 11q^{9} + 3q^{10} + q^{12} + 4q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(10)ω^{1} & 0 \\ 0 & u(4)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−512q^{2}+−13092q^{3}+ 262144q^{4} + 6546750q^{5} + 6703104q^{6} + 96674264q^{7} + −134217728q^{8} + −990861003q^{9} + −3351936000q^{10} + 907668804 ∗ 13q^{11} + −3431989248q^{12} + 2646286766 ∗ 13q^{13} + −49497223168q^{14} + −85710051000q^{15} + 68719476736q^{16} + −400697609166q^{17} + 507320833536q^{18} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q + 10q^{2} + 10q^{3} + 9q^{4} + 3q^{5} + 9q^{6} + q^{7} + 12q^{8} + 8q^{9} + 4q^{10} + 12q^{12} + 10q^{14} + 4q^{15} + 3q^{16} + 12q^{17} + 2q^{18} + ⋯ ∈ S_{22}(Γ_{0}(2); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-8 newform g = q + 5q^{2} + 12q^{3} + 12q^{4} + 11q^{5} + 8q^{6} + 2q^{7} + 8q^{8} + 11q^{9} + 3q^{10} + q^{12} + 4q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(10)ω^{8} & 0 \\ 0 & u(4)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 1024q^{2} + 59316q^{3} + 1048576q^{4}+4975350q^{5}+60739584q^{6}+1427425832q^{7}+1073741824q^{8}+−6941965347q^{9}+5094758400q^{10}+ −631762692∗13^{2}q^{11}+62197334016q^{12}+−11550043498∗13q^{13}+1461684051968q^{14}+295117860600q^{15}+ 1099511627776q^{16} + −11203980739758q^{17} + −7108572515328q^{18} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

4. Consider f = q + 6q^{2} + 10q^{4} + 8q^{5} + 5q^{7} + 8q^{8} + 4q^{9} + 9q^{10} + 2q^{11} + 4q^{14} + 9q^{16} + 4q^{17} + 11q^{18} + ⋯ ∈ S_{24}(Γ_{0}(2); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-10 newform g = q + 3q^{2} + 9q^{4} + 12q^{5} + 10q^{7} + q^{8} + 12q^{9} + 10q^{10} + 12q^{11} + 7q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{10} & 0 \\ 0 & u(7)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −2048q^{2} + −38916 ∗ 13q^{3} + 4194304q^{4} + −90135570q^{5} + 79699968 ∗ 13q^{6} + 6872255096q^{7} + −8589934592q^{8} + 161799725637q^{9} + 184597647360q^{10} + −965328798588q^{11} + −163225534464 ∗ 13q^{12} + 41719999934 ∗ 13q^{13} + −14074378436608q^{14} + 3507715842120 ∗ 13q^{15} + 17592186044416q^{16} + 82083537265266q^{17} + −331365838104576q^{18} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

5. Consider f = q + 2q2 + 12q3 + 4q4 + 4q5 + 11q6 + 4q7 + 8q8 + 8q10 + 2q11 + 9q12 + 8q14 + 9q15 + 3q16 + 7q17 + ⋯ ∈ S16(Γ0(2); \overline{F}13). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

6. Consider f = q + 5q2 + 12q3 + 12q4 + 11q5 + 8q6 + 2q7 + 8q8 + 11q9 + 3q10 + q12 + 4q13 + 10q14 + 2q15 + q16 + 3q17 + 3q18 + ⋯ ∈ S20(Γ0(2); \overline{F}13). Since f is ordinary, we do not continue our analysis.

7. Consider f = q + 3q2 + 9q4 + 12q5 + 10q7 + q8 + 12q9 + 10q10 + 12q11 + 7q13 + 4q14 + 3q16 + q17 + 10q18 + ⋯ ∈ S22(Γ0(2); \overline{F}13). Since f is ordinary, we do not continue our analysis.

8. Consider f = q +12q2 +q3 +q4 +10q5 +12q6 +12q7 +12q8 +11q9 +3q10 +6q11 +q12 +q13 +q14 +10q15 +q16 +10q17 +2q18 +⋯ ∈ S26(Γ0(2); \overline{F}13). Since f is ordinary, we do not continue our analysis.

9. Consider f = q +q2 +10q3 +q4 +12q5 +10q6 +q7 +q8 +6q9 +12q10 +11q11 +10q12 +12q13 +q14 +3q15 +q16 +10q17 +6q18 +⋯ ∈ S26(Γ0(2); \overline{F}13). Since f is ordinary, we do not continue our analysis.

10. Consider f = q + q2 + 4q3 + q4 + 6q5 + 4q6 + 8q7 + q8 + 6q10 + 12q11 + 4q12 + 8q14 + 11q15 + q16 + 5q17 + ⋯ ∈ S26(Γ0(2); \overline{F}13). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

13.2 N = 3, ℓ = 13 (20 forms)

1. Consider f = q + 10q^{3} + 5q^{4} + q^{5} + 2q^{7} + 9q^{9} + 3q^{11} + 11q^{12} + 10q^{15} + 12q^{16} + 9q^{19} + 5q^{20} + 7q^{21} + ⋯ ∈ S_{16}(Γ_{0}(3); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{10} ∼ ρ_{g}, where g is the weight-12 newform g = q + 4q^{3} + 6q^{4} + 12q^{5} + 8q^{7} + 3q^{9} + 4q^{11} + 11q^{12} + 7q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(7)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−18∗13q^{2}+−2187q^{3}+ 21988q^{4} + 280710q^{5} + 39366 ∗ 13q^{6} + −1373344q^{7} + 194040 ∗ 13q^{8} + 4782969q^{9} + −5052780 ∗ 13q^{10} + 34031052q^{11}+−48087756q^{12}+29540174∗13q^{13}+24720192∗13q^{14}+−613912770q^{15}+−1310772464q^{16}+ 96862122 ∗ 13q^{17} + −86093442 ∗ 13q^{18} + −2499071020q^{19} + 6172251480q^{20} + 3003503328q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q+9q^{2} +4q^{3} +10q^{4} +12q^{5} +10q^{6} +6q^{7} +10q^{8} +3q^{9} +4q^{10} +8q^{11} +q^{12} +2q^{14} +9q^{15} +4q^{16} + 5q^{17} + q^{18} + 8q^{19} + 3q^{20} + 11q^{21} + ⋯ ∈ S_{18}(Γ_{0}(3); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 3q^{2} + 10q^{3} + 4q^{4} + 12q^{5} + 4q^{6} + 5q^{7} + 10q^{8} + 9q^{9} + 10q^{10} + 7q^{11} + q^{12} + 6q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(11)ω^{1} & 0 \\ 0 & u(6)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 204q^{2} + −6561q^{3} + −89456q^{4} + −163554q^{5} + −1338444q^{6} + −20846560q^{7} + −44987712q^{8} + 43046721q^{9} + −33365016q^{10} + 817372356q^{11} +586920816q^{12} +23045366∗13q^{13} +−4252698240q^{14} +1073077794q^{15} +2547683584q^{16} + −44775606078q^{17} + 8781531084q^{18} + 78748651964q^{19} + 14630886624q^{20} + 136774280160q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q + 2q^{2} + 9q^{3} + 11q^{4} + 4q^{5} + 5q^{6} + 5q^{7} + 10q^{8} + 3q^{9} + 8q^{10} + 8q^{11} + 8q^{12} + 10q^{14} + 10q^{15} + 6q^{16} + 5q^{17} + 6q^{18} + 2q^{19} + 5q^{20} + 6q^{21} + ⋯ ∈ S_{18}(Γ_{0}(3); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q +5q^{2} +3q^{3} +7q^{4} +4q^{5} +2q^{6} +2q^{7} +10q^{8} +9q^{9} +7q^{10} +7q^{11} +8q^{12} +q^{13} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+6533394632131479889766+ O(13^{2}0)q^{2}+6561q^{3}+3823801410415968917440+O(13^{2}0)q^{4}+−1542937497041889790240+O(13^{2}0)q^{5}+ 9408869058436822258471+O(13^{2}0)q^{6}+6128449026840311645739+O(13^{2}0)q^{7}+8675214794001793095263+ O(13^{2}0)q^{8}+43046721q^{9}+6945025549243229718486+O(13^{2}0)q^{10}+56162924892324301556032+O(13^{2}1)q^{11}+ 1408870896516808106520+O(13^{2}0)q^{12}+−52642131997196450296∗13+O(13^{2}0)q^{13}+−5111726847205962718946+ O(13^{2}0)q^{14}+6432773919627187116293+O(13^{2}0)q^{15}+6447501208163510051225+O(13^{2}0)q^{16}+7815365607506701953090+ O(13^{2}0)q^{17}+3467551591154260602583+O(13^{2}0)q^{18}+−3038300688986688448893+O(13^{2}0)q^{19}+8112039492467298620510+ O(13^{2}0)q^{20} + −5749282548486904809337 + O(13^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

4. Consider f = q + q^{2} + q^{3} + 3q^{4} + 4q^{5} + q^{6} + 6q^{7} + 5q^{8} + q^{9} + 4q^{10} + 10q^{11} + 3q^{12} + 6q^{14} + 4q^{15} + 11q^{16} + 5q^{17} + q^{18} + 5q^{19} + 12q^{20} + 6q^{21} + ⋯ ∈ S_{20}(Γ_{0}(3); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-6 newform g = q + 7q^{2} + 9q^{3} + 4q^{4} + 6q^{5} + 11q^{6} + 12q^{7} + 12q^{8} + 3q^{9} + 3q^{10} + 8q^{11} + 10q^{12} + q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{6} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −1104q^{2} + 19683q^{3} + 694528q^{4} + 3516270q^{5} + −21730032q^{6} + −195590584q^{7} + −187944960q^{8} + 387420489q^{9} + −3881962080q^{10} + −2746857948q^{11} + 13670394624q^{12} + −3415418866 ∗ 13q^{13} + 215932004736q^{14} + 69210742410q^{15} + −156641460224q^{16} + −785982517614q^{17} + −427712219856q^{18} + 315410465180q^{19} + 2442147970560q^{20} + −3849809464872q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

5. Consider f = q + 7q^{2} + 12q^{3} + 12q^{4} + 6q^{6} + 12q^{7} + 7q^{8} + q^{9} + 12q^{11} + q^{12} + 6q^{14} + 8q^{16} + 7q^{17} + 7q^{18} + q^{19} + q^{21} + ⋯ ∈ S_{20}(Γ_{0}(3); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + 6q^{2} + 12q^{3} + 12q^{4} + 7q^{6} + q^{7} + 6q^{8} + q^{9} + q^{11} + q^{12} + 11q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{1} & 0 \\ 0 & u(11)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−6747558085137094074388+ O(13^{2}0)q^{2} + −19683q^{3} + −42559723370682578696689 + O(13^{2}1)q^{4} + 7347283092541032338224 ∗ 13 + O(13^{2}1)q^{5}+5498930886396187837616+O(13^{2}0)q^{6}+−803958468900441323964+O(13^{2}0)q^{7}+1557510324270151944999+ O(13^{2}0)q^{8}+387420489q^{9}+−8455029991767779053689∗13+O(13^{2}1)q^{10}+−3326006659352169235353+ O(13^{2}0)q^{11} +−92782982925085173734896+O(13^{2}1)q^{12} +−324133456109013333717∗13+O(13^{2}0)q^{13} + −79050543617869195250119+O(13^{2}1)q^{14}+−7803747417136583426183∗13+O(13^{2}1)q^{15}+584967883585102597254605+ O(13^{2}2)q^{16} + 118829380733358305297788 + O(13^{2}1)q^{17} + −2187938990012403824033 + O(13^{2}0)q^{18} + 2732136477488455389+O(13^{2}0)q^{19}+−8166293125988553999268∗13+O(13^{2}1)q^{20}+−6820281108319352937821+ O(13^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

6. Consider f = q + 12q^{2} + 10q^{3} + 9q^{4} + 3q^{6} + 7q^{7} + 9q^{8} + 9q^{9} + 11q^{11} + 12q^{12} + 6q^{14} + 11q^{16} + 2q^{17} + 4q^{18} + 7q^{19} + 5q^{21} + ⋯ ∈ S_{22}(Γ_{0}(3); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-8 newform g = q + 6q^{2} + 12q^{3} + 12q^{4} + 7q^{6} + q^{7} + 6q^{8} + q^{9} + q^{11} + q^{12} + 11q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{8} & 0 \\ 0 & u(11)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+1728q^{2} +−59049q^{3} + 888832q^{4}+−3193290∗13q^{5}+−102036672q^{6}+538429808q^{7}+−2087976960q^{8}+3486784401q^{9}+−5518005120∗ 13q^{10}+−64113040188q^{11}+−52484640768q^{12}+−10075392922∗13q^{13}+930406708224q^{14}+188560581210∗ 13q^{15}+−5472039993344q^{16}+8242029723618q^{17}+6025163444928q^{18}+13492101753020q^{19}+−2838298337280∗ 13q^{20} + −31793741732592q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

7. Consider f = q + 11q^{2} + 3q^{3} + 12q^{4} + 6q^{5} + 7q^{6} + 10q^{7} + 12q^{8} + 9q^{9} + q^{10} + 7q^{11} + 10q^{12} + 6q^{14} + 5q^{15} + 7q^{16} + 7q^{17} + 8q^{18} + 9q^{19} + 7q^{20} + 4q^{21} + ⋯ ∈ S_{22}(Γ_{0}(3); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q+7q^{2}+9q^{3}+4q^{4}+6q^{5}+11q^{6}+12q^{7}+12q^{8}+3q^{9}+3q^{10}+8q^{11}+10q^{12}+q^{13}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+5466151690893158575087+ O(13^{2}0)q^{2}+59049q^{3}+−8496018642269880874010+O(13^{2}0)q^{4}+7377492453496132559208+O(13^{2}0)q^{5}+ 9491406749503831154880+O(13^{2}0)q^{6}+1792409175580618195143+O(13^{2}0)q^{7}+−6704706229958762072295+ O(13^{2}0)q^{8} + 3486784401q^{9} + −8825601452609899112765 + O(13^{2}0)q^{10} + −25009939063780530177428 + O(13^{2}1)q^{11} + −7376041865732943386493 + O(13^{2}0)q^{12} + 379367062299107008261 ∗ 13 + O(13^{2}0)q^{13} + −4550457299176469102143+O(13^{2}0)q^{14}+1772238675446752476670+O(13^{2}0)q^{15}+1577906784961885665021+ O(13^{2}0)q^{16}+1045474998125928907635+O(13^{2}0)q^{17}+1695430216950414267630+O(13^{2}0)q^{18}+4460561882471771995967+ O(13^{2}0)q^{19} + −6275045986857869274852 + O(13^{2}0)q^{20} + 1326146548751730316238 + O(13^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

8. Consider f = q + 10q^{2} + 9q^{3} + 2q^{4} + 7q^{5} + 12q^{6} + q^{7} + 2q^{8} + 3q^{9} + 5q^{10} + 12q^{11} + 5q^{12} + 10q^{14} + 11q^{15} + 6q^{16} + 8q^{17} + 4q^{18} + 10q^{19} + q^{20} + 9q^{21} + ⋯ ∈ S_{24}(Γ_{0}(3); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-10 newform g = q +5q^{2} +3q^{3} +7q^{4} +4q^{5} +2q^{6} +2q^{7} +10q^{8} +9q^{9} +7q^{10} +7q^{11} +8q^{12} +q^{13} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{10} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +1128q^{2} +177147q^{3} + −7116224q^{4} + −48863730q^{5} + 199821816q^{6} + −1723688680q^{7} + −17489450496q^{8} + 31381059609q^{9} + −55118287440q^{10}+−1428263180124q^{11}+−1260617732928q^{12}+−632381849602∗13q^{13}+−1944320831040q^{14}+ −8656063178310q^{15}+39967113416704q^{16}+−5989210330446q^{17}+35397835238952q^{18}+680005481275676q^{19}+ 347725248155520q^{20} + −305346278595960q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

9. Consider f = q+6q^{2} +4q^{3} +3q^{4} +8q^{5} +11q^{6} +9q^{7} +2q^{8} +3q^{9} +9q^{10} +12q^{11} +12q^{12} +2q^{14} +6q^{15} +4q^{16} + 8q^{17} +5q^{18} +q^{19} +11q^{20} +10q^{21} +⋯ ∈ S_{24}(Γ_{0}(3); \overline{F}_{13}). We have ρ_{f} ⊗χ^{11} ∼ ρ_{g}, where g is the weight-10 newform g = q + 3q^{2} + 10q^{3} + 4q^{4} + 12q^{5} + 4q^{6} + 5q^{7} + 10q^{8} + 9q^{9} + 10q^{10} + 7q^{11} + q^{12} + 6q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(11)ω^{10} & 0 \\ 0 & u(6)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+7999190284076529919946+ O(13^{2}0)q^{2}+−177147q^{3}+4601721439607941919351+O(13^{2}0)q^{4}+−2824075798267790349769+O(13^{2}0)q^{5}+ −3457234417758772232701+O(13^{2}0)q^{6}+−6855991533239110890125+O(13^{2}0)q^{7}+2201679203470269506050+ O(13^{2}0)q^{8} + 31381059609q^{9} + −3047782701909282676616 + O(13^{2}0)q^{10} + 3625538007373198488259 + O(13^{2}0)q^{11} + −1236666265956858780304 + O(13^{2}0)q^{12} + −641482261434181347247 ∗ 13 + O(13^{2}0)q^{13} + −6145206873657502223558+O(13^{2}0)q^{14}+8893989556973462970320+O(13^{2}0)q^{15}+−5296669873604408233255+ O(13^{2}0)q^{16}+4361411687787850103520+O(13^{2}0)q^{17}+3747757179462790921822+O(13^{2}0)q^{18}+9110235842523878111492+ O(13^{2}0)q^{19} + 5085087935422918591751 + O(13^{2}0)q^{20} + 6122104951288716395470 + O(13^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

10. Consider f = q + 12q^{3} + 11q^{4} + 8q^{5} + 4q^{7} + q^{9} + 5q^{11} + 2q^{12} + 5q^{15} + 4q^{16} + 8q^{19} + 10q^{20} + 9q^{21} + ⋯ ∈ S_{26}(Γ_{0}(3); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-12 newform g = q + 4q^{3} + 6q^{4} + 12q^{5} + 8q^{7} + 3q^{9} + 4q^{11} + 11q^{12} + 7q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{0} & 0 \\ 0 & u(7)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+644367760420693768319∗ 13 + O(13^{2}0)q^{2} + −531441q^{3} + 3632812915992180340275 + O(13^{2}0)q^{4} + 2325514655030757746369 + O(13^{2}0)q^{5} + −314307608669072015845 ∗ 13 + O(13^{2}0)q^{6} + 1166740301787361687104 + O(13^{2}0)q^{7} + −702100192985395702497∗13+O(13^{2}0)q^{8}+282429536481q^{9}+−562062436975633545577∗13+O(13^{2}0)q^{10}+ 2613135682245105670905+O(13^{2}0)q^{11}+−6483816534301225486690+O(13^{2}0)q^{12}+−682759059343875729066∗ 13+O(13^{2}0)q^{13} +134779380892237488549∗13+O(13^{2}0)q^{14} +−44467477420782297500+O(13^{2}0)q^{15} + −9087606788716508980768+O(13^{2}0)q^{16}+195076649034533838412∗13+O(13^{2}0)q^{17}+−138679017420541050021∗ 13 + O(13^{2}0)q^{18} + −239624546601476546697 + O(13^{2}0)q^{19} + 2355175398120143292938 + O(13^{2}0)q^{20} + 2315397083680134084562 + O(13^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

11. Consider f = q +6q2 +3q3 +2q4 +4q5 +5q6 +4q7 +3q8 +9q9 +11q10 +2q11 +6q12 +11q14 +12q15 +2q16 +7q17 +2q18 +3q19 +8q20 +12q21 +⋯ ∈ S16(Γ0(3); \overline{F}13). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

12. Consider f = q+7q2+9q3+4q4+6q5+11q6+12q7+12q8+3q9+3q10+8q11+10q12+q13+6q14+2q15+8q16+11q17+8q18+3q19+11q20+4q21+⋯ ∈ S18(Γ0(3); \overline{F}13). Since f is ordinary, we do not continue our analysis.

13. Consider f = q + 6q2 + 12q3 + 12q4 + 7q6 + q7 + 6q8 + q9 + q11 + q12 + 11q13 + 6q14 + 8q16 + 7q17 + 6q18 + 12q19 + 12q21 + ⋯ ∈ S20(Γ0(3); \overline{F}13). Since f is ordinary, we do not continue our analysis.

14. Consider f = q+3q2+10q3+4q4+12q5+4q6+5q7+10q8+9q9+10q10+7q11+q12+6q13+2q14+3q15+10q16+2q17+q18+11q19+9q20+11q21+⋯ ∈ S22(Γ0(3); \overline{F}13). Since f is ordinary, we do not continue our analysis.

15. Consider f = q+5q2 +3q3 +7q4 +4q5 +2q6 +2q7 +10q8 +9q9 +7q10 +7q11 +8q12 +q13 +10q14 +12q15 +2q16 +2q17 +6q18 +6q19 +2q20 +6q21 +⋯ ∈ S22(Γ0(3); \overline{F}13). Since f is ordinary, we do not continue our analysis.

16. Consider f = q + 4q3 + 6q4 + 12q5 + 8q7 + 3q9 + 4q11 + 11q12 + 7q13 + 9q15 + 10q16 + 10q19 + 7q20 + 6q21 + ⋯ ∈ S24(Γ0(3); \overline{F}13). Since f is ordinary, we do not continue our analysis.

17. Consider f = q + q2 + 12q3 + 12q4 + 2q5 + 12q6 + 9q7 + 10q8 + q9 + 2q10 + 4q11 + q12 + q13 + 9q14 + 11q15 + 12q16 + 2q17 + q18 + 11q20 + 4q21 + ⋯ ∈ S26(Γ0(3); \overline{F}13). Since f is ordinary, we do not continue our analysis.

18. Consider f = q + z102q2 + q3 + z89q4 + z21q5 + z102q6 + z105q7 + z167q8 + q9 + z123q10 + 11q11 + z89q12 + 12q13 + z39q14 + z21q15 + 3q16 + z163q17 + z102q18 + z21q19 + z110q20 + z105q21 + ⋯ ∈ S26(Γ0(3); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

19. Consider f = q + z150q2 + q3 + z149q4 + z105q5 + z150q6 + z21q7 + z155q8 + q9 + z87q10 + 11q11 + z149q12 + 12q13 + z3q14 + z105q15 + 3q16 + z103q17 + z150q18 + z105q19 + z86q20 + z21q21 + ⋯ ∈ S26(Γ0(3); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

20. Consider f = q + 3q2 + q3 + 7q4 + 6q5 + 3q6 + 8q7 + 2q8 + q9 + 5q10 + 12q11 + 7q12 + 11q14 + 6q15 + 5q16 + 5q17 + 3q18 + 7q19 + 3q20 + 8q21 + ⋯ ∈ S26(Γ0(3); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

13.3 N = 4, ℓ = 13 (10 forms)

1. Consider f = q + 10q^{3} + 2q^{7} + 8q^{9} + 11q^{11} + q^{17} + 7q^{19} + 7q^{21} + 9q^{23} + 5q^{25} + ⋯ ∈ S_{16}(Γ_{0}(4); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{10} ∼ ρ_{g}, where g is the weight-12 newform g = q + 4q^{3} + 8q^{7} + 7q^{9} + 6q^{11} + 11q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{1} & 0 \\ 0 & u(11)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −276q^{3} + −10170 ∗ 13q^{5}+−3585736q^{7}+−14272731q^{9}+47801700q^{11}+19060382∗13q^{13}+2806920∗13q^{15}+−2127682062q^{17}+ −1074862756q^{19} + 989663136q^{21} + 24982896168q^{23} + −13038094025q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 8q^{3} + 10q^{5} + q^{7} + 3q^{9} + 2q^{15} + 9q^{17} + 6q^{19} + 8q^{21} + 11q^{23} + 4q^{25} + ⋯ ∈ S_{18}(Γ_{0}(4); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 7q^{3} + 10q^{5} + 3q^{7} + 9q^{9} + 6q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(11)ω^{1} & 0 \\ 0 & u(6)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+1286516409746090632052+ O(13^{2}0)q^{3}+−8304663201097663680088+O(13^{2}0)q^{5}+3992196393945872308288+O(13^{2}0)q^{7}+−740906904454532522821+ O(13^{2}0)q^{9} + 212801406137125418280 ∗ 13 + O(13^{2}0)q^{11} + 648160538609243686552 ∗ 13 + O(13^{2}0)q^{13} + −43903840148628846031332+O(13^{2}1)q^{15}+−105754244764442019312571+O(13^{2}1)q^{17}+617212345240815903988+ O(13^{2}0)q^{19}+−8612919168897456738590+O(13^{2}0)q^{21}+3874938670922714935421+O(13^{2}0)q^{23}+−2785300275674232115146+ O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q + 3q^{3} + 10q^{5} + 8q^{7} + 6q^{9} + 12q^{11} + 4q^{15} + 10q^{17} + 11q^{19} + 11q^{21} + q^{23} + q^{25} + ⋯ ∈ S_{20}(Γ_{0}(4); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-6 newform g = q + q^{3} + 2q^{5} + 3q^{7} + 5q^{9} + 7q^{11} + 11q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{6} & 0 \\ 0 & u(11)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −36q^{3} + −196290q^{5} +−35905576q^{7} +−1162260171q^{9} +−12016099980q^{11} +−3502281298∗13q^{13} +7066440q^{15} + 496563248178q^{17}+1410273986444q^{19}+1292600736q^{21}+−7039745388792q^{23}+−19034956564025q^{25}+⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

4. Consider f = q + 9q^{3} + 2q^{5} + 9q^{7} + 2q^{9} + 11q^{11} + 5q^{15} + q^{17} + 12q^{19} + 3q^{21} + 10q^{23} + 12q^{25} + ⋯ ∈ S_{22}(Γ_{0}(4); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + q^{3} + 2q^{5} + 3q^{7} + 5q^{9} + 7q^{11} + 11q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{1} & 0 \\ 0 & u(11)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+8462545186296674284210+ O(13^{2}0)q^{3}+−3092485509631194802669+O(13^{2}0)q^{5}+1469087420350555161841+O(13^{2}0)q^{7}+4384816297698094731553+ O(13^{2}0)q^{9} + −6013201347648037186074 + O(13^{2}0)q^{11} + 575156692355116302526 ∗ 13 + O(13^{2}0)q^{13} + 19023796711690116237481+O(13^{2}2)q^{15}+20837583580866031349800+O(13^{2}1)q^{17}+−557610737800364088400+ O(13^{2}0)q^{19} + −3279839831451634945920 + O(13^{2}0)q^{21} + −5482614947438987743349 + O(13^{2}0)q^{23} + −4174005067286584637056 + O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

5. Consider f = q+8q^{3} +11q^{5} +8q^{7} +3q^{9} +10q^{15} +4q^{17} +4q^{19} +12q^{21} +2q^{23} +9q^{25} +⋯ ∈ S_{24}(Γ_{0}(4); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-10 newform g = q + 7q^{3} + 10q^{5} + 3q^{7} + 9q^{9} + 6q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(11)ω^{10} & 0 \\ 0 & u(6)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+1331450926767302681374+ O(13^{2}0)q^{3}+3205122991126930639868+O(13^{2}0)q^{5}+86325831963985043709261+O(13^{2}1)q^{7}+5714777634412893382443+ O(13^{2}0)q^{9} + 336883763590512412823 ∗ 13 + O(13^{2}0)q^{11} + 654417486999956679088 ∗ 13 + O(13^{2}0)q^{13} + −8641147596384797105290+O(13^{2}0)q^{15}+−9039199475879134148739+O(13^{2}0)q^{17}+8769950732729624650528+ O(13^{2}0)q^{19}+7450228016826599530785+O(13^{2}0)q^{21}+2244806087364882828971+O(13^{2}0)q^{23}+3087593104344696290768+ O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

6. Consider f = q + 12q^{3} + 4q^{7} + 11q^{9} + q^{11} + 10q^{17} + 12q^{19} + 9q^{21} + 10q^{23} + 8q^{25} + ⋯ ∈ S_{26}(Γ_{0}(4); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-12 newform g = q + 4q^{3} + 8q^{7} + 7q^{9} + 6q^{11} + 11q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{0} & 0 \\ 0 & u(11)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−8650085295057409311837+ O(13^{2}0)q^{3} + −125783729197557520988 ∗ 13 + O(13^{2}0)q^{5} + −3764039380953877673150 + O(13^{2}0)q^{7} + 217945006786812127431+O(13^{2}0)q^{9}+7481104591237751251454+O(13^{2}0)q^{11}+−470320238802562194668∗ 13+O(13^{2}0)q^{13}+−2059659876732524918447∗13+O(13^{2}1)q^{15}+4886493467816888617011+O(13^{2}0)q^{17}+ −8424716420627594589815+O(13^{2}0)q^{19}+193938970300302612693+O(13^{2}0)q^{21}+−4991984407306657099420+ O(13^{2}0)q^{23} + 3911837401795252551279 + O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

7. Consider f = q + q3 + 2q5 + 3q7 + 5q9 + 7q11 + 11q13 + 2q15 + 9q17 + 4q19 + 3q21 + 4q23 + 12q25 + ⋯ ∈ S18(Γ0(4); \overline{F}13). Since f is ordinary, we do not continue our analysis.

8. Consider f = q + 7q3 + 10q5 + 3q7 + 9q9 + 6q13 + 5q15 + q17 + 5q19 + 8q21 + 8q23 + 4q25 + ⋯ ∈ S22(Γ0(4); \overline{F}13). Since f is ordinary, we do not continue our analysis.

9. Consider f = q + 4q3 + 8q7 + 7q9 + 6q11 + 11q13 + 9q17 + 2q19 + 6q21 + q23 + 5q25 + ⋯ ∈ S24(Γ0(4); \overline{F}13). Since f is ordinary, we do not continue our analysis.

10. Consider f = q + 2q5 + 11q7 + 10q9 + 11q11 + 12q13 + 6q17 + 7q19 + 8q23 + 12q25 + ⋯ ∈ S26(Γ0(4); \overline{F}13). Since f is ordinary, we do not continue our analysis.

1. Consider f = q+6q^{2}+9q^{3}+2q^{4}+8q^{5}+2q^{6}+5q^{7}+3q^{8}+2q^{9}+9q^{10}+7q^{11}+5q^{12}+4q^{14}+7q^{15}+2q^{16}+5q^{17}+ 12q^{18}+9q^{19}+3q^{20}+6q^{21}+3q^{22}+3q^{23}+q^{24}+12q^{25}+⋯ ∈ S_{16}(Γ_{0}(5); \overline{F}_{13}). We have ρ_{f} ⊗χ^{10} ∼ ρ_{g}, where g is the weight-12 newform g = q+8q^{2}+q^{3}+5q^{4}+5q^{5}+8q^{6}+7q^{7}+10q^{8}+5q^{9}+q^{10}+5q^{11}+5q^{12}+7q^{13}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(7)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−7317555275418938706865+ O(13^{2}0)q^{2} +4369853224042922026251+O(13^{2}0)q^{3} +6851446169055865882559+O(13^{2}0)q^{4} +78125q^{5} + 7241221139647002945608+O(13^{2}0)q^{6}+−8884759449514947803865+O(13^{2}0)q^{7}+−5481959072079439705961+ O(13^{2}0)q^{8}+1559099882364535164617+O(13^{2}0)q^{9}+4309420084741444744756+O(13^{2}0)q^{10}+6337551149897323847069+ O(13^{2}0)q^{11} + 4586322503949046117067 + O(13^{2}0)q^{12} + 310809223956986403598 ∗ 13 + O(13^{2}0)q^{13} + −5851625339545966354513+O(13^{2}0)q^{14}+8618840169482981677012+O(13^{2}0)q^{15}+2470238417902660021437+ O(13^{2}0)q^{16}+1650161398412153928384+O(13^{2}0)q^{17}+−9346877808115782954670+O(13^{2}0)q^{18}+−93774431108904567930458+ O(13^{2}1)q^{19} + −5572762028194118908290 + O(13^{2}0)q^{20} + −4979397372529051051574 + O(13^{2}0)q^{21} + −3151619181712356784465+O(13^{2}0)q^{22}+−9204305260627694932452+O(13^{2}0)q^{23}+3018445152997214475432+ O(13^{2}0)q^{24} + 6103515625q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q+z^{9}9q^{2}+z^{7}9q^{3}+z^{6}4q^{4}+5q^{5}+z^{1}0q^{6}+z^{6}7q^{7}+z^{4}8q^{8}+z^{1}27q^{9}+z^{5}7q^{10}+z^{1}19q^{11}+z^{1}43q^{12}+ z^{1}66q^{14} +z^{3}7q^{15} +z^{8}6q^{16} +z^{7}q^{17} +z^{5}8q^{18} +z^{2}2q^{20} +z^{1}46q^{21} +z^{5}0q^{22} +z^{5}4q^{23} +z^{1}27q^{24} +12q^{25} +⋯ ∈ S_{16}(Γ_{0}(5); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{10} ∼ ρ_{g}, where g is the weight-12 newform g = q + z^{7}1q^{2} + z^{1}35q^{3} + z^{8}q^{4} + 8q^{5} + z^{3}8q^{6} + z^{9}5q^{7} + z^{1}32q^{8} + z^{7}1q^{9} + z^{1}13q^{10} + z^{9}1q^{11} + z^{1}43q^{12} + z^{1}23q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{4}5)ω^{1} & 0 \\ 0 & u(z^{1}23)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(5220066930185672060144+ O(13^{2}0)) ∗ a − 3930958870456881499590 + O(13^{2}0)q^{3} + −(5872575296458881067501 + O(13^{2}0)) ∗ a + 4422328729263991694938 + O(13^{2}0)q^{4} + −78125q^{5} + −(2337065719276137568291 + O(13^{2}0)) ∗ a + 9001701865544814262000+O(13^{2}0)q^{6}+−(8187241859314354706443+O(13^{2}0))∗a−8490095368981084080962+ O(13^{2}0)q^{7} + −(4485337410954724837403 + O(13^{2}0)) ∗ a − 1315648857824827119721 + O(13^{2}0)q^{8} + −(2266125169566136746825+O(13^{2}0))∗a+455518494574115769770∗13+O(13^{2}0)q^{9} +−78125∗aq^{10} + (8617762588546044810657+O(13^{2}0))∗a−7431882862003163593015+O(13^{2}0)q^{11}+−(2152228184105116914181+ O(13^{2}0)) ∗ a + 3094366649193076852094 + O(13^{2}0)q^{12} + (165960082856164194028 ∗ 13 + O(13^{2}1)) ∗ a + 4306357358057521888581∗13+O(13^{2}1)q^{13}+−(8658264380083863399096+O(13^{2}0))∗a−4249744541549648999608+ O(13^{2}0)q^{14} + −(9216239363435340958142 + O(13^{2}0)) ∗ a + 4952116145029023883391 + O(13^{2}0)q^{15} + (4808953201033301535716+O(13^{2}0))∗a+2442993228561431063000+O(13^{2}0)q^{16}+−(5954622859256493986977+ O(13^{2}0)) ∗ a − 1868043271838626439554 + O(13^{2}0)q^{17} + (5459188477617129926663 + O(13^{2}0)) ∗ a + 732326720394207323054+O(13^{2}0)q^{18}+−(42377726526926250859∗13+O(13^{2}0))∗a+491889676940628709447∗ 13+O(13^{2}0)q^{19}+−(3885453547295853579316+O(13^{2}0))∗a−3195510191298169067871+O(13^{2}0)q^{20}+ (2860399369675169900761+O(13^{2}0))∗a+34669688693115936688+O(13^{2}0)q^{21}+−(12595736001393929554795+ O(13^{2}1)) ∗ a − 112823376207622871642925 + O(13^{2}1)q^{22} + (1588112852478991517499 + O(13^{2}0)) ∗ a − 4539486969322660788060+O(13^{2}0)q^{23}+(9045159991055509217667+O(13^{2}0))∗a−685631717776809004374∗ 13 + O(13^{2}0)q^{24} + 6103515625q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2}+5872575296458881067501+O(13^{2}0)x+−4422328729263991727706+ O(13^{2}0). The slope of f_{0} is 1.

3. Consider f = q + z^{1}11q^{2} + z^{1}9q^{3} + z^{1}60q^{4} + 5q^{5} + z^{1}30q^{6} + z^{3}1q^{7} + z^{1}20q^{8} + z^{1}39q^{9} + z^{6}9q^{10} + z^{3}5q^{11} + z^{1}1q^{12} +z^{1}42q^{14} +z^{1}45q^{15} +z^{1}10q^{16} +z^{9}1q^{17} +z^{8}2q^{18} +z^{1}18q^{20} +z^{5}0q^{21} +z^{1}46q^{22} +z^{3}0q^{23} +z^{1}39q^{24} + 12q^{25} + ⋯ ∈ S_{16}(Γ_{0}(5); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{10} ∼ ρ_{g}, where g is the weight-12 newform g = q + z^{8}3q^{2} + z^{7}5q^{3} + z^{1}04q^{4} + 8q^{5} + z^{1}58q^{6} + z^{5}9q^{7} + z^{3}6q^{8} +

z^{8}3q^{9} + z^{1}25q^{10} + z^{7}q^{11} + z^{1}1q^{12} + z^{8}7q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{8}1)ω^{1} & 0 \\ 0 & u(z^{8}7)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(5220066930185672060144+ O(13^{2}0)) ∗ a − 3930958870456881499590 + O(13^{2}0)q^{3} + −(5872575296458881067501 + O(13^{2}0)) ∗ a + 4422328729263991694938 + O(13^{2}0)q^{4} + −78125q^{5} + −(2337065719276137568291 + O(13^{2}0)) ∗ a + 9001701865544814262000+O(13^{2}0)q^{6}+−(8187241859314354706443+O(13^{2}0))∗a−8490095368981084080962+ O(13^{2}0)q^{7} + −(4485337410954724837403 + O(13^{2}0)) ∗ a − 1315648857824827119721 + O(13^{2}0)q^{8} + −(2266125169566136746825+O(13^{2}0))∗a+455518494574115769770∗13+O(13^{2}0)q^{9} +−78125∗aq^{10} + (8617762588546044810657+O(13^{2}0))∗a−7431882862003163593015+O(13^{2}0)q^{11}+−(2152228184105116914181+ O(13^{2}0)) ∗ a + 3094366649193076852094 + O(13^{2}0)q^{12} + (165960082856164194028 ∗ 13 + O(13^{2}1)) ∗ a + 4306357358057521888581∗13+O(13^{2}1)q^{13}+−(8658264380083863399096+O(13^{2}0))∗a−4249744541549648999608+ O(13^{2}0)q^{14} + −(9216239363435340958142 + O(13^{2}0)) ∗ a + 4952116145029023883391 + O(13^{2}0)q^{15} + (4808953201033301535716+O(13^{2}0))∗a+2442993228561431063000+O(13^{2}0)q^{16}+−(5954622859256493986977+ O(13^{2}0)) ∗ a − 1868043271838626439554 + O(13^{2}0)q^{17} + (5459188477617129926663 + O(13^{2}0)) ∗ a + 732326720394207323054+O(13^{2}0)q^{18}+−(42377726526926250859∗13+O(13^{2}0))∗a+491889676940628709447∗ 13+O(13^{2}0)q^{19}+−(3885453547295853579316+O(13^{2}0))∗a−3195510191298169067871+O(13^{2}0)q^{20}+ (2860399369675169900761+O(13^{2}0))∗a+34669688693115936688+O(13^{2}0)q^{21}+−(12595736001393929554795+ O(13^{2}1)) ∗ a − 112823376207622871642925 + O(13^{2}1)q^{22} + (1588112852478991517499 + O(13^{2}0)) ∗ a − 4539486969322660788060+O(13^{2}0)q^{23}+(9045159991055509217667+O(13^{2}0))∗a−685631717776809004374∗ 13 + O(13^{2}0)q^{24} + 6103515625q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2}+5872575296458881067501+O(13^{2}0)x+−4422328729263991727706+ O(13^{2}0). The slope of f_{0} is 1.

4. Consider f = q+2q^{2}+9q^{3}+11q^{4}+12q^{5}+5q^{6}+10q^{7}+10q^{8}+7q^{9}+11q^{10}+8q^{11}+8q^{12}+7q^{14}+4q^{15}+6q^{16}+ 12q^{17}+q^{18}+7q^{19}+2q^{20}+12q^{21}+3q^{22}+12q^{23}+12q^{24}+q^{25}+⋯ ∈ S_{18}(Γ_{0}(5); \overline{F}_{13}). We have ρ_{f} ⊗χ^{12} ∼ ρ_{g}, where g is the weight-6 newform g = q+2q^{2} +9q^{3} +11q^{4} +12q^{5} +5q^{6} +10q^{7} +10q^{8} +7q^{9} +11q^{10} +8q^{11} + 8q^{12} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{5}. The form f_{0} = q +aq^{2} +−4∗13∗a+12190q^{3} +680∗a−196128q^{4} +−390625q^{5} +−23170∗a+260224∗13q^{6} + −3668∗13∗a+4802210q^{7} +800∗13^{2} ∗a−44238080q^{8} +43920∗13∗a−156455487q^{9} +−390625∗aq^{10} + −895000∗a−222377728q^{11} +−5556944∗a−90420160q^{12} +19322608∗a−7388399550q^{13} +−27622910∗ a+238625408∗13q^{14}+1562500∗13∗a−4761718750q^{15}+−41431040∗a+16911318016q^{16}+49420496∗a+ 5839310330q^{17} +231797313∗a−2857259520∗13q^{18} +−356859680∗a+124815536700q^{19} +−265625000∗ a+76612500000q^{20}+65778720∗13∗a−102771835908q^{21}+−830977728∗a+58225120000q^{22}+6400036∗ 13 ∗ a + 7811623910q^{23} + −64015680 ∗ 13 ∗ a − 81892492800q^{24} + 152587890625q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −680x + 65056. The slope of f_{0} is 1/2.

5. Consider f = q + 5q^{2} + 6q^{3} + 6q^{4} + q^{5} + 4q^{6} + 3q^{7} + q^{9} + 5q^{10} + q^{11} + 10q^{12} + 2q^{14} + 6q^{15} + 3q^{16} + 5q^{18} + 2q^{19} + 6q^{20} + 5q^{21} + 5q^{22} + q^{25} + ⋯ ∈ S_{18}(Γ_{0}(5); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-4 newform g = q + 9q^{2} + 2q^{3} + 8q^{4} + 8q^{5} + 5q^{6} + 6q^{7} + 3q^{9} + 7q^{10} + 6q^{11} + 3q^{12} + q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{4} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−3628868016030687910891+ O(13^{2}0)q^{2}+−5504779284713785909918+O(13^{2}0)q^{3}+1023525606569183131550+O(13^{2}0)q^{4}+390625q^{5}+ 7488231323307237180773+O(13^{2}0)q^{6}+−2966270903751734366148+O(13^{2}0)q^{7}+519765099132085556010∗ 13 + O(13^{2}0)q^{8} + −8600398090149394577073 + O(13^{2}0)q^{9} + −3335684953277449946688 + O(13^{2}0)q^{10} + −4963304452297732829508+O(13^{2}0)q^{11}+−3391291486846302714223+O(13^{2}0)q^{12}+8916898538001502833158∗ 13+O(13^{2}1)q^{13}+−2023094226239506021964+O(13^{2}0)q^{14}+−6786746209449358018406+O(13^{2}0)q^{15}+ 6396786447117712380062+O(13^{2}0)q^{16}+25588113189823371428∗13^{2}+O(13^{2}0)q^{17}+−8267642963816146769750+ O(13^{2}0)q^{18}+195426920231191621273+O(13^{2}0)q^{19}+7267133919782967662113+O(13^{2}0)q^{20}+6019343683582895058567+ O(13^{2}0)q^{21} + 3408641427768813938915 + O(13^{2}0)q^{22} + −693084030206458736165 ∗ 13 + O(13^{2}0)q^{23} + −370345835700916827148 ∗ 13 + O(13^{2}0)q^{24} + 152587890625q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

6. Consider f = q+z^{1}42q^{2}+z^{9}1q^{3}+z^{2}6q^{4}+q^{5}+z^{6}5q^{6}+z^{5}0q^{7}+z^{1}47q^{8}+6q^{9}+z^{1}42q^{10}+z^{3}4q^{11}+z^{1}17q^{12}+ z^{2}4q^{14}+z^{9}1q^{15}+z^{1}14q^{16}+z^{3}1q^{17}+z^{4}4q^{18}+z^{5}0q^{19}+z^{2}6q^{20}+z^{1}41q^{21}+z^{8}q^{22}+z^{2}5q^{23}+6q^{24}+q^{25}+⋯ ∈ S_{18}(Γ_{0}(5); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{8}6q^{2} + z^{3}5q^{3} + z^{8}2q^{4} + q^{5} + z^{1}21q^{6} + z^{1}06q^{7} + z^{1}47q^{8} + 5q^{9} + z^{8}6q^{10} + z^{1}46q^{11} + z^{1}17q^{12} + 11q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{1} & 0 \\ 0 & u(11)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(8897670551435285067606+ O(13^{2}0)) ∗ a + 4481529849824804981044 + O(13^{2}0)q^{3} + (3628868016030687911009 + O(13^{2}0)) ∗ a + 7884215324068030411695+O(13^{2}0)q^{4}+390625q^{5}+(1023249434888980944818+O(13^{2}0))∗a−4565751906732634548755+ O(13^{2}0)q^{6}+(3686433389151871135188+O(13^{2}0))∗a+8628463238335466319478+O(13^{2}0)q^{7}+−(8907740930637213657401+ O(13^{2}0)) ∗ a − 905816729131560384215 + O(13^{2}0)q^{8} + (456938160840024630069 ∗ 13 + O(13^{2}0)) ∗ a − 1870646085682006081638 + O(13^{2}0)q^{9} + 390625 ∗ aq^{10} + (5893308489695306725060 + O(13^{2}0)) ∗ a − 2664167686264396340838+O(13^{2}0)q^{11}+−(7050007251147785820815+O(13^{2}0))∗a−2338166414383750692459+ O(13^{2}0)q^{12}+−(502213623340343758699∗13^{2}+O(13^{2}1))∗a−4529083645657792988525∗13+O(13^{2}1)q^{13}+ −(5662192334583729814022+O(13^{2}0))∗a−1735998064830012149610+O(13^{2}0)q^{14}+−(5779040432747366228069+ O(13^{2}0)) ∗ a − 6630607780632985964013 + O(13^{2}0)q^{15} + (7393630982736995746305 + O(13^{2}0)) ∗ a + 146228111682366633866∗13+O(13^{2}0)q^{16}+−(2017892231264477554405+O(13^{2}0))∗a+1326919908701455567470+ O(13^{2}0)q^{17} + −(8533919599049074887651 + O(13^{2}0)) ∗ a − 477002432080175539617 ∗ 13 + O(13^{2}0)q^{18} + −(9010191186037033431899+O(13^{2}0))∗a+8771420192295613525227+O(13^{2}0)q^{19}+(3335684953277496040438+ O(13^{2}0)) ∗ a − 1773719134050288781476 + O(13^{2}0)q^{20} + −(72521969090526922888 + O(13^{2}0)) ∗ a − 4678739695580296151890+O(13^{2}0)q^{21}+(7627472138563918917862+O(13^{2}0))∗a+6756498618780231762603+ O(13^{2}0)q^{22} + −(4135906859789091867998 + O(13^{2}0)) ∗ a − 169602222772110909996 + O(13^{2}0)q^{23} + (339145189732646616887∗13+O(13^{2}0))∗a+2104964063673080765096+O(13^{2}0)q^{24}+152587890625q^{25}+ ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −3628868016030687911009 + O(13^{2}0)x + −7884215324068030542767 + O(13^{2}0). The slope of f_{0} is 1.

7. Consider f = q + z^{1}66q^{2} + z^{7}q^{3} + z^{2}q^{4} + q^{5} + z^{5}q^{6} + z^{1}46q^{7} + z^{6}3q^{8} + 6q^{9} + z^{1}66q^{10} + z^{1}06q^{11} + z^{9}q^{12} + z^{1}44q^{14}+z^{7}q^{15}+z^{1}38q^{16}+z^{6}7q^{17}+z^{6}8q^{18}+z^{1}46q^{19}+z^{2}q^{20}+z^{1}53q^{21}+z^{1}04q^{22}+z^{1}57q^{23}+6q^{24}+q^{25}+⋯ ∈ S_{18}(Γ_{0}(5); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{1}10q^{2} + z^{1}19q^{3} + z^{5}8q^{4} + q^{5} + z^{6}1q^{6} + z^{3}4q^{7} + z^{6}3q^{8} + 5q^{9} + z^{1}10q^{10} + z^{5}0q^{11} + z^{9}q^{12} + 11q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{1} & 0 \\ 0 & u(11)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(8897670551435285067606+ O(13^{2}0)) ∗ a + 4481529849824804981044 + O(13^{2}0)q^{3} + (3628868016030687911009 + O(13^{2}0)) ∗ a + 7884215324068030411695+O(13^{2}0)q^{4}+390625q^{5}+(1023249434888980944818+O(13^{2}0))∗a−4565751906732634548755+ O(13^{2}0)q^{6}+(3686433389151871135188+O(13^{2}0))∗a+8628463238335466319478+O(13^{2}0)q^{7}+−(8907740930637213657401+ O(13^{2}0)) ∗ a − 905816729131560384215 + O(13^{2}0)q^{8} + (456938160840024630069 ∗ 13 + O(13^{2}0)) ∗ a − 1870646085682006081638 + O(13^{2}0)q^{9} + 390625 ∗ aq^{10} + (5893308489695306725060 + O(13^{2}0)) ∗ a − 2664167686264396340838+O(13^{2}0)q^{11}+−(7050007251147785820815+O(13^{2}0))∗a−2338166414383750692459+ O(13^{2}0)q^{12}+−(502213623340343758699∗13^{2}+O(13^{2}1))∗a−4529083645657792988525∗13+O(13^{2}1)q^{13}+ −(5662192334583729814022+O(13^{2}0))∗a−1735998064830012149610+O(13^{2}0)q^{14}+−(5779040432747366228069+ O(13^{2}0)) ∗ a − 6630607780632985964013 + O(13^{2}0)q^{15} + (7393630982736995746305 + O(13^{2}0)) ∗ a + 146228111682366633866∗13+O(13^{2}0)q^{16}+−(2017892231264477554405+O(13^{2}0))∗a+1326919908701455567470+ O(13^{2}0)q^{17} + −(8533919599049074887651 + O(13^{2}0)) ∗ a − 477002432080175539617 ∗ 13 + O(13^{2}0)q^{18} + −(9010191186037033431899+O(13^{2}0))∗a+8771420192295613525227+O(13^{2}0)q^{19}+(3335684953277496040438+ O(13^{2}0)) ∗ a − 1773719134050288781476 + O(13^{2}0)q^{20} + −(72521969090526922888 + O(13^{2}0)) ∗ a − 4678739695580296151890+O(13^{2}0)q^{21}+(7627472138563918917862+O(13^{2}0))∗a+6756498618780231762603+ O(13^{2}0)q^{22} + −(4135906859789091867998 + O(13^{2}0)) ∗ a − 169602222772110909996 + O(13^{2}0)q^{23} + (339145189732646616887∗13+O(13^{2}0))∗a+2104964063673080765096+O(13^{2}0)q^{24}+152587890625q^{25}+ ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −3628868016030687911009 + O(13^{2}0)x + −7884215324068030542767 + O(13^{2}0). The slope of f_{0} is 1.

8. Consider f = q + q^{2} + 4q^{3} + 3q^{4} + 5q^{5} + 4q^{6} + 6q^{7} + 5q^{8} + 5q^{10} + 10q^{11} + 12q^{12} + 6q^{14} + 7q^{15} + 11q^{16} + 5q^{17} + 5q^{19} + 2q^{20} + 11q^{21} + 10q^{22} + 11q^{23} + 7q^{24} + 12q^{25} + ⋯ ∈ S_{20}(Γ_{0}(5); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + 12q^{2} + 4q^{3} + 3q^{4} + 8q^{5} + 9q^{6} + 7q^{7} + 8q^{8} + 5q^{10} + 3q^{11} + 12q^{12} + q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−3997202943981199501626+ O(13^{2}0)q^{2}+−3815922344508578977188+O(13^{2}0)q^{3}+830545956429987925520+O(13^{2}0)q^{4}+1953125q^{5}+ 3703085128421191167555+O(13^{2}0)q^{6}+7213553689463891027446+O(13^{2}0)q^{7}+9019033054024800675107+ O(13^{2}0)q^{8} + 704528137250708010859 ∗ 13 + O(13^{2}0)q^{9} + −6935843771555947657261 + O(13^{2}0)q^{10} + 1332029656160716152988+O(13^{2}0)q^{11}+−1906772986591111193270+O(13^{2}0)q^{12}+59663741439041826656∗ 13+O(13^{2}0)q^{13}+−1839997685510947869628+O(13^{2}0)q^{14}+−5740124838887699551141+O(13^{2}0)q^{15}+ −8982213938502413755343+O(13^{2}0)q^{16}+−1692752069010160812246+O(13^{2}0)q^{17}+−665652996891829027619∗ 13+O(13^{2}0)q^{18}+−4394520728234124610026+O(13^{2}0)q^{19}+−8611852630469067609355+O(13^{2}0)q^{20}+ 2075544623961906833708+O(13^{2}0)q^{21}+2870473235565059563241+O(13^{2}0)q^{22}+5531446455694287795067+ O(13^{2}0)q^{23} + −8582109836927375781623 + O(13^{2}0)q^{24} + 3814697265625q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

9. Consider f = q +4q^{2} +q^{3} +5q^{4} +8q^{5} +4q^{6} +5q^{7} +2q^{8} +11q^{9} +6q^{10} +10q^{11} +5q^{12} +7q^{14} +8q^{15} +5q^{16} + 9q^{17} + 5q^{18} + 3q^{19} + q^{20} + 5q^{21} + q^{22} + 3q^{23} + 2q^{24} + 12q^{25} + ⋯ ∈ S_{20}(Γ_{0}(5); \overline{F}_{13}). Also, z satisfies the equation x^{2} +12x+2 = 0. We have ρ_{f} ⊗χ^{11} ∼ ρ_{g}, where g is the weight-6 newform g = q +2q^{2} +9q^{3} + 11q^{4}+12q^{5}+5q^{6}+10q^{7}+10q^{8}+7q^{9}+11q^{10}+8q^{11}+8q^{12}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{19}. The form f_{0} = q+590916624805747362321+O(13^{2}0)q^{2}+ −5620525212896564496018+O(13^{2}0)q^{3}+2275749366404628727722+O(13^{2}0)q^{4}+−1953125q^{5}+1082101236754686018342+ O(13^{2}0)q^{6}+−5225716572032899906010+O(13^{2}0)q^{7}+−3225429355888714982356+O(13^{2}0)q^{8}+−1557602863956814317866+ O(13^{2}0)q^{9}+−592702764128713695997+O(13^{2}0)q^{10}+2920048328496090125708+O(13^{2}0)q^{11}+−1926296774862491398284+ O(13^{2}0)q^{12} + −54726844501180126467 ∗ 13^{2} + O(13^{2}0)q^{13} + 4618759515079135482245 + O(13^{2}0)q^{14} + −1854316720198156760967+O(13^{2}0)q^{15}+4641252644652018795795+O(13^{2}0)q^{16}+−3126970046856470374707+ O(13^{2}0)q^{17} + −1236823702288867187776 + O(13^{2}0)q^{18} + −5993116319547924990719 + O(13^{2}0)q^{19} + 931518756246516430227+O(13^{2}0)q^{20}+−1909955612149109668879+O(13^{2}0)q^{21}+−3104831619741225211694+ O(13^{2}0)q^{22}+−3108168701517293773343+O(13^{2}0)q^{23}+8750837984708492009768+O(13^{2}0)q^{24}+3814697265625q^{25}+ ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 2.

10. Consider f = q + z^{3}1q^{2} + z^{1}55q^{3} + z^{1}0q^{4} + 8q^{5} + z^{1}8q^{6} + z^{1}42q^{7} + z^{1}10q^{8} + z^{6}9q^{9} + z^{7}3q^{10} + z^{1}32q^{11} + z^{1}65q^{12} + z^{5}q^{14} + z^{2}9q^{15} + z^{4}8q^{16} + z^{1}14q^{17} + z^{1}00q^{18} + z^{7}1q^{19} + z^{5}2q^{20} + z^{1}29q^{21} + z^{1}63q^{22} + z^{1}06q^{23} + z^{9}7q^{24} + 12q^{25} + ⋯ ∈ S_{20}(Γ_{0}(5); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q +z^{1}15q^{2} +z^{1}55q^{3} +z^{1}0q^{4} +5q^{5} +z^{1}02q^{6} +z^{5}8q^{7} + z^{2}6q^{8} + z^{6}9q^{9} + z^{7}3q^{10} + z^{4}8q^{11} + z^{1}65q^{12} + z^{8}0q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{8}8)ω^{1} & 0 \\ 0 & u(z^{8}0)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(8849054544638271778316+ O(13^{2}0)) ∗ a + 7393445151983381691030 + O(13^{2}0)q^{3} + (665468732674315613452 + O(13^{2}0)) ∗ a − 8612220786347579753582 + O(13^{2}0)q^{4} + −1953125q^{5} + −(7165144223036931759022 + O(13^{2}0)) ∗ a − 1451518083788802409923+O(13^{2}0)q^{6}+−(8438625943682890625367+O(13^{2}0))∗a−6612631716691744846493+ O(13^{2}0)q^{7} + −(5270232639785987670248 + O(13^{2}0)) ∗ a + 1625854945821294370753 + O(13^{2}0)q^{8} + −(5254859995511055602617+O(13^{2}0))∗a+7318334516511865494137+O(13^{2}0)q^{9} +−1953125∗aq^{10} + −(1748929593451908841899+O(13^{2}0))∗a+4607457042881780549097+O(13^{2}0)q^{11}+−(254691389356904123648+ O(13^{2}0)) ∗ a + 5204795227336880142202 + O(13^{2}0)q^{12} + (260234160304804397174 ∗ 13 + O(13^{2}0)) ∗ a − 479634133704909184154∗13+O(13^{2}0)q^{13}+(5182820113049397322540+O(13^{2}0))∗a+3649983210569739410778+ O(13^{2}0)q^{14} + −(5550982276749383420090 + O(13^{2}0)) ∗ a − 8996965163097302070532 + O(13^{2}0)q^{15} + (8061152260853551789803+O(13^{2}0))∗a+3729859948412920961133+O(13^{2}0)q^{16}+−(3583351279466826108786+ O(13^{2}0)) ∗ a − 2925297003712872186434 + O(13^{2}0)q^{17} + (1686985168227981882046 + O(13^{2}0)) ∗ a + 5007917712907574784834+O(13^{2}0)q^{18}+(1476366789493103655230+O(13^{2}0))∗a−44355410741935931715+ O(13^{2}0)q^{19} + (5854059575191096162890 + O(13^{2}0)) ∗ a + 1430137592666115803879 + O(13^{2}0)q^{20} + −(88619093300289800099464+O(13^{2}1))∗a−8747386396886377526729∗13+O(13^{2}1)q^{21}+(7659469304783353745214+ O(13^{2}0)) ∗ a + 909695618483026268329 + O(13^{2}0)q^{22} + (3884137569396310681265 + O(13^{2}0)) ∗ a − 3024443866355099996051+O(13^{2}0)q^{23}+(8112648615256407171147+O(13^{2}0))∗a−3222010919335750576232+ O(13^{2}0)q^{24} + 3814697265625q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} +−665468732674315613452+O(13^{2}0)x+8612220786347579229294+ O(13^{2}0). The slope of f_{0} is 1.

11. Consider f = q + z^{6}7q^{2} + z^{1}67q^{3} + z^{1}30q^{4} + 8q^{5} + z^{6}6q^{6} + z^{1}66q^{7} + z^{8}6q^{8} + z^{5}7q^{9} + z^{1}09q^{10} + z^{3}6q^{11} + z^{1}29q^{12} + z^{6}5q^{14} + z^{4}1q^{15} + z^{1}20q^{16} + z^{1}38q^{17} + z^{1}24q^{18} + z^{8}3q^{19} + z^{4}q^{20} + z^{1}65q^{21} + z^{1}03q^{22} + z^{3}4q^{23} + z^{8}5q^{24} + 12q^{25} + ⋯ ∈ S_{20}(Γ_{0}(5); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{1}51q^{2} + z^{1}67q^{3} + z^{1}30q^{4} + 5q^{5} + z^{1}50q^{6} + z^{8}2q^{7} +z^{2}q^{8} +z^{5}7q^{9} +z^{1}09q^{10} +z^{1}20q^{11} +z^{1}29q^{12} +z^{3}2q^{13} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}36)ω^{1} & 0 \\ 0 & u(z^{3}2)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(8849054544638271778316+ O(13^{2}0)) ∗ a + 7393445151983381691030 + O(13^{2}0)q^{3} + (665468732674315613452 + O(13^{2}0)) ∗ a − 8612220786347579753582 + O(13^{2}0)q^{4} + −1953125q^{5} + −(7165144223036931759022 + O(13^{2}0)) ∗ a − 1451518083788802409923+O(13^{2}0)q^{6}+−(8438625943682890625367+O(13^{2}0))∗a−6612631716691744846493+ O(13^{2}0)q^{7} + −(5270232639785987670248 + O(13^{2}0)) ∗ a + 1625854945821294370753 + O(13^{2}0)q^{8} + −(5254859995511055602617+O(13^{2}0))∗a+7318334516511865494137+O(13^{2}0)q^{9} +−1953125∗aq^{10} + −(1748929593451908841899+O(13^{2}0))∗a+4607457042881780549097+O(13^{2}0)q^{11}+−(254691389356904123648+ O(13^{2}0)) ∗ a + 5204795227336880142202 + O(13^{2}0)q^{12} + (260234160304804397174 ∗ 13 + O(13^{2}0)) ∗ a − 479634133704909184154∗13+O(13^{2}0)q^{13}+(5182820113049397322540+O(13^{2}0))∗a+3649983210569739410778+ O(13^{2}0)q^{14} + −(5550982276749383420090 + O(13^{2}0)) ∗ a − 8996965163097302070532 + O(13^{2}0)q^{15} + (8061152260853551789803+O(13^{2}0))∗a+3729859948412920961133+O(13^{2}0)q^{16}+−(3583351279466826108786+ O(13^{2}0)) ∗ a − 2925297003712872186434 + O(13^{2}0)q^{17} + (1686985168227981882046 + O(13^{2}0)) ∗ a + 5007917712907574784834+O(13^{2}0)q^{18}+(1476366789493103655230+O(13^{2}0))∗a−44355410741935931715+ O(13^{2}0)q^{19} + (5854059575191096162890 + O(13^{2}0)) ∗ a + 1430137592666115803879 + O(13^{2}0)q^{20} + −(88619093300289800099464+O(13^{2}1))∗a−8747386396886377526729∗13+O(13^{2}1)q^{21}+(7659469304783353745214+ O(13^{2}0)) ∗ a + 909695618483026268329 + O(13^{2}0)q^{22} + (3884137569396310681265 + O(13^{2}0)) ∗ a − 3024443866355099996051+O(13^{2}0)q^{23}+(8112648615256407171147+O(13^{2}0))∗a−3222010919335750576232+ O(13^{2}0)q^{24} + 3814697265625q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} +−665468732674315613452+O(13^{2}0)x+8612220786347579229294+ O(13^{2}0). The slope of f_{0} is 1.

12. Consider f = q + 11q^{2} + 12q^{3} + 12q^{4} + q^{5} + 2q^{6} + 10q^{7} + 12q^{8} + 11q^{10} + 7q^{11} + q^{12} + 6q^{14} + 12q^{15} + 7q^{16} + 7q^{17} + 9q^{19} + 12q^{20} + 3q^{21} + 12q^{22} + 6q^{23} + q^{24} + q^{25} + ⋯ ∈ S_{22}(Γ_{0}(5); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-8 newform g = q + 12q^{2} + 4q^{3} + 3q^{4} + 8q^{5} + 9q^{6} + 7q^{7} + 8q^{8} + 5q^{10} + 3q^{11} + 12q^{12} + q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{8} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−1819554810271021127956+ O(13^{2}0)q^{2}+−8328385053544572625493+O(13^{2}0)q^{3}+−1479175213448935642160+O(13^{2}0)q^{4}+−9765625q^{5}+ 3780268403966995216369+O(13^{2}0)q^{6}+3760260595160442877839+O(13^{2}0)q^{7}+5583720152748733494615+ O(13^{2}0)q^{8} + −24282554209330779441 ∗ 13 + O(13^{2}0)q^{9} + −41511135229399897271 + O(13^{2}0)q^{10} + 1862874499978548942347+O(13^{2}0)q^{11}+3330858244866728329807+O(13^{2}0)q^{12}+−517015767100422140247∗ 13 + O(13^{2}0)q^{13} + 6781005475680848174925 + O(13^{2}0)q^{14} + −9225791363198914311783 + O(13^{2}0)q^{15} + 4987446201851936178597+O(13^{2}0)q^{16}+606595826770047085082+O(13^{2}0)q^{17}+140821471560980825916∗ 13 + O(13^{2}0)q^{18} + −4080218807728687044347 + O(13^{2}0)q^{19} + 5637391162662618151532 + O(13^{2}0)q^{20} + −8147337207873222479494+O(13^{2}0)q^{21}+−7520349561707154082860+O(13^{2}0)q^{22}+3709709954636947570549+ O(13^{2}0)q^{23} + −7074878713167804721233 + O(13^{2}0)q^{24} + 95367431640625q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

13. Consider f = q + 5q^{2} + 3q^{3} + 7q^{4} + 12q^{5} + 2q^{6} + 4q^{7} + 10q^{8} + 8q^{9} + 8q^{10} + 7q^{11} + 8q^{12} + 7q^{14} + 10q^{15} + 2q^{16} + 10q^{17} + q^{18} + 8q^{19} + 6q^{20} + 12q^{21} + 9q^{22} + 4q^{23} + 4q^{24} + q^{25} + ⋯ ∈ S_{22}(Γ_{0}(5); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 2q^{2} + 9q^{3} + 11q^{4} + 12q^{5} + 5q^{6} + 10q^{7} + 10q^{8} + 7q^{9} + 11q^{10} + 8q^{11} + 8q^{12} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{117}. The form f_{0} = q + aq^{2} + (387707808244872145507 ∗ 13 + O(13^{2}0)) ∗ a + 4495394509652086036363 + O(13^{2}0)q^{3} + (768285134085983183771 + O(13^{2}0)) ∗ a − 7674192862009659033624 + O(13^{2}0)q^{4} + 9765625q^{5} + −(956715252372160439792 + O(13^{2}0)) ∗ a + 279567864209877498264∗13+O(13^{2}0)q^{6}+−(113222794681017468079∗13+O(13^{2}0))∗a+3383181363578729779629+ O(13^{2}0)q^{7} + −(39536137854109273472 ∗ 13 + O(13^{2}0)) ∗ a + 4940991148828087424211 + O(13^{2}0)q^{8} + (234573022357938683876 ∗ 13 + O(13^{2}0)) ∗ a + 5740774548035120505738 + O(13^{2}0)q^{9} + 9765625 ∗ aq^{10} + −(1177592394701296536965+O(13^{2}0))∗a−3977164628725001028259+O(13^{2}0)q^{11}+−(8158551407268608359854+ O(13^{2}0)) ∗ a − 6944010523164507118143 + O(13^{2}0)q^{12} + (2906631575255586571970 + O(13^{2}0)) ∗ a + 326152819318163766449+O(13^{2}0)q^{13}+−(6960200860352474494158+O(13^{2}0))∗a−464989925717657470243∗ 13+O(13^{2}0)q^{14}+(709811736544723873376∗13+O(13^{2}0))∗a−8018790776037605998866+O(13^{2}0)q^{15}+ −(8190904040628646290123+O(13^{2}0))∗a+4818473929426668122047+O(13^{2}0)q^{16}+−(8187779803144707527820+ O(13^{2}0)) ∗ a + 7564928197715657707953 + O(13^{2}0)q^{17} + (2281404053059584564678 + O(13^{2}0)) ∗ a − 25127213539461629464∗13+O(13^{2}0)q^{18}+−(2652805818812333321781+O(13^{2}0))∗a+2258093222292059159622+ O(13^{2}0)q^{19} + (4913510987526563813095 + O(13^{2}0)) ∗ a − 8751495551390840135247 + O(13^{2}0)q^{20} + (572239007082756901546∗13+O(13^{2}0))∗a−7436632825165734833930+O(13^{2}0)q^{21}+−(4064576869278099049489+ O(13^{2}0)) ∗ a + 3119186254389819782140 + O(13^{2}0)q^{22} + (43271267814950951678 ∗ 13^{2} + O(13^{2}0)) ∗ a − 2466755811907446199643+O(13^{2}0)q^{23}+(3970451548593424143∗13^{2}+O(13^{2}0))∗a+1519630202791409810001+ O(13^{2}0)q^{24} + 95367431640625q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} +−768285134085983183771+O(13^{2}0)x+7674192862009656936472+ O(13^{2}0). The slope of f_{0} is 1/2.

14. Consider f = q+z^{1}29q^{2}+z^{4}3q^{3}+z^{3}8q^{4}+12q^{5}+z^{4}q^{6}+z^{4}4q^{7}+z^{6}8q^{8}+z^{1}3q^{9}+z^{4}5q^{10}+z^{1}46q^{11}+z^{8}1q^{12}+ z^{5}q^{14} +z^{1}27q^{15} +z^{1}04q^{16} +z^{1}42q^{17} +z^{1}42q^{18} +z^{5}7q^{19} +z^{1}22q^{20} +z^{8}7q^{21} +z^{1}07q^{22} +z^{7}8q^{23} +z^{1}11q^{24} + q^{25} + ⋯ ∈ S_{22}(Γ_{0}(5); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{1}15q^{2} + z^{1}55q^{3} + z^{1}0q^{4} + 5q^{5} + z^{1}02q^{6} + z^{5}8q^{7} + z^{2}6q^{8} + z^{6}9q^{9} + z^{7}3q^{10} + z^{4}8q^{11} + z^{1}65q^{12} + z^{8}0q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{8}8)ω^{8} & 0 \\ 0 & u(z^{8}0)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(3137076025398198629864+ O(13^{2}0)) ∗ a − 5847327514049658064236 + O(13^{2}0)q^{3} + −(768285134085983180861 + O(13^{2}0)) ∗ a − 6362015453619850198340+O(13^{2}0)q^{4}+9765625q^{5}+(2308648256769732550905+O(13^{2}0))∗a−747039781059666053835+ O(13^{2}0)q^{6}+(2096593242060881076046+O(13^{2}0))∗a−8418047796034689543945+O(13^{2}0)q^{7}+−(4454785667147864680737+ O(13^{2}0)) ∗ a − 8640037411865395963302 + O(13^{2}0)q^{8} + (3597117634209207942738 + O(13^{2}0)) ∗ a − 8651629180601800073301 + O(13^{2}0)q^{9} + 9765625 ∗ aq^{10} + −(7498846361094114374306 + O(13^{2}0)) ∗ a + 5604332821613891952091+O(13^{2}0)q^{11}+(7366717784328672026128+O(13^{2}0))∗a+3972222772258492526423+ O(13^{2}0)q^{12}+−(377935360980679455966∗13+O(13^{2}0))∗a+500422341392994369885∗13+O(13^{2}0)q^{13}+ −(7009896482071852566527+O(13^{2}0))∗a−1836964407886041728507+O(13^{2}0)q^{14}+(584479581700203821826+ O(13^{2}0)) ∗ a + 532673804861052205926 + O(13^{2}0)q^{15} + −(7115013471215779958827 + O(13^{2}0)) ∗ a − 4814969253528803579688+O(13^{2}0)q^{16}+(6326566897482629201454+O(13^{2}0))∗a−3717389997005075258076+ O(13^{2}0)q^{17}+(629450579528827891417+O(13^{2}0))∗a+5224710588537691081336+O(13^{2}0)q^{18}+(8791340539968633971521+ O(13^{2}0)) ∗ a − 5446783639111509805987 + O(13^{2}0)q^{19} + −(4913510987498145844345 + O(13^{2}0)) ∗ a − 6107648627670977468604+O(13^{2}0)q^{20}+−(8275423512695867990132+O(13^{2}0))∗a+705717360377302954496∗ 13 + O(13^{2}0)q^{21} + (2437408676422935202105 + O(13^{2}0)) ∗ a + 9358959256554706022235 + O(13^{2}0)q^{22} + (2688239783282227518293+O(13^{2}0))∗a−8127304641654319266167+O(13^{2}0)q^{23}+−(8197392670674795105521+ O(13^{2}0)) ∗ a − 6455597542939726826079 + O(13^{2}0)q^{24} + 95367431640625q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} +768285134085983180861+ O(13^{2}0)x + 6362015453619848101188 + O(13^{2}0). The slope of f_{0} is 1.

15. Consider f = q+10q^{2} +9q^{3} +2q^{4} +8q^{5} +12q^{6} +2q^{7} +2q^{8} +7q^{9} +2q^{10} +12q^{11} +5q^{12} +7q^{14} +7q^{15} +6q^{16} + q^{17}+5q^{18}+9q^{19}+3q^{20}+5q^{21}+3q^{22}+q^{23}+5q^{24}+12q^{25}+⋯ ∈ S_{24}(Γ_{0}(5); \overline{F}_{13}). Also, z satisfies the equation x^{2} +12x+2 = 0. We have ρ_{f} ⊗χ^{3} ∼ ρ_{g}, where g is the weight-6 newform g = q +2q^{2} +9q^{3} +11q^{4} + 12q^{5}+5q^{6}+10q^{7}+10q^{8}+7q^{9}+11q^{10}+8q^{11}+8q^{12}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{131}. The form f_{0} = q + 5556620875575284670152 + O(13^{2}0)q^{2} + 9238283260276156424087+O(13^{2}0)q^{3}+1113444115782933846771+O(13^{2}0)q^{4}+48828125q^{5}+7935989305103170261924+ O(13^{2}0)q^{6}+−4069000574030330601388+O(13^{2}0)q^{7}+1171628736252281708624+O(13^{2}0)q^{8}+3999272949549660055943+ O(13^{2}0)q^{9}+−7341377267633823885638+O(13^{2}0)q^{10}+8000908883867868159788+O(13^{2}0)q^{11}+7132552052703385138992+ O(13^{2}0)q^{12} + −27540609652591807658 ∗ 13^{2} + O(13^{2}0)q^{13} + 5365945308395518530704 + O(13^{2}0)q^{14} + 8256337833495085641600+O(13^{2}0)q^{15}+8267523246998781784414+O(13^{2}0)q^{16}+−4107047099541136295869+ O(13^{2}0)q^{17}+2161137051689337762162+O(13^{2}0)q^{18}+−1318367855324606287621+O(13^{2}0)q^{19}+2624944713067083846481+ O(13^{2}0)q^{20}+−284238672550134506251+O(13^{2}0)q^{21}+4620690358626070341486+O(13^{2}0)q^{22}+7406530942783175415915+ O(13^{2}0)q^{23}+−2266761216723468566447+O(13^{2}0)q^{24}+2384185791015625q^{25}+⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 2.

16. Consider f = q+12q^{2} +6q^{3} +7q^{4} +5q^{5} +7q^{6} +11q^{7} +q^{9} +8q^{10} +8q^{11} +3q^{12} +2q^{14} +4q^{15} +3q^{16} +12q^{18} + 10q^{19} + 9q^{20} + q^{21} + 5q^{22} + 12q^{25} + ⋯ ∈ S_{24}(Γ_{0}(5); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 9q^{2} + 2q^{3} + 8q^{4} + 8q^{5} + 5q^{6} + 6q^{7} + 3q^{9} + 7q^{10} + 6q^{11} + 3q^{12} + q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{10} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−8820213318452672500794+ O(13^{2}0)q^{2}+−9344900042246398709541+O(13^{2}0)q^{3}+3031703063370289829268+O(13^{2}0)q^{4}+−48828125q^{5}+ −4543042019055590619055+O(13^{2}0)q^{6}+6629563409341601387189+O(13^{2}0)q^{7}+502858814083977613316∗ 13 + O(13^{2}0)q^{8} + −1374063142323900544437 + O(13^{2}0)q^{9} + −8446778327796148760308 + O(13^{2}0)q^{10} + −1401377418737575946932+O(13^{2}0)q^{11}+−1518848010150012371931+O(13^{2}0)q^{12}+−382920769198010729045∗ 13 + O(13^{2}0)q^{13} + −9073266018160952708956 + O(13^{2}0)q^{14} + 9121371896864844549027 + O(13^{2}0)q^{15} + −9059604011389120720736+O(13^{2}0)q^{16}+29166865604482629927∗13^{2}+O(13^{2}0)q^{17}+6730260095178245597022+ O(13^{2}0)q^{18}+363919564844925632982+O(13^{2}0)q^{19}+4411238921713414175043+O(13^{2}0)q^{20}+−3133577635154775789434+ O(13^{2}0)q^{21} + −5157785890030562021969 + O(13^{2}0)q^{22} + 403194846798276346975 ∗ 13 + O(13^{2}0)q^{23} + 439686534899962398632∗13+O(13^{2}0)q^{24}+2384185791015625q^{25}+⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

17. Consider f = q+z^{1}00q^{2} +z^{9}1q^{3} +z^{1}10q^{4} +5q^{5} +z^{2}3q^{6} +z^{9}2q^{7} +z^{2}1q^{8} +6q^{9} +z^{5}8q^{10} +z^{7}6q^{11} +z^{3}3q^{12} + z^{2}4q^{14}+z^{4}9q^{15}+z^{1}14q^{16}+z^{1}15q^{17}+z^{2}q^{18}+z^{8}q^{19}+z^{6}8q^{20}+z^{1}5q^{21}+z^{8}q^{22}+z^{1}09q^{23}+9q^{24}+12q^{25}+⋯ ∈ S_{24}(Γ_{0}(5); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{8}6q^{2} + z^{3}5q^{3} + z^{8}2q^{4} + q^{5} + z^{1}21q^{6} + z^{1}06q^{7} + z^{1}47q^{8} + 5q^{9} + z^{8}6q^{10} + z^{1}46q^{11} + z^{1}17q^{12} + 11q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{10} & 0 \\ 0 & u(11)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(5588173740060664468381+ O(13^{2}0)) ∗ a + 8038334126079748995929 + O(13^{2}0)q^{3} + (1575543532242978676449 + O(13^{2}0)) ∗ a − 5335928827825482419660 + O(13^{2}0)q^{4} + −48828125q^{5} + −(3921101660051101246587 + O(13^{2}0)) ∗ a + 1164530698400942332963+O(13^{2}0)q^{6}+(4910211264593652835524+O(13^{2}0))∗a−8590218479878900641159+ O(13^{2}0)q^{7} + −(5290286604155839233045 + O(13^{2}0)) ∗ a + 7457351713744050046274 + O(13^{2}0)q^{8} + −(50460910847679574831∗13+O(13^{2}0))∗a−4600656185715835764487+O(13^{2}0)q^{9}+−48828125∗aq^{10}+ −(7336647298535474881561+O(13^{2}0))∗a−8520575535835728608028+O(13^{2}0)q^{11}+(5686030920839623525329+ O(13^{2}0)) ∗ a + 8645279762916755767266 + O(13^{2}0)q^{12} + (41700739326444700069 ∗ 13^{2} + O(13^{2}0)) ∗ a − 716299325687200141011∗13+O(13^{2}0)q^{13}+−(7162258075540135413848+O(13^{2}0))∗a+6861723610770643660849+ O(13^{2}0)q^{14}+(3476448430939270663121+O(13^{2}0))∗a+16896236732087353205+O(13^{2}0)q^{15}+(8696384008184048350007+ O(13^{2}0)) ∗ a + 275923906225627905 ∗ 13 + O(13^{2}0)q^{16} + (4146291709543845910384 + O(13^{2}0)) ∗ a − 648632023576352356759+O(13^{2}0)q^{17}+(2275583519248206939219+O(13^{2}0))∗a−680243047620055579297∗ 13 + O(13^{2}0)q^{18} + (4874408188065808379346 + O(13^{2}0)) ∗ a − 5387612182488668407321 + O(13^{2}0)q^{19} + (2497806640809517159009+O(13^{2}0))∗a−730379193437133456929+O(13^{2}0)q^{20}+(8813182732470542764857+ O(13^{2}0)) ∗ a − 7893560194467883325763 + O(13^{2}0)q^{21} + (8152822500530779621246 + O(13^{2}0)) ∗ a + 4516013678862253613545+O(13^{2}0)q^{22}+(602394723840443283760+O(13^{2}0))∗a−2196015909510745073524+ O(13^{2}0)q^{23} + −(362267696735467124230 ∗ 13 + O(13^{2}0)) ∗ a − 8569672676481301813510 + O(13^{2}0)q^{24} + 2384185791015625q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −1575543532242978676449 + O(13^{2}0)x + 5335928827825474031052 + O(13^{2}0). The slope of f_{0} is 1.

18. Consider f = q+z^{1}24q^{2}+z^{7}q^{3}+z^{8}6q^{4}+5q^{5}+z^{1}31q^{6}+z^{2}0q^{7}+z^{1}05q^{8}+6q^{9}+z^{8}2q^{10}+z^{1}48q^{11}+z^{9}3q^{12}+ z^{1}44q^{14} + z^{1}33q^{15} + z^{1}38q^{16} + z^{1}51q^{17} + z^{2}6q^{18} + z^{1}04q^{19} + z^{4}4q^{20} + z^{2}7q^{21} + z^{1}04q^{22} + z^{7}3q^{23} + 9q^{24} + 12q^{25} + ⋯ ∈ S_{24}(Γ_{0}(5); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{1}10q^{2} + z^{1}19q^{3} + z^{5}8q^{4} + q^{5} + z^{6}1q^{6} + z^{3}4q^{7} + z^{6}3q^{8} + 5q^{9} + z^{1}10q^{10} + z^{5}0q^{11} + z^{9}q^{12} + 11q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{10} & 0 \\ 0 & u(11)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(5588173740060664468381+ O(13^{2}0)) ∗ a + 8038334126079748995929 + O(13^{2}0)q^{3} + (1575543532242978676449 + O(13^{2}0)) ∗ a − 5335928827825482419660 + O(13^{2}0)q^{4} + −48828125q^{5} + −(3921101660051101246587 + O(13^{2}0)) ∗ a + 1164530698400942332963+O(13^{2}0)q^{6}+(4910211264593652835524+O(13^{2}0))∗a−8590218479878900641159+ O(13^{2}0)q^{7} + −(5290286604155839233045 + O(13^{2}0)) ∗ a + 7457351713744050046274 + O(13^{2}0)q^{8} + −(50460910847679574831∗13+O(13^{2}0))∗a−4600656185715835764487+O(13^{2}0)q^{9}+−48828125∗aq^{10}+ −(7336647298535474881561+O(13^{2}0))∗a−8520575535835728608028+O(13^{2}0)q^{11}+(5686030920839623525329+ O(13^{2}0)) ∗ a + 8645279762916755767266 + O(13^{2}0)q^{12} + (41700739326444700069 ∗ 13^{2} + O(13^{2}0)) ∗ a − 716299325687200141011∗13+O(13^{2}0)q^{13}+−(7162258075540135413848+O(13^{2}0))∗a+6861723610770643660849+ O(13^{2}0)q^{14}+(3476448430939270663121+O(13^{2}0))∗a+16896236732087353205+O(13^{2}0)q^{15}+(8696384008184048350007+ O(13^{2}0)) ∗ a + 275923906225627905 ∗ 13 + O(13^{2}0)q^{16} + (4146291709543845910384 + O(13^{2}0)) ∗ a − 648632023576352356759+O(13^{2}0)q^{17}+(2275583519248206939219+O(13^{2}0))∗a−680243047620055579297∗ 13 + O(13^{2}0)q^{18} + (4874408188065808379346 + O(13^{2}0)) ∗ a − 5387612182488668407321 + O(13^{2}0)q^{19} + (2497806640809517159009+O(13^{2}0))∗a−730379193437133456929+O(13^{2}0)q^{20}+(8813182732470542764857+ O(13^{2}0)) ∗ a − 7893560194467883325763 + O(13^{2}0)q^{21} + (8152822500530779621246 + O(13^{2}0)) ∗ a + 4516013678862253613545+O(13^{2}0)q^{22}+(602394723840443283760+O(13^{2}0))∗a−2196015909510745073524+ O(13^{2}0)q^{23} + −(362267696735467124230 ∗ 13 + O(13^{2}0)) ∗ a − 8569672676481301813510 + O(13^{2}0)q^{24} + 2384185791015625q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −1575543532242978676449 + O(13^{2}0)x + 5335928827825474031052 + O(13^{2}0). The slope of f_{0} is 1.

19. Consider f = q + 3q^{2} + 3q^{3} + 7q^{4} + 12q^{5} + 9q^{6} + 10q^{7} + 2q^{8} + 6q^{9} + 10q^{10} + 3q^{11} + 8q^{12} + 4q^{14} + 10q^{15} + 5q^{16} + 11q^{17} + 5q^{18} + 8q^{19} + 6q^{20} + 4q^{21} + 9q^{22} + 12q^{23} + 6q^{24} + q^{25} + ⋯ ∈ S_{26}(Γ_{0}(5); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-12 newform g = q + 8q^{2} + q^{3} + 5q^{4} + 5q^{5} + 8q^{6} + 7q^{7} + 10q^{8} + 5q^{9} + q^{10} + 5q^{11} + 5q^{12} + 7q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{0} & 0 \\ 0 & u(7)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−4029624225301874793877+ O(13^{2}0)q^{2}+−5454035615511091695109+O(13^{2}0)q^{3}+−7904822684897798928885+O(13^{2}0)q^{4}+−244140625q^{5}+ −6286927990675357244782+O(13^{2}0)q^{6}+−8690356711864411733555+O(13^{2}0)q^{7}+8493843782681485983341+ O(13^{2}0)q^{8}+−4429716151182386651204+O(13^{2}0)q^{9}+5284395736852232418766+O(13^{2}0)q^{10}+−5819814160387020461717+ O(13^{2}0)q^{11} + −8504347407260693384401 + O(13^{2}0)q^{12} + −444812471269482632063 ∗ 13 + O(13^{2}0)q^{13} + 3193379884152866087489+O(13^{2}0)q^{14}+7186777662946109394163+O(13^{2}0)q^{15}+−2497576911307946566093+ O(13^{2}0)q^{16}+3532538172932740805741+O(13^{2}0)q^{17}+−4022197183744891757965+O(13^{2}0)q^{18}+3084360946839732161298+ O(13^{2}0)q^{19}+−1369284953583467723017+O(13^{2}0)q^{20}+4611384805646377984983+O(13^{2}0)q^{21}+−5894917916848539085329+ O(13^{2}0)q^{22}+1587895890471923082764+O(13^{2}0)q^{23}+3112114168494502055787+O(13^{2}0)q^{24}+59604644775390625q^{25}+ ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

20. Consider f = q +z^{8}5q^{2} +z^{2}3q^{3} +z^{3}6q^{4} +q^{5} +z^{1}08q^{6} +z^{8}1q^{7} +z^{6}q^{8} +z^{1}5q^{9} +z^{8}5q^{10} +z^{2}1q^{11} +z^{5}9q^{12} + z^{1}66q^{14} +z^{2}3q^{15} +z^{3}0q^{16} +z^{1}47q^{17} +z^{1}00q^{18} +z^{3}6q^{20} +z^{1}04q^{21} +z^{1}06q^{22} +z^{8}2q^{23} +z^{2}9q^{24} +q^{25} +⋯ ∈ S_{26}(Γ_{0}(5); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-12 newform g = q + z^{7}1q^{2} + z^{1}35q^{3} + z^{8}q^{4} + 8q^{5} + z^{3}8q^{6} + z^{9}5q^{7} + z^{1}32q^{8} + z^{7}1q^{9} + z^{1}13q^{10} + z^{9}1q^{11} + z^{1}43q^{12} + z^{1}23q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{4}5)ω^{0} & 0 \\ 0 & u(z^{1}23)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(4817024871679759910491+ O(13^{2}0)) ∗ a − 5238730844434464704939 + O(13^{2}0)q^{3} + (3225649235121353490722 + O(13^{2}0)) ∗ a − 6471259024495946884285 + O(13^{2}0)q^{4} + 244140625q^{5} + −(6493714410467951448895 + O(13^{2}0)) ∗ a − 3347797951523560593204+O(13^{2}0)q^{6}+−(2497547174853538828288+O(13^{2}0))∗a−6564615942529186529024+ O(13^{2}0)q^{7}+(4009325613210611031034+O(13^{2}0))∗a+568654097995765401388+O(13^{2}0)q^{8}+(2732960379137993655558+ O(13^{2}0))∗a+701135667072764056141∗13+O(13^{2}0)q^{9}+244140625∗aq^{10}+−(7224751876456490397373+ O(13^{2}0)) ∗ a + 3259318721005238008929 + O(13^{2}0)q^{11} + −(79211798230931188664 + O(13^{2}0)) ∗ a + 9319433509915452225959+O(13^{2}0)q^{12}+−(485552150180362342871∗13+O(13^{2}0))∗a+464792176838410971625∗ 13 + O(13^{2}0)q^{13} + (7928020701746861218357 + O(13^{2}0)) ∗ a − 5639968579109909497905 + O(13^{2}0)q^{14} + (9301004392063537747849+O(13^{2}0))∗a+2197126987648448220650+O(13^{2}0)q^{15}+(8375979133029538977722+ O(13^{2}0)) ∗ a + 8920441091876197701693 + O(13^{2}0)q^{16} + (434277189605808982233 + O(13^{2}0)) ∗ a + 4345445697441727193072+O(13^{2}0)q^{17}+−(6298916763814019474920+O(13^{2}0))∗a−7421195438995738762267+ O(13^{2}0)q^{18}+−(585319522320448659722∗13+O(13^{2}0))∗a−582026080329355693541∗13+O(13^{2}0)q^{19}+ −(8520060873295497484726+O(13^{2}0))∗a−7364274591375445887758+O(13^{2}0)q^{20}+(2824734236630031092624+ O(13^{2}0)) ∗ a + 9029005023699013092762 + O(13^{2}0)q^{21} + (7908832809482419460952 + O(13^{2}0)) ∗ a − 3215224639950137163490+O(13^{2}0)q^{22}+(2185984208655910488374+O(13^{2}0))∗a+3053387386610046729409+ O(13^{2}0)q^{23} + (4124987653296844901784 + O(13^{2}0)) ∗ a − 4418910412511518567 ∗ 13^{2} + O(13^{2}0)q^{24} + 59604644775390625q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −3225649235121353490722 + O(13^{2}0)x + 6471259024495913329853 + O(13^{2}0). The slope of f_{0} is 1.

21. Consider f = q+z^{9}7q^{2}+z^{1}31q^{3}+z^{1}32q^{4}+q^{5}+z^{6}0q^{6}+z^{4}5q^{7}+z^{7}8q^{8}+z^{2}7q^{9}+z^{9}7q^{10}+z^{1}05q^{11}+z^{9}5q^{12}+ z^{1}42q^{14} + z^{1}31q^{15} + z^{5}4q^{16} + z^{6}3q^{17} + z^{1}24q^{18} + z^{1}32q^{20} + z^{8}q^{21} + z^{3}4q^{22} + z^{5}8q^{23} + z^{4}1q^{24} + q^{25} + ⋯ ∈ S_{26}(Γ_{0}(5); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-12 newform g = q + z^{8}3q^{2} + z^{7}5q^{3} + z^{1}04q^{4} + 8q^{5} + z^{1}58q^{6} + z^{5}9q^{7} + z^{3}6q^{8} + z^{8}3q^{9} + z^{1}25q^{10} + z^{7}q^{11} + z^{1}1q^{12} + z^{8}7q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{8}1)ω^{0} & 0 \\ 0 & u(z^{8}7)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(4817024871679759910491+ O(13^{2}0)) ∗ a − 5238730844434464704939 + O(13^{2}0)q^{3} + (3225649235121353490722 + O(13^{2}0)) ∗ a − 6471259024495946884285 + O(13^{2}0)q^{4} + 244140625q^{5} + −(6493714410467951448895 + O(13^{2}0)) ∗ a − 3347797951523560593204+O(13^{2}0)q^{6}+−(2497547174853538828288+O(13^{2}0))∗a−6564615942529186529024+ O(13^{2}0)q^{7}+(4009325613210611031034+O(13^{2}0))∗a+568654097995765401388+O(13^{2}0)q^{8}+(2732960379137993655558+ O(13^{2}0))∗a+701135667072764056141∗13+O(13^{2}0)q^{9}+244140625∗aq^{10}+−(7224751876456490397373+ O(13^{2}0)) ∗ a + 3259318721005238008929 + O(13^{2}0)q^{11} + −(79211798230931188664 + O(13^{2}0)) ∗ a + 9319433509915452225959+O(13^{2}0)q^{12}+−(485552150180362342871∗13+O(13^{2}0))∗a+464792176838410971625∗ 13 + O(13^{2}0)q^{13} + (7928020701746861218357 + O(13^{2}0)) ∗ a − 5639968579109909497905 + O(13^{2}0)q^{14} + (9301004392063537747849+O(13^{2}0))∗a+2197126987648448220650+O(13^{2}0)q^{15}+(8375979133029538977722+ O(13^{2}0)) ∗ a + 8920441091876197701693 + O(13^{2}0)q^{16} + (434277189605808982233 + O(13^{2}0)) ∗ a + 4345445697441727193072+O(13^{2}0)q^{17}+−(6298916763814019474920+O(13^{2}0))∗a−7421195438995738762267+ O(13^{2}0)q^{18}+−(585319522320448659722∗13+O(13^{2}0))∗a−582026080329355693541∗13+O(13^{2}0)q^{19}+ −(8520060873295497484726+O(13^{2}0))∗a−7364274591375445887758+O(13^{2}0)q^{20}+(2824734236630031092624+ O(13^{2}0)) ∗ a + 9029005023699013092762 + O(13^{2}0)q^{21} + (7908832809482419460952 + O(13^{2}0)) ∗ a − 3215224639950137163490+O(13^{2}0)q^{22}+(2185984208655910488374+O(13^{2}0))∗a+3053387386610046729409+ O(13^{2}0)q^{23} + (4124987653296844901784 + O(13^{2}0)) ∗ a − 4418910412511518567 ∗ 13^{2} + O(13^{2}0)q^{24} + 59604644775390625q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −3225649235121353490722 + O(13^{2}0)x + 6471259024495913329853 + O(13^{2}0). The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

22. Consider f = q+9q2 +2q3 +8q4 +8q5 +5q6 +6q7 +3q9 +7q10 +6q11 +3q12 +q13 +2q14 +3q15 +q16 +q18 +9q19 +12q20 +12q21 +2q22 +12q25 +⋯ ∈ S16(Γ0(5); \overline{F}13). Since f is ordinary, we do not continue our analysis.

23. Consider f = q + 6q2 + 12q3 + 2q4 + 5q5 + 7q6 + 4q7 + 3q8 + 4q10 + 2q11 + 11q12 + 11q14 + 8q15 + 2q16 + 7q17 + 3q19 + 10q20 + 9q21 + 12q22 + 6q23 + 10q24 + 12q25 + ⋯ ∈ S16(Γ0(5); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

24. Consider f = q + z115q2 + z155q3 + z10q4 + 5q5 + z102q6 + z58q7 + z26q8 + z69q9 + z73q10 + z48q11 + z165q12 + z80q13 + z5q14 + z113q15 + z48q16 + z114q17 + z16q18 + z155q19 + z136q20 + z45q21 + z163q22 + z106q23 + z13q24 + 12q25 + ⋯ ∈ S20(Γ0(5); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

25. Consider f = q + z151q2 + z167q3 + z130q4 + 5q5 + z150q6 + z82q7 + z2q8 + z57q9 + z109q10 + z120q11 + z129q12 + z32q13 + z65q14 + z125q15 + z120q16 + z138q17 + z40q18 + z167q19 + z88q20 + z81q21 + z103q22 + z34q23 + zq24 + 12q25 + ⋯ ∈ S20(Γ0(5); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

26. Consider f = q + 12q2 + 4q3 + 3q4 + 8q5 + 9q6 + 7q7 + 8q8 + 5q10 + 3q11 + 12q12 + q13 + 6q14 + 6q15 + 11q16 + 5q17 + 8q19 + 11q20 + 2q21 + 10q22 + 11q23 + 6q24 + 12q25 + ⋯ ∈ S20(Γ0(5); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

27. Consider f = q + z86q2 + z35q3 + z82q4 + q5 + z121q6 + z106q7 + z147q8 + 5q9 + z86q10 + z146q11 + z117q12 + 11q13 + z24q14 + z35q15 + z58q16 + z87q17 + z44q18 + z106q19 + z82q20 + z141q21 + z64q22 + z137q23 + 2q24 + q25 + ⋯ ∈ S22(Γ0(5); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

28. Consider f = q + z110q2 + z119q3 + z58q4 + q5 + z61q6 + z34q7 + z63q8 + 5q9 + z110q10 + z50q11 + z9q12 + 11q13 + z144q14 + z119q15 + z82q16 + z123q17 + z68q18 + z34q19 + z58q20 + z153q21 + z160q22 + z101q23 + 2q24 + q25 + ⋯ ∈ S22(Γ0(5); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

29. Consider f = q + z71q2 + z135q3 + z8q4 + 8q5 + z38q6 + z95q7 + z132q8 + z71q9 + z113q10 + z91q11 + z143q12 + z123q13 + z166q14 + z9q15 + z142q16 + z119q17 + z142q18 + z50q20 + z62q21 + z162q22 + z110q23 + z99q24 + 12q25 + ⋯ ∈ S24(Γ0(5); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

30. Consider f = q + z83q2 + z75q3 + z104q4 + 8q5 + z158q6 + z59q7 + z36q8 + z83q9 + z125q10 + z7q11 + z11q12 + z87q13 + z142q14 + z117q15 + z166q16 + z35q17 + z166q18 + z146q20 + z134q21 + z90q22 + z86q23 + z111q24 + 12q25 + ⋯ ∈ S24(Γ0(5); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

31. Consider f = q + 8q2 + q3 + 5q4 + 5q5 + 8q6 + 7q7 + 10q8 + 5q9 + q10 + 5q11 + 5q12 + 7q13 + 4q14 + 5q15 + 6q16 + 6q17 + q18 + 10q19 + 12q20 + 7q21 + q22 + 9q23 + 10q24 + 12q25 + ⋯ ∈ S24(Γ0(5); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

32. Consider f = q + 12q2 + 11q3 + 12q4 + 12q5 + 2q6 + 9q7 + 3q8 + q9 + q10 + 2q11 + 2q12 + 12q13 + 4q14 + 2q15 + 12q16 + 2q17 + 12q18 + 7q19 + q20 + 8q21 + 11q22 + 7q23 + 7q24 + q25 + ⋯ ∈ S26(Γ0(5); \overline{F}13). Since f is ordinary, we do not continue our analysis.

33. Consider f = q + 9q2 + 5q3 + q4 + 12q5 + 6q6 + 2q7 + 4q8 + 9q9 + 4q10 + 6q11 + 5q12 + q13 + 5q14 + 8q15 + 8q16 + 5q17 + 3q18 + 12q20 + 10q21 + 2q22 + 12q23 + 7q24 + q25 + ⋯ ∈ S26(Γ0(5); \overline{F}13). Since f is ordinary, we do not continue our analysis.

34. Consider f = q + 4q2 + 10q3 + q4 + 12q5 + q6 + 2q7 + 9q8 + 6q9 + 9q10 + q11 + 10q12 + q13 + 8q14 + 3q15 + 8q16 + 8q17 + 11q18 + 11q19 + 12q20 + 7q21 + 4q22 + 7q23 + 12q24 + q25 + ⋯ ∈ S26(Γ0(5); \overline{F}13). Since f is ordinary, we do not continue our analysis.

35. Consider f = q + z102q2 + z91q3 + z89q4 + q5 + z25q6 + z32q7 + z167q8 + 12q9 + z102q10 + z25q11 + z12q12 + 12q13 + z134q14 + z91q15 + 3q16 + z164q17 + z18q18 + z157q19 + z89q20 + z123q21 + z127q22 + z7q23 + z90q24 + q25 + ⋯ ∈ S26(Γ0(5); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

36. Consider f = q + z150q2 + z7q3 + z149q4 + q5 + z157q6 + z80q7 + z155q8 + 12q9 + z150q10 + z157q11 + z156q12 + 12q13 + z62q14 + z7q15 + 3q16 + z116q17 + z66q18 + z25q19 + z149q20 + z87q21 + z139q22 + z91q23 + z162q24 + q25 + ⋯ ∈ S26(Γ0(5); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

37. Consider f = q + 3q2 + 4q3 + 7q4 + q5 + 12q6 + 8q7 + 2q8 + 3q10 + 12q11 + 2q12 + 11q14 + 4q15 + 5q16 + 5q17 + 7q19 + 7q20 + 6q21 + 10q22 + 11q23 + 8q24 + q25 + ⋯ ∈ S26(Γ0(5); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

13.5 N = 6, ℓ = 13 (20 forms)

1. Consider f = q +11q^{2} +3q^{3} +4q^{4} +10q^{5} +7q^{6} +5q^{7} +5q^{8} +9q^{9} +6q^{10} +q^{11} +12q^{12} +3q^{14} +4q^{15} +3q^{16} + 8q^{18}+6q^{19}+q^{20}+2q^{21}+11q^{22}+2q^{24}+q^{25}+⋯ ∈ S_{16}(Γ_{0}(6); \overline{F}_{13}). We have ρ_{f} ⊗χ^{10} ∼ ρ_{g}, where g is the weight-12 newform g = q+6q^{2}+9q^{3}+10q^{4}+3q^{5}+2q^{6}+7q^{7}+8q^{8}+3q^{9}+5q^{10}+10q^{11}+12q^{12}+12q^{13}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(12)ω^{1} & 0 \\ 0 & u(12)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 128q^{2} + 2187q^{3} + 16384q^{4} + 77646q^{5} + 279936q^{6} + 762104q^{7} + 2097152q^{8} + 4782969q^{9} + 9938688q^{10} + 48011172q^{11} + 35831808q^{12} + 21933086 ∗ 13q^{13} + 97549312q^{14} + 169811802q^{15} + 268435456q^{16} + −244128582 ∗ 13q^{17} + 612220032q^{18} + −5895116260q^{19} + 1272152064q^{20} + 1666721448q^{21} + 6145430016q^{22} + −25616184 ∗ 13q^{23} + 4586471424q^{24} + −24488676809q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 2q^{2} + 10q^{3} + 4q^{4} + 6q^{5} + 7q^{6} + 7q^{7} + 8q^{8} + 9q^{9} + 12q^{10} + 11q^{11} + q^{12} + q^{14} + 8q^{15} + 3q^{16} + 9q^{17} + 5q^{18} + 10q^{19} + 11q^{20} + 5q^{21} + 9q^{22} + 6q^{23} + 2q^{24} + 2q^{25} + ⋯ ∈ S_{16}(Γ_{0}(6); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{10} ∼ ρ_{g}, where g is the weight-12 newform g = q + 7q^{2} + 4q^{3} + 10q^{4} + 7q^{5} + 2q^{6} + 2q^{7} + 5q^{8} + 3q^{9} + 10q^{10} + 6q^{11} + q^{12} + 10q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{1} & 0 \\ 0 & u(10)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +−128q^{2} +−2187q^{3} + 16384q^{4}+−314490q^{5}+279936q^{6}+2025056q^{7}+−2097152q^{8}+4782969q^{9}+40254720q^{10}+110255052q^{11}+ −35831808q^{12} + 4311374 ∗ 13q^{13} + −259207168q^{14} + 687789630q^{15} + 268435456q^{16} + −1930104414q^{17} + −612220032q^{18}+2163188180q^{19}+−5152604160q^{20}+−4428797472q^{21}+−14112646656q^{22}+6228974472q^{23}+ 4586471424q^{24} + 68386381975q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q+4q^{2}+9q^{3}+3q^{4}+3q^{5}+10q^{6}+6q^{7}+12q^{8}+3q^{9}+12q^{10}+2q^{11}+q^{12}+11q^{14}+q^{15}+9q^{16}+4q^{17}+ 12q^{18}+6q^{19}+9q^{20}+2q^{21}+8q^{22}+11q^{23}+4q^{24}+4q^{25}+⋯ ∈ S_{18}(Γ_{0}(6); \overline{F}_{13}). We have ρ_{f} ⊗χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q+10q^{2}+3q^{3}+9q^{4}+3q^{5}+4q^{6}+5q^{7}+12q^{8}+9q^{9}+4q^{10}+5q^{11}+q^{12}+6q^{13}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(11)ω^{1} & 0 \\ 0 & u(6)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −256q^{2} + 6561q^{3} +65536q^{4} +−72186q^{5} +−1679616q^{6} +−8640184q^{7} +−16777216q^{8} +43046721q^{9} +18479616q^{10} + 1159304460q^{11} +429981696q^{12} +215466374∗13q^{13} +2211887104q^{14} +−473612346q^{15} +4294967296q^{16} + 32979662226q^{17}+−11019960576q^{18}+5778498836q^{19}+−4730781696q^{20}+−56688247224q^{21}+−296781941760q^{22}+ 169116994200q^{23} +−110075314176q^{24} +−757728634529q^{25} +⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

4. Consider f = q + 9q^{2} + 4q^{3} + 3q^{4} + 4q^{5} + 10q^{6} + 5q^{7} + q^{8} + 3q^{9} + 10q^{10} + 2q^{11} + 12q^{12} + 6q^{14} + 3q^{15} + 9q^{16} + 3q^{17} + q^{18} + 3q^{19} + 12q^{20} + 7q^{21} + 5q^{22} + 3q^{23} + 4q^{24} + 11q^{25} + ⋯ ∈ S_{18}(Γ_{0}(6); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-4 newform g = q + 11q^{2} + 10q^{3} + 4q^{4} + 6q^{5} + 6q^{6} + 10q^{7} + 5q^{8} + 9q^{9} + q^{10} + 12q^{11} + q^{12} + 12q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(12)ω^{4} & 0 \\ 0 & u(12)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 256q^{2} + −6561q^{3} + 65536q^{4}+−199650q^{5}+−1679616q^{6}+24959264q^{7}+16777216q^{8}+43046721q^{9}+−51110400q^{10}+125556420q^{11}+ −429981696q^{12}+325168886∗13q^{13}+6389571584q^{14}+1309903650q^{15}+4294967296q^{16}+35551782594q^{17}+ 11019960576q^{18}+−64354589764q^{19}+−13084262400q^{20}+−163757731104q^{21}+32142443520q^{22}+−245819296200q^{23}+ −110075314176q^{24} + −723079330625q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

5. Consider f = q+5q^{2} +q^{3} +12q^{4} +10q^{5} +5q^{6} +3q^{7} +8q^{8} +q^{9} +11q^{10} +12q^{12} +2q^{14} +10q^{15} +q^{16} +11q^{17} + 5q^{18} + 9q^{19} + 3q^{20} + 3q^{21} + 4q^{23} + 8q^{24} + q^{25} + ⋯ ∈ S_{20}(Γ_{0}(6); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + 8q^{2} + q^{3} + 12q^{4} + 3q^{5} + 8q^{6} + 10q^{7} + 5q^{8} + q^{9} + 11q^{10} + 12q^{12} + 7q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(7)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 512q^{2} + 19683q^{3} + 262144q^{4} + 1953390q^{5} + 10077696q^{6} + 40488776q^{7} + 134217728q^{8} + 387420489q^{9} + 1000135680q^{10} + 147450804∗13q^{11}+5159780352q^{12}+240960014∗13q^{13}+20730253312q^{14}+38448575370q^{15}+68719476736q^{16}+ 607659965586q^{17}+198359290368q^{18}+2507511106460q^{19}+512069468160q^{20}+796940578008q^{21}+75494811648∗ 13q^{22} + −13588841327928q^{23} + 2641807540224q^{24} + −15257753836025q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

6. Consider f = q + 8q^{2} + 12q^{3} + 12q^{4} + 8q^{5} + 5q^{6} + 10q^{7} + 5q^{8} + q^{9} + 12q^{10} + 3q^{11} + q^{12} + 2q^{14} + 5q^{15} + q^{16} + 8q^{17} + 8q^{18} + 3q^{19} + 5q^{20} + 3q^{21} + 11q^{22} + 7q^{23} + 8q^{24} + 4q^{25} + ⋯ ∈ S_{20}(Γ_{0}(6); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-6 newform g = q + 4q^{2} + 4q^{3} + 3q^{4} + 12q^{5} + 3q^{6} + 7q^{7} + 12q^{8} + 3q^{9} + 9q^{10} + 5q^{11} + 12q^{12} + 5q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{6} & 0 \\ 0 & u(5)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−512q^{2}+−19683q^{3}+ 262144q^{4}+−3732474q^{5}+10077696q^{6}+−149672656q^{7}+−134217728q^{8}+387420489q^{9}+1911026688q^{10}+ −7459672308q^{11}+−5159780352q^{12}+4556804606∗13q^{13}+76632399872q^{14}+73466285742q^{15}+68719476736q^{16}+ 523110429954q^{17} + −198359290368q^{18} + 969502037780q^{19} + −978445664256q^{20} + 2946006888048q^{21} + 3819352221696q^{22} + −1368374071512q^{23} + 2641807540224q^{24} + −5142124167449q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

7. Consider f = q + 10q^{2} + 10q^{3} + 9q^{4} + 12q^{5} + 9q^{6} + 8q^{7} + 12q^{8} + 9q^{9} + 3q^{10} + 6q^{11} + 12q^{12} + 2q^{14} + 3q^{15} + 3q^{16} + 6q^{17} + 12q^{18} + 8q^{19} + 4q^{20} + 2q^{21} + 8q^{22} + 5q^{23} + 3q^{24} + 9q^{25} + ⋯ ∈ S_{22}(Γ_{0}(6); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 4q^{2} + 4q^{3} + 3q^{4} + 12q^{5} + 3q^{6} + 7q^{7} + 12q^{8} + 3q^{9} + 9q^{10} + 5q^{11} + 12q^{12} + 5q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{1} & 0 \\ 0 & u(5)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 1024q^{2} + −59049q^{3} +1048576q^{4} +12954174q^{5} +−60466176q^{6} +−479513104q^{7} +1073741824q^{8} +3486784401q^{9} + 13265074176q^{10} + 115657781700q^{11} + −61917364224q^{12} + 22742942054 ∗ 13q^{13} + −491021418496q^{14} + −764931020526q^{15}+1099511627776q^{16}+6626983431906q^{17}+3570467226624q^{18}+28576184164796q^{19}+ 13583435956224q^{20}+28314769278096q^{21}+118433568460800q^{22}+335385196791000q^{23}+−63403380965376q^{24}+ −309026534180849q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

8. Consider f = q + 3q^{2} + 3q^{3} + 9q^{4} + 2q^{5} + 9q^{6} + 5q^{7} + q^{8} + 9q^{9} + 6q^{10} + q^{12} + 2q^{14} + 6q^{15} + 3q^{16} + 5q^{17} + q^{18} + 11q^{19} + 5q^{20} + 2q^{21} + q^{23} + 3q^{24} + 12q^{25} + ⋯ ∈ S_{22}(Γ_{0}(6); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-8 newform g = q + 8q^{2} + q^{3} + 12q^{4} + 3q^{5} + 8q^{6} + 10q^{7} + 5q^{8} + q^{9} + 11q^{10} + 12q^{12} + 7q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{8} & 0 \\ 0 & u(7)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −1024q^{2} + 59049q^{3} + 1048576q^{4} + 26444550q^{5} + −60466176q^{6} + 166115864q^{7} + −1073741824q^{8} + 3486784401q^{9} + −27079219200q^{10} +−8067597060∗13q^{11} +61917364224q^{12} +25814717366∗13q^{13} +−170102644736q^{14} + 1561524232950q^{15}+1099511627776q^{16}+14596144763634q^{17}+−3570467226624q^{18}+3569529974996q^{19}+ 27729120460800q^{20}+9808975653336q^{21}+8261219389440∗13q^{22}+222369240588600q^{23}+−63403380965376q^{24}+ 222477066499375q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

9. Consider f = q + 6q^{2} + 4q^{3} + 10q^{4} + 7q^{5} + 11q^{6} + q^{7} + 8q^{8} + 3q^{9} + 3q^{10} + 3q^{11} + q^{12} + 6q^{14} + 2q^{15} + 9q^{16} + 10q^{17} + 5q^{18} + 2q^{19} + 5q^{20} + 4q^{21} + 5q^{22} + 10q^{23} + 6q^{24} + 2q^{25} + ⋯ ∈ S_{24}(Γ_{0}(6); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 11q^{2} + 10q^{3} + 4q^{4} + 6q^{5} + 6q^{6} + 10q^{7} + 5q^{8} + 9q^{9} + q^{10} + 12q^{11} + q^{12} + 12q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(12)ω^{1} & 0 \\ 0 & u(12)ω^{10} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −2048q^{2} + −177147q^{3}+4194304q^{4}+196251270q^{5}+362797056q^{6}+−8131131904q^{7}+−8589934592q^{8}+31381059609q^{9}+ −401922600960q^{10}+413280901452q^{11}+−743008370688q^{12}+−455779544146∗13q^{13}+16652558139392q^{14}+ −34765323726690q^{15}+17592186044416q^{16}+44831041228386q^{17}+−64268410079232q^{18}+869707616627540q^{19}+ 823137486766080q^{20}+1440405623397888q^{21}+−846399286173696q^{22}+−3916239181103928q^{23}+1521681143169024q^{24}+ 26593632021534775q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

10. Consider f = q + 7q^{2} + 9q^{3} + 10q^{4} + 2q^{5} + 11q^{6} + 9q^{7} + 5q^{8} + 3q^{9} + q^{10} + 3q^{11} + 12q^{12} + 11q^{14} + 5q^{15} + 9q^{16} + 9q^{17} + 8q^{18} + 4q^{19} + 7q^{20} + 3q^{21} + 8q^{22} + 2q^{23} + 6q^{24} + 9q^{25} + ⋯ ∈ S_{24}(Γ_{0}(6); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-10 newform g = q + 10q^{2} + 3q^{3} + 9q^{4} + 3q^{5} + 4q^{6} + 5q^{7} + 12q^{8} + 9q^{9} + 4q^{10} + 5q^{11} + q^{12} + 6q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(11)ω^{10} & 0 \\ 0 & u(6)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +2048q^{2} +177147q^{3} + 4194304q^{4} + −678040920585597483315 + O(13^{2}0)q^{5} + 362797056q^{6} + −2053940760237664522592 + O(13^{2}0)q^{7}+8589934592q^{8}+31381059609q^{9}+−1265449793005286796647+O(13^{2}0)q^{10}+−2924605830802968318354+ O(13^{2}0)q^{11}+743008370688q^{12}+324506150306473008911∗13+O(13^{2}0)q^{13}+−6373682718080266293395+ O(13^{2}0)q^{14}+−1543901730184923579985+O(13^{2}0)q^{15}+17592186044416q^{16}+−1249174841287578647173+ O(13^{2}0)q^{17}+64268410079232q^{18}+1078342756058768380639+O(13^{2}0)q^{19}+−6966102691038635856120+ O(13^{2}0)q^{20}+587616271348072240121+O(13^{2}0)q^{21}+−3029152397027292766677+O(13^{2}0)q^{22}+9144697135906431425211+ O(13^{2}0)q^{23} +1521681143169024q^{24} +8331831115340023375015+O(13^{2}0)q^{25} +⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

11. Consider f = q+12q^{2}+q^{3}+q^{4}+2q^{5}+12q^{6}+10q^{7}+12q^{8}+q^{9}+11q^{10}+6q^{11}+q^{12}+3q^{14}+2q^{15}+q^{16}+12q^{18}+ q^{19} + 2q^{20} + 10q^{21} + 7q^{22} + 12q^{24} + 12q^{25} + ⋯ ∈ S_{26}(Γ_{0}(6); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-12 newform g = q+6q^{2}+9q^{3}+10q^{4}+3q^{5}+2q^{6}+7q^{7}+8q^{8}+3q^{9}+5q^{10}+10q^{11}+12q^{12}+12q^{13}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(12)ω^{0} & 0 \\ 0 & u(12)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−4096q^{2}+531441q^{3}+ 16777216q^{4} + 590425734q^{5} + −2176782336q^{6} + 57857417576q^{7} + −68719476736q^{8} + 282429536481q^{9} + −2418383806464q^{10}+9494266240140q^{11}+8916100448256q^{12}+−10382155466266∗13q^{13}+−236983982391296q^{14}+ 313776442502694q^{15} + 281474976710656q^{16} + −194316462831078 ∗ 13q^{17} + −1156831381426176q^{18} + 11468758872260756q^{19} + 9905700071276544q^{20} + 30747803854007016q^{21} + −38888514519613440q^{22} + 8718663907846200∗13q^{23}+−36520347436056576q^{24}+50579323492485631q^{25}+⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

12. Consider f = q+q^{2}+12q^{3}+q^{4}+9q^{5}+12q^{6}+q^{7}+q^{8}+q^{9}+9q^{10}+q^{11}+12q^{12}+q^{14}+4q^{15}+q^{16}+12q^{17}+q^{18}+ 6q^{19}+9q^{20}+12q^{21}+q^{22}+11q^{23}+12q^{24}+11q^{25}+⋯ ∈ S_{26}(Γ_{0}(6); \overline{F}_{13}). We have ρ_{f} ⊗χ^{11} ∼ ρ_{g}, where g is the weight-12 newform g = q+7q^{2}+4q^{3}+10q^{4}+7q^{5}+2q^{6}+2q^{7}+5q^{8}+3q^{9}+10q^{10}+6q^{11}+q^{12}+10q^{13}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{0} & 0 \\ 0 & u(10)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+4096q^{2}+−531441q^{3}+ 16777216q^{4} + −799327650q^{5} + −2176782336q^{6} + 7962409664q^{7} + 68719476736q^{8} + 282429536481q^{9} + −3274046054400q^{10}+−1760169438780q^{11}+−8916100448256q^{12}+12630597778646∗13q^{13}+32614029983744q^{14}+ 424795485643650q^{15}+281474976710656q^{16}+−2377485783158526q^{17}+1156831381426176q^{18}+10628455411412156q^{19}+ −13410492638822400q^{20} +−4231550954245824q^{21} +−7209654021242880q^{22} +76636270038478200q^{23} + −36520347436056576q^{24} + 340901468177569375q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

13. Consider f = q + 11q2 + 10q3 + 4q4 + 6q5 + 6q6 + 10q7 + 5q8 + 9q9 + q10 + 12q11 + q12 + 12q13 + 6q14 + 8q15 + 3q16 + 4q17 + 8q18 + 7q19 + 11q20 + 9q21 + 2q22 + 12q23 + 11q24 + 2q25 + ⋯ ∈ S16(Γ0(6); \overline{F}13). Since f is ordinary, we do not continue our analysis.

14. Consider f = q + 4q2 + 4q3 + 3q4 + 12q5 + 3q6 + 7q7 + 12q8 + 3q9 + 9q10 + 5q11 + 12q12 + 5q13 + 2q14 + 9q15 + 9q16 + 2q17 + 12q18 + 7q19 + 10q20 + 2q21 + 7q22 + 2q23 + 9q24 + 9q25 + ⋯ ∈ S18(Γ0(6); \overline{F}13). Since f is ordinary, we do not continue our analysis.

15. Consider f = q + 8q2 + q3 + 12q4 + 3q5 + 8q6 + 10q7 + 5q8 + q9 + 11q10 + 12q12 + 7q13 + 2q14 + 3q15 + q16 + 11q17 + 8q18 + 4q19 + 10q20 + 10q21 + 4q23 + 5q24 + q25 + ⋯ ∈ S20(Γ0(6); \overline{F}13). Since f is ordinary, we do not continue our analysis.

16. Consider f = q + 10q2 + 3q3 + 9q4 + 3q5 + 4q6 + 5q7 + 12q8 + 9q9 + 4q10 + 5q11 + q12 + 6q13 + 11q14 + 9q15 + 3q16 + 12q17 + 12q18 + 5q19 + q20 + 2q21 + 11q22 + 8q23 + 10q24 + 4q25 + ⋯ ∈ S22(Γ0(6); \overline{F}13). Since f is ordinary, we do not continue our analysis.

17. Consider f = q + 7q2 + 4q3 + 10q4 + 7q5 + 2q6 + 2q7 + 5q8 + 3q9 + 10q10 + 6q11 + q12 + 10q13 + q14 + 2q15 + 9q16 + 3q17 + 8q18 + q19 + 5q20 + 8q21 + 3q22 + 5q23 + 7q24 + 2q25 + ⋯ ∈ S24(Γ0(6); \overline{F}13). Since f is ordinary, we do not continue our analysis.

18. Consider f = q + 6q2 + 9q3 + 10q4 + 3q5 + 2q6 + 7q7 + 8q8 + 3q9 + 5q10 + 10q11 + 12q12 + 12q13 + 3q14 + q15 + 9q16 + 5q18 + 11q19 + 4q20 + 11q21 + 8q22 + 7q24 + q25 + ⋯ ∈ S24(Γ0(6); \overline{F}13). Since f is ordinary, we do not continue our analysis.

19. Consider f = q + 7q2 + 9q3 + 10q4 + 9q5 + 11q6 + 3q7 + 5q8 + 3q9 + 11q10 + 7q11 + 12q12 + q13 + 8q14 + 3q15 + 9q16 + 11q17 + 8q18 + 12q19 + 12q20 + q21 + 10q22 + 5q23 + 6q24 + 8q25 + ⋯ ∈ S24(Γ0(6); \overline{F}13). Since f is ordinary, we do not continue our analysis.

20. Consider f = q + 12q2 + 12q3 + q4 + 2q5 + q6 + 4q7 + 12q8 + q9 + 11q10 + 9q11 + 12q12 + q13 + 9q14 + 11q15 + q16 + 2q17 + 12q18 + 5q19 + 2q20 + 9q21 + 4q22 + q24 + 12q25 + ⋯ ∈ S26(Γ0(6); \overline{F}13). Since f is ordinary, we do not continue our analysis.

13.6 N = 7, ℓ = 13 (56 forms)

1. Consider f = q + z^{1}30q^{2} + zq^{3} + z^{1}43q^{4} + z^{1}22q^{5} + z^{1}31q^{6} + 6q^{7} + z^{5}0q^{8} + z^{3}9q^{9} + 12q^{10} + 11q^{11} + z^{1}44q^{12} + z^{3}2q^{14} + z^{1}23q^{15} + z^{4}4q^{16} + z^{1}33q^{17} + zq^{18} + z^{9}7q^{19} + z^{9}7q^{20} + z^{7}1q^{21} + z^{6}0q^{22} + z^{1}64q^{23} + z^{5}1q^{24} + z^{1}38q^{25} + ⋯ ∈ S_{16}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{10} ∼ ρ_{g}, where g is the weight-12 newform g = q + z^{1}02q^{2} + z^{5}7q^{3} + z^{8}7q^{4} + z^{3}8q^{5} + z^{1}59q^{6} + 11q^{7} + z^{1}34q^{8} + z^{1}51q^{9} + 10q^{10} + 6q^{11} + z^{1}44q^{12} + z^{1}67q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z)ω^{1} & 0 \\ 0 & u(z^{1}67)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(9329958858288907968487+ O(13^{2}0)) ∗ a + 7562562216112441623311 + O(13^{2}0)q^{3} + −(2760368466423868014502 + O(13^{2}0)) ∗ a − 6971212808514269314810+O(13^{2}0)q^{4}+(8267390115163250621502+O(13^{2}0))∗a−259606510531290228534+ O(13^{2}0)q^{5}+−(207893217219449301505+O(13^{2}0))∗a+6557687179708102195828+O(13^{2}0)q^{6}+823543q^{7}+ −(7276293298716973815740+O(13^{2}0))∗a−6407957626558759662389+O(13^{2}0)q^{8}+−(4612632675558796866387+ O(13^{2}0)) ∗ a + 3657550997986767747634 + O(13^{2}0)q^{9} + −(127045723787799152879 ∗ 13 + O(13^{2}0)) ∗ a − 1313135677938326435025+O(13^{2}0)q^{10}+−(35477406620544946425∗13^{2}+O(13^{2}0))∗a+511221811079119917931+ O(13^{2}0)q^{11} + −(2870244219471936583651 + O(13^{2}0)) ∗ a − 510752989406308633064 ∗ 13 + O(13^{2}0)q^{12} + (453054560525487782930∗13+O(13^{2}0))∗a−636101102354146201010∗13+O(13^{2}0)q^{13}+823543∗aq^{14}+ (4045000997094555040251+O(13^{2}0))∗a+9091857072255346124734+O(13^{2}0)q^{15}+(33108491273964405228991+ O(13^{2}1)) ∗ a − 104878616586830885839068 + O(13^{2}1)q^{16} + (4002448281428558961755 + O(13^{2}0)) ∗ a − 8052215476668073979351+O(13^{2}0)q^{17}+(4327911188296000936169+O(13^{2}0))∗a+3606810869568952698385+ O(13^{2}0)q^{18} + −(2420627208219898283522 + O(13^{2}0)) ∗ a + 4938669912157861915975 + O(13^{2}0)q^{19} + −(7684293478524480068193+O(13^{2}0))∗a+1008598992274449635128+O(13^{2}0)q^{20}+(8526297385554817799655+ O(13^{2}0)) ∗ a − 2498558523391503626036 + O(13^{2}0)q^{21} + −(9288199545084343608467 + O(13^{2}0)) ∗ a + 6185546384500848704∗13^{2}+O(13^{2}0)q^{22}+(2468986169217895590399+O(13^{2}0))∗a+7607690842342525786757+ O(13^{2}0)q^{23} + −(5259270433181216630409 + O(13^{2}0)) ∗ a − 2191531923848513372433 + O(13^{2}0)q^{24} + −(3220354663910245615171 + O(13^{2}0)) ∗ a − 9224363017955273273802 + O(13^{2}0)q^{25} + ⋯ is an ℓadic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 2760368466423868014502 + O(13^{2}0)x + 6971212808514269282042 + O(13^{2}0). The slope of f_{0} is 1.

2. Consider f = q + z^{1}0q^{2} + z^{1}3q^{3} + z^{1}1q^{4} + z^{7}4q^{5} + z^{2}3q^{6} + 6q^{7} + z^{1}46q^{8} + z^{3}q^{9} + 12q^{10} + 11q^{11} + z^{2}4q^{12} + z^{8}0q^{14} + z^{8}7q^{15} + z^{6}8q^{16} + z^{4}9q^{17} + z^{1}3q^{18} + z^{8}5q^{19} + z^{8}5q^{20} + z^{8}3q^{21} + z^{1}08q^{22} + z^{1}16q^{23} + z^{1}59q^{24} + z^{1}14q^{25} + ⋯ ∈ S_{16}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{10} ∼ ρ_{g}, where g is the weight-12 newform g = q + z^{1}50q^{2} + z^{6}9q^{3} + z^{1}23q^{4} + z^{1}58q^{5} + z^{5}1q^{6} + 11q^{7} + z^{6}2q^{8} + z^{1}15q^{9} + 10q^{10} + 6q^{11} + z^{2}4q^{12} + z^{1}55q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}3)ω^{1} & 0 \\ 0 & u(z^{1}55)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(9329958858288907968487+ O(13^{2}0)) ∗ a + 7562562216112441623311 + O(13^{2}0)q^{3} + −(2760368466423868014502 + O(13^{2}0)) ∗ a − 6971212808514269314810+O(13^{2}0)q^{4}+(8267390115163250621502+O(13^{2}0))∗a−259606510531290228534+ O(13^{2}0)q^{5}+−(207893217219449301505+O(13^{2}0))∗a+6557687179708102195828+O(13^{2}0)q^{6}+823543q^{7}+ −(7276293298716973815740+O(13^{2}0))∗a−6407957626558759662389+O(13^{2}0)q^{8}+−(4612632675558796866387+ O(13^{2}0)) ∗ a + 3657550997986767747634 + O(13^{2}0)q^{9} + −(127045723787799152879 ∗ 13 + O(13^{2}0)) ∗ a − 1313135677938326435025+O(13^{2}0)q^{10}+−(35477406620544946425∗13^{2}+O(13^{2}0))∗a+511221811079119917931+ O(13^{2}0)q^{11} + −(2870244219471936583651 + O(13^{2}0)) ∗ a − 510752989406308633064 ∗ 13 + O(13^{2}0)q^{12} + (453054560525487782930∗13+O(13^{2}0))∗a−636101102354146201010∗13+O(13^{2}0)q^{13}+823543∗aq^{14}+ (4045000997094555040251+O(13^{2}0))∗a+9091857072255346124734+O(13^{2}0)q^{15}+(33108491273964405228991+ O(13^{2}1)) ∗ a − 104878616586830885839068 + O(13^{2}1)q^{16} + (4002448281428558961755 + O(13^{2}0)) ∗ a − 8052215476668073979351+O(13^{2}0)q^{17}+(4327911188296000936169+O(13^{2}0))∗a+3606810869568952698385+ O(13^{2}0)q^{18} + −(2420627208219898283522 + O(13^{2}0)) ∗ a + 4938669912157861915975 + O(13^{2}0)q^{19} + −(7684293478524480068193+O(13^{2}0))∗a+1008598992274449635128+O(13^{2}0)q^{20}+(8526297385554817799655+ O(13^{2}0)) ∗ a − 2498558523391503626036 + O(13^{2}0)q^{21} + −(9288199545084343608467 + O(13^{2}0)) ∗ a + 6185546384500848704∗13^{2}+O(13^{2}0)q^{22}+(2468986169217895590399+O(13^{2}0))∗a+7607690842342525786757+ O(13^{2}0)q^{23} + −(5259270433181216630409 + O(13^{2}0)) ∗ a − 2191531923848513372433 + O(13^{2}0)q^{24} + −(3220354663910245615171 + O(13^{2}0)) ∗ a − 9224363017955273273802 + O(13^{2}0)q^{25} + ⋯ is an ℓadic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 2760368466423868014502 + O(13^{2}0)x + 6971212808514269282042 + O(13^{2}0). The slope of f_{0} is 1.

3. Consider f = q + z^{1}376q^{2} + z^{8}10q^{3} + z^{7}40q^{4} + z^{1}871q^{5} + z^{2}186q^{6} + 7q^{7} + z^{1}790q^{8} + z^{2}92q^{9} + z^{1}051q^{10} + z^{8}21q^{11} +z^{1}550q^{12} +z^{1}193q^{14} +z^{4}85q^{15} +z^{5}51q^{16} +z^{9}74q^{17} +z^{1}668q^{18} +z^{6}43q^{19} +z^{4}15q^{20} +z^{6}27q^{21} + zq^{22}+z^{3}91q^{23}+z^{4}04q^{24}+z^{1}066q^{25}+⋯ ∈ S_{16}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{3}+2x+11 = 0. We have ρ_{f} ⊗ χ^{10} ∼ ρ_{g}, where g is the weight-12 newform g = q + z^{1}010q^{2} + z^{1}542q^{3} + z^{8}q^{4} + z^{7}73q^{5} + z^{3}56q^{6} + 2q^{7} + z^{6}92q^{8} + z^{1}756q^{9} + z^{1}783q^{10} + z^{4}55q^{11} + z^{1}550q^{12} + z^{2}006q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}90)ω^{1} & 0 \\ 0 & u(z^{2}006)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(9008477182565175429812+ O(13^{2}0)) ∗ a^{2} − (6930333744638621806994 + O(13^{2}0)) ∗ a + 4300258995866419843421 + O(13^{2}0)q^{3} + a^{2} −32768q^{4} +−(2964359928287402304331+O(13^{2}0))∗a^{2} −(1757721686401842603645+O(13^{2}0))∗a− 4443798190065103286486+O(13^{2}0)q^{5}+(4998574995193868571287+O(13^{2}0))∗a^{2}−(7637852155287736050564+ O(13^{2}0)) ∗ a + 9299819375120993796321 + O(13^{2}0)q^{6} + −823543q^{7} + −(1906876503905461345339 + O(13^{2}0)) ∗ a^{2} − (6129181235575036689214 + O(13^{2}0)) ∗ a + 6841595529072991540694 + O(13^{2}0)q^{8} + (1793137601382775662480+O(13^{2}0))∗a^{2}−(40762911108046055139∗13^{2}+O(13^{2}0))∗a−973902014274334146851+ O(13^{2}0)q^{9} + −(5959567866928065320929 + O(13^{2}0)) ∗ a^{2} − (5057469441796534747190 + O(13^{2}0)) ∗ a + 867459049674325195240+O(13^{2}0)q^{10}+(6351981275299786845756+O(13^{2}0))∗a^{2}−(2103830659479674766374+ O(13^{2}0)) ∗ a + 2519601504799296911512 + O(13^{2}0)q^{11} + (6954427907184748615104 + O(13^{2}0)) ∗ a^{2} + (873694955974738345564+O(13^{2}0))∗a−6886480903641220665519+O(13^{2}0)q^{12}+−(1030615802461359048306∗ 13 + O(13^{2}1)) ∗ a^{2} − (79167870065788685880 ∗ 13^{2} + O(13^{2}1)) ∗ a + 553311921083753685440 ∗ 13^{2} + O(13^{2}1)q^{13}+−823543∗aq^{14}+−(4664743818236403190836+O(13^{2}0))∗a^{2}−(7092915083513485855520+ O(13^{2}0))∗a+7658356677567277172343+O(13^{2}0)q^{15} +−(37247579226513077920∗13^{2} +O(13^{2}0))∗a^{2} + (135058337945574250392+O(13^{2}0))∗a+9104579632722349766733+O(13^{2}0)q^{16}+−(6143200634935647101434+ O(13^{2}0)) ∗ a^{2} − (571853563683940298298 ∗ 13 + O(13^{2}0)) ∗ a + 8207864993725851737170 + O(13^{2}0)q^{17} + (4292732895182297414103+O(13^{2}0))∗a^{2}−(6144362799009061339864+O(13^{2}0))∗a−8258885600001175582669+ O(13^{2}0)q^{18} + (4775097506603671335669 + O(13^{2}0)) ∗ a^{2} − (9002886334337849257354 + O(13^{2}0)) ∗ a + 444957876281915954470∗13+O(13^{2}0)q^{19}+−(5308630941184266793278+O(13^{2}0))∗a^{2}+(3871200439242191073082+ O(13^{2}0)) ∗ a + 3373448483642680676881 + O(13^{2}0)q^{20} + −(4359620071003933888449 + O(13^{2}0)) ∗ a^{2} + (9161898921877732053830+O(13^{2}0))∗a−6229530206187285965860+O(13^{2}0)q^{21}+−(1296639899694981309427+ O(13^{2}0)) ∗ a^{2} − (2108420887004687990167 + O(13^{2}0)) ∗ a + 1404619961762949145807 + O(13^{2}0)q^{22} + (1936349912447558024678+O(13^{2}0))∗a^{2}+(3824381964405959177977+O(13^{2}0))∗a−7344721669079266968810+ O(13^{2}0)q^{23} + −(5706444204834258561613 + O(13^{2}0)) ∗ a^{2} + (4108826849791958911736 + O(13^{2}0)) ∗ a + 2770118309626649129566+O(13^{2}0)q^{24}+−(6653555267272187151544+O(13^{2}0))∗a^{2}−(536848235660997762200+ O(13^{2}0)) ∗ a − 493592215178014659534 ∗ 13 + O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 1906876503905461345339 + O(13^{2}0)x^{2} + 6129181235575036623678 + O(13^{2}0)x + −6841595529072991540694 + O(13^{2}0). The slope of f_{0} is 1.

4. Consider f = q +z^{1}964q^{2} +z^{7}38q^{3} +z^{2}084q^{4} +z^{2}171q^{5} +z^{5}06q^{6} +7q^{7} +z^{1}658q^{8} +z^{1}036q^{9} +z^{1}939q^{10} + z^{4}01q^{11} +z^{6}26q^{12} +z^{1}781q^{14} +z^{7}13q^{15} +z^{8}87q^{16} +z^{2}102q^{17} +z^{8}04q^{18} +z^{1}063q^{19} +z^{2}059q^{20} +z^{5}55q^{21} + z^{1}69q^{22}+z^{1}99q^{23}+z^{2}00q^{24}+z^{8}2q^{25}+⋯ ∈ S_{16}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{3}+2x+11 = 0. We have ρ_{f} ⊗ χ^{10} ∼ ρ_{g}, where g is the weight-12 newform g = q + z^{1}598q^{2} + z^{1}470q^{3} + z^{1}352q^{4} + z^{1}073q^{5} + z^{8}72q^{6} + 2q^{7} + z^{5}60q^{8} + z^{3}04q^{9} + z^{4}75q^{10} + z^{3}5q^{11} + z^{6}26q^{12} + z^{8}30q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}366)ω^{1} & 0 \\ 0 & u(z^{8}30)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(9008477182565175429812+ O(13^{2}0)) ∗ a^{2} − (6930333744638621806994 + O(13^{2}0)) ∗ a + 4300258995866419843421 + O(13^{2}0)q^{3} + a^{2} −32768q^{4} +−(2964359928287402304331+O(13^{2}0))∗a^{2} −(1757721686401842603645+O(13^{2}0))∗a− 4443798190065103286486+O(13^{2}0)q^{5}+(4998574995193868571287+O(13^{2}0))∗a^{2}−(7637852155287736050564+ O(13^{2}0)) ∗ a + 9299819375120993796321 + O(13^{2}0)q^{6} + −823543q^{7} + −(1906876503905461345339 + O(13^{2}0)) ∗ a^{2} − (6129181235575036689214 + O(13^{2}0)) ∗ a + 6841595529072991540694 + O(13^{2}0)q^{8} + (1793137601382775662480+O(13^{2}0))∗a^{2}−(40762911108046055139∗13^{2}+O(13^{2}0))∗a−973902014274334146851+ O(13^{2}0)q^{9} + −(5959567866928065320929 + O(13^{2}0)) ∗ a^{2} − (5057469441796534747190 + O(13^{2}0)) ∗ a + 867459049674325195240+O(13^{2}0)q^{10}+(6351981275299786845756+O(13^{2}0))∗a^{2}−(2103830659479674766374+ O(13^{2}0)) ∗ a + 2519601504799296911512 + O(13^{2}0)q^{11} + (6954427907184748615104 + O(13^{2}0)) ∗ a^{2} + (873694955974738345564+O(13^{2}0))∗a−6886480903641220665519+O(13^{2}0)q^{12}+−(1030615802461359048306∗ 13 + O(13^{2}1)) ∗ a^{2} − (79167870065788685880 ∗ 13^{2} + O(13^{2}1)) ∗ a + 553311921083753685440 ∗ 13^{2} + O(13^{2}1)q^{13}+−823543∗aq^{14}+−(4664743818236403190836+O(13^{2}0))∗a^{2}−(7092915083513485855520+ O(13^{2}0))∗a+7658356677567277172343+O(13^{2}0)q^{15} +−(37247579226513077920∗13^{2} +O(13^{2}0))∗a^{2} + (135058337945574250392+O(13^{2}0))∗a+9104579632722349766733+O(13^{2}0)q^{16}+−(6143200634935647101434+ O(13^{2}0)) ∗ a^{2} − (571853563683940298298 ∗ 13 + O(13^{2}0)) ∗ a + 8207864993725851737170 + O(13^{2}0)q^{17} + (4292732895182297414103+O(13^{2}0))∗a^{2}−(6144362799009061339864+O(13^{2}0))∗a−8258885600001175582669+ O(13^{2}0)q^{18} + (4775097506603671335669 + O(13^{2}0)) ∗ a^{2} − (9002886334337849257354 + O(13^{2}0)) ∗ a + 444957876281915954470∗13+O(13^{2}0)q^{19}+−(5308630941184266793278+O(13^{2}0))∗a^{2}+(3871200439242191073082+ O(13^{2}0)) ∗ a + 3373448483642680676881 + O(13^{2}0)q^{20} + −(4359620071003933888449 + O(13^{2}0)) ∗ a^{2} + (9161898921877732053830+O(13^{2}0))∗a−6229530206187285965860+O(13^{2}0)q^{21}+−(1296639899694981309427+ O(13^{2}0)) ∗ a^{2} − (2108420887004687990167 + O(13^{2}0)) ∗ a + 1404619961762949145807 + O(13^{2}0)q^{22} + (1936349912447558024678+O(13^{2}0))∗a^{2}+(3824381964405959177977+O(13^{2}0))∗a−7344721669079266968810+ O(13^{2}0)q^{23} + −(5706444204834258561613 + O(13^{2}0)) ∗ a^{2} + (4108826849791958911736 + O(13^{2}0)) ∗ a + 2770118309626649129566+O(13^{2}0)q^{24}+−(6653555267272187151544+O(13^{2}0))∗a^{2}−(536848235660997762200+ O(13^{2}0)) ∗ a − 493592215178014659534 ∗ 13 + O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 1906876503905461345339 + O(13^{2}0)x^{2} + 6129181235575036623678 + O(13^{2}0)x + −6841595529072991540694 + O(13^{2}0). The slope of f_{0} is 1.

5. Consider f = q + z^{3}20q^{2} + z^{1}746q^{3} + z^{8}36q^{4} + z^{1}67q^{5} + z^{2}066q^{6} + 7q^{7} + z^{1}310q^{8} + z^{1}600q^{9} + z^{4}87q^{10} + z^{1}889q^{11} + z^{3}86q^{12} + z^{1}37q^{14} + z^{1}913q^{15} + z^{5}75q^{16} + z^{1}682q^{17} + z^{1}920q^{18} + z^{1}771q^{19} + z^{1}003q^{20} + z^{1}563q^{21} + z^{1}3q^{22} + z^{6}91q^{23} + z^{8}60q^{24} + z^{6}82q^{25} + ⋯ ∈ S_{16}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{3} + 2x + 11 = 0. We have ρ_{f} ⊗ χ^{10} ∼ ρ_{g}, where g is the weight-12 newform g = q + z^{2}150q^{2} + z^{2}82q^{3} + z^{1}04q^{4} + z^{1}265q^{5} + z^{2}36q^{6} + 2q^{7} + z^{2}12q^{8} + z^{8}68q^{9} + z^{1}219q^{10} + z^{1}523q^{11} + z^{3}86q^{12} + z^{1}922q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}74)ω^{1} & 0 \\ 0 & u(z^{1}922)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(9008477182565175429812+ O(13^{2}0)) ∗ a^{2} − (6930333744638621806994 + O(13^{2}0)) ∗ a + 4300258995866419843421 + O(13^{2}0)q^{3} + a^{2} −32768q^{4} +−(2964359928287402304331+O(13^{2}0))∗a^{2} −(1757721686401842603645+O(13^{2}0))∗a− 4443798190065103286486+O(13^{2}0)q^{5}+(4998574995193868571287+O(13^{2}0))∗a^{2}−(7637852155287736050564+ O(13^{2}0)) ∗ a + 9299819375120993796321 + O(13^{2}0)q^{6} + −823543q^{7} + −(1906876503905461345339 + O(13^{2}0)) ∗ a^{2} − (6129181235575036689214 + O(13^{2}0)) ∗ a + 6841595529072991540694 + O(13^{2}0)q^{8} + (1793137601382775662480+O(13^{2}0))∗a^{2}−(40762911108046055139∗13^{2}+O(13^{2}0))∗a−973902014274334146851+ O(13^{2}0)q^{9} + −(5959567866928065320929 + O(13^{2}0)) ∗ a^{2} − (5057469441796534747190 + O(13^{2}0)) ∗ a + 867459049674325195240+O(13^{2}0)q^{10}+(6351981275299786845756+O(13^{2}0))∗a^{2}−(2103830659479674766374+ O(13^{2}0)) ∗ a + 2519601504799296911512 + O(13^{2}0)q^{11} + (6954427907184748615104 + O(13^{2}0)) ∗ a^{2} + (873694955974738345564+O(13^{2}0))∗a−6886480903641220665519+O(13^{2}0)q^{12}+−(1030615802461359048306∗ 13 + O(13^{2}1)) ∗ a^{2} − (79167870065788685880 ∗ 13^{2} + O(13^{2}1)) ∗ a + 553311921083753685440 ∗ 13^{2} + O(13^{2}1)q^{13}+−823543∗aq^{14}+−(4664743818236403190836+O(13^{2}0))∗a^{2}−(7092915083513485855520+ O(13^{2}0))∗a+7658356677567277172343+O(13^{2}0)q^{15} +−(37247579226513077920∗13^{2} +O(13^{2}0))∗a^{2} + (135058337945574250392+O(13^{2}0))∗a+9104579632722349766733+O(13^{2}0)q^{16}+−(6143200634935647101434+ O(13^{2}0)) ∗ a^{2} − (571853563683940298298 ∗ 13 + O(13^{2}0)) ∗ a + 8207864993725851737170 + O(13^{2}0)q^{17} + (4292732895182297414103+O(13^{2}0))∗a^{2}−(6144362799009061339864+O(13^{2}0))∗a−8258885600001175582669+ O(13^{2}0)q^{18} + (4775097506603671335669 + O(13^{2}0)) ∗ a^{2} − (9002886334337849257354 + O(13^{2}0)) ∗ a + 444957876281915954470∗13+O(13^{2}0)q^{19}+−(5308630941184266793278+O(13^{2}0))∗a^{2}+(3871200439242191073082+ O(13^{2}0)) ∗ a + 3373448483642680676881 + O(13^{2}0)q^{20} + −(4359620071003933888449 + O(13^{2}0)) ∗ a^{2} + (9161898921877732053830+O(13^{2}0))∗a−6229530206187285965860+O(13^{2}0)q^{21}+−(1296639899694981309427+ O(13^{2}0)) ∗ a^{2} − (2108420887004687990167 + O(13^{2}0)) ∗ a + 1404619961762949145807 + O(13^{2}0)q^{22} + (1936349912447558024678+O(13^{2}0))∗a^{2}+(3824381964405959177977+O(13^{2}0))∗a−7344721669079266968810+ O(13^{2}0)q^{23} + −(5706444204834258561613 + O(13^{2}0)) ∗ a^{2} + (4108826849791958911736 + O(13^{2}0)) ∗ a + 2770118309626649129566+O(13^{2}0)q^{24}+−(6653555267272187151544+O(13^{2}0))∗a^{2}−(536848235660997762200+ O(13^{2}0)) ∗ a − 493592215178014659534 ∗ 13 + O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 1906876503905461345339 + O(13^{2}0)x^{2} + 6129181235575036623678 + O(13^{2}0)x + −6841595529072991540694 + O(13^{2}0). The slope of f_{0} is 1.

6. Consider f = q + z^{3}q^{2} + z^{2}7q^{3} + z^{4}7q^{4} + z^{9}5q^{5} + z^{3}0q^{6} + 10q^{7} + z^{1}31q^{8} + z^{1}13q^{9} + 11q^{10} + z^{1}11q^{11} + z^{7}4q^{12} + z^{1}43q^{14} + z^{1}22q^{15} + z^{8}3q^{16} + z^{1}04q^{17} + z^{1}16q^{18} + z^{8}3q^{19} + z^{1}42q^{20} + z^{1}67q^{21} + z^{1}14q^{22} + z^{3}5q^{23} + z^{1}58q^{24} + z^{8}3q^{25} + ⋯ ∈ S_{18}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q +z^{1}15q^{2} +z^{1}39q^{3} +z^{1}03q^{4} +z^{9}5q^{5} +z^{8}6q^{6} + 4q^{7} + z^{1}31q^{8} + zq^{9} + 8q^{10} + z^{5}5q^{11} + z^{7}4q^{12} + z^{1}18q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{5}0)ω^{1} & 0 \\ 0 & u(z^{1}18)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(4135691119088469399136+ O(13^{2}0)) ∗ a + 5615421483611304172489 + O(13^{2}0)q^{3} + −(8953618332239728730471 + O(13^{2}0)) ∗ a + 167263672957981951779+O(13^{2}0)q^{4}+(97040373503270447720454+O(13^{2}1))∗a+121187770553216030323515+ O(13^{2}1)q^{5} + −(23126313581007614138893 + O(13^{2}1)) ∗ a − 28561551487026734918472 + O(13^{2}1)q^{6} + −5764801q^{7} + (4090575135417907098201 + O(13^{2}0)) ∗ a − 3256671757963281841429 + O(13^{2}0)q^{8} + (1203441815515054614581+O(13^{2}0))∗a+894543179245889120553+O(13^{2}0)q^{9}+(12103911822807321977∗ 13+O(13^{2}0))∗a−7068627447387227516043+O(13^{2}0)q^{10} +(3182180005789565585868+O(13^{2}0))∗a− 7650712860188665902082+O(13^{2}0)q^{11}+−(7799936456098822206647+O(13^{2}0))∗a−4048875828633923707073+ O(13^{2}0)q^{12}+−(334823407601103396372∗13+O(13^{2}0))∗a−425389944685188984257∗13+O(13^{2}0)q^{13}+ −5764801 ∗ aq^{14} + (8296034556876916673741 + O(13^{2}0)) ∗ a − 2940860462411217097534 + O(13^{2}0)q^{15} + −(5038973355310959427058+O(13^{2}0))∗a−7857079557472973136681+O(13^{2}0)q^{16}+(4676889142546149647297+ O(13^{2}0)) ∗ a + 4948958855141927397983 + O(13^{2}0)q^{17} + −(5434069693852536120994 + O(13^{2}0)) ∗ a + 3297700911199274468922+O(13^{2}0)q^{18}+−(623651140214688620548+O(13^{2}0))∗a−3891405350598755450911+ O(13^{2}0)q^{19} + −(4107806623478057653943 + O(13^{2}0)) ∗ a + 8836909102253628070102 + O(13^{2}0)q^{20} + (5682119012203375660549+O(13^{2}0))∗a−5422584893154377535956+O(13^{2}0)q^{21}+−(111645267936654784147045+ O(13^{2}1)) ∗ a − 56360267065638431812394 + O(13^{2}1)q^{22} + −(7443594906951534726176 + O(13^{2}0)) ∗ a + 5676537174290327554516+O(13^{2}0)q^{23}+(7933365861727221063351+O(13^{2}0))∗a+4419559733912660929908+ O(13^{2}0)q^{24} +−(9441201803695877715325+O(13^{2}0))∗a+9004167038026633551772+O(13^{2}0)q^{25} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 8953618332239728730471 + O(13^{2}0)x + −167263672957982082851 + O(13^{2}0). The slope of f_{0} is 1.

7. Consider f = q + z^{3}9q^{2} + z^{1}5q^{3} + z^{1}07q^{4} + z^{5}9q^{5} + z^{5}4q^{6} + 10q^{7} + z^{2}3q^{8} + z^{1}25q^{9} + 11q^{10} + z^{9}9q^{11} + z^{1}22q^{12} + z^{1}1q^{14} + z^{7}4q^{15} + z^{7}1q^{16} + z^{8}q^{17} + z^{1}64q^{18} + z^{7}1q^{19} + z^{1}66q^{20} + z^{1}55q^{21} + z^{1}38q^{22} + z^{1}19q^{23} + z^{3}8q^{24} + z^{7}1q^{25} + ⋯ ∈ S_{18}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{1}51q^{2} + z^{1}27q^{3} + z^{1}63q^{4} + z^{5}9q^{5} + z^{1}10q^{6} + 4q^{7} + z^{2}3q^{8} + z^{1}3q^{9} + 8q^{10} + z^{4}3q^{11} + z^{1}22q^{12} + z^{2}2q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}46)ω^{1} & 0 \\ 0 & u(z^{2}2)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(4135691119088469399136+ O(13^{2}0)) ∗ a + 5615421483611304172489 + O(13^{2}0)q^{3} + −(8953618332239728730471 + O(13^{2}0)) ∗ a + 167263672957981951779+O(13^{2}0)q^{4}+(97040373503270447720454+O(13^{2}1))∗a+121187770553216030323515+ O(13^{2}1)q^{5} + −(23126313581007614138893 + O(13^{2}1)) ∗ a − 28561551487026734918472 + O(13^{2}1)q^{6} + −5764801q^{7} + (4090575135417907098201 + O(13^{2}0)) ∗ a − 3256671757963281841429 + O(13^{2}0)q^{8} + (1203441815515054614581+O(13^{2}0))∗a+894543179245889120553+O(13^{2}0)q^{9}+(12103911822807321977∗ 13+O(13^{2}0))∗a−7068627447387227516043+O(13^{2}0)q^{10} +(3182180005789565585868+O(13^{2}0))∗a− 7650712860188665902082+O(13^{2}0)q^{11}+−(7799936456098822206647+O(13^{2}0))∗a−4048875828633923707073+ O(13^{2}0)q^{12}+−(334823407601103396372∗13+O(13^{2}0))∗a−425389944685188984257∗13+O(13^{2}0)q^{13}+ −5764801 ∗ aq^{14} + (8296034556876916673741 + O(13^{2}0)) ∗ a − 2940860462411217097534 + O(13^{2}0)q^{15} + −(5038973355310959427058+O(13^{2}0))∗a−7857079557472973136681+O(13^{2}0)q^{16}+(4676889142546149647297+ O(13^{2}0)) ∗ a + 4948958855141927397983 + O(13^{2}0)q^{17} + −(5434069693852536120994 + O(13^{2}0)) ∗ a + 3297700911199274468922+O(13^{2}0)q^{18}+−(623651140214688620548+O(13^{2}0))∗a−3891405350598755450911+ O(13^{2}0)q^{19} + −(4107806623478057653943 + O(13^{2}0)) ∗ a + 8836909102253628070102 + O(13^{2}0)q^{20} + (5682119012203375660549+O(13^{2}0))∗a−5422584893154377535956+O(13^{2}0)q^{21}+−(111645267936654784147045+ O(13^{2}1)) ∗ a − 56360267065638431812394 + O(13^{2}1)q^{22} + −(7443594906951534726176 + O(13^{2}0)) ∗ a + 5676537174290327554516+O(13^{2}0)q^{23}+(7933365861727221063351+O(13^{2}0))∗a+4419559733912660929908+ O(13^{2}0)q^{24} +−(9441201803695877715325+O(13^{2}0))∗a+9004167038026633551772+O(13^{2}0)q^{25} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 8953618332239728730471 + O(13^{2}0)x + −167263672957982082851 + O(13^{2}0). The slope of f_{0} is 1.

8. Consider f = q + 11q^{2} + 7q^{3} + 11q^{4} + 2q^{5} + 12q^{6} + 3q^{7} + 3q^{8} + q^{9} + 9q^{10} + 3q^{11} + 12q^{12} + 7q^{14} + q^{15} + 6q^{16} + 8q^{17} + 11q^{18} + 3q^{19} + 9q^{20} + 8q^{21} + 7q^{22} + 12q^{23} + 8q^{24} + 12q^{25} + ⋯ ∈ S_{18}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-4 newform g = q + 12q^{2} + 11q^{3} + 6q^{4} + 3q^{5} + 2q^{6} + 6q^{7} + 2q^{8} + 3q^{9} + 10q^{10} + 5q^{11} + q^{12} + 2q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(7)ω^{4} & 0 \\ 0 & u(2)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−7589705310983969802878+ O(13^{2}0)q^{2}+9496844757500014553022+O(13^{2}0)q^{3}+1301808711064250167868+O(13^{2}0)q^{4}+1936044398929893245190+ O(13^{2}0)q^{5}+−6491584577940418311577+O(13^{2}0)q^{6}+5764801q^{7}+−5092380269740505224935+O(13^{2}0)q^{8}+ 3694405039637708567172+O(13^{2}0)q^{9}+−1401206984063299569056+O(13^{2}0)q^{10}+2749260702215776144698+ O(13^{2}0)q^{11} + 6757301000209897025017 + O(13^{2}0)q^{12} + 726449554905506171214 ∗ 13 + O(13^{2}0)q^{13} + −8188753998039590429083+O(13^{2}0)q^{14}+8705132151021197056621+O(13^{2}0)q^{15}+1115678824487780459068+ O(13^{2}0)q^{16}+4380420129206394093569+O(13^{2}0)q^{17}+2555710853112133418089+O(13^{2}0)q^{18}+−4196621061797246836179+ O(13^{2}0)q^{19} + −921028695926306309918 + O(13^{2}0)q^{20} + −7946744303620883312869 + O(13^{2}0)q^{21} + 2404518539904964413990+O(13^{2}0)q^{22}+2229492392173875172889+O(13^{2}0)q^{23}+8720538151427934910892+ O(13^{2}0)q^{24} + −5445931347197257854289 + O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

9. Consider f = q + 10q^{2} + 6q^{3} + 3q^{4} + 4q^{5} + 8q^{6} + 3q^{7} + 9q^{8} + q^{9} + q^{10} + 4q^{11} + 5q^{12} + 4q^{14} + 11q^{15} + 7q^{16} + 9q^{17} + 10q^{18} + 6q^{19} + 12q^{20} + 5q^{21} + q^{22} + 11q^{23} + 2q^{24} + 11q^{25} + ⋯ ∈ S_{18}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 12q^{2} + 2q^{3} + 9q^{4} + 4q^{5} + 11q^{6} + 9q^{7} + 9q^{8} + 3q^{9} + 9q^{10} + 10q^{11} + 5q^{12} + q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+4146701349320501209131+ O(13^{2}0)q^{2}+−5844042283421131351569+O(13^{2}0)q^{3}+6131931283082185036667+O(13^{2}0)q^{4}+3278965742150834903106+ O(13^{2}0)q^{5}+−409441523009699730425+O(13^{2}0)q^{6}+5764801q^{7}+7718178980283142588967+O(13^{2}0)q^{8}+ 1646684027308414496351+O(13^{2}0)q^{9}+7853916524245650791531+O(13^{2}0)q^{10}+−411935890250139660271+ O(13^{2}0)q^{11} + 9063689550472113508502 + O(13^{2}0)q^{12} + 677400404968384019725 ∗ 13 + O(13^{2}0)q^{13} + 8530088508017589168907+O(13^{2}0)q^{14}+−163916779067699773377+O(13^{2}0)q^{15}+6897192902352350308445+ O(13^{2}0)q^{16}+8886986675357533161727+O(13^{2}0)q^{17}+6447154017443695724243+O(13^{2}0)q^{18}+2181852432686819508284+ O(13^{2}0)q^{19}+1034273767976679886629+O(13^{2}0)q^{20}+−2610088951006683127288+O(13^{2}0)q^{21}+2772324120817075158491+ O(13^{2}0)q^{22}+4009349276712152104132+O(13^{2}0)q^{23}+8431623641689530254650+O(13^{2}0)q^{24}+1460912948783451919429+ O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

10. Consider f = q + z^{4}q^{2} + z^{8}8q^{3} + z^{1}42q^{4} + z^{6}0q^{5} + z^{9}2q^{6} + 3q^{7} + z^{1}45q^{8} + z^{6}9q^{9} + z^{6}4q^{10} + z^{1}38q^{11} + z^{6}2q^{12} + z^{6}0q^{14} + z^{1}48q^{15} + z^{4}9q^{16} + z^{9}3q^{17} + z^{7}3q^{18} + z^{1}66q^{19} + z^{3}4q^{20} + z^{1}44q^{21} + z^{1}42q^{22} + z^{1}05q^{23} + z^{6}5q^{24} + z^{2}6q^{25} + ⋯ ∈ S_{18}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{1}16q^{2} + z^{3}2q^{3} + z^{3}0q^{4} + z^{6}0q^{5} + z^{1}48q^{6} + 9q^{7} + z^{1}45q^{8} + z^{1}25q^{9} + z^{8}q^{10} + z^{8}2q^{11} + z^{6}2q^{12} + 8q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{1} & 0 \\ 0 & u(8)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(2570396566428796525915+ O(13^{2}0)) ∗ a + 392002808417544896697 ∗ 13 + O(13^{2}0)q^{3} + −(4514430979968735704241 + O(13^{2}0)) ∗ a − 6477076594262049597220+O(13^{2}0)q^{4}+−(5943012507313321990521+O(13^{2}0))∗a+242341498929553476520∗ 13 + O(13^{2}0)q^{5} + −(4107435925560635292847 + O(13^{2}0)) ∗ a − 3974370009624533539641 + O(13^{2}0)q^{6} + 5764801q^{7} + −(6435775507995285863646 + O(13^{2}0)) ∗ a + 4928970314920578979114 + O(13^{2}0)q^{8} + (6889737696415391095006+O(13^{2}0))∗a−3866375508760252743423+O(13^{2}0)q^{9}+−(5350209181146482907511+ O(13^{2}0)) ∗ a − 6182023586381934995661 + O(13^{2}0)q^{10} + (6920265432789589561801 + O(13^{2}0)) ∗ a + 7835805491182961679346+O(13^{2}0)q^{11}+−(5780489575609420228822+O(13^{2}0))∗a+5500383635175291765295+ O(13^{2}0)q^{12} +(22149822445825371832∗13^{2} +O(13^{2}0))∗a+661753559260912600454∗13+O(13^{2}0)q^{13} + 5764801 ∗ aq^{14} + −(7095945208609697332389 + O(13^{2}0)) ∗ a + 7766619775981325906825 + O(13^{2}0)q^{15} + −(9033485667371355428144+O(13^{2}0))∗a+1104300059905202840087+O(13^{2}0)q^{16}+(4724076280629758378524+ O(13^{2}0)) ∗ a − 2103779191348849033798 + O(13^{2}0)q^{17} + (3580003643237780993834 + O(13^{2}0)) ∗ a + 419779844477921773778+O(13^{2}0)q^{18}+−(2657918183029079955518+O(13^{2}0))∗a−7577330183713367275765+ O(13^{2}0)q^{19} + −(1007645979786069908893 + O(13^{2}0)) ∗ a + 1932111968659936270926 + O(13^{2}0)q^{20} + (3469381318826651923367+O(13^{2}0))∗a−659850608854507416179∗13+O(13^{2}0)q^{21}+−(4262952089315456196776+ O(13^{2}0)) ∗ a − 6320527862483861943068 + O(13^{2}0)q^{22} + (3131995806227488750377 + O(13^{2}0)) ∗ a + 3588990220516143171874+O(13^{2}0)q^{23}+(4861501604080001765876+O(13^{2}0))∗a−7482528747764644443106+ O(13^{2}0)q^{24} +−(6426105391242690556056+O(13^{2}0))∗a−3384448183678945241878+O(13^{2}0)q^{25} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 4514430979968735704241 + O(13^{2}0)x + 6477076594262049466148 + O(13^{2}0). The slope of f_{0} is 1.

11. Consider f = q + z^{5}2q^{2} + z^{1}36q^{3} + z^{1}66q^{4} + z^{1}08q^{5} + z^{2}0q^{6} + 3q^{7} + z^{3}7q^{8} + z^{5}7q^{9} + z^{1}60q^{10} + z^{1}14q^{11} + z^{1}34q^{12} + z^{1}08q^{14} + z^{7}6q^{15} + z^{1}33q^{16} + z^{3}3q^{17} + z^{1}09q^{18} + z^{1}42q^{19} + z^{1}06q^{20} + z^{2}4q^{21} + z^{1}66q^{22} + z^{2}1q^{23} + z^{5}q^{24} + z^{2}q^{25} + ⋯ ∈ S_{18}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{1}64q^{2} + z^{8}0q^{3} + z^{5}4q^{4} + z^{1}08q^{5} + z^{7}6q^{6} + 9q^{7} + z^{3}7q^{8} + z^{1}13q^{9} + z^{1}04q^{10} + z^{5}8q^{11} + z^{1}34q^{12} + 8q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{1} & 0 \\ 0 & u(8)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(2570396566428796525915+ O(13^{2}0)) ∗ a + 392002808417544896697 ∗ 13 + O(13^{2}0)q^{3} + −(4514430979968735704241 + O(13^{2}0)) ∗ a − 6477076594262049597220+O(13^{2}0)q^{4}+−(5943012507313321990521+O(13^{2}0))∗a+242341498929553476520∗ 13 + O(13^{2}0)q^{5} + −(4107435925560635292847 + O(13^{2}0)) ∗ a − 3974370009624533539641 + O(13^{2}0)q^{6} + 5764801q^{7} + −(6435775507995285863646 + O(13^{2}0)) ∗ a + 4928970314920578979114 + O(13^{2}0)q^{8} + (6889737696415391095006+O(13^{2}0))∗a−3866375508760252743423+O(13^{2}0)q^{9}+−(5350209181146482907511+ O(13^{2}0)) ∗ a − 6182023586381934995661 + O(13^{2}0)q^{10} + (6920265432789589561801 + O(13^{2}0)) ∗ a + 7835805491182961679346+O(13^{2}0)q^{11}+−(5780489575609420228822+O(13^{2}0))∗a+5500383635175291765295+ O(13^{2}0)q^{12} +(22149822445825371832∗13^{2} +O(13^{2}0))∗a+661753559260912600454∗13+O(13^{2}0)q^{13} + 5764801 ∗ aq^{14} + −(7095945208609697332389 + O(13^{2}0)) ∗ a + 7766619775981325906825 + O(13^{2}0)q^{15} + −(9033485667371355428144+O(13^{2}0))∗a+1104300059905202840087+O(13^{2}0)q^{16}+(4724076280629758378524+ O(13^{2}0)) ∗ a − 2103779191348849033798 + O(13^{2}0)q^{17} + (3580003643237780993834 + O(13^{2}0)) ∗ a + 419779844477921773778+O(13^{2}0)q^{18}+−(2657918183029079955518+O(13^{2}0))∗a−7577330183713367275765+ O(13^{2}0)q^{19} + −(1007645979786069908893 + O(13^{2}0)) ∗ a + 1932111968659936270926 + O(13^{2}0)q^{20} + (3469381318826651923367+O(13^{2}0))∗a−659850608854507416179∗13+O(13^{2}0)q^{21}+−(4262952089315456196776+ O(13^{2}0)) ∗ a − 6320527862483861943068 + O(13^{2}0)q^{22} + (3131995806227488750377 + O(13^{2}0)) ∗ a + 3588990220516143171874+O(13^{2}0)q^{23}+(4861501604080001765876+O(13^{2}0))∗a−7482528747764644443106+ O(13^{2}0)q^{24} +−(6426105391242690556056+O(13^{2}0))∗a−3384448183678945241878+O(13^{2}0)q^{25} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 4514430979968735704241 + O(13^{2}0)x + 6477076594262049466148 + O(13^{2}0). The slope of f_{0} is 1.

12. Consider f = q + 6q^{2} + 10q^{3} + 12q^{4} + 6q^{5} + 8q^{6} + 8q^{7} + 6q^{8} + 6q^{9} + 10q^{10} + 4q^{11} + 3q^{12} + 9q^{14} + 8q^{15} + 8q^{16} + 2q^{17} + 10q^{18} + 10q^{19} + 7q^{20} + 2q^{21} + 11q^{22} + q^{23} + 8q^{24} + 2q^{25} + ⋯ ∈ S_{20}(Γ_{0}(7); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-6 newform g = q + 3q^{2} + 12q^{3} + 3q^{4} + 9q^{5} + 10q^{6} + 3q^{7} + 4q^{8} + 5q^{9} + q^{10} + 11q^{11} + 10q^{12} + 3q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(9)ω^{6} & 0 \\ 0 & u(3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(7228824951412772606963+ O(13^{2}0)) ∗ a − 4775185671740066072565 + O(13^{2}0)q^{3} + −(2367254378849365366986 + O(13^{2}0)) ∗ a − 3803533158444676676469+O(13^{2}0)q^{4}+(2086922470246777233264+O(13^{2}0))∗a−5741648551522319951321+ O(13^{2}0)q^{5} + −(2581738870247643536998 + O(13^{2}0)) ∗ a − 7022323958188996077143 + O(13^{2}0)q^{6} + 40353607q^{7} + −(8794204325987758329579 + O(13^{2}0)) ∗ a − 9059584671223619031671 + O(13^{2}0)q^{8} + (7634151328480426543813+O(13^{2}0))∗a−8030175103902700977492+O(13^{2}0)q^{9}+−(587895175743608402005∗ 13+O(13^{2}0))∗a−5991935404556924371045+O(13^{2}0)q^{10} +(7904888577755133221952+O(13^{2}0))∗a+ 795817166505871077835+O(13^{2}0)q^{11}+−(1049210289108971353347+O(13^{2}0))∗a+7784180409042472770639+ O(13^{2}0)q^{12}+−(478916432566721772478∗13+O(13^{2}0))∗a−454229562539508832510∗13+O(13^{2}0)q^{13}+ 40353607∗aq^{14} +(9056531739212281281448+O(13^{2}0))∗a+60152788040048203114∗13+O(13^{2}0)q^{15} + −(675340741761910362833∗13+O(13^{2}0))∗a−3441932357680177430919+O(13^{2}0)q^{16}+(6687787588690137274369+ O(13^{2}0)) ∗ a − 9282262281452397594785 + O(13^{2}0)q^{17} + −(1374726887090579200045 + O(13^{2}0)) ∗ a − 154421770993848360062+O(13^{2}0)q^{18}+−(718115822222708327726+O(13^{2}0))∗a+4183934987488832675020+ O(13^{2}0)q^{19} + −(4343970765284934774359 + O(13^{2}0)) ∗ a − 7798015039354394443006 + O(13^{2}0)q^{20} + −(4335772857627547818763+O(13^{2}0))∗a−1024752136021255085511+O(13^{2}0)q^{21}+−(3100799129286022654502+ O(13^{2}0)) ∗ a + 5881952028274274954877 + O(13^{2}0)q^{22} + −(1190752807730778423 + O(13^{2}0)) ∗ a + 7835053682447102747214+O(13^{2}0)q^{23}+(4367075384953729341929+O(13^{2}0))∗a−3303185871607957458481+ O(13^{2}0)q^{24} +−(1856520308904325004355+O(13^{2}0))∗a+7032929404127710677408+O(13^{2}0)q^{25} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 2367254378849365366986 + O(13^{2}0)x + 3803533158444676152181 + O(13^{2}0). The slope of f_{0} is 1.

13. Consider f = q + z^{1}27q^{2} + z^{9}6q^{3} + z^{1}20q^{4} + z^{1}29q^{5} + z^{5}5q^{6} + 5q^{7} + z^{1}32q^{8} + z^{2}9q^{9} + z^{8}8q^{10} + z^{5}4q^{11} + z^{4}8q^{12} + z^{8}5q^{14} + z^{5}7q^{15} + z^{3}0q^{16} + z^{1}50q^{17} + z^{1}56q^{18} + z^{1}04q^{19} + z^{8}1q^{20} + z^{5}4q^{21} + z^{1}3q^{22} + z^{7}6q^{23} + z^{6}0q^{24} + z^{1}29q^{25} + ⋯ ∈ S_{20}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗χ^{11} ∼ ρ_{g}, where g is the weight-6 newform g = q +z^{1}13q^{2} +z^{4}0q^{3} +z^{9}2q^{4} +z^{3}q^{5} +z^{1}53q^{6} +10q^{7} + z^{9}0q^{8} + z^{8}5q^{9} + z^{1}16q^{10} + z^{1}24q^{11} + z^{1}32q^{12} + z^{5}2q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}16)ω^{6} & 0 \\ 0 & u(z^{5}2)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(1429185604795825049463+ O(13^{2}0)) ∗ a + 3706085701922263396601 + O(13^{2}0)q^{3} + (3640113536828162924080 + O(13^{2}0)) ∗ a + 6181494794288393561492+O(13^{2}0)q^{4}+(2747412779011195026588+O(13^{2}0))∗a+2582347564517853155780+ O(13^{2}0)q^{5}+(9490781035704896840706+O(13^{2}0))∗a+2953066046271830579423+O(13^{2}0)q^{6}+−40353607q^{7}+ −(4098345411965926696492+O(13^{2}0))∗a+8885576288168160971244+O(13^{2}0)q^{8}+−(2490064776027194336328+ O(13^{2}0)) ∗ a − 133768578650421935975 ∗ 13 + O(13^{2}0)q^{9} + −(2477071460456747803859 + O(13^{2}0)) ∗ a − 7156371631592903896841+O(13^{2}0)q^{10}+−(1572163062030478558582+O(13^{2}0))∗a+608594466611543237029+ O(13^{2}0)q^{11} + −(265760977531702812684 + O(13^{2}0)) ∗ a + 5743463471951740345203 + O(13^{2}0)q^{12} + −(56652534622872533445∗13+O(13^{2}0))∗a+395696841117786909027∗13+O(13^{2}0)q^{13} +−40353607∗ aq^{14}+(6486527667418528946721+O(13^{2}0))∗a−339241858593452642831∗13+O(13^{2}0)q^{15}+−(9180714684975962815408+ O(13^{2}0)) ∗ a + 8493877082876865526813 + O(13^{2}0)q^{16} + (7735985780488794682104 + O(13^{2}0)) ∗ a − 9086601107156692188990+O(13^{2}0)q^{17}+(9241770057762816162008+O(13^{2}0))∗a+440315971743493599181+ O(13^{2}0)q^{18} + −(4399449679209673659573 + O(13^{2}0)) ∗ a − 3325531310451946402408 + O(13^{2}0)q^{19} + −(3869308524720202035161+O(13^{2}0))∗a−4540163276780186500066+O(13^{2}0)q^{20}+−(8070350592336209488824+ O(13^{2}0)) ∗ a − 7971506850070710744204 + O(13^{2}0)q^{21} + (6371287289390826710523 + O(13^{2}0)) ∗ a + 4257383793104925683422+O(13^{2}0)q^{22}+−(1679983369101519558967+O(13^{2}0))∗a+16015279493206874126+ O(13^{2}0)q^{23} + (8252791140543674761120 + O(13^{2}0)) ∗ a + 8221975605861215033439 + O(13^{2}0)q^{24} + (3464370552500859034305 + O(13^{2}0)) ∗ a + 26922290289021975539 + O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2}+−3640113536828162924080+ O(13^{2}0)x + −6181494794288394085780 + O(13^{2}0). The slope of f_{0} is 1.

14. Consider f = q+z^{1}31q^{2}+z^{5}7q^{3}+z^{1}46q^{4}+z^{9}4q^{5}+z^{2}0q^{6}+5q^{7}+z^{1}48q^{8}+z^{3}1q^{9}+z^{5}7q^{10}+z^{2}3q^{11}+z^{3}5q^{12}+ z^{8}9q^{14}+z^{1}51q^{15}+z^{1}9q^{16}+z^{1}58q^{17}+z^{1}62q^{18}+q^{19}+z^{7}2q^{20}+z^{1}5q^{21}+7q^{22}+z^{1}8q^{23}+z^{3}7q^{24}+z^{1}1q^{25}+⋯ ∈ S_{20}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{4}7q^{2} + z^{5}7q^{3} + z^{1}46q^{4} + z^{1}0q^{5} + z^{1}04q^{6} + 8q^{7} + z^{6}4q^{8} + z^{3}1q^{9} + z^{5}7q^{10} + z^{1}07q^{11} + z^{3}5q^{12} + z^{8}8q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{8}0)ω^{1} & 0 \\ 0 & u(z^{8}8)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(1362735429840992962549+ O(13^{2}0)) ∗ a + 9152311934162367676568 + O(13^{2}0)q^{3} + (9347234290010263570121 + O(13^{2}0)) ∗ a − 4138029506060468467649+O(13^{2}0)q^{4}+(1823865870242621126444+O(13^{2}0))∗a−8828544679813861177910+ O(13^{2}0)q^{5}+(450986930186645562513+O(13^{2}0))∗a−4184969928380714076956+O(13^{2}0)q^{6}+−40353607q^{7}+ (6237780592425004122649+O(13^{2}0))∗a−8736139397476563949300+O(13^{2}0)q^{8}+(4048169531845250792584+ O(13^{2}0)) ∗ a + 4919292136431095946608 + O(13^{2}0)q^{9} + −(5530021511774533838582 + O(13^{2}0)) ∗ a − 5433040073830401650977+O(13^{2}0)q^{10}+(426276559096753555462+O(13^{2}0))∗a−864540268696844471755+ O(13^{2}0)q^{11}+(78282186373111394359+O(13^{2}0))∗a+3243551210010266873937+O(13^{2}0)q^{12}+−(722899158790690835203∗ 13+O(13^{2}0))∗a−187428189272485489152∗13+O(13^{2}0)q^{13}+−40353607∗aq^{14}+(5665685971038964953972+ O(13^{2}0)) ∗ a + 7648596074527063315557 + O(13^{2}0)q^{15} + (7368926560490288362598 + O(13^{2}0)) ∗ a − 499016375656843233035∗13+O(13^{2}0)q^{16}+−(7427215053566423514075+O(13^{2}0))∗a−6089490605610847153792+ O(13^{2}0)q^{17} + −(3460592049467995679693 + O(13^{2}0)) ∗ a − 7692369873150462965299 + O(13^{2}0)q^{18} + −(714498685061749900744∗13+O(13^{2}0))∗a+8837171171004040079684+O(13^{2}0)q^{19}+−(16840614321536997494+ O(13^{2}0)) ∗ a + 5219536015955587725618 + O(13^{2}0)q^{20} + −(9188958861946709717121 + O(13^{2}0)) ∗ a + 2499481257589133955305+O(13^{2}0)q^{21}+(612042365933511269905∗13+O(13^{2}0))∗a+3297238637976307265677+ O(13^{2}0)q^{22} + (1581423023948898040673 + O(13^{2}0)) ∗ a + 9478059672733049209182 + O(13^{2}0)q^{23} + −(5073686614522904036635+O(13^{2}0))∗a−2943378294820649893266+O(13^{2}0)q^{24}+−(3529601006709718385512+ O(13^{2}0)) ∗ a − 5691926378318935727930 + O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −9347234290010263570121 + O(13^{2}0)x + 4138029506060467943361 + O(13^{2}0). The slope of f_{0} is 1.

15. Consider f = q + z^{1}39q^{2} + z^{7}2q^{3} + z^{4}8q^{4} + z^{1}65q^{5} + z^{4}3q^{6} + 5q^{7} + z^{3}6q^{8} + z^{4}1q^{9} + z^{1}36q^{10} + z^{3}0q^{11} + z^{1}20q^{12} + z^{9}7q^{14} + z^{6}9q^{15} + z^{5}4q^{16} + z^{1}02q^{17} + z^{1}2q^{18} + z^{8}q^{19} + z^{4}5q^{20} + z^{3}0q^{21} + zq^{22} + z^{1}48q^{23} + z^{1}08q^{24} + z^{1}65q^{25} + ⋯ ∈ S_{20}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{1}25q^{2} + z^{1}6q^{3} + z^{2}0q^{4} + z^{3}9q^{5} + z^{1}41q^{6} + 10q^{7} +z^{1}62q^{8} +z^{9}7q^{9} +z^{1}64q^{10} +z^{1}00q^{11} +z^{3}6q^{12} +z^{4}q^{13} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}64)ω^{6} & 0 \\ 0 & u(z^{4})ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(1429185604795825049463+ O(13^{2}0)) ∗ a + 3706085701922263396601 + O(13^{2}0)q^{3} + (3640113536828162924080 + O(13^{2}0)) ∗ a + 6181494794288393561492+O(13^{2}0)q^{4}+(2747412779011195026588+O(13^{2}0))∗a+2582347564517853155780+ O(13^{2}0)q^{5}+(9490781035704896840706+O(13^{2}0))∗a+2953066046271830579423+O(13^{2}0)q^{6}+−40353607q^{7}+ −(4098345411965926696492+O(13^{2}0))∗a+8885576288168160971244+O(13^{2}0)q^{8}+−(2490064776027194336328+ O(13^{2}0)) ∗ a − 133768578650421935975 ∗ 13 + O(13^{2}0)q^{9} + −(2477071460456747803859 + O(13^{2}0)) ∗ a − 7156371631592903896841+O(13^{2}0)q^{10}+−(1572163062030478558582+O(13^{2}0))∗a+608594466611543237029+ O(13^{2}0)q^{11} + −(265760977531702812684 + O(13^{2}0)) ∗ a + 5743463471951740345203 + O(13^{2}0)q^{12} + −(56652534622872533445∗13+O(13^{2}0))∗a+395696841117786909027∗13+O(13^{2}0)q^{13} +−40353607∗ aq^{14}+(6486527667418528946721+O(13^{2}0))∗a−339241858593452642831∗13+O(13^{2}0)q^{15}+−(9180714684975962815408+ O(13^{2}0)) ∗ a + 8493877082876865526813 + O(13^{2}0)q^{16} + (7735985780488794682104 + O(13^{2}0)) ∗ a − 9086601107156692188990+O(13^{2}0)q^{17}+(9241770057762816162008+O(13^{2}0))∗a+440315971743493599181+ O(13^{2}0)q^{18} + −(4399449679209673659573 + O(13^{2}0)) ∗ a − 3325531310451946402408 + O(13^{2}0)q^{19} + −(3869308524720202035161+O(13^{2}0))∗a−4540163276780186500066+O(13^{2}0)q^{20}+−(8070350592336209488824+ O(13^{2}0)) ∗ a − 7971506850070710744204 + O(13^{2}0)q^{21} + (6371287289390826710523 + O(13^{2}0)) ∗ a + 4257383793104925683422+O(13^{2}0)q^{22}+−(1679983369101519558967+O(13^{2}0))∗a+16015279493206874126+ O(13^{2}0)q^{23} + (8252791140543674761120 + O(13^{2}0)) ∗ a + 8221975605861215033439 + O(13^{2}0)q^{24} + (3464370552500859034305 + O(13^{2}0)) ∗ a + 26922290289021975539 + O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2}+−3640113536828162924080+ O(13^{2}0)x + −6181494794288394085780 + O(13^{2}0). The slope of f_{0} is 1.

16. Consider f = q+z^{2}3q^{2}+z^{6}9q^{3}+z^{5}0q^{4}+z^{4}6q^{5}+z^{9}2q^{6}+5q^{7}+z^{7}6q^{8}+z^{6}7q^{9}+z^{6}9q^{10}+z^{1}31q^{11}+z^{1}19q^{12}+ z^{1}49q^{14}+z^{1}15q^{15}+z^{7}9q^{16}+z^{3}8q^{17}+z^{9}0q^{18}+q^{19}+z^{9}6q^{20}+z^{2}7q^{21}+7q^{22}+z^{6}6q^{23}+z^{1}45q^{24}+z^{1}43q^{25}+⋯ ∈ S_{20}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{1}07q^{2} + z^{6}9q^{3} + z^{5}0q^{4} + z^{1}30q^{5} + z^{8}q^{6} + 8q^{7} + z^{1}60q^{8} + z^{6}7q^{9} + z^{6}9q^{10} + z^{4}7q^{11} + z^{1}19q^{12} + z^{1}36q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{3}2)ω^{1} & 0 \\ 0 & u(z^{1}36)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(1362735429840992962549+ O(13^{2}0)) ∗ a + 9152311934162367676568 + O(13^{2}0)q^{3} + (9347234290010263570121 + O(13^{2}0)) ∗ a − 4138029506060468467649+O(13^{2}0)q^{4}+(1823865870242621126444+O(13^{2}0))∗a−8828544679813861177910+ O(13^{2}0)q^{5}+(450986930186645562513+O(13^{2}0))∗a−4184969928380714076956+O(13^{2}0)q^{6}+−40353607q^{7}+ (6237780592425004122649+O(13^{2}0))∗a−8736139397476563949300+O(13^{2}0)q^{8}+(4048169531845250792584+ O(13^{2}0)) ∗ a + 4919292136431095946608 + O(13^{2}0)q^{9} + −(5530021511774533838582 + O(13^{2}0)) ∗ a − 5433040073830401650977+O(13^{2}0)q^{10}+(426276559096753555462+O(13^{2}0))∗a−864540268696844471755+ O(13^{2}0)q^{11}+(78282186373111394359+O(13^{2}0))∗a+3243551210010266873937+O(13^{2}0)q^{12}+−(722899158790690835203∗ 13+O(13^{2}0))∗a−187428189272485489152∗13+O(13^{2}0)q^{13}+−40353607∗aq^{14}+(5665685971038964953972+ O(13^{2}0)) ∗ a + 7648596074527063315557 + O(13^{2}0)q^{15} + (7368926560490288362598 + O(13^{2}0)) ∗ a − 499016375656843233035∗13+O(13^{2}0)q^{16}+−(7427215053566423514075+O(13^{2}0))∗a−6089490605610847153792+ O(13^{2}0)q^{17} + −(3460592049467995679693 + O(13^{2}0)) ∗ a − 7692369873150462965299 + O(13^{2}0)q^{18} + −(714498685061749900744∗13+O(13^{2}0))∗a+8837171171004040079684+O(13^{2}0)q^{19}+−(16840614321536997494+ O(13^{2}0)) ∗ a + 5219536015955587725618 + O(13^{2}0)q^{20} + −(9188958861946709717121 + O(13^{2}0)) ∗ a + 2499481257589133955305+O(13^{2}0)q^{21}+(612042365933511269905∗13+O(13^{2}0))∗a+3297238637976307265677+ O(13^{2}0)q^{22} + (1581423023948898040673 + O(13^{2}0)) ∗ a + 9478059672733049209182 + O(13^{2}0)q^{23} + −(5073686614522904036635+O(13^{2}0))∗a−2943378294820649893266+O(13^{2}0)q^{24}+−(3529601006709718385512+ O(13^{2}0)) ∗ a − 5691926378318935727930 + O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −9347234290010263570121 + O(13^{2}0)x + 4138029506060467943361 + O(13^{2}0). The slope of f_{0} is 1.

17. Consider f = q + q^{2} + 4q^{3} + 9q^{4} + 9q^{5} + 4q^{6} + 9q^{7} + 4q^{8} + 2q^{9} + 9q^{10} + 8q^{11} + 10q^{12} + 9q^{14} + 10q^{15} + 11q^{16} + 8q^{17} + 2q^{18} + 5q^{19} + 3q^{20} + 10q^{21} + 8q^{22} + 10q^{23} + 3q^{24} + 11q^{25} + ⋯ ∈ S_{22}(Γ_{0}(7); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 3q^{2} + 12q^{3} + 3q^{4} + 9q^{5} + 10q^{6} + 3q^{7} + 4q^{8} + 5q^{9} + q^{10} + 11q^{11} + 10q^{12} + 3q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(9)ω^{1} & 0 \\ 0 & u(3)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(9063697277764992442666+ O(13^{2}0)) ∗ a + 565911594807616672195 + O(13^{2}0)q^{3} + −(3604294880271540498795 + O(13^{2}0)) ∗ a − 6775330269216092372717+O(13^{2}0)q^{4}+(1867861771759339896479+O(13^{2}0))∗a−7173264497122891164759+ O(13^{2}0)q^{5}+(1295487771912059241450+O(13^{2}0))∗a−7849740054396028251458+O(13^{2}0)q^{6}+−282475249q^{7}+ −(3339211555862849752963+O(13^{2}0))∗a−6025404340986741640997+O(13^{2}0)q^{8}+(7112566699653068840120+ O(13^{2}0)) ∗ a + 8199197089310758877076 + O(13^{2}0)q^{9} + (472391659809778830898 ∗ 13 + O(13^{2}0)) ∗ a + 3400813668900733653174+O(13^{2}0)q^{10}+(601710112539431937503+O(13^{2}0))∗a+6280852027271989828662+ O(13^{2}0)q^{11} + (7709129259084088268604 + O(13^{2}0)) ∗ a − 4409883993332636089614 + O(13^{2}0)q^{12} + −(592229249125125467636∗13+O(13^{2}0))∗a−72050329986747787122∗13+O(13^{2}0)q^{13}+−282475249∗ aq^{14}+−(8089468388966895597046+O(13^{2}0))∗a−248372871777637270173∗13+O(13^{2}0)q^{15}+−(23605512742000116489∗ 13^{2} +O(13^{2}0))∗a−246730942702362896702+O(13^{2}0)q^{16} +(7763081968210252859621+O(13^{2}0))∗a− 3830684842773896653854+O(13^{2}0)q^{17}+(1776154548986388743836+O(13^{2}0))∗a−6421467670242009485944+ O(13^{2}0)q^{18} + (3711883119363329478879 + O(13^{2}0)) ∗ a + 5521297720812821954155 + O(13^{2}0)q^{19} + −(6593123648423169596360+O(13^{2}0))∗a−2676045970549317895411+O(13^{2}0)q^{20}+(7834407539445194142832+ O(13^{2}0)) ∗ a − 3969780580513634242678 + O(13^{2}0)q^{21} + (217180303001569802975 + O(13^{2}0)) ∗ a + 2036746802993517277994+O(13^{2}0)q^{22}+(3995539859973101349265+O(13^{2}0))∗a−7997194862629258743023+ O(13^{2}0)q^{23} + (4210651487672439303303 + O(13^{2}0)) ∗ a − 5539878886744296667496 + O(13^{2}0)q^{24} + (6805273334773101517791+O(13^{2}0))∗a−2180179475156225188174+O(13^{2}0)q^{25}+⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2}+3604294880271540498795+ O(13^{2}0)x + 6775330269216090275565 + O(13^{2}0). The slope of f_{0} is 1.

18. Consider f = q + z^{1}21q^{2} + z^{1}25q^{3} + z^{7}8q^{4} + z^{8}8q^{5} + z^{7}8q^{6} + 4q^{7} + z^{3}4q^{8} + z^{1}1q^{9} + z^{4}1q^{10} + z^{1}45q^{11} + z^{3}5q^{12} + z^{1}49q^{14} + z^{4}5q^{15} + z^{1}35q^{16} + z^{6}6q^{17} + z^{1}32q^{18} + 7q^{19} + z^{1}66q^{20} + z^{1}53q^{21} + 11q^{22} + z^{3}8q^{23} + z^{1}59q^{24} + z^{5}9q^{25} + ⋯ ∈ S_{22}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{1}07q^{2} + z^{6}9q^{3} + z^{5}0q^{4} + z^{1}30q^{5} + z^{8}q^{6} + 8q^{7} + z^{1}60q^{8} + z^{6}7q^{9} + z^{6}9q^{10} + z^{4}7q^{11} + z^{1}19q^{12} + z^{1}36q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{3}2)ω^{8} & 0 \\ 0 & u(z^{1}36)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(6338462514291800738602+ O(13^{2}0)) ∗ a − 2936298529252824860629 + O(13^{2}0)q^{3} + (117418922662178849966 + O(13^{2}0)) ∗ a + 2654016535870675167549+O(13^{2}0)q^{4}+(4125788853084514780126+O(13^{2}0))∗a−1953372626488373504808+ O(13^{2}0)q^{5}+−(250425636828318709699+O(13^{2}0))∗a+1929667579357029818754+O(13^{2}0)q^{6}+282475249q^{7}+ −(2730840662325065010229+O(13^{2}0))∗a+6030462336220569751178+O(13^{2}0)q^{8}+−(8382359772791656743911+ O(13^{2}0)) ∗ a + 6868570725067761935938 + O(13^{2}0)q^{9} + (8180557631046004422910 + O(13^{2}0)) ∗ a + 6162811422643396690661+O(13^{2}0)q^{10}+(7404952390280043339976+O(13^{2}0))∗a−1483773880911319299049+ O(13^{2}0)q^{11} + (7419964401482414179117 + O(13^{2}0)) ∗ a − 4604411513617388790510 + O(13^{2}0)q^{12} + −(390174381584682561872∗13+O(13^{2}0))∗a+368636905942728880121∗13+O(13^{2}0)q^{13} +282475249∗ aq^{14}+−(8593979123549836028812+O(13^{2}0))∗a+2317898952700723084808+O(13^{2}0)q^{15}+−(3609232429711977163743+ O(13^{2}0)) ∗ a + 240836734412194724526 ∗ 13 + O(13^{2}0)q^{16} + (2237943049428996957036 + O(13^{2}0)) ∗ a + 1949851086332077822589+O(13^{2}0)q^{17}+−(5953291249291185936691+O(13^{2}0))∗a+7059525261018518248820+ O(13^{2}0)q^{18} + −(175573549944232119309 ∗ 13 + O(13^{2}0)) ∗ a − 3063549372407886876949 + O(13^{2}0)q^{19} + −(451102636263353058347+O(13^{2}0))∗a+491227280843436528925+O(13^{2}0)q^{20}+(527604135682424676162+ O(13^{2}0)) ∗ a − 678135344155116663925 + O(13^{2}0)q^{21} + −(181915756111550238876 ∗ 13 + O(13^{2}0)) ∗ a + 7980565409986365894709+O(13^{2}0)q^{22}+−(4395466866198499379463+O(13^{2}0))∗a−1284755934875684847402+ O(13^{2}0)q^{23} + −(1369646702787961938949 + O(13^{2}0)) ∗ a − 1074664601520307799717 + O(13^{2}0)q^{24} + −(8087910713174069877074 + O(13^{2}0)) ∗ a + 1538624134943289219575 + O(13^{2}0)q^{25} + ⋯ is an ℓadic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −117418922662178849966 + O(13^{2}0)x + −2654016535870677264701 + O(13^{2}0). The slope of f_{0} is 1.

19. Consider f = q + z^{5}7q^{2} + z^{1}52q^{3} + z^{1}48q^{4} + z^{3}q^{5} + z^{4}1q^{6} + 4q^{7} + z^{9}0q^{8} + z^{1}41q^{9} + z^{6}0q^{10} + z^{6}8q^{11} + z^{1}32q^{12} + z^{8}5q^{14} + z^{1}55q^{15} + z^{8}6q^{16} + z^{1}0q^{17} + z^{3}0q^{18} + z^{9}0q^{19} + z^{1}51q^{20} + z^{1}2q^{21} + z^{1}25q^{22} + z^{4}8q^{23} + z^{7}4q^{24} + z^{4}5q^{25} + ⋯ ∈ S_{22}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{1}13q^{2} + z^{4}0q^{3} + z^{9}2q^{4} + z^{3}q^{5} + z^{1}53q^{6} + 10q^{7} + z^{9}0q^{8} + z^{8}5q^{9} + z^{1}16q^{10} + z^{1}24q^{11} + z^{1}32q^{12} + z^{5}2q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}16)ω^{1} & 0 \\ 0 & u(z^{5}2)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(8609516376120470298040+ O(13^{2}0)) ∗ a − 8552765710906928785422 + O(13^{2}0)q^{3} + (490571435843240426925 + O(13^{2}0)) ∗ a + 6620324333100541920002+O(13^{2}0)q^{4}+−(1025369575977938970333+O(13^{2}0))∗a−2538612799517533224137+ O(13^{2}0)q^{5} + −(2042427507412128050705 + O(13^{2}0)) ∗ a + 7041760987752472046693 + O(13^{2}0)q^{6} + 282475249q^{7} + (1449595854086163334311 + O(13^{2}0)) ∗ a + 1879733201412653226014 + O(13^{2}0)q^{8} + (2778192264144707461622+O(13^{2}0))∗a−171913196196520880611∗13+O(13^{2}0)q^{9}+−(2250029696061300622888+ O(13^{2}0)) ∗ a − 9018755153996786747088 + O(13^{2}0)q^{10} + (9008968999950141804432 + O(13^{2}0)) ∗ a + 3105978521650385475125+O(13^{2}0)q^{11}+−(7847635134851409395611+O(13^{2}0))∗a−5124057156314135094175+ O(13^{2}0)q^{12}+−(171545067346144333042∗13+O(13^{2}0))∗a−295990137821346555757∗13+O(13^{2}0)q^{13}+ 282475249∗aq^{14}+(5981031426096399801373+O(13^{2}0))∗a+262203910009693440041∗13+O(13^{2}0)q^{15}+ −(223165704710190848628+O(13^{2}0))∗a+9450229173842842388408+O(13^{2}0)q^{16}+(5823825405948678138600+ O(13^{2}0)) ∗ a + 8969402406860828798692 + O(13^{2}0)q^{17} + (8920234727796113704837 + O(13^{2}0)) ∗ a − 845135776781835854440+O(13^{2}0)q^{18}+(3394218327282254197265+O(13^{2}0))∗a−3816856036196443036422+ O(13^{2}0)q^{19} + (6132857166483476308506 + O(13^{2}0)) ∗ a + 2842528079267737113187 + O(13^{2}0)q^{20} + −(668705666804466351708+O(13^{2}0))∗a−8983331735692756661109+O(13^{2}0)q^{21}+(3296568564766914483407+ O(13^{2}0)) ∗ a − 1739007335554001499900 + O(13^{2}0)q^{22} + −(975072512521381108189 + O(13^{2}0)) ∗ a + 7185374090701288171979+O(13^{2}0)q^{23}+(2960325932463899420766+O(13^{2}0))∗a+9231216277259167685844+ O(13^{2}0)q^{24} + (6296113091504134697233 + O(13^{2}0)) ∗ a + 3606582224584630871447 + O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −490571435843240426925 + O(13^{2}0)x + −6620324333100544017154 + O(13^{2}0). The slope of f_{0} is 1.

20. Consider f = q + z^{6}1q^{2} + z^{1}13q^{3} + z^{6}q^{4} + z^{1}36q^{5} + z^{6}q^{6} + 4q^{7} + z^{1}06q^{8} + z^{1}43q^{9} + z^{2}9q^{10} + z^{3}7q^{11} + z^{1}19q^{12} + z^{8}9q^{14} + z^{8}1q^{15} + z^{7}5q^{16} + z^{1}8q^{17} + z^{3}6q^{18} + 7q^{19} + z^{1}42q^{20} + z^{1}41q^{21} + 11q^{22} + z^{1}58q^{23} + z^{5}1q^{24} + z^{9}5q^{25} + ⋯ ∈ S_{22}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-8 newform g = q + z^{4}7q^{2} + z^{5}7q^{3} + z^{1}46q^{4} + z^{1}0q^{5} + z^{1}04q^{6} + 8q^{7} + z^{6}4q^{8} + z^{3}1q^{9} + z^{5}7q^{10} + z^{1}07q^{11} + z^{3}5q^{12} + z^{8}8q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{8}0)ω^{8} & 0 \\ 0 & u(z^{8}8)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(6338462514291800738602+ O(13^{2}0)) ∗ a − 2936298529252824860629 + O(13^{2}0)q^{3} + (117418922662178849966 + O(13^{2}0)) ∗ a + 2654016535870675167549+O(13^{2}0)q^{4}+(4125788853084514780126+O(13^{2}0))∗a−1953372626488373504808+ O(13^{2}0)q^{5}+−(250425636828318709699+O(13^{2}0))∗a+1929667579357029818754+O(13^{2}0)q^{6}+282475249q^{7}+ −(2730840662325065010229+O(13^{2}0))∗a+6030462336220569751178+O(13^{2}0)q^{8}+−(8382359772791656743911+ O(13^{2}0)) ∗ a + 6868570725067761935938 + O(13^{2}0)q^{9} + (8180557631046004422910 + O(13^{2}0)) ∗ a + 6162811422643396690661+O(13^{2}0)q^{10}+(7404952390280043339976+O(13^{2}0))∗a−1483773880911319299049+ O(13^{2}0)q^{11} + (7419964401482414179117 + O(13^{2}0)) ∗ a − 4604411513617388790510 + O(13^{2}0)q^{12} + −(390174381584682561872∗13+O(13^{2}0))∗a+368636905942728880121∗13+O(13^{2}0)q^{13} +282475249∗ aq^{14}+−(8593979123549836028812+O(13^{2}0))∗a+2317898952700723084808+O(13^{2}0)q^{15}+−(3609232429711977163743+ O(13^{2}0)) ∗ a + 240836734412194724526 ∗ 13 + O(13^{2}0)q^{16} + (2237943049428996957036 + O(13^{2}0)) ∗ a + 1949851086332077822589+O(13^{2}0)q^{17}+−(5953291249291185936691+O(13^{2}0))∗a+7059525261018518248820+ O(13^{2}0)q^{18} + −(175573549944232119309 ∗ 13 + O(13^{2}0)) ∗ a − 3063549372407886876949 + O(13^{2}0)q^{19} + −(451102636263353058347+O(13^{2}0))∗a+491227280843436528925+O(13^{2}0)q^{20}+(527604135682424676162+ O(13^{2}0)) ∗ a − 678135344155116663925 + O(13^{2}0)q^{21} + −(181915756111550238876 ∗ 13 + O(13^{2}0)) ∗ a + 7980565409986365894709+O(13^{2}0)q^{22}+−(4395466866198499379463+O(13^{2}0))∗a−1284755934875684847402+ O(13^{2}0)q^{23} + −(1369646702787961938949 + O(13^{2}0)) ∗ a − 1074664601520307799717 + O(13^{2}0)q^{24} + −(8087910713174069877074 + O(13^{2}0)) ∗ a + 1538624134943289219575 + O(13^{2}0)q^{25} + ⋯ is an ℓadic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −117418922662178849966 + O(13^{2}0)x + −2654016535870677264701 + O(13^{2}0). The slope of f_{0} is 1.

21. Consider f = q + z^{6}9q^{2} + z^{1}28q^{3} + z^{7}6q^{4} + z^{3}9q^{5} + z^{2}9q^{6} + 4q^{7} + z^{1}62q^{8} + z^{1}53q^{9} + z^{1}08q^{10} + z^{4}4q^{11} + z^{3}6q^{12} + z^{9}7q^{14} + z^{1}67q^{15} + z^{1}10q^{16} + z^{1}30q^{17} + z^{5}4q^{18} + z^{1}62q^{19} + z^{1}15q^{20} + z^{1}56q^{21} + z^{1}13q^{22} + z^{1}20q^{23} + z^{1}22q^{24} + z^{8}1q^{25} + ⋯ ∈ S_{22}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + z^{1}25q^{2} + z^{1}6q^{3} + z^{2}0q^{4} + z^{3}9q^{5} + z^{1}41q^{6} + 10q^{7} +z^{1}62q^{8} +z^{9}7q^{9} +z^{1}64q^{10} +z^{1}00q^{11} +z^{3}6q^{12} +z^{4}q^{13} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}64)ω^{1} & 0 \\ 0 & u(z^{4})ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(8609516376120470298040+ O(13^{2}0)) ∗ a − 8552765710906928785422 + O(13^{2}0)q^{3} + (490571435843240426925 + O(13^{2}0)) ∗ a + 6620324333100541920002+O(13^{2}0)q^{4}+−(1025369575977938970333+O(13^{2}0))∗a−2538612799517533224137+ O(13^{2}0)q^{5} + −(2042427507412128050705 + O(13^{2}0)) ∗ a + 7041760987752472046693 + O(13^{2}0)q^{6} + 282475249q^{7} + (1449595854086163334311 + O(13^{2}0)) ∗ a + 1879733201412653226014 + O(13^{2}0)q^{8} + (2778192264144707461622+O(13^{2}0))∗a−171913196196520880611∗13+O(13^{2}0)q^{9}+−(2250029696061300622888+ O(13^{2}0)) ∗ a − 9018755153996786747088 + O(13^{2}0)q^{10} + (9008968999950141804432 + O(13^{2}0)) ∗ a + 3105978521650385475125+O(13^{2}0)q^{11}+−(7847635134851409395611+O(13^{2}0))∗a−5124057156314135094175+ O(13^{2}0)q^{12}+−(171545067346144333042∗13+O(13^{2}0))∗a−295990137821346555757∗13+O(13^{2}0)q^{13}+ 282475249∗aq^{14}+(5981031426096399801373+O(13^{2}0))∗a+262203910009693440041∗13+O(13^{2}0)q^{15}+ −(223165704710190848628+O(13^{2}0))∗a+9450229173842842388408+O(13^{2}0)q^{16}+(5823825405948678138600+ O(13^{2}0)) ∗ a + 8969402406860828798692 + O(13^{2}0)q^{17} + (8920234727796113704837 + O(13^{2}0)) ∗ a − 845135776781835854440+O(13^{2}0)q^{18}+(3394218327282254197265+O(13^{2}0))∗a−3816856036196443036422+ O(13^{2}0)q^{19} + (6132857166483476308506 + O(13^{2}0)) ∗ a + 2842528079267737113187 + O(13^{2}0)q^{20} + −(668705666804466351708+O(13^{2}0))∗a−8983331735692756661109+O(13^{2}0)q^{21}+(3296568564766914483407+ O(13^{2}0)) ∗ a − 1739007335554001499900 + O(13^{2}0)q^{22} + −(975072512521381108189 + O(13^{2}0)) ∗ a + 7185374090701288171979+O(13^{2}0)q^{23}+(2960325932463899420766+O(13^{2}0))∗a+9231216277259167685844+ O(13^{2}0)q^{24} + (6296113091504134697233 + O(13^{2}0)) ∗ a + 3606582224584630871447 + O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −490571435843240426925 + O(13^{2}0)x + −6620324333100544017154 + O(13^{2}0). The slope of f_{0} is 1.

22. Consider f = q+11∗z^{5}+7∗z^{4}+4∗z^{3}+5∗z+8q^{2}+7∗z^{5}+8∗z^{4}+12∗z^{3}+2∗z+4q^{3}+11∗z^{5}+7∗z^{4}+4∗z^{3}+5∗ z+9q^{4}+6∗z^{5}+5∗z^{4}+z^{3}+11∗z+5q^{5}+8∗z^{5}+11∗z^{4}+10∗z^{3}+6∗z+5q^{6}+2q^{7}+10∗z^{5}+4∗z^{4}+6∗z^{3}+z+7q^{8}+ z^{5}+3∗z^{4}+11∗z^{3}+4∗z+4q^{9}+2q^{10}+9∗z^{5}+z^{4}+8∗z^{3}+10∗z+7q^{11}+2∗z^{5}+6∗z^{4}+9∗z^{3}+8∗z+9q^{12}+9∗z^{5}+ z^{4}+8∗z^{3}+10∗z+3q^{14}+10∗z^{5}+4∗z^{4}+6∗z^{3}+z+10q^{15}+8∗z^{5}+11∗z^{4}+10∗z^{3}+6∗z+12q^{16}+4∗z^{5}+12∗z^{4}+ 5∗z^{3}+3∗z+7q^{17}+11∗z^{5}+7∗z^{4}+4∗z^{3}+5∗z+4q^{18}+z^{5}+3∗z^{4}+11∗z^{3}+4∗z+8q^{19}+6∗z^{5}+5∗z^{4}+z^{3}+11∗ z+7q^{20}+z^{5}+3∗z^{4}+11∗z^{3}+4∗z+8q^{21}+z^{5}+3∗z^{4}+11∗z^{3}+4∗z+12q^{22}+7∗z^{5}+8∗z^{4}+12∗z^{3}+2∗z+3q^{23}+ 3∗z^{5}+9∗z^{4}+7∗z^{3}+12∗z+7q^{24}+5∗z^{5}+2∗z^{4}+3∗z^{3}+7∗z+1q^{25}+⋯ ∈ S_{24}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{6} + 10x^{3} + 11x^{2} + 11x + 2 = 0. We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-10 newform g = q+z^{1}15q^{2}+z^{1}39q^{3}+z^{1}03q^{4}+z^{9}5q^{5}+z^{8}6q^{6}+4q^{7}+z^{1}31q^{8}+zq^{9}+8q^{10}+z^{5}5q^{11}+z^{7}4q^{12}+z^{1}18q^{13}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{5}0)ω^{10} & 0 \\ 0 & u(z^{1}18)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(2484469554688361540102+ O(13^{2}0)) ∗ a − 2426438264827323355187 + O(13^{2}0)q^{3} + (2266014815912095228111 + O(13^{2}0)) ∗ a − 6568182989375040620053+O(13^{2}0)q^{4}+−(174891855798365764828+O(13^{2}0))∗a−9217118824557607083611+ O(13^{2}0)q^{5} + −(9042954874921806536299 + O(13^{2}0)) ∗ a − 2591969385754446010640 + O(13^{2}0)q^{6} + 1977326743q^{7} + (4837695667204699399355 + O(13^{2}0)) ∗ a + 8310234335664692465237 + O(13^{2}0)q^{8} + (2103571677344298550135+O(13^{2}0))∗a−1001931302080202153717+O(13^{2}0)q^{9}+(721678517129837379649∗ 13+O(13^{2}0))∗a+4003100912554191074642+O(13^{2}0)q^{10} +(1934170090008768726701+O(13^{2}0))∗a+ 5152669570252992438226+O(13^{2}0)q^{11}+(471631730867369776573+O(13^{2}0))∗a−7281498357509785458006+ O(13^{2}0)q^{12} +(321535953361027313717∗13+O(13^{2}0))∗a+587201597917145269498∗13+O(13^{2}0)q^{13} + 1977326743∗aq^{14}+−(122687731113632400877+O(13^{2}0))∗a+2151654550084183351725+O(13^{2}0)q^{15}+ −(456201307269262092841+O(13^{2}0))∗a−4817379210706668487851+O(13^{2}0)q^{16}+(8414074193746020990916+ O(13^{2}0)) ∗ a + 2998371464231092989426 + O(13^{2}0)q^{17} + (8301718318763842136217 + O(13^{2}0)) ∗ a − 1307422697226620719541+O(13^{2}0)q^{18}+−(1513946518849323607588+O(13^{2}0))∗a+408129597473082842291+ O(13^{2}0)q^{19} + (5153956550237925597060 + O(13^{2}0)) ∗ a − 1345799876292780993794 + O(13^{2}0)q^{20} + (7435721698670875595004+O(13^{2}0))∗a+7881189981132651753920+O(13^{2}0)q^{21}+(239925361974681532246+ O(13^{2}0)) ∗ a − 1685771079295205713951 + O(13^{2}0)q^{22} + (2248566798810745555717 + O(13^{2}0)) ∗ a − 650096464982845261974+O(13^{2}0)q^{23}+−(7910578067318396725432+O(13^{2}0))∗a+3337412891101860316141+ O(13^{2}0)q^{24} + (3091042885070506999364 + O(13^{2}0)) ∗ a + 4524946181334462424588 + O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −2266014815912095228111 + O(13^{2}0)x + 6568182989375032231445 + O(13^{2}0). The slope of f_{0} is 1.

23. Consider f = q+2∗z^{5}+6∗z^{4}+9∗z^{3}+8∗z+6q^{2}+6∗z^{5}+5∗z^{4}+z^{3}+11∗z+11q^{3}+2∗z^{5}+6∗z^{4}+9∗z^{3}+8∗z+ 7q^{4}+7∗z^{5}+8∗z^{4}+12∗z^{3}+2∗z+11q^{5}+5∗z^{5}+2∗z^{4}+3∗z^{3}+7∗zq^{6}+2q^{7}+3∗z^{5}+9∗z^{4}+7∗z^{3}+12∗z+4q^{8}+ 12∗z^{5}+10∗z^{4}+2∗z^{3}+9∗z+5q^{9}+2q^{10}+4∗z^{5}+12∗z^{4}+5∗z^{3}+3∗z+3q^{11}+11∗z^{5}+7∗z^{4}+4∗z^{3}+5∗z+11q^{12}+ 4∗z^{5}+12∗z^{4}+5∗z^{3}+3∗z+12q^{14}+3∗z^{5}+9∗z^{4}+7∗z^{3}+12∗z+7q^{15}+5∗z^{5}+2∗z^{4}+3∗z^{3}+7∗z+7q^{16}+9∗ z^{5}+z^{4}+8∗z^{3}+10∗z+11q^{17}+2∗z^{5}+6∗z^{4}+9∗z^{3}+8∗z+2q^{18}+12∗z^{5}+10∗z^{4}+2∗z^{3}+9∗z+9q^{19}+7∗z^{5}+8∗ z^{4}+12∗z^{3}+2∗zq^{20}+12∗z^{5}+10∗z^{4}+2∗z^{3}+9∗z+9q^{21}+12∗z^{5}+10∗z^{4}+2∗z^{3}+9∗zq^{22}+6∗z^{5}+5∗z^{4}+z^{3}+ 11∗z+10q^{23}+10∗z^{5}+4∗z^{4}+6∗z^{3}+z+10q^{24}+8∗z^{5}+11∗z^{4}+10∗z^{3}+6∗z+6q^{25}+⋯ ∈ S_{24}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{6} + 10x^{3} + 11x^{2} + 11x + 2 = 0. We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{1}51q^{2} + z^{1}27q^{3} + z^{1}63q^{4} + z^{5}9q^{5} + z^{1}10q^{6} + 4q^{7} + z^{2}3q^{8} + z^{1}3q^{9} + 8q^{10} + z^{4}3q^{11} + z^{1}22q^{12} + z^{2}2q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}46)ω^{10} & 0 \\ 0 & u(z^{2}2)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(2484469554688361540102+ O(13^{2}0)) ∗ a − 2426438264827323355187 + O(13^{2}0)q^{3} + (2266014815912095228111 + O(13^{2}0)) ∗ a − 6568182989375040620053+O(13^{2}0)q^{4}+−(174891855798365764828+O(13^{2}0))∗a−9217118824557607083611+ O(13^{2}0)q^{5} + −(9042954874921806536299 + O(13^{2}0)) ∗ a − 2591969385754446010640 + O(13^{2}0)q^{6} + 1977326743q^{7} + (4837695667204699399355 + O(13^{2}0)) ∗ a + 8310234335664692465237 + O(13^{2}0)q^{8} + (2103571677344298550135+O(13^{2}0))∗a−1001931302080202153717+O(13^{2}0)q^{9}+(721678517129837379649∗ 13+O(13^{2}0))∗a+4003100912554191074642+O(13^{2}0)q^{10} +(1934170090008768726701+O(13^{2}0))∗a+ 5152669570252992438226+O(13^{2}0)q^{11}+(471631730867369776573+O(13^{2}0))∗a−7281498357509785458006+ O(13^{2}0)q^{12} +(321535953361027313717∗13+O(13^{2}0))∗a+587201597917145269498∗13+O(13^{2}0)q^{13} + 1977326743∗aq^{14}+−(122687731113632400877+O(13^{2}0))∗a+2151654550084183351725+O(13^{2}0)q^{15}+ −(456201307269262092841+O(13^{2}0))∗a−4817379210706668487851+O(13^{2}0)q^{16}+(8414074193746020990916+ O(13^{2}0)) ∗ a + 2998371464231092989426 + O(13^{2}0)q^{17} + (8301718318763842136217 + O(13^{2}0)) ∗ a − 1307422697226620719541+O(13^{2}0)q^{18}+−(1513946518849323607588+O(13^{2}0))∗a+408129597473082842291+ O(13^{2}0)q^{19} + (5153956550237925597060 + O(13^{2}0)) ∗ a − 1345799876292780993794 + O(13^{2}0)q^{20} + (7435721698670875595004+O(13^{2}0))∗a+7881189981132651753920+O(13^{2}0)q^{21}+(239925361974681532246+ O(13^{2}0)) ∗ a − 1685771079295205713951 + O(13^{2}0)q^{22} + (2248566798810745555717 + O(13^{2}0)) ∗ a − 650096464982845261974+O(13^{2}0)q^{23}+−(7910578067318396725432+O(13^{2}0))∗a+3337412891101860316141+ O(13^{2}0)q^{24} + (3091042885070506999364 + O(13^{2}0)) ∗ a + 4524946181334462424588 + O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −2266014815912095228111 + O(13^{2}0)x + 6568182989375032231445 + O(13^{2}0). The slope of f_{0} is 1.

24. Consider f = q + 11q^{2} + 6q^{3} + 10q^{4} + 7q^{5} + q^{6} + 11q^{7} + 7q^{8} + q^{9} + 12q^{10} + 6q^{11} + 8q^{12} + 4q^{14} + 3q^{15} + 7q^{16} + 4q^{17} + 11q^{18} + 4q^{19} + 5q^{20} + q^{21} + q^{22} + 2q^{23} + 3q^{24} + 2q^{25} + ⋯ ∈ S_{24}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-10 newform g = q + 12q^{2} + 2q^{3} + 9q^{4} + 4q^{5} + 11q^{6} + 9q^{7} + 9q^{8} + 3q^{9} + 9q^{10} + 10q^{11} + 5q^{12} + q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{10} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−388687521006345835159+ O(13^{2}0)q^{2}+−7887892961217996030883+O(13^{2}0)q^{3}+−7387382903314854403200+O(13^{2}0)q^{4}+7053564359396607756308+ O(13^{2}0)q^{5} + 1271343183854795113011 + O(13^{2}0)q^{6} + −1977326743q^{7} + 9381280059389800216766 + O(13^{2}0)q^{8}+6569616062147708199072+O(13^{2}0)q^{9}+8229886528165811624504+O(13^{2}0)q^{10}+6373866355667102552461+ O(13^{2}0)q^{11} + 1047218889364948470100 + O(13^{2}0)q^{12} + 278363248466367294507 ∗ 13 + O(13^{2}0)q^{13} + −6492095852321707748142+O(13^{2}0)q^{14}+7693349567919629243373+O(13^{2}0)q^{15}+−7957871984848145469700+ O(13^{2}0)q^{16}+−1633208696885182361538+O(13^{2}0)q^{17}+1844434858629423883041+O(13^{2}0)q^{18}+−8922218229900813221185+ O(13^{2}0)q^{19}+−8642198572227525802960+O(13^{2}0)q^{20}+5944555220284070244804+O(13^{2}0)q^{21}+4973248512722168334423+ O(13^{2}0)q^{22}+−4139103080511913229090+O(13^{2}0)q^{23}+1364645124925758908365+O(13^{2}0)q^{24}+−4356084353409501493155+ O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

25. Consider f = q + z^{1}30q^{2} + z^{8}8q^{3} + z^{5}8q^{4} + z^{1}8q^{5} + z^{5}0q^{6} + 11q^{7} + z^{1}9q^{8} + z^{6}9q^{9} + z^{1}48q^{10} + z^{1}2q^{11} + z^{1}46q^{12} + z^{6}0q^{14} + z^{1}06q^{15} + z^{4}9q^{16} + z^{9}q^{17} + z^{3}1q^{18} + z^{1}24q^{19} + z^{7}6q^{20} + z^{1}8q^{21} + z^{1}42q^{22} + z^{2}1q^{23} + z^{1}07q^{24} + z^{1}10q^{25} + ⋯ ∈ S_{24}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{1}16q^{2} + z^{3}2q^{3} + z^{3}0q^{4} + z^{6}0q^{5} + z^{1}48q^{6} + 9q^{7} + z^{1}45q^{8} + z^{1}25q^{9} + z^{8}q^{10} + z^{8}2q^{11} + z^{6}2q^{12} + 8q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{10} & 0 \\ 0 & u(8)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(7565195705774706336706+ O(13^{2}0)) ∗ a − 71152947862039535138 ∗ 13 + O(13^{2}0)q^{3} + (6002942855152480763908 + O(13^{2}0)) ∗ a + 8614554764139447613746+O(13^{2}0)q^{4}+−(4555777184357246114888+O(13^{2}0))∗a+168957637946900843659∗ 13 + O(13^{2}0)q^{5} + (673799667896749470946 + O(13^{2}0)) ∗ a + 4701734188615662045863 + O(13^{2}0)q^{6} + −1977326743q^{7} +−(4642066967643649850396+O(13^{2}0))∗a−3939192942132924097282+O(13^{2}0)q^{8} + (9189022361663378645991+O(13^{2}0))∗a+7219928727971967399133+O(13^{2}0)q^{9}+−(7518471386094576059917+ O(13^{2}0)) ∗ a − 4601229581419292636975 + O(13^{2}0)q^{10} + (2996240457725132625784 + O(13^{2}0)) ∗ a − 6056121175744615668449+O(13^{2}0)q^{11}+−(4085962185954943203376+O(13^{2}0))∗a−2920011730972300414206+ O(13^{2}0)q^{12} +(40256674268881518116∗13^{2} +O(13^{2}0))∗a−502492283300214938517∗13+O(13^{2}0)q^{13} + −1977326743∗aq^{14}+−(2996323767569134425231+O(13^{2}0))∗a+9233338815653306904816+O(13^{2}0)q^{15}+ (2904450313674481554409+O(13^{2}0))∗a+2568204828914590365258+O(13^{2}0)q^{16}+(6379748732354267454160+ O(13^{2}0)) ∗ a + 3749311238780085124613 + O(13^{2}0)q^{17} + (1573315339353364208773 + O(13^{2}0)) ∗ a − 9433821462743873558976+O(13^{2}0)q^{18}+(5383685237920199502655+O(13^{2}0))∗a−3828343039969167527809+ O(13^{2}0)q^{19} + −(5121644181344550807638 + O(13^{2}0)) ∗ a − 1739477387326797575594 + O(13^{2}0)q^{20} + (7988313984449453347631+O(13^{2}0))∗a−343604683302067428884∗13+O(13^{2}0)q^{21}+(2238817553380487487819+ O(13^{2}0)) ∗ a − 8330633530223349643736 + O(13^{2}0)q^{22} + (6197795593765404631502 + O(13^{2}0)) ∗ a + 3922348867049274559675+O(13^{2}0)q^{23}+(5955304593720815046411+O(13^{2}0))∗a−8362332486101546423326+ O(13^{2}0)q^{24} +−(6116687506885547401791+O(13^{2}0))∗a+1082980164056780717916+O(13^{2}0)q^{25} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −6002942855152480763908 + O(13^{2}0)x + −8614554764139456002354 + O(13^{2}0). The slope of f_{0} is 1.

26. Consider f = q + z^{1}0q^{2} + z^{1}36q^{3} + z^{8}2q^{4} + z^{6}6q^{5} + z^{1}46q^{6} + 11q^{7} + z^{7}9q^{8} + z^{5}7q^{9} + z^{7}6q^{10} + z^{1}56q^{11} + z^{5}0q^{12} + z^{1}08q^{14} + z^{3}4q^{15} + z^{1}33q^{16} + z^{1}17q^{17} + z^{6}7q^{18} + z^{1}00q^{19} + z^{1}48q^{20} + z^{6}6q^{21} + z^{1}66q^{22} + z^{1}05q^{23} + z^{4}7q^{24} + z^{8}6q^{25} + ⋯ ∈ S_{24}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗χ^{11} ∼ ρ_{g}, where g is the weight-10 newform g = q +z^{1}64q^{2} +z^{8}0q^{3} +z^{5}4q^{4} +z^{1}08q^{5} +z^{7}6q^{6} + 9q^{7} + z^{3}7q^{8} + z^{1}13q^{9} + z^{1}04q^{10} + z^{5}8q^{11} + z^{1}34q^{12} + 8q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{10} & 0 \\ 0 & u(8)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(7565195705774706336706+ O(13^{2}0)) ∗ a − 71152947862039535138 ∗ 13 + O(13^{2}0)q^{3} + (6002942855152480763908 + O(13^{2}0)) ∗ a + 8614554764139447613746+O(13^{2}0)q^{4}+−(4555777184357246114888+O(13^{2}0))∗a+168957637946900843659∗ 13 + O(13^{2}0)q^{5} + (673799667896749470946 + O(13^{2}0)) ∗ a + 4701734188615662045863 + O(13^{2}0)q^{6} + −1977326743q^{7} +−(4642066967643649850396+O(13^{2}0))∗a−3939192942132924097282+O(13^{2}0)q^{8} + (9189022361663378645991+O(13^{2}0))∗a+7219928727971967399133+O(13^{2}0)q^{9}+−(7518471386094576059917+ O(13^{2}0)) ∗ a − 4601229581419292636975 + O(13^{2}0)q^{10} + (2996240457725132625784 + O(13^{2}0)) ∗ a − 6056121175744615668449+O(13^{2}0)q^{11}+−(4085962185954943203376+O(13^{2}0))∗a−2920011730972300414206+ O(13^{2}0)q^{12} +(40256674268881518116∗13^{2} +O(13^{2}0))∗a−502492283300214938517∗13+O(13^{2}0)q^{13} + −1977326743∗aq^{14}+−(2996323767569134425231+O(13^{2}0))∗a+9233338815653306904816+O(13^{2}0)q^{15}+ (2904450313674481554409+O(13^{2}0))∗a+2568204828914590365258+O(13^{2}0)q^{16}+(6379748732354267454160+ O(13^{2}0)) ∗ a + 3749311238780085124613 + O(13^{2}0)q^{17} + (1573315339353364208773 + O(13^{2}0)) ∗ a − 9433821462743873558976+O(13^{2}0)q^{18}+(5383685237920199502655+O(13^{2}0))∗a−3828343039969167527809+ O(13^{2}0)q^{19} + −(5121644181344550807638 + O(13^{2}0)) ∗ a − 1739477387326797575594 + O(13^{2}0)q^{20} + (7988313984449453347631+O(13^{2}0))∗a−343604683302067428884∗13+O(13^{2}0)q^{21}+(2238817553380487487819+ O(13^{2}0)) ∗ a − 8330633530223349643736 + O(13^{2}0)q^{22} + (6197795593765404631502 + O(13^{2}0)) ∗ a + 3922348867049274559675+O(13^{2}0)q^{23}+(5955304593720815046411+O(13^{2}0))∗a−8362332486101546423326+ O(13^{2}0)q^{24} +−(6116687506885547401791+O(13^{2}0))∗a+1082980164056780717916+O(13^{2}0)q^{25} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −6002942855152480763908 + O(13^{2}0)x + −8614554764139456002354 + O(13^{2}0). The slope of f_{0} is 1.

27. Consider f = q + 3q^{2} + 7q^{3} + 2q^{4} + 10q^{5} + 8q^{6} + 11q^{7} + 11q^{8} + q^{9} + 4q^{10} + 11q^{11} + q^{12} + 7q^{14} + 5q^{15} + 6q^{16} + 5q^{17} + 3q^{18} + 2q^{19} + 7q^{20} + 12q^{21} + 7q^{22} + q^{23} + 12q^{24} + q^{25} + ⋯ ∈ S_{24}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 12q^{2} + 11q^{3} + 6q^{4} + 3q^{5} + 2q^{6} + 6q^{7} + 2q^{8} + 3q^{9} + 10q^{10} + 5q^{11} + q^{12} + 2q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(7)ω^{1} & 0 \\ 0 & u(2)ω^{10} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+7767572476037191028630+ O(13^{2}0)q^{2}+−378752615724652980908+O(13^{2}0)q^{3}+670962647984731631661+O(13^{2}0)q^{4}+8719590777421576599544+ O(13^{2}0)q^{5} + −5829277750064609415281 + O(13^{2}0)q^{6} + −1977326743q^{7} + −1308879840488868890881 + O(13^{2}0)q^{8}+5147893415563500540732+O(13^{2}0)q^{9}+−590832832670688849770+O(13^{2}0)q^{10}+8118522826670180918264+ O(13^{2}0)q^{11} + −5267959897681971443199 + O(13^{2}0)q^{12} + 322988989244927038503 ∗ 13 + O(13^{2}0)q^{13} + 9311927633515509701914+O(13^{2}0)q^{14}+5666273607423262816702+O(13^{2}0)q^{15}+−4665372819062696716463+ O(13^{2}0)q^{16}+7951821020899648415060+O(13^{2}0)q^{17}+4206769311341298701249+O(13^{2}0)q^{18}+−263267274957698355680+ O(13^{2}0)q^{19}+6808095805776681999850+O(13^{2}0)q^{20}+−3568998443481096873781+O(13^{2}0)q^{21}+2685000942670434414930+ O(13^{2}0)q^{22}+−5801743553020820006094+O(13^{2}0)q^{23}+2877557937676930568410+O(13^{2}0)q^{24}+−5131166949184775754215+ O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

28. Consider f = q+10∗z^{5}+4∗z^{4}+6∗z^{3}+z+6q^{2}+z^{5}+3∗z^{4}+11∗z^{3}+4∗z+4q^{3}+12∗z^{5}+10∗z^{4}+2∗z^{3}+9∗ z+4q^{4}+3∗z^{5}+9∗z^{4}+7∗z^{3}+12∗z+3q^{5}+4∗z^{5}+12∗z^{4}+5∗z^{3}+3∗z+8q^{6}+12q^{7}+4∗z^{5}+12∗z^{4}+5∗z^{3}+3∗ z+2q^{8}+9∗z^{5}+z^{4}+8∗z^{3}+10∗z+1q^{9}+9q^{10}+q^{11}+12∗z^{5}+10∗z^{4}+2∗z^{3}+9∗z+2q^{12}+3∗z^{5}+9∗z^{4}+7∗z^{3}+ 12∗z+7q^{14}+5∗z^{5}+2∗z^{4}+3∗z^{3}+7∗z+2q^{15}+8∗z^{5}+11∗z^{4}+10∗z^{3}+6∗z+5q^{16}+10∗z^{5}+4∗z^{4}+6∗z^{3}+z+ 8q^{17}+11∗z^{5}+7∗z^{4}+4∗z^{3}+5∗z+5q^{18}+7∗z^{5}+8∗z^{4}+12∗z^{3}+2∗z+4q^{19}+6∗z^{5}+5∗z^{4}+z^{3}+11∗z+9q^{20}+ 12∗z^{5}+10∗z^{4}+2∗z^{3}+9∗z+9q^{21}+10∗z^{5}+4∗z^{4}+6∗z^{3}+z+6q^{22}+12∗z^{5}+10∗z^{4}+2∗z^{3}+9∗z+12q^{23}+ 9∗z^{5}+z^{4}+8∗z^{3}+10∗z+12q^{24}+z^{5}+3∗z^{4}+11∗z^{3}+4∗zq^{25}+⋯ ∈ S_{26}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{6} +10x^{3} +11x^{2} +11x+2 = 0. We have ρ_{f} ⊗χ^{11} ∼ ρ_{g}, where g is the weight-12 newform g = q+z^{1}02q^{2}+z^{5}7q^{3}+z^{8}7q^{4}+z^{3}8q^{5}+z^{1}59q^{6}+11q^{7}+z^{1}34q^{8}+z^{1}51q^{9}+10q^{10}+6q^{11}+z^{1}44q^{12}+z^{1}67q^{13}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z)ω^{0} & 0 \\ 0 & u(z^{1}67)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(6868203939942869102713+ O(13^{2}0)) ∗ a − 2417489604866421290966 + O(13^{2}0)q^{3} + −(3262346035873847354280 + O(13^{2}0)) ∗ a + 2984682408751974139172+O(13^{2}0)q^{4}+(2308363067631335429740+O(13^{2}0))∗a−9215197892227872283093+ O(13^{2}0)q^{5} + −(2330094127289992445652 + O(13^{2}0)) ∗ a + 1360516616433532114664 + O(13^{2}0)q^{6} + −13841287201q^{7}+−(1734712368946825942957+O(13^{2}0))∗a+8891486627348655534344+O(13^{2}0)q^{8}+ (6730107098902027451647+O(13^{2}0))∗a−4493322247978729952937+O(13^{2}0)q^{9}+−(17925171230528823641∗ 13^{2}+O(13^{2}0))∗a−4457442854746392585663+O(13^{2}0)q^{10}+−(511172828167001722643∗13+O(13^{2}0))∗ a+4556594525098247182097+O(13^{2}0)q^{11}+(2113493567645176350500+O(13^{2}0))∗a−185354641839921248684∗ 13 + O(13^{2}0)q^{12} + −(251071530962138977552 ∗ 13 + O(13^{2}0)) ∗ a + 263955075499393623594 ∗ 13 + O(13^{2}0)q^{13}+−13841287201∗aq^{14}+(264625519746244852215+O(13^{2}0))∗a+2817582471587238129852+ O(13^{2}0)q^{15} + −(4914752365803884991199 + O(13^{2}0)) ∗ a + 7624896221778696805225 + O(13^{2}0)q^{16} + (5149766065831690297762+O(13^{2}0))∗a+1023341744195450250339+O(13^{2}0)q^{17}+−(3006200237097265312424+ O(13^{2}0)) ∗ a − 9302254109851928384197 + O(13^{2}0)q^{18} + −(7749663649566594311372 + O(13^{2}0)) ∗ a − 6181188096425340914339+O(13^{2}0)q^{19}+−(5655776223315784952908+O(13^{2}0))∗a−8584316048962613793891+ O(13^{2}0)q^{20} + (3644324526577886457328 + O(13^{2}0)) ∗ a + 3653802089456528705638 + O(13^{2}0)q^{21} + −(3288277883664295556254+O(13^{2}0))∗a+400481881367523861676∗13+O(13^{2}0)q^{22}+(1175623423210093952038+ O(13^{2}0)) ∗ a − 7197256146781466801077 + O(13^{2}0)q^{23} + (8753714433951936898875 + O(13^{2}0)) ∗ a + 2600386535654299689301+O(13^{2}0)q^{24}+−(8582215288155023270817+O(13^{2}0))∗a+5852189346620799282622+ O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 3262346035873847354280 + O(13^{2}0)x + −2984682408752007693604 + O(13^{2}0). The slope of f_{0} is 1.

29. Consider f = q+3∗z^{5}+9∗z^{4}+7∗z^{3}+12∗z+3q^{2}+12∗z^{5}+10∗z^{4}+2∗z^{3}+9∗z+5q^{3}+z^{5}+3∗z^{4}+11∗z^{3}+ 4∗z+3q^{4}+10∗z^{5}+4∗z^{4}+6∗z^{3}+z+6q^{5}+9∗z^{5}+z^{4}+8∗z^{3}+10∗z+12q^{6}+12q^{7}+9∗z^{5}+z^{4}+8∗z^{3}+10∗z+ 6q^{8}+4∗z^{5}+12∗z^{4}+5∗z^{3}+3∗z+10q^{9}+9q^{10}+q^{11}+z^{5}+3∗z^{4}+11∗z^{3}+4∗z+1q^{12}+10∗z^{5}+4∗z^{4}+6∗z^{3}+ z+10q^{14}+8∗z^{5}+11∗z^{4}+10∗z^{3}+6∗z+7q^{15}+5∗z^{5}+2∗z^{4}+3∗z^{3}+7∗zq^{16}+3∗z^{5}+9∗z^{4}+7∗z^{3}+12∗z+ 5q^{17}+2∗z^{5}+6∗z^{4}+9∗z^{3}+8∗z+3q^{18}+6∗z^{5}+5∗z^{4}+z^{3}+11∗z+11q^{19}+7∗z^{5}+8∗z^{4}+12∗z^{3}+2∗z+2q^{20}+ z^{5}+3∗z^{4}+11∗z^{3}+4∗z+8q^{21}+3∗z^{5}+9∗z^{4}+7∗z^{3}+12∗z+3q^{22}+z^{5}+3∗z^{4}+11∗z^{3}+4∗z+11q^{23}+4∗z^{5}+ 12∗z^{4}+5∗z^{3}+3∗z+8q^{24}+12∗z^{5}+10∗z^{4}+2∗z^{3}+9∗z+1q^{25}+⋯ ∈ S_{26}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{6} +10x^{3} +11x^{2} +11x+2 = 0. We have ρ_{f} ⊗χ^{11} ∼ ρ_{g}, where g is the weight-12 newform g = q+z^{1}50q^{2}+z^{6}9q^{3}+z^{1}23q^{4}+z^{1}58q^{5}+z^{5}1q^{6}+11q^{7}+z^{6}2q^{8}+z^{1}15q^{9}+10q^{10}+6q^{11}+z^{2}4q^{12}+z^{1}55q^{13}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}3)ω^{0} & 0 \\ 0 & u(z^{1}55)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(6868203939942869102713+ O(13^{2}0)) ∗ a − 2417489604866421290966 + O(13^{2}0)q^{3} + −(3262346035873847354280 + O(13^{2}0)) ∗ a + 2984682408751974139172+O(13^{2}0)q^{4}+(2308363067631335429740+O(13^{2}0))∗a−9215197892227872283093+ O(13^{2}0)q^{5} + −(2330094127289992445652 + O(13^{2}0)) ∗ a + 1360516616433532114664 + O(13^{2}0)q^{6} + −13841287201q^{7}+−(1734712368946825942957+O(13^{2}0))∗a+8891486627348655534344+O(13^{2}0)q^{8}+ (6730107098902027451647+O(13^{2}0))∗a−4493322247978729952937+O(13^{2}0)q^{9}+−(17925171230528823641∗ 13^{2}+O(13^{2}0))∗a−4457442854746392585663+O(13^{2}0)q^{10}+−(511172828167001722643∗13+O(13^{2}0))∗ a+4556594525098247182097+O(13^{2}0)q^{11}+(2113493567645176350500+O(13^{2}0))∗a−185354641839921248684∗ 13 + O(13^{2}0)q^{12} + −(251071530962138977552 ∗ 13 + O(13^{2}0)) ∗ a + 263955075499393623594 ∗ 13 + O(13^{2}0)q^{13}+−13841287201∗aq^{14}+(264625519746244852215+O(13^{2}0))∗a+2817582471587238129852+ O(13^{2}0)q^{15} + −(4914752365803884991199 + O(13^{2}0)) ∗ a + 7624896221778696805225 + O(13^{2}0)q^{16} + (5149766065831690297762+O(13^{2}0))∗a+1023341744195450250339+O(13^{2}0)q^{17}+−(3006200237097265312424+ O(13^{2}0)) ∗ a − 9302254109851928384197 + O(13^{2}0)q^{18} + −(7749663649566594311372 + O(13^{2}0)) ∗ a − 6181188096425340914339+O(13^{2}0)q^{19}+−(5655776223315784952908+O(13^{2}0))∗a−8584316048962613793891+ O(13^{2}0)q^{20} + (3644324526577886457328 + O(13^{2}0)) ∗ a + 3653802089456528705638 + O(13^{2}0)q^{21} + −(3288277883664295556254+O(13^{2}0))∗a+400481881367523861676∗13+O(13^{2}0)q^{22}+(1175623423210093952038+ O(13^{2}0)) ∗ a − 7197256146781466801077 + O(13^{2}0)q^{23} + (8753714433951936898875 + O(13^{2}0)) ∗ a + 2600386535654299689301+O(13^{2}0)q^{24}+−(8582215288155023270817+O(13^{2}0))∗a+5852189346620799282622+ O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 3262346035873847354280 + O(13^{2}0)x + −2984682408752007693604 + O(13^{2}0). The slope of f_{0} is 1.

30. Consider f = q+10∗z^{5}+10∗z^{4}+z^{3}+4∗z^{2}+8∗z+9q^{2}+4∗z^{5}+2∗z^{4}+7∗z^{3}+10∗z^{2}+11∗z+7q^{3}+8∗z^{5}+6∗ z^{4}+10∗z^{3}+9∗z^{2}+9∗z+12q^{4}+4∗z^{5}+8∗z^{4}+8∗z^{3}+3∗z^{2}+11∗z+10q^{5}+11∗z^{5}+8∗z^{4}+11∗z^{3}+4∗z^{2}+z+ 4q^{6}+q^{7}+3∗z^{5}+7∗z^{4}+4∗z^{3}+5∗z+12q^{8}+3∗z^{5}+12∗z^{4}+7∗z^{3}+5∗z^{2}+5∗zq^{9}+z^{4}+11∗z^{3}+z^{2}+8q^{10}+z^{5}+ 6∗z^{4}+7∗z^{3}+8∗z^{2}+6∗zq^{11}+5∗z^{4}+3∗z^{3}+5∗z^{2}+4q^{12}+10∗z^{5}+10∗z^{4}+z^{3}+4∗z^{2}+8∗z+9q^{14}+8∗z^{5}+12∗ z^{4}+11∗z^{3}+2∗z^{2}+9∗z+8q^{15}+7∗z^{5}+7∗z^{4}+2∗z^{3}+8∗z^{2}+3∗z+10q^{16}+12∗z^{5}+9∗z^{4}+2∗z^{3}+7∗z^{2}+7∗z+ 9q^{17}+z^{5}+9∗z^{4}+z^{3}+11∗z^{2}+6∗zq^{18}+7∗z^{5}+8∗z^{4}+9∗z^{2}+3∗z+12q^{19}+11∗z^{5}+8∗z^{4}+11∗z^{3}+4∗z^{2}+z+ 7q^{20}+4∗z^{5}+2∗z^{4}+7∗z^{3}+10∗z^{2}+11∗z+7q^{21}+4∗z^{5}+8∗z^{4}+8∗z^{3}+3∗z^{2}+11∗z+1q^{22}+2∗z^{5}+9∗z^{4}+7∗ z^{3}+12∗z+9q^{23}+12∗z^{5}+4∗z^{4}+12∗z^{3}+2∗z^{2}+7∗z+11q^{24}+3∗z^{5}+10∗z^{4}+11∗z^{3}+3∗z^{2}+5∗z+1q^{25}+⋯ ∈ S_{26}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{6} + 10x^{3} + 11x^{2} + 11x + 2 = 0. We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-12 newform g = q +z^{2}150q^{2} +z^{2}82q^{3} +z^{1}04q^{4} +z^{1}265q^{5} +z^{2}36q^{6} +2q^{7} +z^{2}12q^{8} +

(\begin{matrix}u(z^{2}74)ω^{0} & 0 \\ 0 & u(z^{1}922)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(2396543941948207676652+ O(13^{2}0))∗a^{2} −(5697646522923485946351+O(13^{2}0))∗a+7932205930256236586549+O(13^{2}0)q^{3} +a^{2} − 33554432q^{4} + −(5585263190040512908906 + O(13^{2}0)) ∗ a^{2} − (4144445679497291104805 + O(13^{2}0)) ∗ a + 5953275276069564606937+O(13^{2}0)q^{5}+(5639364833520142838017+O(13^{2}0))∗a^{2}+(5971301614166209571103+ O(13^{2}0)) ∗ a − 5414086546776123920789 + O(13^{2}0)q^{6} + 13841287201q^{7} + −(1696686836016970566810 + O(13^{2}0))∗a^{2}−(66176604252901816598+O(13^{2}0))∗a−243392728349102967707+O(13^{2}0)q^{8}+−(5484880286265336210486+ O(13^{2}0)) ∗ a^{2} + (146747570722518480152 ∗ 13 + O(13^{2}0)) ∗ a − 7465989049058692613242 + O(13^{2}0)q^{9} + −(2087740243676731506363+O(13^{2}0))∗a^{2}−(3744900075353240383850+O(13^{2}0))∗a−8709289006204567433301+ O(13^{2}0)q^{10} + (4250159309773095909701 + O(13^{2}0)) ∗ a^{2} − (2576422466448228256307 + O(13^{2}0)) ∗ a − 2376933568871454086188+O(13^{2}0)q^{11}+(5535632113305989362789+O(13^{2}0))∗a^{2}−(7480098596077999651108+ O(13^{2}0))∗a+6029614681387868554329+O(13^{2}0)q^{12} +−(204047194576133986966∗13+O(13^{2}0))∗a^{2} + (261593762922430052275∗13+O(13^{2}0))∗a−345719795038303484590∗13+O(13^{2}0)q^{13}+13841287201∗ aq^{14}+(1872713315870832960497+O(13^{2}0))∗a^{2}+(5244678611134631842269+O(13^{2}0))∗a+4460374562541302431784+ O(13^{2}0)q^{15} +(219838901257956543315∗13+O(13^{2}0))∗a^{2} +(5760172980400363844815+O(13^{2}0))∗a+ 3819113443395491975804+O(13^{2}0)q^{16}+−(591582155063645448526+O(13^{2}0))∗a^{2}+(639370804094891869907∗ 13+O(13^{2}0))∗a−9181853739433512805043+O(13^{2}0)q^{17}+−(2390139118881523741346+O(13^{2}0))∗a^{2}− (909564889304577260142+O(13^{2}0))∗a−455679467719837546316+O(13^{2}0)q^{18}+(9199660173573127779345+ O(13^{2}0)) ∗ a^{2} + (201322706509641035883 + O(13^{2}0)) ∗ a + 201085981504202429380 ∗ 13 + O(13^{2}0)q^{19} + (3740670699755844806206+O(13^{2}0))∗a^{2}−(5017357742814535889346+O(13^{2}0))∗a+4763375826514411550063+ O(13^{2}0)q^{20} + −(6540673063336709468781 + O(13^{2}0)) ∗ a^{2} − (7903188726361476557558 + O(13^{2}0)) ∗ a + 8535074992981441277822+O(13^{2}0)q^{21}+−(5404445543051774920040+O(13^{2}0))∗a^{2}+(3130475777706186896495+ O(13^{2}0)) ∗ a + 6951818485847528794993 + O(13^{2}0)q^{22} + (1764168578003036537647 + O(13^{2}0)) ∗ a^{2} − (3582846681979085551768+O(13^{2}0))∗a+4598561710724475273590+O(13^{2}0)q^{23}+(8381766193820494934918+ O(13^{2}0)) ∗ a^{2} − (3842793603144208643127 + O(13^{2}0)) ∗ a − 5316866631803382241613 + O(13^{2}0)q^{24} + (6666237493156699614471+O(13^{2}0))∗a^{2}−(2527998001691491363170+O(13^{2}0))∗a−520028730086071562383∗ 13 + O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 1696686836016970566810 + O(13^{2}0)x^{2} + 66176604252834707734 + O(13^{2}0)x + 243392728349102967707 + O(13^{2}0). The slope of f_{0} is 1.

31. Consider f = q+z^{5}+10∗z^{4}+12∗z^{3}+12∗z^{2}+6∗z+1q^{2}+2∗z^{5}+12∗z^{3}+4∗z^{2}+12∗z+12q^{3}+5∗z^{5}+4∗z^{4}+9∗ z^{3}+z^{2}+4∗z+6q^{4}+5∗z^{5}+12∗z^{4}+6∗z^{3}+9∗z^{2}+4∗z+6q^{5}+10∗z^{5}+5∗z^{4}+11∗z^{3}+12∗z^{2}+8∗z+1q^{6}+q^{7}+z^{5}+ 11∗z^{4}+10∗z^{3}+6∗z+1q^{8}+10∗z^{5}+8∗z^{4}+5∗z^{3}+2∗z^{2}+8∗z+1q^{9}+7∗z^{5}+2∗z^{4}+12∗z^{3}+3∗z^{2}+3∗zq^{10}+11∗ z^{4}+4∗z^{3}+11∗z^{2}+6q^{11}+9∗z^{5}+10∗z^{4}+8∗z^{3}+2∗z^{2}+2∗z+3q^{12}+z^{5}+10∗z^{4}+12∗z^{3}+12∗z^{2}+6∗z+1q^{14}+ 8∗z^{5}+3∗z^{4}+3∗z^{3}+6∗z^{2}+9∗z+6q^{15}+2∗z^{5}+7∗z^{4}+11∗z^{3}+11∗z^{2}+12∗z+7q^{16}+z^{5}+6∗z^{4}+7∗z^{3}+8∗z^{2}+ 6∗zq^{17}+8∗z^{5}+4∗z^{4}+z^{3}+7∗z^{2}+9∗z+8q^{18}+9∗z^{5}+9∗z^{4}+10∗z^{3}+z^{2}+2∗z+1q^{19}+10∗z^{5}+5∗z^{4}+11∗z^{3}+ 12∗z^{2}+8∗z+4q^{20}+2∗z^{5}+12∗z^{3}+4∗z^{2}+12∗z+12q^{21}+5∗z^{5}+12∗z^{4}+6∗z^{3}+9∗z^{2}+4∗z+10q^{22}+5∗z^{5}+3∗ z^{4}+11∗z^{3}+4∗z+6q^{23}+5∗z^{5}+9∗z^{4}+12∗z^{3}+6∗z^{2}+4∗z+3q^{24}+9∗z^{5}+4∗z^{4}+7∗z^{3}+9∗z^{2}+2∗z+5q^{25}+⋯ ∈ S_{26}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{6} + 10x^{3} + 11x^{2} + 11x + 2 = 0. We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-12 newform g = q + z^{1}010q^{2} + z^{1}542q^{3} + z^{8}q^{4} + z^{7}73q^{5} + z^{3}56q^{6} + 2q^{7} + z^{6}92q^{8} + z^{1}756q^{9} + z^{1}783q^{10} + z^{4}55q^{11} + z^{1}550q^{12} + z^{2}006q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}90)ω^{0} & 0 \\ 0 & u(z^{2}006)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(2396543941948207676652+ O(13^{2}0))∗a^{2} −(5697646522923485946351+O(13^{2}0))∗a+7932205930256236586549+O(13^{2}0)q^{3} +a^{2} − 33554432q^{4} + −(5585263190040512908906 + O(13^{2}0)) ∗ a^{2} − (4144445679497291104805 + O(13^{2}0)) ∗ a + 5953275276069564606937+O(13^{2}0)q^{5}+(5639364833520142838017+O(13^{2}0))∗a^{2}+(5971301614166209571103+ O(13^{2}0)) ∗ a − 5414086546776123920789 + O(13^{2}0)q^{6} + 13841287201q^{7} + −(1696686836016970566810 + O(13^{2}0))∗a^{2}−(66176604252901816598+O(13^{2}0))∗a−243392728349102967707+O(13^{2}0)q^{8}+−(5484880286265336210486+ O(13^{2}0)) ∗ a^{2} + (146747570722518480152 ∗ 13 + O(13^{2}0)) ∗ a − 7465989049058692613242 + O(13^{2}0)q^{9} + −(2087740243676731506363+O(13^{2}0))∗a^{2}−(3744900075353240383850+O(13^{2}0))∗a−8709289006204567433301+ O(13^{2}0)q^{10} + (4250159309773095909701 + O(13^{2}0)) ∗ a^{2} − (2576422466448228256307 + O(13^{2}0)) ∗ a − 2376933568871454086188+O(13^{2}0)q^{11}+(5535632113305989362789+O(13^{2}0))∗a^{2}−(7480098596077999651108+ O(13^{2}0))∗a+6029614681387868554329+O(13^{2}0)q^{12} +−(204047194576133986966∗13+O(13^{2}0))∗a^{2} + (261593762922430052275∗13+O(13^{2}0))∗a−345719795038303484590∗13+O(13^{2}0)q^{13}+13841287201∗ aq^{14}+(1872713315870832960497+O(13^{2}0))∗a^{2}+(5244678611134631842269+O(13^{2}0))∗a+4460374562541302431784+ O(13^{2}0)q^{15} +(219838901257956543315∗13+O(13^{2}0))∗a^{2} +(5760172980400363844815+O(13^{2}0))∗a+ 3819113443395491975804+O(13^{2}0)q^{16}+−(591582155063645448526+O(13^{2}0))∗a^{2}+(639370804094891869907∗ 13+O(13^{2}0))∗a−9181853739433512805043+O(13^{2}0)q^{17}+−(2390139118881523741346+O(13^{2}0))∗a^{2}− (909564889304577260142+O(13^{2}0))∗a−455679467719837546316+O(13^{2}0)q^{18}+(9199660173573127779345+ O(13^{2}0)) ∗ a^{2} + (201322706509641035883 + O(13^{2}0)) ∗ a + 201085981504202429380 ∗ 13 + O(13^{2}0)q^{19} + (3740670699755844806206+O(13^{2}0))∗a^{2}−(5017357742814535889346+O(13^{2}0))∗a+4763375826514411550063+ O(13^{2}0)q^{20} + −(6540673063336709468781 + O(13^{2}0)) ∗ a^{2} − (7903188726361476557558 + O(13^{2}0)) ∗ a + 8535074992981441277822+O(13^{2}0)q^{21}+−(5404445543051774920040+O(13^{2}0))∗a^{2}+(3130475777706186896495+ O(13^{2}0)) ∗ a + 6951818485847528794993 + O(13^{2}0)q^{22} + (1764168578003036537647 + O(13^{2}0)) ∗ a^{2} − (3582846681979085551768+O(13^{2}0))∗a+4598561710724475273590+O(13^{2}0)q^{23}+(8381766193820494934918+ O(13^{2}0)) ∗ a^{2} − (3842793603144208643127 + O(13^{2}0)) ∗ a − 5316866631803382241613 + O(13^{2}0)q^{24} + (6666237493156699614471+O(13^{2}0))∗a^{2}−(2527998001691491363170+O(13^{2}0))∗a−520028730086071562383∗ 13 + O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 1696686836016970566810 + O(13^{2}0)x^{2} + 66176604252834707734 + O(13^{2}0)x + 243392728349102967707 + O(13^{2}0). The slope of f_{0} is 1.

32. Consider f = q+2∗z^{5}+6∗z^{4}+10∗z^{2}+12∗z+1q^{2}+7∗z^{5}+11∗z^{4}+7∗z^{3}+12∗z^{2}+3∗z+3q^{3}+3∗z^{4}+7∗z^{3}+ 3∗z^{2}+10q^{4}+4∗z^{5}+6∗z^{4}+12∗z^{3}+z^{2}+11∗z+11q^{5}+5∗z^{5}+4∗z^{3}+10∗z^{2}+4∗z+7q^{6}+q^{7}+9∗z^{5}+8∗z^{4}+ 12∗z^{3}+2∗z+6q^{8}+6∗z^{4}+z^{3}+6∗z^{2}+9q^{9}+6∗z^{5}+10∗z^{4}+3∗z^{3}+9∗z^{2}+10∗z+11q^{10}+12∗z^{5}+9∗z^{4}+2∗ z^{3}+7∗z^{2}+7∗z+9q^{11}+4∗z^{5}+11∗z^{4}+2∗z^{3}+6∗z^{2}+11∗z+6q^{12}+2∗z^{5}+6∗z^{4}+10∗z^{2}+12∗z+1q^{14}+10∗ z^{5}+11∗z^{4}+12∗z^{3}+5∗z^{2}+8∗z+11q^{15}+4∗z^{5}+12∗z^{4}+7∗z^{2}+11∗z+7q^{16}+11∗z^{4}+4∗z^{3}+11∗z^{2}+6q^{17}+ 4∗z^{5}+11∗z^{3}+8∗z^{2}+11∗z+5q^{18}+10∗z^{5}+9∗z^{4}+3∗z^{3}+3∗z^{2}+8∗z+12q^{19}+5∗z^{5}+4∗z^{3}+10∗z^{2}+4∗z+ 10q^{20}+7∗z^{5}+11∗z^{4}+7∗z^{3}+12∗z^{2}+3∗z+3q^{21}+4∗z^{5}+6∗z^{4}+12∗z^{3}+z^{2}+11∗z+2q^{22}+6∗z^{5}+z^{4}+8∗z^{3}+ 10∗z+5q^{23}+9∗z^{5}+2∗z^{3}+5∗z^{2}+2∗z+6q^{24}+z^{5}+12∗z^{4}+8∗z^{3}+z^{2}+6∗z+4q^{25}+⋯ ∈ S_{26}(Γ_{0}(7); \overline{F}_{13}). Also, z satisfies the equation x^{6} + 10x^{3} + 11x^{2} + 11x + 2 = 0. We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-12 newform g = q + z^{1}598q^{2} + z^{1}470q^{3} + z^{1}352q^{4} + z^{1}073q^{5} + z^{8}72q^{6} + 2q^{7} + z^{5}60q^{8} + z^{3}04q^{9} + z^{4}75q^{10} + z^{3}5q^{11} + z^{6}26q^{12} + z^{8}30q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}366)ω^{0} & 0 \\ 0 & u(z^{8}30)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(2396543941948207676652+ O(13^{2}0))∗a^{2} −(5697646522923485946351+O(13^{2}0))∗a+7932205930256236586549+O(13^{2}0)q^{3} +a^{2} − 33554432q^{4} + −(5585263190040512908906 + O(13^{2}0)) ∗ a^{2} − (4144445679497291104805 + O(13^{2}0)) ∗ a + 5953275276069564606937+O(13^{2}0)q^{5}+(5639364833520142838017+O(13^{2}0))∗a^{2}+(5971301614166209571103+ O(13^{2}0)) ∗ a − 5414086546776123920789 + O(13^{2}0)q^{6} + 13841287201q^{7} + −(1696686836016970566810 + O(13^{2}0))∗a^{2}−(66176604252901816598+O(13^{2}0))∗a−243392728349102967707+O(13^{2}0)q^{8}+−(5484880286265336210486+ O(13^{2}0)) ∗ a^{2} + (146747570722518480152 ∗ 13 + O(13^{2}0)) ∗ a − 7465989049058692613242 + O(13^{2}0)q^{9} + −(2087740243676731506363+O(13^{2}0))∗a^{2}−(3744900075353240383850+O(13^{2}0))∗a−8709289006204567433301+ O(13^{2}0)q^{10} + (4250159309773095909701 + O(13^{2}0)) ∗ a^{2} − (2576422466448228256307 + O(13^{2}0)) ∗ a − 2376933568871454086188+O(13^{2}0)q^{11}+(5535632113305989362789+O(13^{2}0))∗a^{2}−(7480098596077999651108+ O(13^{2}0))∗a+6029614681387868554329+O(13^{2}0)q^{12} +−(204047194576133986966∗13+O(13^{2}0))∗a^{2} + (261593762922430052275∗13+O(13^{2}0))∗a−345719795038303484590∗13+O(13^{2}0)q^{13}+13841287201∗ aq^{14}+(1872713315870832960497+O(13^{2}0))∗a^{2}+(5244678611134631842269+O(13^{2}0))∗a+4460374562541302431784+ O(13^{2}0)q^{15} +(219838901257956543315∗13+O(13^{2}0))∗a^{2} +(5760172980400363844815+O(13^{2}0))∗a+ 3819113443395491975804+O(13^{2}0)q^{16}+−(591582155063645448526+O(13^{2}0))∗a^{2}+(639370804094891869907∗ 13+O(13^{2}0))∗a−9181853739433512805043+O(13^{2}0)q^{17}+−(2390139118881523741346+O(13^{2}0))∗a^{2}− (909564889304577260142+O(13^{2}0))∗a−455679467719837546316+O(13^{2}0)q^{18}+(9199660173573127779345+ O(13^{2}0)) ∗ a^{2} + (201322706509641035883 + O(13^{2}0)) ∗ a + 201085981504202429380 ∗ 13 + O(13^{2}0)q^{19} + (3740670699755844806206+O(13^{2}0))∗a^{2}−(5017357742814535889346+O(13^{2}0))∗a+4763375826514411550063+ O(13^{2}0)q^{20} + −(6540673063336709468781 + O(13^{2}0)) ∗ a^{2} − (7903188726361476557558 + O(13^{2}0)) ∗ a + 8535074992981441277822+O(13^{2}0)q^{21}+−(5404445543051774920040+O(13^{2}0))∗a^{2}+(3130475777706186896495+ O(13^{2}0)) ∗ a + 6951818485847528794993 + O(13^{2}0)q^{22} + (1764168578003036537647 + O(13^{2}0)) ∗ a^{2} − (3582846681979085551768+O(13^{2}0))∗a+4598561710724475273590+O(13^{2}0)q^{23}+(8381766193820494934918+ O(13^{2}0)) ∗ a^{2} − (3842793603144208643127 + O(13^{2}0)) ∗ a − 5316866631803382241613 + O(13^{2}0)q^{24} + (6666237493156699614471+O(13^{2}0))∗a^{2}−(2527998001691491363170+O(13^{2}0))∗a−520028730086071562383∗ 13 + O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 1696686836016970566810 + O(13^{2}0)x^{2} + 66176604252834707734 + O(13^{2}0)x + 243392728349102967707 + O(13^{2}0). The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

33. Consider f = q + 12q2 + 11q3 + 6q4 + 3q5 + 2q6 + 6q7 + 2q8 + 3q9 + 10q10 + 5q11 + q12 + 2q13 + 7q14 + 7q15 + 2q16 + 2q17 + 10q18 + 7q19 + 5q20 + q21 + 8q22 + 9q23 + 9q24 + q25 + ⋯ ∈ S16(Γ0(7); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

34. Consider f = q + 6q2 + 12q3 + 2q4 + 4q5 + 7q6 + 7q7 + 3q8 + 11q10 + 2q11 + 11q12 + 3q14 + 9q15 + 2q16 + 7q17 + 3q19 + 8q20 + 6q21 + 12q22 + 6q23 + 10q24 + 8q25 + ⋯ ∈ S16(Γ0(7); \overline{F}13). Also, z satisfies the equation x3 + 2x + 11 = 0. I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

35. Consider f = q + z113q2 + z40q3 + z92q4 + z3q5 + z153q6 + 10q7 + z90q8 + z85q9 + z116q10 + z124q11 + z132q12 + z52q13 + z85q14 + z43q15 + z142q16 + z122q17 + z30q18 + z34q19 + z95q20 + z12q21 + z69q22 + z104q23 + z130q24 + z45q25 + ⋯ ∈ S18(Γ0(7); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

36. Consider f = q + z125q2 + z16q3 + z20q4 + z39q5 + z141q6 + 10q7 + z162q8 + z97q9 + z164q10 + z100q11 + z36q12 + z4q13 + z97q14 + z55q15 + z166q16 + z74q17 + z54q18 + z106q19 + z59q20 + z156q21 + z57q22 + z8q23 + z10q24 + z81q25 + ⋯ ∈ S18(Γ0(7); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

37. Consider f = q + 3q2 + 12q3 + 3q4 + 9q5 + 10q6 + 3q7 + 4q8 + 5q9 + q10 + 11q11 + 10q12 + 3q13 + 9q14 + 4q15 + 7q16 + 7q17 + 2q18 + 6q19 + q20 + 10q21 + 7q22 + 4q23 + 9q24 + 11q25 + ⋯ ∈ S18(Γ0(7); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

38. Consider f = q + z47q2 + z57q3 + z146q4 + z10q5 + z104q6 + 8q7 + z64q8 + z31q9 + z57q10 + z107q11 + z35q12 + z88q13 + z89q14 + z67q15 + z19q16 + z158q17 + z78q18 + 12q19 + z156q20 + z99q21 + 7q22 + z18q23 + z121q24 + z11q25 + ⋯ ∈ S20(Γ0(7); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

39. Consider f = q + 7q2 + 10q3 + 12q4 + 7q5 + 5q6 + 5q7 + 7q8 + 6q9 + 10q10 + 9q11 + 3q12 + 9q13 + 9q14 + 5q15 + 8q16 + 2q17 + 3q18 + 3q19 + 6q20 + 11q21 + 11q22 + q23 + 5q24 + 2q25 + ⋯ ∈ S20(Γ0(7); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

40. Consider f = q + 12q2 + 2q3 + 9q4 + 4q5 + 11q6 + 9q7 + 9q8 + 3q9 + 9q10 + 10q11 + 5q12 + q13 + 4q14 + 8q15 + 11q16 + q17 + 10q18 + 5q19 + 10q20 + 5q21 + 3q22 + 8q23 + 5q24 + 11q25 + ⋯ ∈ S22(Γ0(7); \overline{F}13). Since f is ordinary, we do not continue our analysis.

41. Consider f = q + z116q2 + z32q3 + z30q4 + z60q5 + z148q6 + 9q7 + z145q8 + z125q9 + z8q10 + z82q11 + z62q12 + 8q13 + z60q14 + z92q15 + z161q16 + z149q17 + z73q18 + z54q19 + z90q20 + z144q21 + z30q22 + z49q23 + z9q24 + z26q25 + ⋯ ∈ S22(Γ0(7); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

42. Consider f = q + z115q2 + z139q3 + z103q4 + z95q5 + z86q6 + 4q7 + z131q8 + zq9 + 8q10 + z55q11 + z74q12 + z118q13 + z143q14 + z66q15 + z27q16 + z160q17 + z116q18 + z139q19 + z30q20 + z167q21 + z2q22 + z147q23 + z102q24 + z83q25 + ⋯ ∈ S22(Γ0(7); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

43. Consider f = q + z151q2 + z127q3 + z163q4 + z59q5 + z110q6 + 4q7 + z23q8 + z13q9 + 8q10 + z43q11 + z122q12 + z22q13 + z11q14 + z18q15 + z15q16 + z64q17 + z164q18 + z127q19 + z54q20 + z155q21 + z26q22 + z63q23 + z150q24 + z71q25 + ⋯ ∈ S22(Γ0(7); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

44. Consider f = q+7∗z5+5∗z4+6∗z3+6∗z2+3∗z+7q2+5∗z5+4∗z3+10∗z2+4∗z+4q3+11∗z5+z4+12∗z3+10∗z2+z+8q4+z5+5∗z4+9∗z3+7∗z2+6∗ z+9q5+6∗z5+3∗z4+4∗z3+2∗z2+10∗z+11q6+2q7+5∗z5+3∗z4+11∗z3+4∗z+5q8+4∗z5+11∗z4+2∗z3+6∗z2+11∗z+3q9+2∗z5+8∗z4+9∗z3+12∗ z2+12∗zq10+z4+11∗z3+z2+10q11+4∗z5+3∗z4+5∗z3+11∗z2+11∗z+10q12+7∗z5+9∗z4+11∗z3+10∗z2+3∗z+8q13+z5+10∗z4+12∗z3+12∗z2+ 6∗z+1q14 +4∗z5 +8∗z4 +8∗z3 +3∗z2 +11∗z+3q15 +5∗z5 +11∗z4 +8∗z3 +8∗z2 +4∗z+11q16 +10∗z5 +8∗z4 +5∗z3 +2∗z2 +8∗zq17 +12∗z5 +6∗z4 +8∗ z3+4∗z2+7∗z+12q18+8∗z5+8∗z4+6∗z3+11∗z2+9∗z+11q19+7∗z5+10∗z4+9∗z3+11∗z2+3∗z+8q20+10∗z5+8∗z3+7∗z2+8∗z+8q21+2∗z5+10∗ z4+5∗z3+z2+12∗z+4q22+7∗z5+12∗z4+5∗z3+3∗z+11q23+4∗z5+2∗z4+7∗z3+10∗z2+11∗z+5q24+4∗z5+9∗z4+6∗z3+4∗z2+11∗z+8q25+⋯ ∈ S24(Γ0(7); \overline{F}13). Also, z satisfies the equation x6 + 10x3 + 11x2 + 11x + 2 = 0. Since f is ordinary, we do not continue our analysis.

45. Consider f = q+z5 +3∗z4 +5∗z2 +6∗z+7q2 +11∗z5 +8∗z4 +11∗z3 +4∗z2 +z+1q3 +4∗z4 +5∗z3 +4∗z2 +9q4 +6∗z5 +9∗z4 +5∗z3 +8∗z2 +10∗z+ 10q5 +3∗z5 +5∗z3 +6∗z2 +5∗z+12q6 +2q7 +6∗z5 +z4 +8∗z3 +10∗z+4q8 +5∗z4 +3∗z3 +5∗z2 +1q9 +11∗z5 +z4 +12∗z3 +10∗z2 +z+5q10 +7∗z5 + 2∗z4 +12∗z3 +3∗z2 +3∗z+2q11 +9∗z5 +2∗z4 +11∗z3 +7∗z2 +2∗z+7q12 +3∗z5 +11∗z4 +9∗z3 +4∗z2 +5∗z+2q13 +2∗z5 +6∗z4 +10∗z2 +12∗z+ 1q14 +5∗z5 +12∗z4 +6∗z3 +9∗z2 +4∗z+12q15 +10∗z5 +4∗z4 +11∗z2 +8∗z+11q16 +6∗z4 +z3 +6∗z2 +8q17 +6∗z5 +10∗z3 +12∗z2 +10∗z+1q18 + 6∗z5 +8∗z4 +7∗z3 +7∗z2 +10∗z+2q19 +10∗z5 +8∗z3 +7∗z2 +8∗z+7q20 +9∗z5 +3∗z4 +9∗z3 +8∗z2 +2∗z+2q21 +12∗z5 +5∗z4 +10∗z3 +3∗z2 + 7∗z +6q22 +11∗z5 +4∗z4 +6∗z3 +z +7q23 +2∗z5 +12∗z3 +4∗z2 +12∗z +10q24 +12∗z5 +z4 +5∗z3 +12∗z2 +7∗z +9q25 +⋯ ∈ S24(Γ0(7); \overline{F}13). Also, z satisfies the equation x6 + 10x3 + 11x2 + 11x + 2 = 0. Since f is ordinary, we do not continue our analysis.

46. Consider f = q + 5 ∗ z5 + 5 ∗ z4 + 7 ∗ z3 + 2 ∗ z2 + 4 ∗ z + 11q2 + 10 ∗ z5 + 5 ∗ z4 + 11 ∗ z3 + 12 ∗ z2 + 8 ∗ z + 11q3 + 2 ∗ z5 + 8 ∗ z4 + 9 ∗ z3 + 12 ∗ z2 + 12 ∗ z + 3q4 + 6 ∗ z5 + 12 ∗ z4 + 12 ∗ z3 + 11 ∗ z2 + 10 ∗ z + 2q5 + 4 ∗ z5 + 10 ∗ z4 + 4 ∗ z3 + 5 ∗ z2 + 11 ∗ z + 5q6 + 2q7 + 2 ∗ z5 + 9 ∗ z4 + 7 ∗ z3 + 12 ∗ z + 8q8 + 9 ∗ z5 + 10 ∗ z4 + 8 ∗ z3 + 2 ∗ z2 + 2 ∗ zq9 + 4 ∗ z4 + 5 ∗ z3 + 4 ∗ z2 + 6q10 + 6 ∗ z5 + 10 ∗ z4 + 3 ∗ z3 + 9 ∗ z2 + 10 ∗ zq11 + 8 ∗ z4 + 10 ∗ z3 + 8 ∗ z2 + 9q12 + 3 ∗ z5 + 6 ∗ z4 + 6 ∗ z3 + 12 ∗ z2 + 5 ∗ z + 11q13 + 10 ∗ z5 + 10 ∗ z4 + z3 + 4 ∗ z2 + 8 ∗ z + 9q14 + 4 ∗ z5 + 6 ∗ z4 + 12 ∗ z3 + z2 + 11 ∗ z + 4q15 + 11 ∗ z5 + 11 ∗ z4 + 5 ∗ z3 + 7 ∗ z2 + z + 12q16 + 3 ∗ z5 + 12 ∗ z4 + 7 ∗ z3 + 5 ∗ z2 + 5 ∗ z + 12q17 + 8 ∗ z5 + 7 ∗ z4 + 8 ∗ z3 + 10 ∗ z2 + 9 ∗ zq18 + 12 ∗ z5 + 10 ∗ z4 + 8 ∗ z2 + 7 ∗ z + 2q19 + 9 ∗ z5 + 3 ∗ z4 + 9 ∗ z3 + 8 ∗ z2 + 2 ∗ z + 1q20 + 7 ∗ z5 + 10 ∗ z4 + 9 ∗ z3 + 11 ∗ z2 + 3 ∗ z + 9q21 + 12 ∗ z5 + 11 ∗ z4 + 11 ∗ z3 + 9 ∗ z2 + 7 ∗ z + 3q22 + 8 ∗ z5 + 10 ∗ z4 + 2 ∗ z3 + 9 ∗ z + 10q23 + 7 ∗ z5 + 11 ∗ z4 + 7 ∗ z3 + 12 ∗ z2 + 3 ∗ z + 1q24 + 10 ∗ z5 + 3 ∗ z4 + 2 ∗ z3 + 10 ∗ z2 + 8 ∗ z + 12q25 + ⋯ ∈ S24(Γ0(7); \overline{F}13). Also, z satisfies the equation x6 + 10x3 + 11x2 + 11x + 2 = 0. Since f is ordinary, we do not continue our analysis.

47. Consider f = q + z102q2 + z57q3 + z87q4 + z38q5 + z159q6 + 11q7 + z134q8 + z151q9 + 10q10 + 6q11 + z144q12 + z167q13 + z32q14 + z95q15 + z100q16 + z77q17 + z85q18 + z125q19 + z125q20 + z155q21 + z4q22 + z52q23 + z23q24 + z138q25 + ⋯ ∈ S24(Γ0(7); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

48. Consider f = q + z150q2 + z69q3 + z123q4 + z158q5 + z51q6 + 11q7 + z62q8 + z115q9 + 10q10 + 6q11 + z24q12 + z155q13 + z80q14 + z59q15 + z124q16 + z161q17 + z97q18 + z113q19 + z113q20 + z167q21 + z52q22 + z4q23 + z131q24 + z114q25 + ⋯ ∈ S24(Γ0(7); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

49. Consider f = q + 11q2 + 2q4 + 10q5 + 12q7 + 10q9 + 6q10 + 7q11 + 12q13 + 2q14 + 9q16 + 4q17 + 6q18 + 5q19 + 7q20 + 12q22 + 3q23 + 4q25 + ⋯ ∈ S26(Γ0(7); \overline{F}13). Also, z satisfies the equation x6 + 10x3 + 11x2 + 11x + 2 = 0. Since f is ordinary, we do not continue our analysis.

50. Consider f = q+5∗z5+3∗z4+11∗z3+4∗z+4q2+7∗z5+3∗z3+z2+3∗z+7q3+11∗z5+3∗z4+8∗z3+12∗z2+z+10q4+8∗z5+10∗z4+2∗z3+9∗z+10q5+ 3∗z5 +7∗z4 +4∗z3 +5∗z+7q6 +12q7 +11∗z5 +3∗z4 +8∗z3 +12∗z2 +z+10q8 +3∗z5 +7∗z4 +4∗z3 +5∗z+8q9 +7∗z5 +3∗z3 +z2 +3∗z+5q10 +6∗z5 +10∗ z3 +12∗z2 +10∗z+6q11 +9q12 +q13 +8∗z5 +10∗z4 +2∗z3 +9∗z+9q14 +4∗z5 +6∗z4 +12∗z3 +z2 +11∗zq15 +12∗z5 +3∗z4 +z3 +z2 +7∗z+11q16 +3∗ z5+6∗z4+6∗z3+12∗z2+5∗z+1q17+6∗z5+3∗z4+4∗z3+2∗z2+10∗z+1q18+8∗z5+10∗z4+2∗z3+9∗z+8q19+3∗z5+7∗z4+4∗z3+5∗z+5q20+6∗z5+ 10∗z3 +12∗z2 +10∗z+6q21 +10∗z5 +6∗z4 +9∗z3 +8∗z+6q22 +5∗z5 +4∗z4 +9∗z3 +z2 +4∗zq23 +9q24 +z5 +10∗z4 +12∗z3 +12∗z2 +6∗zq25 +⋯ ∈ S26(Γ0(7); \overline{F}13). Also, z satisfies the equation x6 + 10x3 + 11x2 + 11x + 2 = 0. Since f is ordinary, we do not continue our analysis.

51. Consider f = q+6∗z5+z4+8∗z3+10∗z+3q2+5∗z5+6∗z4+5∗z3+3∗z2+4∗z+8q3+z5+8∗z4+3∗z3+10∗z2+6∗z+8q4+7∗z5+12∗z4+5∗z3+3∗z+ 11q5 +z5 +11∗z4 +10∗z3 +6∗z+9q6 +12q7 +z5 +8∗z4 +3∗z3 +10∗z2 +6∗z+8q8 +z5 +11∗z4 +10∗z3 +6∗z+10q9 +5∗z5 +6∗z4 +5∗z3 +3∗z2 +4∗z+ 6q10+8∗z5+7∗z4+8∗z3+10∗z2+9∗z+5q11+9q12+q13+7∗z5+12∗z4+5∗z3+3∗z+10q14+4∗z5+8∗z4+8∗z3+3∗z2+11∗z+12q15+11∗z5+7∗z4+ 3∗z2+z+11q16+7∗z5+9∗z4+11∗z3+10∗z2+3∗z+11q17+3∗z5+5∗z3+6∗z2+5∗z+2q18+7∗z5+12∗z4+5∗z3+3∗z+9q19+z5+11∗z4+10∗z3+6∗z+ 7q20+8∗z5+7∗z4+8∗z3+10∗z2+9∗z+5q21+12∗z5+2∗z4+3∗z3+7∗z+4q22+3∗z4+7∗z3+3∗z2+4q23+9q24+2∗z5+6∗z4+10∗z2+12∗zq25+⋯ ∈ S26(Γ0(7); \overline{F}13). Also, z satisfies the equation x6 + 10x3 + 11x2 + 11x + 2 = 0. Since f is ordinary, we do not continue our analysis.

52. Consider f = q + 2 ∗ z5 + 9 ∗ z4 + 7 ∗ z3 + 12 ∗ z + 7q2 + z5 + 7 ∗ z4 + 5 ∗ z3 + 9 ∗ z2 + 6 ∗ z + 9q3 + z5 + 2 ∗ z4 + 2 ∗ z3 + 4 ∗ z2 + 6 ∗ z + 11q4 + 11 ∗ z5 + 4 ∗ z4 + 6 ∗ z3 + z + 7q5 + 9 ∗ z5 + 8 ∗ z4 + 12 ∗ z3 + 2 ∗ z + 1q6 + 12q7 + z5 + 2 ∗ z4 + 2 ∗ z3 + 4 ∗ z2 + 6 ∗ z + 11q8 + 9 ∗ z5 + 8 ∗ z4 + 12 ∗ z3 + 2 ∗ z + 2q9 + z5 + 7 ∗ z4 + 5 ∗ z3 + 9 ∗ z2 + 6 ∗ z + 7q10 + 12 ∗ z5 + 6 ∗ z4 + 8 ∗ z3 + 4 ∗ z2 + 7 ∗ z + 4q11 + 9q12 + q13 + 11 ∗ z5 + 4 ∗ z4 + 6 ∗ z3 + z + 6q14 + 5 ∗ z5 + 12 ∗ z4 + 6 ∗ z3 + 9 ∗ z2 + 4 ∗ z + 8q15 + 3 ∗ z5 + 3 ∗ z4 + 12 ∗ z3 + 9 ∗ z2 + 5 ∗ z + 3q16 + 3 ∗ z5 + 11 ∗ z4 + 9 ∗ z3 + 4 ∗ z2 + 5 ∗ z + 5q17 + 4 ∗ z5 + 10 ∗ z4 + 4 ∗ z3 + 5 ∗ z2 + 11 ∗ z + 8q18 + 11 ∗ z5 + 4 ∗ z4 + 6 ∗ z3 + z + 5q19 + 9 ∗ z5 + 8 ∗ z4 + 12 ∗ z3 + 2 ∗ z + 12q20 + 12 ∗ z5 + 6 ∗ z4 + 8 ∗ z3 + 4 ∗ z2 + 7 ∗ z + 4q21 + 4 ∗ z5 + 5 ∗ z4 + z3 + 11 ∗ z + 12q22 + 8 ∗ z5 + 6 ∗ z4 + 10 ∗ z3 + 9 ∗ z2 + 9 ∗ z + 6q23 + 9q24 + 10 ∗ z5 + 10 ∗ z4 + z3 + 4 ∗ z2 + 8 ∗ z + 8q25 + ⋯ ∈ S26(Γ0(7); \overline{F}13). Also, z satisfies the equation x6 + 10x3 + 11x2 + 11x + 2 = 0. Since f is ordinary, we do not continue our analysis.

53. Consider f = q + 11q3 + 11q4 + 10q5 + q7 + q9 + 4q12 + q13 + 6q15 + 4q16 + 7q17 + 6q19 + 6q20 + 11q21 + 3q23 + 4q25 + ⋯ ∈ S26(Γ0(7); \overline{F}13). Also, z satisfies the equation x6 + 10x3 + 11x2 + 11x + 2 = 0. Since f is ordinary, we do not continue our analysis.

54. Consider f = q + 3q2 + 4q3 + 7q4 + 6q5 + 12q6 + q7 + 2q8 + 5q10 + 12q11 + 2q12 + 3q14 + 11q15 + 5q16 + 5q17 + 7q19 + 3q20 + 4q21 + 10q22 + 11q23 + 8q24 + 5q25 + ⋯ ∈ S26(Γ0(7); \overline{F}13). Also, z satisfies the equation x6 + 10x3 + 11x2 + 11x + 2 = 0. I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

55. Consider f = q +5∗z5 +2∗z4 +3∗z3 +7∗z +4q2 +8∗z5 +11∗z4 +10∗z3 +6∗z +9q3 +5∗z5 +2∗z4 +3∗z3 +7∗z +7q5 +11q6 +q7 +3∗z5 +9∗z4 +7∗ z3 + 12 ∗ z + 5q8 + 12q9 + 2 ∗ z5 + 6 ∗ z4 + 9 ∗ z3 + 8 ∗ z + 1q10 + 11 ∗ z5 + 7 ∗ z4 + 4 ∗ z3 + 5 ∗ z + 1q11 + 12q13 + 5 ∗ z5 + 2 ∗ z4 + 3 ∗ z3 + 7 ∗ z + 4q14 + 11 ∗ z5 + 7 ∗ z4 + 4 ∗ z3 + 5 ∗ z + 12q15 + 9q16 + 8 ∗ z5 + 11 ∗ z4 + 10 ∗ z3 + 6 ∗ z + 9q17 + 8 ∗ z5 + 11 ∗ z4 + 10 ∗ z3 + 6 ∗ z + 9q18 + 2 ∗ z5 + 6 ∗ z4 + 9 ∗ z3 + 8 ∗ z + 9q19 + 8 ∗ z5 + 11 ∗ z4 + 10 ∗ z3 + 6 ∗ z + 9q21 + 7q22 + 10 ∗ z5 + 4 ∗ z4 + 6 ∗ z3 + z + 5q23 + 4q24 + 4 ∗ z5 + 12 ∗ z4 + 5 ∗ z3 + 3 ∗ z + 4q25 + ⋯ ∈ S26(Γ0(7); \overline{F}13). Also, z satisfies the equation x6 + 10x3 + 11x2 + 11x + 2 = 0. Since f is ordinary, we do not continue our analysis.

56. Consider f = q + 8 ∗ z5 + 11 ∗ z4 + 10 ∗ z3 + 6 ∗ z + 9q2 + 5 ∗ z5 + 2 ∗ z4 + 3 ∗ z3 + 7 ∗ z + 4q3 + 8 ∗ z5 + 11 ∗ z4 + 10 ∗ z3 + 6 ∗ z + 12q5 + 11q6 + q7 + 10 ∗ z5 + 4 ∗ z4 +6∗z3 +z +8q8 +12q9 +11∗z5 +7∗z4 +4∗z3 +5∗z +3q10 +2∗z5 +6∗z4 +9∗z3 +8∗z +12q11 +12q13 +8∗z5 +11∗z4 +10∗z3 +6∗z +9q14 +2∗ z5 +6∗z4 +9∗z3 +8∗z +10q15 +9q16 +5∗z5 +2∗z4 +3∗z3 +7∗z +4q17 +5∗z5 +2∗z4 +3∗z3 +7∗z +4q18 +11∗z5 +7∗z4 +4∗z3 +5∗z +11q19 + 5 ∗ z5 + 2 ∗ z4 + 3 ∗ z3 + 7 ∗ z + 4q21 + 7q22 + 3 ∗ z5 + 9 ∗ z4 + 7 ∗ z3 + 12 ∗ z + 2q23 + 4q24 + 9 ∗ z5 + z4 + 8 ∗ z3 + 10 ∗ z + 8q25 + ⋯ ∈ S26(Γ0(7); \overline{F}13). Also, z satisfies the equation x6 + 10x3 + 11x2 + 11x + 2 = 0. Since f is ordinary, we do not continue our analysis.

13.7 N = 8, ℓ = 13 (30 forms)

1. Consider f = q + q^{3} + 6q^{5} + 5q^{7} + 6q^{11} + 6q^{15} + 2q^{17} + 6q^{19} + 5q^{21} + q^{23} + 2q^{25} + ⋯ ∈ S_{16}(Γ_{0}(8); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{10} ∼ ρ_{g}, where g is the weight-12 newform g = q + 3q^{3} + 7q^{5} + 7q^{7} + 8q^{11} + 12q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(12)ω^{1} & 0 \\ 0 & u(12)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−3444q^{3}+313358q^{5}+ −2324616q^{7}+−191367∗13q^{9}+−55249084q^{11}+−8481506∗13q^{13}+−1079204952q^{15}+−2601428750q^{17}+ 1952124284q^{19} + 8005977504q^{21} + −25430340376q^{23} + 67675658039q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q+z^{1}06q^{3}+z^{1}07q^{5}+z^{7}5q^{7}+z^{6}3q^{9}+z^{1}9q^{11}+z^{4}5q^{15}+z^{1}61q^{17}+z^{3}1q^{19}+z^{1}3q^{21}+z^{1}06q^{23}+ z^{1}0q^{25} + ⋯ ∈ S_{16}(Γ_{0}(8); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{10} ∼ ρ_{g}, where g is the weight-12 newform g = q + z^{1}62q^{3} + z^{2}3q^{5} + z^{1}03q^{7} + z^{7}q^{9} + z^{1}59q^{11} + z^{1}52q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}6)ω^{1} & 0 \\ 0 & u(z^{1}52)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{3} + 36 ∗ a + 3166q^{5} + 90 ∗ a + 246336q^{7} + −4072 ∗ a + 5262597q^{9} + 17307 ∗ a + 45578368q^{11} + 180 ∗ 13 ∗ a + 19589806 ∗ 13q^{13} + −143426 ∗ a + 706014144q^{15} + 175896 ∗ a + 1927882706q^{17} + 285813 ∗ a + 344580992q^{19} + −120144 ∗ a + 1765035360q^{21} + −407250 ∗ a − 1632089408 ∗ 13q^{23} + −5049360 ∗ a − 5091045385q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 4072x + −19611504. The slope of f_{0} is 1.

3. Consider f = q + z^{3}4q^{3} + z^{4}7q^{5} + z^{1}35q^{7} + z^{1}47q^{9} + z^{7}9q^{11} + z^{8}1q^{15} + z^{7}7q^{17} + z^{6}7q^{19} + zq^{21} + z^{3}4q^{23} + z^{1}30q^{25} + ⋯ ∈ S_{16}(Γ_{0}(8); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{10} ∼ ρ_{g}, where g is the weight-12 newform g = q + z^{9}0q^{3} + z^{1}31q^{5} + z^{1}63q^{7} + z^{9}1q^{9} + z^{5}1q^{11} + z^{1}28q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{4}0)ω^{1} & 0 \\ 0 & u(z^{1}28)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{3} + 36 ∗ a + 3166q^{5} + 90 ∗ a + 246336q^{7} + −4072 ∗ a + 5262597q^{9} + 17307 ∗ a + 45578368q^{11} + 180 ∗ 13 ∗ a + 19589806 ∗ 13q^{13} + −143426 ∗ a + 706014144q^{15} + 175896 ∗ a + 1927882706q^{17} + 285813 ∗ a + 344580992q^{19} + −120144 ∗ a + 1765035360q^{21} + −407250 ∗ a − 1632089408 ∗ 13q^{23} + −5049360 ∗ a − 5091045385q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 4072x + −19611504. The slope of f_{0} is 1.

4. Consider f = q + 9q^{3} + 2q^{5} + 10q^{7} + 7q^{9} + 9q^{11} + 5q^{15} + 9q^{17} + 8q^{19} + 12q^{21} + 9q^{23} + 12q^{25} + ⋯ ∈ S_{18}(Γ_{0}(8); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 3q^{3} + 2q^{5} + 4q^{7} + 8q^{9} + 3q^{11} + 11q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{1} & 0 \\ 0 & u(11)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+2245903632940260511734+ O(13^{2}0)q^{3}+1719471552250034634183+O(13^{2}0)q^{5}+−8082044870597954126120+O(13^{2}0)q^{7}+9460648002402340687262+ O(13^{2}0)q^{9} + −4054696283996507404233 + O(13^{2}0)q^{11} + 8261047290370531545962 ∗ 13 + O(13^{2}1)q^{13} + −6206176857145381815837+O(13^{2}0)q^{15}+3362509203017680087277+O(13^{2}0)q^{17}+9329809162409085874064+ O(13^{2}0)q^{19} + −29853169087858789066070 + O(13^{2}1)q^{21} + 7487718150108861694896 + O(13^{2}0)q^{23} + −4985359700356360576434 + O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

5. Consider f = q + q^{3} + 3q^{5} + 12q^{7} + 5q^{9} + 10q^{11} + 3q^{15} + 5q^{17} + 4q^{19} + 12q^{21} + 12q^{23} + 4q^{25} + ⋯ ∈ S_{18}(Γ_{0}(8); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-4 newform g = q + 9q^{3} + 11q^{5} + 11q^{7} + 2q^{9} + 8q^{11} + 9q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(3)ω^{4} & 0 \\ 0 & u(9)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−2245903632940260512686+ O(13^{2}0)q^{3}+−1719471552250034687803+O(13^{2}0)q^{5}+8082044870597953792952+O(13^{2}0)q^{7}+−9460648002402317233924+ O(13^{2}0)q^{9} + 4054696283996938378913 + O(13^{2}0)q^{11} + −8261047290370326351966 ∗ 13 + O(13^{2}1)q^{13} + 6206176857162284169677+O(13^{2}0)q^{15}+−3362509202957006584009+O(13^{2}0)q^{17}+−9329809162230455914024+ O(13^{2}0)q^{19} + 29853169088134039994774 + O(13^{2}1)q^{21} + −7487718149580105100288 + O(13^{2}0)q^{23} + 4985359699844529065584 + O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

6. Consider f = q + 2q^{3} + 6q^{5} + 8q^{7} + 8q^{9} + 2q^{11} + 12q^{15} + 11q^{17} + 8q^{19} + 3q^{21} + 4q^{23} + 5q^{25} + ⋯ ∈ S_{18}(Γ_{0}(8); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 5q^{3} + 6q^{5} + 11q^{7} + 11q^{9} + 5q^{11} + 4q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(10)ω^{1} & 0 \\ 0 & u(4)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+8559363604084486866246+ O(13^{2}0)q^{3}+7117151943559675696269+O(13^{2}0)q^{5}+−9379013946529022271884+O(13^{2}0)q^{7}+−4772970105282127900312+ O(13^{2}0)q^{9} + 1852608480101599107955 + O(13^{2}0)q^{11} + 494860066797313705107 ∗ 13 + O(13^{2}0)q^{13} + 1704811146793748984977+O(13^{2}0)q^{15}+9842571865383051852315+O(13^{2}1)q^{17}+2441421667862002197913+ O(13^{2}0)q^{19}+1467041692120228787189+O(13^{2}0)q^{21}+3690495298374311187310+O(13^{2}0)q^{23}+1656073521210913700186+ O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

7. Consider f = q+7q^{3}+4q^{5}+q^{7}+7q^{9}+2q^{11}+2q^{15}+4q^{17}+10q^{19}+7q^{21}+12q^{23}+8q^{25}+⋯ ∈ S_{20}(Γ_{0}(8); \overline{F}_{13}). We have ρ_{f} ⊗χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q+7q^{3}+9q^{5}+12q^{7}+7q^{9}+11q^{11}+12q^{13}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(12)ω^{1} & 0 \\ 0 & u(12)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−5831063428316393944098+ O(13^{2}0)q^{3}+9502261772470940835145+O(13^{2}0)q^{5}+−4767199279005969177943+O(13^{2}0)q^{7}+−1867356911978644174991+ O(13^{2}0)q^{9} + 6085125347526753015646 + O(13^{2}0)q^{11} + −625634242411791569513 ∗ 13 + O(13^{2}0)q^{13} + −6385921794298429512088+O(13^{2}0)q^{15}+−5534797471758433514866+O(13^{2}0)q^{17}+−102269410145085986968818+ O(13^{2}1)q^{19}+3536828891284719542961+O(13^{2}0)q^{21}+5417653403276962758234+O(13^{2}0)q^{23}+−3927931006182580649621+ O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

8. Consider f = q+8q^{3}+7q^{5}+q^{7}+9q^{9}+12q^{11}+4q^{15}+5q^{17}+12q^{19}+8q^{21}+7q^{23}+2q^{25}+⋯ ∈ S_{20}(Γ_{0}(8); \overline{F}_{13}). We have ρ_{f} ⊗χ^{11} ∼ ρ_{g}, where g is the weight-6 newform g = q+7q^{3} +4q^{5} +2q^{7} +q^{9} +7q^{11} +10q^{13} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{6} & 0 \\ 0 & u(10)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−616092432286464997999+ O(13^{2}0)q^{3}+−7214397011083619884702+O(13^{2}0)q^{5}+−619630028155594491320+O(13^{2}0)q^{7}+−5989960287303822025863+ O(13^{2}0)q^{9} + −3179618541464242932562 + O(13^{2}0)q^{11} + 511781107083617554982 ∗ 13 + O(13^{2}0)q^{13} + 2143385744739514886992+O(13^{2}0)q^{15}+487045365514565779706+O(13^{2}0)q^{17}+3153274539537441335951+ O(13^{2}0)q^{19}+−4687601148511351468260+O(13^{2}0)q^{21}+6323873427840747189522+O(13^{2}0)q^{23}+−4351502157599543881738+ O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

9. Consider f = q + 5q^{3} + 12q^{5} + 2q^{7} + 9q^{9} + 5q^{11} + 8q^{15} + 6q^{17} + 3q^{19} + 10q^{21} + 3q^{23} + 6q^{25} + ⋯ ∈ S_{20}(Γ_{0}(8); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + 5q^{3} + q^{5} + 11q^{7} + 9q^{9} + 8q^{11} + 8q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{1} & 0 \\ 0 & u(8)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+2336769506467221422472+ O(13^{2}0)q^{3}+4029067750337994777243+O(13^{2}0)q^{5}+−7914228113915309554587+O(13^{2}0)q^{7}+7243792810836658501395+ O(13^{2}0)q^{9} + 156407070777162307133 + O(13^{2}0)q^{11} + −143246676393506922896 ∗ 13 + O(13^{2}0)q^{13} + 2385564341536121602943+O(13^{2}0)q^{15}+−7012349246385728905017+O(13^{2}0)q^{17}+−19344497996441093473+ O(13^{2}0)q^{19}+2516755447163894110028+O(13^{2}0)q^{21}+−3826898132750684056296+O(13^{2}0)q^{23}+−2561691060160367174761+ O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

10. Consider f = q+8q^{3}+6q^{5}+6q^{7}+11q^{9}+4q^{11}+9q^{15}+3q^{17}+5q^{19}+9q^{21}+3q^{23}+5q^{25}+⋯ ∈ S_{22}(Γ_{0}(8); \overline{F}_{13}). We have ρ_{f} ⊗χ^{11} ∼ ρ_{g}, where g is the weight-8 newform g = q+7q^{3}+9q^{5}+12q^{7}+7q^{9}+11q^{11}+12q^{13}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(12)ω^{8} & 0 \\ 0 & u(12)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−4636211866528331146244+ O(13^{2}0)q^{3}+−2300581543837374269375+O(13^{2}0)q^{5}+−1026187644609322703918+O(13^{2}0)q^{7}+−2578778119152176943835+ O(13^{2}0)q^{9} + 8446703501810092822424 + O(13^{2}0)q^{11} + −134633869393471375829 ∗ 13 + O(13^{2}0)q^{13} + −5689152577454045709624+O(13^{2}0)q^{15}+5473649087011340178024+O(13^{2}0)q^{17}+−8527521491414919015134+ O(13^{2}0)q^{19}+492673228003768606427+O(13^{2}0)q^{21}+118476115331385631906557+O(13^{2}1)q^{23}+4749741135878543536269+ O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

11. Consider f = q + 11q^{3} + 4q^{5} + 6q^{7} + 3q^{9} + 11q^{11} + 5q^{15} + 7q^{17} + 6q^{19} + q^{21} + 5q^{23} + 11q^{25} + ⋯ ∈ S_{22}(Γ_{0}(8); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 7q^{3} + 4q^{5} + 2q^{7} + q^{9} + 7q^{11} + 10q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{1} & 0 \\ 0 & u(10)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+9097162316174554235495+ O(13^{2}0)q^{3}+5542337889900579074376+O(13^{2}0)q^{5}+−3952991202203941041294+O(13^{2}0)q^{7}+−1412816896715330889533+ O(13^{2}0)q^{9} + 7482709925500831911966 + O(13^{2}0)q^{11} + 216460064871572465054 ∗ 13 + O(13^{2}0)q^{13} + 7918334463357751357085+O(13^{2}0)q^{15}+−7587522546482530910381+O(13^{2}0)q^{17}+1295658595378041107583+ O(13^{2}0)q^{19}+6941580840982073318631+O(13^{2}0)q^{21}+1724170230154533886912+O(13^{2}0)q^{23}+−4895555806879575439929+ O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

12. Consider f = q + 2q^{3} + 5q^{5} + 12q^{7} + 3q^{9} + 10q^{11} + 10q^{15} + 11q^{17} + 8q^{19} + 11q^{21} + 4q^{23} + 7q^{25} + ⋯ ∈ S_{22}(Γ_{0}(8); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-8 newform g = q + 5q^{3} + q^{5} + 11q^{7} + 9q^{9} + 8q^{11} + 8q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{8} & 0 \\ 0 & u(8)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−2270354936150882765598+ O(13^{2}0)q^{3}+−4824044652769069118269+O(13^{2}0)q^{5}+3173760795036250853901+O(13^{2}0)q^{7}+5340587487154742498913+ O(13^{2}0)q^{9} + −4394853077742973259131 + O(13^{2}0)q^{11} + 240779329686646359727 ∗ 13 + O(13^{2}0)q^{13} + 2687591415475876044925+O(13^{2}0)q^{15}+−859809395125916980747+O(13^{2}0)q^{17}+1935552034808463945193+ O(13^{2}0)q^{19}+−3422344357071451137705+O(13^{2}0)q^{21}+8851886561920646296290+O(13^{2}0)q^{23}+9188243423017146821717+ O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

13. Consider f = q+9q^{3}+10q^{5}+2q^{7}+7q^{9}+7q^{11}+12q^{15}+4q^{17}+q^{19}+5q^{21}+4q^{23}+q^{25}+⋯ ∈ S_{24}(Γ_{0}(8); \overline{F}_{13}). We have ρ_{f} ⊗χ^{11} ∼ ρ_{g}, where g is the weight-10 newform g = q+3q^{3}+2q^{5}+4q^{7}+8q^{9}+3q^{11}+11q^{13}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{10} & 0 \\ 0 & u(11)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−8943128514685727487376+ O(13^{2}0)q^{3}+−3400203640338848840513+O(13^{2}0)q^{5}+2242913568231219225058+O(13^{2}0)q^{7}+1244198439300147606763+ O(13^{2}0)q^{9} + −1432054194410729994071 + O(13^{2}0)q^{11} + 69456744368812600012 ∗ 13 + O(13^{2}0)q^{13} + −8865665571063982754282+O(13^{2}0)q^{15}+3310778246486322864958+O(13^{2}0)q^{17}+4894230787444778454754+ O(13^{2}0)q^{19}+−1256386945348052005899+O(13^{2}0)q^{21}+3235334554957479910810+O(13^{2}0)q^{23}+839750745721484342422+ O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

14. Consider f = q+q^{3}+2q^{5}+5q^{7}+5q^{9}+2q^{11}+2q^{15}+8q^{17}+7q^{19}+5q^{21}+q^{23}+9q^{25}+⋯ ∈ S_{24}(Γ_{0}(8); \overline{F}_{13}). We have ρ_{f} ⊗χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q+9q^{3}+11q^{5}+11q^{7}+2q^{9}+8q^{11}+9q^{13}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(3)ω^{1} & 0 \\ 0 & u(9)ω^{10} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+7190452087146604866539+ O(13^{2}0)q^{3}+−1098743093541750956401+O(13^{2}0)q^{5}+−9230220643127349013468+O(13^{2}0)q^{7}+9162228328104142308906+ O(13^{2}0)q^{9} + −6588013562644907825404 + O(13^{2}0)q^{11} + 596288627455527690722 ∗ 13 + O(13^{2}0)q^{13} + 7056999260165574926234+O(13^{2}0)q^{15}+2798655497170585700435+O(13^{2}0)q^{17}+370645593976858035232+ O(13^{2}0)q^{19}+−757512352861964365622+O(13^{2}0)q^{21}+8440751675956143064457+O(13^{2}0)q^{23}+−7642227586038850039733+ O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

15. Consider f = q+2q^{3}+4q^{5}+12q^{7}+8q^{9}+3q^{11}+8q^{15}+2q^{17}+q^{19}+11q^{21}+9q^{23}+8q^{25}+⋯ ∈ S_{24}(Γ_{0}(8); \overline{F}_{13}). Also, z satisfies the equation x^{2}+12x+2 = 0. We have ρ_{f} ⊗χ^{11} ∼ ρ_{g}, where g is the weight-10 newform g = q + 5q^{3} + 6q^{5} + 11q^{7} + 11q^{9} + 5q^{11} + 4q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(10)ω^{10} & 0 \\ 0 & u(4)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+7971170922725648862031+ O(13^{2}0)q^{3}+9080516810636569839038+O(13^{2}0)q^{5}+−1577615153550249507864+O(13^{2}0)q^{7}+−7555864715311336212950+ O(13^{2}0)q^{9} + −3116626392870451788909 + O(13^{2}0)q^{11} + 255772410483363862975 ∗ 13 + O(13^{2}0)q^{13} + −8298539923585305367485+O(13^{2}0)q^{15}+−3997571179237645198577+O(13^{2}0)q^{17}+−7481327353668926941351+ O(13^{2}0)q^{19}+−1800175952467808424147+O(13^{2}0)q^{21}+1111050661023679158857+O(13^{2}0)q^{23}+−4250001294670940529752+ O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

16. Consider f = q+9q^{3}+9q^{5}+10q^{7}+10q^{11}+3q^{15}+7q^{17}+q^{19}+12q^{21}+4q^{23}+11q^{25}+⋯ ∈ S_{26}(Γ_{0}(8); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-12 newform g = q + 3q^{3} + 7q^{5} + 7q^{7} + 8q^{11} + 12q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(12)ω^{0} & 0 \\ 0 & u(12)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−6413855801715347783711+ O(13^{2}0)q^{3}+2563418200817005887000+O(13^{2}0)q^{5}+−3170842747244878469728+O(13^{2}0)q^{7}+187638769849035490470∗ 13+O(13^{2}0)q^{9} +2312456228045051691301+O(13^{2}0)q^{11} +540053367331645865686∗13+O(13^{2}0)q^{13} + −6656790093351995916433+O(13^{2}0)q^{15}+3536947924616541299950+O(13^{2}0)q^{17}+−5207592121693051449618+ O(13^{2}0)q^{19}+−6777288185045085680012+O(13^{2}0)q^{21}+8316273303964871100132+O(13^{2}0)q^{23}+3613584404394170032032+ O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

17. Consider f = q+z^{1}46q^{3}+z^{8}9q^{5}+z^{1}49q^{7}+z^{3}5q^{9}+z^{1}49q^{11}+z^{6}7q^{15}+z^{4}9q^{17}+z^{1}65q^{19}+z^{1}27q^{21}+z^{6}2q^{23}+ z^{4}6q^{25} + ⋯ ∈ S_{26}(Γ_{0}(8); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-12 newform g = q + z^{9}0q^{3} + z^{1}31q^{5} + z^{1}63q^{7} + z^{9}1q^{9} + z^{5}1q^{11} + z^{1}28q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{4}0)ω^{0} & 0 \\ 0 & u(z^{1}28)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{3}+(97025672287777198946+ O(13^{2}0)) ∗ a + 7320296195926660333456 + O(13^{2}0)q^{5} + (5045334958964414347370 + O(13^{2}1)) ∗ a − 113473655958714448024370+O(13^{2}1)q^{7}+−(7531798457876026564855+O(13^{2}0))∗a+8378160045960096318101+ O(13^{2}0)q^{9} + (3774341846197411853228 + O(13^{2}0)) ∗ a − 6528698762767082777930 + O(13^{2}0)q^{11} + −(655566328156921797854∗13+O(13^{2}0))∗a+231891047438583675960∗13+O(13^{2}0)q^{13}+(9078062461937228774025+ O(13^{2}0)) ∗ a + 1991910306348495353472 + O(13^{2}0)q^{15} + −(48635955400595054527336 + O(13^{2}1)) ∗ a + 121355734057992403468979+O(13^{2}1)q^{17}+(459822495716589452334+O(13^{2}0))∗a+1773691681862871878337+ O(13^{2}0)q^{19} + −(6038149883140032244908 + O(13^{2}0)) ∗ a − 7583088658246127310390 + O(13^{2}0)q^{21} + −(7222527488210759409634+O(13^{2}0))∗a+682997330155035350760∗13+O(13^{2}0)q^{23}+−(32428219359680323171253+ O(13^{2}1)) ∗ a − 122310092032305114459993 + O(13^{2}1)q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 7531798457876026564855 + O(13^{2}0)x + −8378160046807384927544 + O(13^{2}0). The slope of f_{0} is 1.

18. Consider f = q+z^{5}0q^{3}+z^{1}49q^{5}+z^{8}9q^{7}+z^{1}19q^{9}+z^{8}9q^{11}+z^{3}1q^{15}+z^{1}33q^{17}+z^{1}29q^{19}+z^{1}39q^{21}+z^{1}34q^{23}+ z^{9}4q^{25} + ⋯ ∈ S_{26}(Γ_{0}(8); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-12 newform g = q + z^{1}62q^{3} + z^{2}3q^{5} + z^{1}03q^{7} + z^{7}q^{9} + z^{1}59q^{11} + z^{1}52q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}6)ω^{0} & 0 \\ 0 & u(z^{1}52)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{3}+(97025672287777198946+ O(13^{2}0)) ∗ a + 7320296195926660333456 + O(13^{2}0)q^{5} + (5045334958964414347370 + O(13^{2}1)) ∗ a − 113473655958714448024370+O(13^{2}1)q^{7}+−(7531798457876026564855+O(13^{2}0))∗a+8378160045960096318101+ O(13^{2}0)q^{9} + (3774341846197411853228 + O(13^{2}0)) ∗ a − 6528698762767082777930 + O(13^{2}0)q^{11} + −(655566328156921797854∗13+O(13^{2}0))∗a+231891047438583675960∗13+O(13^{2}0)q^{13}+(9078062461937228774025+ O(13^{2}0)) ∗ a + 1991910306348495353472 + O(13^{2}0)q^{15} + −(48635955400595054527336 + O(13^{2}1)) ∗ a + 121355734057992403468979+O(13^{2}1)q^{17}+(459822495716589452334+O(13^{2}0))∗a+1773691681862871878337+ O(13^{2}0)q^{19} + −(6038149883140032244908 + O(13^{2}0)) ∗ a − 7583088658246127310390 + O(13^{2}0)q^{21} + −(7222527488210759409634+O(13^{2}0))∗a+682997330155035350760∗13+O(13^{2}0)q^{23}+−(32428219359680323171253+ O(13^{2}1)) ∗ a − 122310092032305114459993 + O(13^{2}1)q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 7531798457876026564855 + O(13^{2}0)x + −8378160046807384927544 + O(13^{2}0). The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

19. Consider f = q + 9q3 + 11q5 + 11q7 + 2q9 + 8q11 + 9q13 + 8q15 + 11q17 + 5q19 + 8q21 + 9q23 + 9q25 + ⋯ ∈ S16(Γ0(8); \overline{F}13). Since f is ordinary, we do not continue our analysis.

20. Consider f = q + 7q3 + 4q5 + 2q7 + q9 + 7q11 + 10q13 + 2q15 + 11q17 + 2q19 + q21 + 2q23 + 11q25 + ⋯ ∈ S18(Γ0(8); \overline{F}13). Since f is ordinary, we do not continue our analysis.

21. Consider f = q + 5q3 + q5 + 11q7 + 9q9 + 8q11 + 8q13 + 5q15 + 6q17 + 10q19 + 3q21 + 3q23 + 6q25 + ⋯ ∈ S20(Γ0(8); \overline{F}13). Since f is ordinary, we do not continue our analysis.

22. Consider f = q + 7q3 + 9q5 + 12q7 + 7q9 + 11q11 + 12q13 + 11q15 + 4q17 + 3q19 + 6q21 + 12q23 + 8q25 + ⋯ ∈ S20(Γ0(8); \overline{F}13). Since f is ordinary, we do not continue our analysis.

23. Consider f = q + 3q3 + 2q5 + 4q7 + 8q9 + 3q11 + 11q13 + 6q15 + q17 + 11q19 + 12q21 + 3q23 + 12q25 + ⋯ ∈ S22(Γ0(8); \overline{F}13). Since f is ordinary, we do not continue our analysis.

24. Consider f = q + 5q3 + 6q5 + 11q7 + 11q9 + 5q11 + 4q13 + 4q15 + 7q17 + 11q19 + 3q21 + 10q23 + 5q25 + ⋯ ∈ S22(Γ0(8); \overline{F}13). Since f is ordinary, we do not continue our analysis.

25. Consider f = q + 3q3 + 7q5 + 7q7 + 8q11 + 12q13 + 8q15 + 5q17 + 11q19 + 8q21 + 3q23 + 2q25 + ⋯ ∈ S24(Γ0(8); \overline{F}13). Since f is ordinary, we do not continue our analysis.

26. Consider f = q+z90q3+z131q5+z163q7+z91q9+z51q11+z128q13+z53q15+z21q17+z95q19+z85q21+z90q23+z130q25+⋯ ∈ S24(Γ0(8); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

27. Consider f = q + z162q3 + z23q5 + z103q7 + z7q9 + z159q11 + z152q13 + z17q15 + z105q17 + z59q19 + z97q21 + z162q23 + z10q25 + ⋯ ∈ S24(Γ0(8); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

28. Consider f = q + 8q3 + 7q5 + 5q7 + 9q9 + 10q11 + q13 + 4q15 + 9q17 + 3q19 + q21 + 5q23 + 5q25 + ⋯ ∈ S26(Γ0(8); \overline{F}13). Since f is ordinary, we do not continue our analysis.

29. Consider f = q + 6q3 + 9q5 + 7q7 + 7q9 + q11 + q13 + 2q15 + 3q17 + 12q19 + 3q21 + 5q23 + 11q25 + ⋯ ∈ S26(Γ0(8); \overline{F}13). Since f is ordinary, we do not continue our analysis.

30. Consider f = q + q3 + 12q5 + 5q7 + 11q9 + 11q11 + 12q13 + 12q15 + 10q17 + 11q19 + 5q21 + 4q23 + 9q25 + ⋯ ∈ S26(Γ0(8); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

13.8 N = 9, ℓ = 13 (42 forms)

1. Consider f = q+5q^{2}+4q^{4}+7q^{5}+6q^{8}+9q^{10}+11q^{16}+q^{17}+4q^{19}+2q^{20}+5q^{23}+2q^{25}+⋯ ∈ S_{16}(Γ_{0}(9); \overline{F}_{13}). We have ρ_{f} ⊗χ^{10} ∼ ρ_{g}, where g is the weight-12 newform g = q+11q^{2}+10q^{4}+6q^{5}+7q^{8}+q^{10}+8q^{13}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{1} & 0 \\ 0 & u(8)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −216q^{2} + 13888q^{4} + −52110q^{5}+217112∗13q^{7}+4078080q^{8}+11255760q^{10}+−1583604∗13q^{11}+−14621026∗13q^{13}+−46896192∗ 13q^{14} + −1335947264q^{16} + −1646527986q^{17} + 1563257180q^{19} + −723703680q^{20} + 342058464 ∗ 13q^{22} + −9451116072q^{23} + −27802126025q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(7)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +18∗13q^{2} +21988q^{4} + −280710q^{5} + −1373344q^{7} + −194040 ∗ 13q^{8} + −5052780 ∗ 13q^{10} + −34031052q^{11} + 29540174 ∗ 13q^{13} + −24720192 ∗ 13q^{14} + −1310772464q^{16} + −96862122 ∗ 13q^{17} + −2499071020q^{19} + −6172251480q^{20} + −612558936 ∗ 13q^{22} + −11284833672q^{23} + 48280525975q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q + z^{1}61q^{2} + 12q^{4} + z^{1}05q^{5} + 4q^{7} + z^{2}1q^{8} + 11q^{10} + z^{9}1q^{11} + z^{2}1q^{14} + 10q^{16} + z^{1}19q^{17} + 4q^{19} + z^{2}1q^{20} + 12q^{22} + z^{7}7q^{23} + ⋯ ∈ S_{16}(Γ_{0}(9); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗χ^{10} ∼ ρ_{g}, where g is the weight-12 newform g = q +z^{1}33q^{2} +4q^{4} +z^{2}1q^{5} +3q^{7} +z^{1}05q^{8} + 7q^{10} + z^{6}3q^{11} + 5q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{1} & 0 \\ 0 & u(5)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +aq^{2} +87112q^{4} +464∗ aq^{5} + −2591260q^{7} + 54344 ∗ aq^{8} + 55624320q^{10} + 109120 ∗ aq^{11} + −5993230 ∗ 13q^{13} + −2591260 ∗ aq^{14} + 3660272704q^{16} + 217248 ∗ aq^{17} + −976566184q^{19} + 40419968 ∗ aq^{20} + 13081305600q^{22} + −60182656 ∗ aq^{23} +−362145665∗13q^{25} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −119880. The slope of f_{0} is 1.

4. Consider f = q + z^{7}7q^{2} + 12q^{4} + z^{2}1q^{5} + 4q^{7} + z^{1}05q^{8} + 11q^{10} + z^{7}q^{11} + z^{1}05q^{14} + 10q^{16} + z^{3}5q^{17} + 4q^{19} + z^{1}05q^{20} + 12q^{22} + z^{1}61q^{23} + ⋯ ∈ S_{16}(Γ_{0}(9); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{10} ∼ ρ_{g}, where g is the weight-12 newform g = q + z^{4}9q^{2} + 4q^{4} + z^{1}05q^{5} + 3q^{7} + z^{2}1q^{8} + 7q^{10} + z^{1}47q^{11} + 5q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{1} & 0 \\ 0 & u(5)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +aq^{2} +87112q^{4} +464∗ aq^{5} + −2591260q^{7} + 54344 ∗ aq^{8} + 55624320q^{10} + 109120 ∗ aq^{11} + −5993230 ∗ 13q^{13} + −2591260 ∗ aq^{14} + 3660272704q^{16} + 217248 ∗ aq^{17} + −976566184q^{19} + 40419968 ∗ aq^{20} + 13081305600q^{22} + −60182656 ∗ aq^{23} +−362145665∗13q^{25} +⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −119880. The slope of f_{0} is 1.

5. Consider f = q + 4q^{2} + 10q^{4} + q^{5} + 6q^{7} + 3q^{8} + 4q^{10} + 5q^{11} + 11q^{14} + 4q^{16} + 8q^{17} + 8q^{19} + 10q^{20} + 7q^{22} + 2q^{23} + 9q^{25} + ⋯ ∈ S_{18}(Γ_{0}(9); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 10q^{2} + 4q^{4} + q^{5} + 5q^{7} + 3q^{8} + 10q^{10} + 6q^{11} + 6q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(11)ω^{1} & 0 \\ 0 & u(6)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −204q^{2} + −89456q^{4} + 163554q^{5} + −20846560q^{7} + 44987712q^{8} + −33365016q^{10} + −817372356q^{11} + 23045366 ∗ 13q^{13} + 4252698240q^{14} + 2547683584q^{16} + 44775606078q^{17} + 78748651964q^{19} + −14630886624q^{20} + 166743960624q^{22} + 704672009160q^{23} + −736189542209q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

6. Consider f = q + 7q^{4} + 10q^{7} + 10q^{16} + 11q^{19} + 8q^{25} + ⋯ ∈ S_{18}(Γ_{0}(9); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-4 newform g = q + 5q^{4} + 7q^{7} + 8q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{4} & 0 \\ 0 & u(8)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2} +−32q^{4} +−280∗ aq^{5} +−196420q^{7} +−131104∗aq^{8} +−2822400∗13q^{10} +−3057440∗aq^{11} +−100663990∗13q^{13} +−196420∗ aq^{14} + −17175673856q^{16} + 96664848 ∗ aq^{17} + −111322541176q^{19} + 8960 ∗ aq^{20} + −30818995200 ∗ 13q^{22} + −954768064 ∗ aq^{23} + −752665917125q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −10080 ∗ 13. The slope of f_{0} is 1.

7. Consider f = q + 11q^{2} + 11q^{4} + 9q^{5} + 5q^{7} + 3q^{8} + 8q^{10} + 5q^{11} + 3q^{14} + 6q^{16} + 8q^{17} + 2q^{19} + 8q^{20} + 3q^{22} + 8q^{23} + 11q^{25} + ⋯ ∈ S_{18}(Γ_{0}(9); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 8q^{2} + 7q^{4} + 9q^{5} + 2q^{7} + 3q^{8} + 7q^{10} + 6q^{11} + q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−6533394632131479889766+ O(13^{2}0)q^{2}+3823801410415968917440+O(13^{2}0)q^{4}+1542937497041889790240+O(13^{2}0)q^{5}+6128449026840311645739+ O(13^{2}0)q^{7}+−8675214794001793095263+O(13^{2}0)q^{8}+6945025549243229718486+O(13^{2}0)q^{10}+−56162924892324301556032+ O(13^{2}1)q^{11} + −52642131997196450296 ∗ 13 + O(13^{2}0)q^{13} + 5111726847205962718946 + O(13^{2}0)q^{14} + 6447501208163510051225+O(13^{2}0)q^{16}+−7815365607506701953090+O(13^{2}0)q^{17}+−3038300688986688448893+ O(13^{2}0)q^{19}+−8112039492467298620510+O(13^{2}0)q^{20}+4543096689268483491499+O(13^{2}0)q^{22}+76970267526337735280895+ O(13^{2}1)q^{23} + 2238897162788921705458 + O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

8. Consider f = q + 12q^{2} + 3q^{4} + 9q^{5} + 6q^{7} + 8q^{8} + 4q^{10} + 3q^{11} + 7q^{14} + 11q^{16} + 8q^{17} + 5q^{19} + q^{20} + 10q^{22} + 2q^{23} + 8q^{25} + ⋯ ∈ S_{20}(Γ_{0}(9); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-6 newform g = q + 6q^{2} + 4q^{4} + 7q^{5} + 12q^{7} + q^{8} + 3q^{10} + 5q^{11} + q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{6} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−456q^{2}+−316352q^{4}+ 2377410q^{5} + −16917544q^{7} + 383331840q^{8} + −1084098960q^{10} + 16212108q^{11} + 3878585774 ∗ 13q^{13} + 7714400064q^{14} + −8939761664q^{16} + −225070099506q^{17} + −1710278572660q^{19} + −752098408320q^{20} + −7392721248q^{22} + −14036534788872q^{23} + −13421408020025q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

9. Consider f = q + 12q^{2} + 3q^{4} + 9q^{5} + 6q^{7} + 8q^{8} + 4q^{10} + 3q^{11} + 7q^{14} + 11q^{16} + 8q^{17} + 5q^{19} + q^{20} + 10q^{22} + 2q^{23} + 8q^{25} + ⋯ ∈ S_{20}(Γ_{0}(9); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-6 newform g = q + 6q^{2} + 4q^{4} + 7q^{5} + 12q^{7} + q^{8} + 3q^{10} + 5q^{11} + q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{6} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +1104q^{2} +694528q^{4} + −3516270q^{5}+−195590584q^{7}+187944960q^{8}+−3881962080q^{10}+2746857948q^{11}+−3415418866∗13q^{13}+ −215932004736q^{14} +−156641460224q^{16} +785982517614q^{17} +315410465180q^{19} +−2442147970560q^{20} + 3032531174592q^{22} + −4900560535752q^{23} + −6709331615225q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

10. Consider f = q+6q^{2}+12q^{4}+12q^{7}+6q^{8}+q^{11}+7q^{14}+8q^{16}+6q^{17}+q^{19}+6q^{22}+5q^{25}+⋯ ∈ S_{20}(Γ_{0}(9); \overline{F}_{13}). We have ρ_{f} ⊗χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q+7q^{2}+12q^{4}+q^{7}+7q^{8}+12q^{11}+11q^{13}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{1} & 0 \\ 0 & u(11)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+6747558085137094074388+ O(13^{2}0)q^{2} + −42559723370682578696689 + O(13^{2}1)q^{4} + −7347283092541032338224 ∗ 13 + O(13^{2}1)q^{5} + −803958468900441323964+O(13^{2}0)q^{7}+−1557510324270151944999+O(13^{2}0)q^{8}+−8455029991767779053689∗ 13+O(13^{2}1)q^{10}+3326006659352169235353+O(13^{2}0)q^{11}+−324133456109013333717∗13+O(13^{2}0)q^{13}+ 79050543617869195250119+O(13^{2}1)q^{14}+584967883585102597254605+O(13^{2}2)q^{16}+−118829380733358305297788+ O(13^{2}1)q^{17}+2732136477488455389+O(13^{2}0)q^{19}+8166293125988553999268∗13+O(13^{2}1)q^{20}+73152956989680761277324+ O(13^{2}1)q^{22}+8977684588642449905677∗13+O(13^{2}1)q^{23}+−405498225659573797438735+O(13^{2}2)q^{25}+ ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

11. Consider f = q + 10q^{2} + 11q^{4} + 9q^{5} + 12q^{10} + 11q^{11} + 9q^{16} + 8q^{20} + 6q^{22} + 8q^{25} + ⋯ ∈ S_{20}(Γ_{0}(9); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{12} ∼ ρ_{g}, where g is the weight-8 newform g = q + 10q^{2} + 11q^{4} + 9q^{5} + 12q^{10} + 11q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{7}. The form f_{0} = q + aq^{2} + −(3514116882626929314040 + O(13^{2}0)) ∗ a − 2564273871959699224319 + O(13^{2}0)q^{4} + −(4340223696147172346287 + O(13^{2}0)) ∗ a + 6970210127561443573263 + O(13^{2}0)q^{5} + (789021202611529291763∗13+O(13^{2}1))∗a−3018454401593588700074∗13+O(13^{2}1)q^{7}+(2564273871959699098295+ O(13^{2}0)) ∗ a + 6266051520754248651495 + O(13^{2}0)q^{8} + −(2266170951792863492695 + O(13^{2}0)) ∗ a + 3583402159534124436854+O(13^{2}0)q^{10}+(73342805947582544669+O(13^{2}0))∗a+7610311402771596299549+ O(13^{2}0)q^{11} + −(4853139917433165902161 + O(13^{2}0)) ∗ a + 7325030090550222472609 + O(13^{2}0)q^{13} + (3018454401593595095154∗13+O(13^{2}1))∗a+5232005457663981229915∗13+O(13^{2}1)q^{14}+(9400133903751349636058+ O(13^{2}0)) ∗ a − 353588223878380454516 + O(13^{2}0)q^{16} + −(123055536298066002827922 ∗ 13 + O(13^{2}2)) ∗ a − 5697326186054601569626 ∗ 13^{2} + O(13^{2}2)q^{17} + −(2222772011290650866159 ∗ 13 + O(13^{2}1)) ∗ a − 1485063248292525987386∗13+O(13^{2}1)q^{19}+(2290145646399873183429+O(13^{2}0))∗a−8944818411915767100111+ O(13^{2}0)q^{20} + −(3412810441971729902564 + O(13^{2}0)) ∗ a − 3915300937573482375905 + O(13^{2}0)q^{22} + (7760213736493487769953∗13+O(13^{2}1))∗a−1834446696126921774977∗13+O(13^{2}1)q^{23}+(7620025334309088545116+ O(13^{2}0)) ∗ a − 1479738155127741345156 + O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 3514116882626929314040 + O(13^{2}0)x + 2564273871959698700031 + O(13^{2}0). The slope of f_{0} is 1/2.

12. Consider f = q + 3q^{2} + 11q^{4} + 4q^{5} + 12q^{10} + 2q^{11} + 9q^{16} + 5q^{20} + 6q^{22} + 8q^{25} + ⋯ ∈ S_{20}(Γ_{0}(9); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + 10q^{2} + 11q^{4} + 9q^{5} + 12q^{10} + 11q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{91}. The form f_{0} = q + aq^{2} + (3514116882626929314040 + O(13^{2}0)) ∗ a − 2564273871959699224319 + O(13^{2}0)q^{4} + −(4340223696147172346287 + O(13^{2}0)) ∗ a − 6970210127561443573263 + O(13^{2}0)q^{5} + −(789021202611529291763∗13+O(13^{2}1))∗a−3018454401593588700074∗13+O(13^{2}1)q^{7}+(2564273871959699098295+ O(13^{2}0)) ∗ a − 6266051520754248651495 + O(13^{2}0)q^{8} + (2266170951792863492695 + O(13^{2}0)) ∗ a + 3583402159534124436854+O(13^{2}0)q^{10}+(73342805947582544669+O(13^{2}0))∗a−7610311402771596299549+ O(13^{2}0)q^{11} + (4853139917433165902161 + O(13^{2}0)) ∗ a + 7325030090550222472609 + O(13^{2}0)q^{13} + (3018454401593595095154∗13+O(13^{2}1))∗a−5232005457663981229915∗13+O(13^{2}1)q^{14}+−(9400133903751349636058+ O(13^{2}0)) ∗ a − 353588223878380454516 + O(13^{2}0)q^{16} + −(123055536298066002827922 ∗ 13 + O(13^{2}2)) ∗ a + 5697326186054601569626 ∗ 13^{2} + O(13^{2}2)q^{17} + (2222772011290650866159 ∗ 13 + O(13^{2}1)) ∗ a − 1485063248292525987386∗13+O(13^{2}1)q^{19}+(2290145646399873183429+O(13^{2}0))∗a+8944818411915767100111+ O(13^{2}0)q^{20} + (3412810441971729902564 + O(13^{2}0)) ∗ a − 3915300937573482375905 + O(13^{2}0)q^{22} + (7760213736493487769953∗13+O(13^{2}1))∗a+1834446696126921774977∗13+O(13^{2}1)q^{23}+−(7620025334309088545116+ O(13^{2}0)) ∗ a − 1479738155127741345156 + O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −3514116882626929314040 + O(13^{2}0)x + 2564273871959698700031 + O(13^{2}0). The slope of f_{0} is 1/2.

13. Consider f = q + 2q^{2} + 12q^{4} + 7q^{5} + 10q^{7} + q^{8} + q^{10} + 6q^{11} + 7q^{14} + 7q^{16} + 6q^{17} + 9q^{19} + 6q^{20} + 12q^{22} + 7q^{23} + 5q^{25} + ⋯ ∈ S_{22}(Γ_{0}(9); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 6q^{2} + 4q^{4} + 7q^{5} + 12q^{7} + q^{8} + 3q^{10} + 5q^{11} + q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+288q^{2}+−2014208q^{4}+ −21640950q^{5} + −768078808q^{7} + −1184071680q^{8} + −6232593600q^{10} + 94724929188q^{11} + −6201676138 ∗ 13q^{13}+−221206696704q^{14}+3883087691776q^{16}+−3052282930002q^{17}+−7920788351740q^{19}+43589374617600q^{20}+ 27280779606144q^{22} + 73845437470344q^{23} + −8506441300625q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

14. Consider f = q+q^{2}+9q^{4}+7q^{7}+4q^{8}+2q^{11}+7q^{14}+11q^{16}+11q^{17}+7q^{19}+2q^{22}+8q^{25}+⋯ ∈ S_{22}(Γ_{0}(9); \overline{F}_{13}). We have ρ_{f} ⊗χ^{11} ∼ ρ_{g}, where g is the weight-8 newform g = q+7q^{2}+12q^{4}+q^{7}+7q^{8}+12q^{11}+11q^{13}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{8} & 0 \\ 0 & u(11)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−1728q^{2}+888832q^{4}+ 3193290∗13q^{5}+538429808q^{7}+2087976960q^{8}+−5518005120∗13q^{10}+64113040188q^{11}+−10075392922∗ 13q^{13}+−930406708224q^{14}+−5472039993344q^{16}+−8242029723618q^{17}+13492101753020q^{19}+2838298337280∗ 13q^{20} + −110787333444864q^{22} + 1379791868904 ∗ 13^{2}q^{23} + 1246472914869775q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

15. Consider f = q + 7q^{2} + 5q^{4} + 6q^{5} + 3q^{10} + 4q^{11} + q^{16} + 4q^{20} + 2q^{22} + 5q^{25} + ⋯ ∈ S_{22}(Γ_{0}(9); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-8 newform g = q + 10q^{2} + 11q^{4} + 9q^{5} + 12q^{10} + 11q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{21}. The form f_{0} = q + 7968061757495416112205 + O(13^{2}0)q^{2} + 1901008q^{4} + 4361913873156945971417 + O(13^{2}0)q^{5} + −78386980∗13q^{7}+3285042805098948906086∗13+O(13^{2}1)q^{8}+−159926400q^{10}+8443539397266546360688+ O(13^{2}0)q^{11} +1865702690∗13^{2}q^{13} +2314020552271676062193∗13+O(13^{2}1)q^{14} +−4770917824256q^{16} + 8468733840021107220037 ∗ 13 + O(13^{2}1)q^{17} + 1405743250568 ∗ 13q^{19} + −3571472127175112337173 + O(13^{2}0)q^{20}+−140417298316800q^{22}+2259386681509956143091∗13+O(13^{2}1)q^{23}+−476830761147125q^{25}+ ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 2.

16. Consider f = q + 6q^{2} + 5q^{4} + 7q^{5} + 3q^{10} + 9q^{11} + q^{16} + 9q^{20} + 2q^{22} + 5q^{25} + ⋯ ∈ S_{22}(Γ_{0}(9); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{5} ∼ ρ_{g}, where g is the weight-8 newform g = q + 10q^{2} + 11q^{4} + 9q^{5} + 12q^{10} + 11q^{11} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is irreducible; it is the induction of ω_{2}^{105}. The form f_{0} = q + −7968061757495416112205 + O(13^{2}0)q^{2} + 1901008q^{4} + −4361913873156945971417 + O(13^{2}0)q^{5} + −78386980 ∗ 13q^{7} + −3285042805098948906086 ∗ 13 + O(13^{2}1)q^{8} + −159926400q^{10} + −8443539397266546360688 + O(13^{2}0)q^{11} + 1865702690 ∗ 13^{2}q^{13} + −2314020552271676062193 ∗ 13 + O(13^{2}1)q^{14} + −4770917824256q^{16} + −8468733840021107220037 ∗ 13 + O(13^{2}1)q^{17} + 1405743250568 ∗ 13q^{19} + 3571472127175112337173 + O(13^{2}0)q^{20} + −140417298316800q^{22} + −2259386681509956143091 ∗ 13 + O(13^{2}1)q^{23} + −476830761147125q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 2.

17. Consider f = q + 2q^{2} + 12q^{4} + 7q^{5} + 10q^{7} + q^{8} + q^{10} + 6q^{11} + 7q^{14} + 7q^{16} + 6q^{17} + 9q^{19} + 6q^{20} + 12q^{22} + 7q^{23} + 5q^{25} + ⋯ ∈ S_{22}(Γ_{0}(9); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 6q^{2} + 4q^{4} + 7q^{5} + 12q^{7} + q^{8} + 3q^{10} + 5q^{11} + q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−5466151690893158575087+ O(13^{2}0)q^{2}+−8496018642269880874010+O(13^{2}0)q^{4}+−7377492453496132559208+O(13^{2}0)q^{5}+1792409175580618195143+ O(13^{2}0)q^{7}+6704706229958762072295+O(13^{2}0)q^{8}+−8825601452609899112765+O(13^{2}0)q^{10}+25009939063780530177428+ O(13^{2}1)q^{11} + 379367062299107008261 ∗ 13 + O(13^{2}0)q^{13} + 4550457299176469102143 + O(13^{2}0)q^{14} + 1577906784961885665021+O(13^{2}0)q^{16}+−1045474998125928907635+O(13^{2}0)q^{17}+4460561882471771995967+ O(13^{2}0)q^{19}+6275045986857869274852+O(13^{2}0)q^{20}+7764373565242871056544+O(13^{2}0)q^{22}+247445423156367082405818+ O(13^{2}2)q^{23} + −689713089611794985992 + O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

18. Consider f = q + 3q^{2} + 2q^{4} + 6q^{5} + q^{7} + 11q^{8} + 5q^{10} + q^{11} + 3q^{14} + 6q^{16} + 5q^{17} + 10q^{19} + 12q^{20} + 3q^{22} + 5q^{23} + 2q^{25} + ⋯ ∈ S_{24}(Γ_{0}(9); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-10 newform g = q + 8q^{2} + 7q^{4} + 9q^{5} + 2q^{7} + 3q^{8} + 7q^{10} + 6q^{11} + q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{10} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −1128q^{2} + −7116224q^{4} +48863730q^{5} +−1723688680q^{7} +17489450496q^{8} +−55118287440q^{10} +1428263180124q^{11} + −632381849602∗13q^{13}+1944320831040q^{14}+39967113416704q^{16}+5989210330446q^{17}+680005481275676q^{19}+ −347725248155520q^{20} + −1611080867179872q^{22} + −15440648191080q^{23} + −9533264845565225q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

19. Consider f = q + 7q^{2} + 3q^{4} + 5q^{5} + 9q^{7} + 11q^{8} + 9q^{10} + q^{11} + 11q^{14} + 4q^{16} + 5q^{17} + q^{19} + 2q^{20} + 7q^{22} + 11q^{23} + 4q^{25} + ⋯ ∈ S_{24}(Γ_{0}(9); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-10 newform g = q + 10q^{2} + 4q^{4} + q^{5} + 5q^{7} + 3q^{8} + 10q^{10} + 6q^{11} + 6q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(11)ω^{10} & 0 \\ 0 & u(6)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−7999190284076529919946+ O(13^{2}0)q^{2}+4601721439607941919351+O(13^{2}0)q^{4}+2824075798267790349769+O(13^{2}0)q^{5}+−6855991533239110890125+ O(13^{2}0)q^{7}+−2201679203470269506050+O(13^{2}0)q^{8}+−3047782701909282676616+O(13^{2}0)q^{10}+−3625538007373198488259+ O(13^{2}0)q^{11} + −641482261434181347247 ∗ 13 + O(13^{2}0)q^{13} + 6145206873657502223558 + O(13^{2}0)q^{14} + −5296669873604408233255+O(13^{2}0)q^{16}+−4361411687787850103520+O(13^{2}0)q^{17}+9110235842523878111492+ O(13^{2}0)q^{19} + −5085087935422918591751 + O(13^{2}0)q^{20} + −1505064013711991728378 + O(13^{2}0)q^{22} + 7898984657638149650638+O(13^{2}0)q^{23} +7486295285953255151917+O(13^{2}0)q^{25} +⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

20. Consider f = q + 6q^{4} + 2q^{7} + 10q^{16} + 3q^{19} + 5q^{25} + ⋯ ∈ S_{24}(Γ_{0}(9); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 5q^{4} + 7q^{7} + 8q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{1} & 0 \\ 0 & u(8)ω^{10} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + O(13^{2}0) ∗ a + 6685797021397122423229 + O(13^{2}0)q^{4} + −(3482489313245962326756 + O(13^{2}0)) ∗ a + O(13^{2}1)q^{5} + O(13^{2}0) ∗ a − 7547210845255465680762 + O(13^{2}0)q^{7} + (6685797021397114034621 + O(13^{2}0)) ∗ a + O(13^{2}1)q^{8} + O(13^{2}0) ∗ a − 476661051906169095835 ∗ 13 + O(13^{2}0)q^{10} + −(5417116805674059282268 + O(13^{2}0))∗a+O(13^{2}1)q^{11}+O(13^{2}0)∗a+567265625940802455076∗13+O(13^{2}0)q^{13}+−(7547210845255465680762+ O(13^{2}0))∗a+O(13^{2}1)q^{14}+O(13^{2}0)∗a−1559407548046393133546+O(13^{2}0)q^{16}+−(452267058364608903342+ O(13^{2}0))∗a+O(13^{2}1)q^{17}+O(13^{2}2)∗a−1317632867851066307411738+O(13^{2}2)q^{19}+−(1485613036645234502944+ O(13^{2}0))∗a+O(13^{2}1)q^{20}+O(13^{2}0)∗a−546718373486624423544∗13+O(13^{2}0)q^{22}+−(1053947365667283639242+ O(13^{2}0)) ∗ a + O(13^{2}1)q^{23} + O(13^{2}0) ∗ a − 423563128156796814363 + O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + O(13^{2}0)x + −514292078569010062449 ∗ 13 + O(13^{2}0). The slope of f_{0} is 1.

21. Consider f = q + 9q^{2} + q^{4} + 4q^{5} + 4q^{8} + 10q^{10} + 8q^{16} + 10q^{17} + 5q^{19} + 4q^{20} + 7q^{23} + 11q^{25} + ⋯ ∈ S_{26}(Γ_{0}(9); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-12 newform g = q + 11q^{2} + 10q^{4} + 6q^{5} + 7q^{8} + q^{10} + 8q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{0} & 0 \\ 0 & u(8)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+48q^{2}+−33552128q^{4}+ 741989850q^{5} + 3006199784 ∗ 13q^{7} + −3221114880q^{8} + 35615512800q^{10} + −647655023004 ∗ 13q^{11} + −6280849641178 ∗ 13q^{13} + 144297589632 ∗ 13q^{14} + 1125667983917056q^{16} + 2519900028948078q^{17} + −6082056370308940q^{19}+−24895338421900800q^{20}+−31087441104192∗13q^{22}+94995280296320424q^{23}+ 252525713626069375q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

(\begin{matrix}u(2)ω^{0} & 0 \\ 0 & u(7)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−644367760420693768319∗ 13 + O(13^{2}0)q^{2} + 3632812915992180340275 + O(13^{2}0)q^{4} + −2325514655030757746369 + O(13^{2}0)q^{5} + 1166740301787361687104+O(13^{2}0)q^{7}+702100192985395702497∗13+O(13^{2}0)q^{8}+−562062436975633545577∗ 13+O(13^{2}0)q^{10}+−2613135682245105670905+O(13^{2}0)q^{11}+−682759059343875729066∗13+O(13^{2}0)q^{13}+ −134779380892237488549∗13+O(13^{2}0)q^{14}+−9087606788716508980768+O(13^{2}0)q^{16}+−195076649034533838412∗ 13+O(13^{2}0)q^{17} +−239624546601476546697+O(13^{2}0)q^{19} +−2355175398120143292938+O(13^{2}0)q^{20} + 4331318099515615782∗13+O(13^{2}0)q^{22}+−4322813063959693010402+O(13^{2}0)q^{23}+−7334610243464978974727+ O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

23. Consider f = q + z^{1}47q^{2} + 3q^{4} + z^{1}47q^{5} + 8q^{7} + z^{1}47q^{8} + 5q^{10} + z^{1}61q^{11} + z^{2}1q^{14} + 12q^{16} + z^{9}1q^{17} + 5q^{19} + z^{3}5q^{20} + 10q^{22} + z^{1}05q^{23} + ⋯ ∈ S_{26}(Γ_{0}(9); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗χ^{11} ∼ ρ_{g}, where g is the weight-12 newform g = q +z^{1}33q^{2} +4q^{4} +z^{2}1q^{5} +3q^{7} +z^{1}05q^{8} + 7q^{10} + z^{6}3q^{11} + 5q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{0} & 0 \\ 0 & u(5)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + O(13^{2}0) ∗ a − 7510955768750553946174 + O(13^{2}0)q^{4} + (4555434276660394328033 + O(13^{2}0)) ∗ a + O(13^{2}0)q^{5} + O(13^{2}0) ∗ a − 337854110889566289465 + O(13^{2}0)q^{7} + −(7510955768750587500606 + O(13^{2}0)) ∗ a + O(13^{2}0)q^{8} +O(13^{2}0)∗a−74809269648344216162+O(13^{2}0)q^{10} +(515551083505083023101+O(13^{2}0))∗ a+O(13^{2}0)q^{11} +O(13^{2}0)∗a+387990634550980987638∗13+O(13^{2}0)q^{13} +−(337854110889566289465+ O(13^{2}0))∗a+O(13^{2}0)q^{14}+O(13^{2}0)∗a−1114296584056394510292+O(13^{2}0)q^{16}+−(105294328503799409232028+ O(13^{2}1))∗a+O(13^{2}1)q^{17}+O(13^{2}0)∗a+8504943356667118849532+O(13^{2}0)q^{19}+−(5601454651295980084319+ O(13^{2}0))∗a+O(13^{2}0)q^{20}+O(13^{2}0)∗a+3753633913360091313680+O(13^{2}0)q^{22}+−(7516769380894553182359+ O(13^{2}0)) ∗ a + O(13^{2}0)q^{23} + O(13^{2}0) ∗ a + 283200636963652598176 ∗ 13 + O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + O(13^{2}0)x + 7510955768750520391742 + O(13^{2}0). The slope of f_{0} is 1.

24. Consider f = q+z^{6}3q^{2}+3q^{4}+z^{6}3q^{5}+8q^{7}+z^{6}3q^{8}+5q^{10}+z^{7}7q^{11}+z^{1}05q^{14}+12q^{16}+z^{7}q^{17}+5q^{19}+z^{1}19q^{20}+ 10q^{22}+z^{2}1q^{23}+⋯ ∈ S_{26}(Γ_{0}(9); \overline{F}_{13}). Also, z satisfies the equation x^{2}+12x+2 = 0. We have ρ_{f} ⊗χ^{11} ∼ ρ_{g}, where g is the weight-12 newform g = q+z^{4}9q^{2}+4q^{4}+z^{1}05q^{5}+3q^{7}+z^{2}1q^{8}+7q^{10}+z^{1}47q^{11}+5q^{13}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{0} & 0 \\ 0 & u(5)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + O(13^{2}0) ∗ a − 7510955768750553946174 + O(13^{2}0)q^{4} + (4555434276660394328033 + O(13^{2}0)) ∗ a + O(13^{2}0)q^{5} + O(13^{2}0) ∗ a − 337854110889566289465 + O(13^{2}0)q^{7} + −(7510955768750587500606 + O(13^{2}0)) ∗ a + O(13^{2}0)q^{8} +O(13^{2}0)∗a−74809269648344216162+O(13^{2}0)q^{10} +(515551083505083023101+O(13^{2}0))∗ a+O(13^{2}0)q^{11} +O(13^{2}0)∗a+387990634550980987638∗13+O(13^{2}0)q^{13} +−(337854110889566289465+ O(13^{2}0))∗a+O(13^{2}0)q^{14}+O(13^{2}0)∗a−1114296584056394510292+O(13^{2}0)q^{16}+−(105294328503799409232028+ O(13^{2}1))∗a+O(13^{2}1)q^{17}+O(13^{2}0)∗a+8504943356667118849532+O(13^{2}0)q^{19}+−(5601454651295980084319+ O(13^{2}0))∗a+O(13^{2}0)q^{20}+O(13^{2}0)∗a+3753633913360091313680+O(13^{2}0)q^{22}+−(7516769380894553182359+ O(13^{2}0)) ∗ a + O(13^{2}0)q^{23} + O(13^{2}0) ∗ a + 283200636963652598176 ∗ 13 + O(13^{2}0)q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + O(13^{2}0)x + 7510955768750520391742 + O(13^{2}0). The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

25. Consider f = q + 5q4 + 7q7 + 8q13 + 12q16 + 4q19 + 5q25 + ⋯ ∈ S16(Γ0(9); \overline{F}13). Since f is ordinary, we do not continue our analysis.

26. Consider f = q + 7q2 + 2q4 + 9q5 + 4q7 + 10q8 + 11q10 + 11q11 + 2q14 + 2q16 + 6q17 + 3q19 + 5q20 + 12q22 + 7q23 + 8q25 + ⋯ ∈ S16(Γ0(9); \overline{F}13). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

27. Consider f = q + 8q2 + 6q4 + 7q5 + 3q7 + 4q10 + 12q11 + 11q14 + 3q16 + 2q19 + 3q20 + 5q22 + 5q25 + ⋯ ∈ S18(Γ0(9); \overline{F}13). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

28. Consider f = q + 6q2 + 4q4 + 7q5 + 12q7 + q8 + 3q10 + 5q11 + q13 + 7q14 + 8q16 + 2q17 + 3q19 + 2q20 + 4q22 + 8q23 + 5q25 + ⋯ ∈ S18(Γ0(9); \overline{F}13). Since f is ordinary, we do not continue our analysis.

29. Consider f = q + 7q2 + 12q4 + q7 + 7q8 + 12q11 + 11q13 + 7q14 + 8q16 + 6q17 + 12q19 + 6q22 + 5q25 + ⋯ ∈ S20(Γ0(9); \overline{F}13). Since f is ordinary, we do not continue our analysis.

30. Consider f = q + 8q4 + 4q7 + 5q13 + 12q16 + 7q19 + 8q25 + ⋯ ∈ S22(Γ0(9); \overline{F}13). Since f is ordinary, we do not continue our analysis.

31. Consider f = q +10q2 +4q4 +q5 +5q7 +3q8 +10q10 +6q11 +6q13 +11q14 +10q16 +11q17 +11q19 +4q20 +8q22 +5q23 +9q25 +⋯ ∈ S22(Γ0(9); \overline{F}13). Since f is ordinary, we do not continue our analysis.

32. Consider f = q + 8q2 + 7q4 + 9q5 + 2q7 + 3q8 + 7q10 + 6q11 + q13 + 3q14 + 2q16 + 11q17 + 6q19 + 11q20 + 9q22 + 7q23 + 11q25 + ⋯ ∈ S22(Γ0(9); \overline{F}13). Since f is ordinary, we do not continue our analysis.

33. Consider f = q + 11q2 + 10q4 + 6q5 + 7q8 + q10 + 8q13 + 7q16 + 9q17 + 3q19 + 8q20 + 2q23 + 2q25 + ⋯ ∈ S24(Γ0(9); \overline{F}13). Since f is ordinary, we do not continue our analysis.

34. Consider f = q + q2 + 7q4 + 9q5 + 11q7 + 9q10 + 5q11 + 11q14 + 3q16 + 10q19 + 11q20 + 5q22 + 8q25 + ⋯ ∈ S24(Γ0(9); \overline{F}13). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

35. Consider f = q + 6q4 + q5 + 8q7 + 9q11 + 7q13 + 10q16 + 10q19 + 6q20 + 2q23 + 6q25 + ⋯ ∈ S24(Γ0(9); \overline{F}13). Since f is ordinary, we do not continue our analysis.

36. Consider f = q + z49q2 + 4q4 + z105q5 + 3q7 + z21q8 + 7q10 + z147q11 + 5q13 + z105q14 + 4q16 + z147q17 + 3q19 + z133q20 + 4q22 + z49q23 + ⋯ ∈ S24(Γ0(9); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

37. Consider f = q + 12q2 + 12q4 + 11q5 + 9q7 + 3q8 + 2q10 + 9q11 + q13 + 4q14 + 12q16 + 11q17 + 2q20 + 4q22 + 12q25 + ⋯ ∈ S26(Γ0(9); \overline{F}13). Since f is ordinary, we do not continue our analysis.

38. Consider f = q + 10q2 + 7q4 + 7q5 + 8q7 + 11q8 + 5q10 + q11 + 2q14 + 5q16 + 8q17 + 7q19 + 10q20 + 10q22 + 2q23 + 5q25 + ⋯ ∈ S26(Γ0(9); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

39. Consider f = q + z18q2 + z89q4 + z105q5 + z105q7 + z83q8 + z123q10 + 2q11 + 12q13 + z123q14 + 3q16 + z79q17 + z21q19 + z26q20 + z32q22 + 4q23 + 3q25 + ⋯ ∈ S26(Γ0(9); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

40. Consider f = q + z66q2 + z149q4 + z21q5 + z21q7 + z71q8 + z87q10 + 2q11 + 12q13 + z87q14 + 3q16 + z19q17 + z105q19 + z2q20 + z80q22 + 4q23 + 3q25 + ⋯ ∈ S26(Γ0(9); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

41. Consider f = q + 9q2 + q4 + 2q7 + 4q8 + 8q11 + q13 + 5q14 + 8q16 + 3q17 + 2q19 + 7q22 + 10q23 + 8q25 + ⋯ ∈ S26(Γ0(9); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

42. Consider f = q + 4q2 + q4 + 2q7 + 9q8 + 5q11 + q13 + 8q14 + 8q16 + 10q17 + 2q19 + 7q22 + 3q23 + 8q25 + ⋯ ∈ S26(Γ0(9); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

13.9 N = 10, ℓ = 13 (40 forms)

1. Consider f = q + 11q^{2} + 11q^{3} + 4q^{4} + 5q^{5} + 4q^{6} + 8q^{7} + 5q^{8} + 3q^{9} + 3q^{10} + 6q^{11} + 5q^{12} + 10q^{14} + 3q^{15} + 3q^{16} + 10q^{17} + 7q^{18} + 12q^{19} + 7q^{20} + 10q^{21} + q^{22} + 7q^{23} + 3q^{24} + 12q^{25} + ⋯ ∈ S_{16}(Γ_{0}(10); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{10} ∼ ρ_{g}, where g is the weight-12 newform g = q + 6q^{2} + 7q^{3} + 10q^{4} + 8q^{5} + 3q^{6} + 6q^{7} + 8q^{8} + q^{9} + 9q^{10} + 8q^{11} + 5q^{12} + 7q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(7)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 128q^{2} + −1302q^{3} + 16384q^{4}+−78125q^{5}+−166656q^{6}+−90706q^{7}+2097152q^{8}+−12653703q^{9}+−10000000q^{10}+−69411648q^{11}+ −21331968q^{12} + −2850574 ∗ 13q^{13} + −11610368q^{14} + 101718750q^{15} + 268435456q^{16} + −1372027686q^{17} + −1619673984q^{18}+−5068760620q^{19}+−1280000000q^{20}+118099212q^{21}+−8884690944q^{22}+−12342632022q^{23}+ −2730491904q^{24} + 6103515625q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q+2q^{2}+9q^{3}+4q^{4}+8q^{5}+5q^{6}+5q^{7}+8q^{8}+2q^{9}+3q^{10}+7q^{11}+10q^{12}+10q^{14}+7q^{15}+3q^{16}+5q^{17}+ 4q^{18}+9q^{19}+6q^{20}+6q^{21}+q^{22}+3q^{23}+7q^{24}+12q^{25}+⋯ ∈ S_{16}(Γ_{0}(10); \overline{F}_{13}). We have ρ_{f} ⊗χ^{10} ∼ ρ_{g}, where g is the weight-12 newform g = q+7q^{2}+q^{3}+10q^{4}+5q^{5}+7q^{6}+7q^{7}+5q^{8}+5q^{9}+9q^{10}+5q^{11}+10q^{12}+7q^{13}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(7)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +−128q^{2} +−5568q^{3} + 16384q^{4}+78125q^{5}+712704q^{6}+2564996q^{7}+−2097152q^{8}+16653717q^{9}+−10000000q^{10}+−81067668q^{11}+ −91226112q^{12}+27031694∗13q^{13}+−328319488q^{14}+−435000000q^{15}+268435456q^{16}+−2157825054q^{17}+ −2131675776q^{18}+−5107458100q^{19}+1280000000q^{20}+−14281897728q^{21}+10376661504q^{22}+11784341052q^{23}+ 11676942336q^{24} + 6103515625q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q + 11q^{2} + z^{1}5q^{3} + 4q^{4} + 8q^{5} + z^{1}13q^{6} + z^{2}1q^{7} + 5q^{8} + z^{1}38q^{9} + 10q^{10} + z^{1}49q^{11} + z^{4}3q^{12} + z^{1}19q^{14}+z^{5}7q^{15}+3q^{16}+z^{5}3q^{17}+z^{6}8q^{18}+z^{2}4q^{19}+6q^{20}+z^{3}6q^{21}+z^{7}9q^{22}+z^{7}2q^{23}+z^{1}41q^{24}+12q^{25}+⋯ ∈ S_{16}(Γ_{0}(10); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{10} ∼ ρ_{g}, where g is the weight-12 newform g = q + 6q^{2} + z^{7}1q^{3} + 10q^{4} + 5q^{5} + z^{1}41q^{6} + z^{4}9q^{7} + 8q^{8} + z^{8}2q^{9} + 4q^{10} + z^{1}21q^{11} + z^{4}3q^{12} + z^{5}3q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}15)ω^{1} & 0 \\ 0 & u(z^{5}3)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 128q^{2} + aq^{3} + 16384q^{4} +78125q^{5} +128∗aq^{6} +423∗a−102460q^{7} +2097152q^{8} +−1844∗a+8757909q^{9} +10000000q^{10} + −12006 ∗ a + 44706540q^{11} + 16384 ∗ aq^{12} + −2196 ∗ 13 ∗ a + 9102734 ∗ 13q^{13} + 54144 ∗ a − 13114880q^{14} + 78125 ∗ aq^{15} + 268435456q^{16} + −13428 ∗ a + 698489058q^{17} + −236032 ∗ a + 1121012352q^{18} + −134964 ∗ a + 2955266252q^{19} + 1280000000q^{20} + −882472 ∗ a + 9774183168q^{21} + −1536768 ∗ a + 5722437120q^{22} + 5204241 ∗ a + 2633227260q^{23} + 2097152 ∗ aq^{24} + 6103515625q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 1844x + −23106816. The slope of f_{0} is 1.

4. Consider f = q + 11q^{2} + z^{2}7q^{3} + 4q^{4} + 8q^{5} + z^{1}25q^{6} + z^{1}05q^{7} + 5q^{8} + z^{1}14q^{9} + 10q^{10} + z^{8}9q^{11} + z^{5}5q^{12} + z^{3}5q^{14}+z^{6}9q^{15}+3q^{16}+z^{1}7q^{17}+z^{4}4q^{18}+z^{1}44q^{19}+6q^{20}+z^{1}32q^{21}+z^{1}9q^{22}+z^{9}6q^{23}+z^{1}53q^{24}+12q^{25}+⋯ ∈ S_{16}(Γ_{0}(10); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{10} ∼ ρ_{g}, where g is the weight-12 newform g = q + 6q^{2} + z^{8}3q^{3} + 10q^{4} + 5q^{5} + z^{1}53q^{6} + z^{1}33q^{7} + 8q^{8} + z^{5}8q^{9} + 4q^{10} + z^{6}1q^{11} + z^{5}5q^{12} + z^{1}7q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}51)ω^{1} & 0 \\ 0 & u(z^{1}7)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 128q^{2} + aq^{3} + 16384q^{4} +78125q^{5} +128∗aq^{6} +423∗a−102460q^{7} +2097152q^{8} +−1844∗a+8757909q^{9} +10000000q^{10} + −12006 ∗ a + 44706540q^{11} + 16384 ∗ aq^{12} + −2196 ∗ 13 ∗ a + 9102734 ∗ 13q^{13} + 54144 ∗ a − 13114880q^{14} + 78125 ∗ aq^{15} + 268435456q^{16} + −13428 ∗ a + 698489058q^{17} + −236032 ∗ a + 1121012352q^{18} + −134964 ∗ a + 2955266252q^{19} + 1280000000q^{20} + −882472 ∗ a + 9774183168q^{21} + −1536768 ∗ a + 5722437120q^{22} + 5204241 ∗ a + 2633227260q^{23} + 2097152 ∗ aq^{24} + 6103515625q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 1844x + −23106816. The slope of f_{0} is 1.

5. Consider f = q + 4q^{2} + 12q^{3} + 3q^{4} + q^{5} + 9q^{6} + 8q^{7} + 12q^{8} + 5q^{9} + 4q^{10} + 3q^{11} + 10q^{12} + 6q^{14} + 12q^{15} + 9q^{16} + 10q^{17} + 7q^{18} + 9q^{19} + 3q^{20} + 5q^{21} + 12q^{22} + 7q^{23} + q^{24} + q^{25} + ⋯ ∈ S_{18}(Γ_{0}(10); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 10q^{2} + 4q^{3} + 9q^{4} + q^{5} + q^{6} + 11q^{7} + 12q^{8} + 2q^{9} + 10q^{10} + q^{11} + 10q^{12} + 3q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(9)ω^{1} & 0 \\ 0 & u(3)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −256q^{2} + 5829490968246822402317 + O(13^{2}0)q^{3} + 65536q^{4} + 390625q^{5} + 9042450344396620672127 + O(13^{2}0)q^{6} + −6097442500160373671873 + O(13^{2}0)q^{7} + −16777216q^{8} + −3983712739643181886414 + O(13^{2}0)q^{9} + −100000000q^{10} + 6103094556588715650461 + O(13^{2}0)q^{11} + 3738292369922639469210 + O(13^{2}0)q^{12} + 151737113603419022158∗13+O(13^{2}0)q^{13}+2538250500830106017806+O(13^{2}0)q^{14}+8159892747373746819907+ O(13^{2}0)q^{15} + 4294967296q^{16} + 2758552993899057786805 + O(13^{2}0)q^{17} + −6437582494908606773270 + O(13^{2}0)q^{18}+−3820736354350780903913+O(13^{2}0)q^{19}+25600000000q^{20}+−1438074313588090213316+ O(13^{2}0)q^{21} + −3985176946485652536334 + O(13^{2}0)q^{22} + −3915340128580304523326 + O(13^{2}0)q^{23} + −6754657956155732177710 + O(13^{2}0)q^{24} + 152587890625q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

6. Consider f = q+4q^{2}+8q^{3}+3q^{4}+12q^{5}+6q^{6}+10q^{7}+12q^{8}+3q^{9}+9q^{10}+q^{11}+11q^{12}+q^{14}+5q^{15}+9q^{16}+8q^{17}+ 12q^{18}+2q^{19}+10q^{20}+2q^{21}+4q^{22}+5q^{24}+q^{25}+⋯ ∈ S_{18}(Γ_{0}(10); \overline{F}_{13}). We have ρ_{f} ⊗χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q+10q^{2}+7q^{3}+9q^{4}+12q^{5}+5q^{6}+4q^{7}+12q^{8}+9q^{9}+3q^{10}+9q^{11}+11q^{12}+5q^{13}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{1} & 0 \\ 0 & u(5)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −256q^{2} + −2637402035200560257570+O(13^{2}0)q^{3}+65536q^{4}+−390625q^{5}+−9003774884365353858916+O(13^{2}0)q^{6}+ −250845986981236595107+O(13^{2}0)q^{7}+−16777216q^{8}+7388735024434687503206+O(13^{2}0)q^{9}+100000000q^{10}+ −5548829604516292344808+O(13^{2}0)q^{11}+5365753636953855787575+O(13^{2}0)q^{12}+7882688434636943951129∗ 13 + O(13^{2}1)q^{13} + 7201681342554170030989 + O(13^{2}0)q^{14} + −4911272294406164682159 + O(13^{2}0)q^{15} + 4294967296q^{16} + −5513109204566230287402 + O(13^{2}0)q^{17} + 8980211232799943059364 + O(13^{2}0)q^{18} + −3741023429470522128126+O(13^{2}0)q^{19}+−25600000000q^{20}+−5935394492964278533657+O(13^{2}0)q^{21}+ −4871904359889117639227+O(13^{2}0)q^{22}+−466714617015615666452∗13+O(13^{2}0)q^{23}+−5275539268769522025528+ O(13^{2}0)q^{24} + 152587890625q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

7. Consider f = q + 4q^{2} + 2q^{3} + 3q^{4} + 12q^{5} + 8q^{6} + 11q^{7} + 12q^{8} + 8q^{9} + 9q^{10} + 2q^{11} + 6q^{12} + 5q^{14} + 11q^{15} + 9q^{16} + 4q^{17} + 6q^{18} + 11q^{19} + 10q^{20} + 9q^{21} + 8q^{22} + 7q^{23} + 11q^{24} + q^{25} + ⋯ ∈ S_{18}(Γ_{0}(10); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-4 newform g = q + 2q^{2} + 5q^{3} + 4q^{4} + 5q^{5} + 10q^{6} + 9q^{7} + 8q^{8} + 11q^{9} + 10q^{10} + 12q^{11} + 7q^{12} + 7q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{4} & 0 \\ 0 & u(7)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −256q^{2} + 2637402035200560251262+O(13^{2}0)q^{3}+65536q^{4}+−390625q^{5}+9003774884365355473764+O(13^{2}0)q^{6}+ 250845986981243138951+O(13^{2}0)q^{7}+−16777216q^{8}+−7388735024434464307300+O(13^{2}0)q^{9}+100000000q^{10}+ 5548829604517481753512+O(13^{2}0)q^{11}+−5365753636954269188663+O(13^{2}0)q^{12}+−7882688434637099175685∗ 13 + O(13^{2}1)q^{13} + −7201681342555845255053 + O(13^{2}0)q^{14} + 4911272294408628744659 + O(13^{2}0)q^{15} + 4294967296q^{16} + 5513109204547474647966 + O(13^{2}0)q^{17} + −8980211232857081211300 + O(13^{2}0)q^{18} + 3741023429333817297526 + O(13^{2}0)q^{19} + −25600000000q^{20} + 5935394492554526635281 + O(13^{2}0)q^{21} + 4871904359584629011003+O(13^{2}0)q^{22}+6067290020553769493168+O(13^{2}0)q^{23}+5275539268875352704056+ O(13^{2}0)q^{24} + 152587890625q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

8. Consider f = q + 9q^{2} + 2q^{3} + 3q^{4} + 12q^{5} + 5q^{6} + 10q^{7} + q^{8} + 8q^{9} + 4q^{10} + 6q^{11} + 6q^{12} + 12q^{14} + 11q^{15} + 9q^{16} + 2q^{17} + 7q^{18} + 3q^{19} + 10q^{20} + 7q^{21} + 2q^{22} + 2q^{23} + 2q^{24} + q^{25} + ⋯ ∈ S_{18}(Γ_{0}(10); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 3q^{2} + 5q^{3} + 9q^{4} + 12q^{5} + 2q^{6} + 4q^{7} + q^{8} + 11q^{9} + 10q^{10} + 2q^{11} + 6q^{12} + 4q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(10)ω^{1} & 0 \\ 0 & u(4)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+256q^{2}+4451687395895446429192+ O(13^{2}0)q^{3}+65536q^{4}+−390625q^{5}+−665853143613680454908+O(13^{2}0)q^{6}+26718367172866459617498+ O(13^{2}1)q^{7}+16777216q^{8}+−92174830512295263631917+O(13^{2}1)q^{9}+−100000000q^{10}+−585074107503091340076+ O(13^{2}0)q^{11} + 586269208824998492761 + O(13^{2}0)q^{12} + −72749048191736988205 ∗ 13 + O(13^{2}0)q^{13} + −77904817802797333644076+O(13^{2}1)q^{14}+−5208583840493552272301+O(13^{2}0)q^{15}+4294967296q^{16}+ 3411905500285664877939+O(13^{2}0)q^{17}+121438179903650209852896+O(13^{2}1)q^{18}+813081256531581099637+ O(13^{2}0)q^{19}+−25600000000q^{20}+−7904080546185133640323+O(13^{2}0)q^{21}+2260738678255012450952+ O(13^{2}0)q^{22} + −12437734986125520758412 + O(13^{2}1)q^{23} + −1954792739846781363592 + O(13^{2}0)q^{24} + 152587890625q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

9. Consider f = q + 5q^{2} + 7q^{3} + 12q^{4} + 8q^{5} + 9q^{6} + 5q^{7} + 8q^{8} + 7q^{9} + q^{10} + 9q^{11} + 6q^{12} + 12q^{14} + 4q^{15} + q^{16} + 9q^{17} + 9q^{18} + 10q^{19} + 5q^{20} + 9q^{21} + 6q^{22} + 7q^{23} + 4q^{24} + 12q^{25} + ⋯ ∈ S_{20}(Γ_{0}(10); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-6 newform g = q + 9q^{2} + 11q^{3} + 3q^{4} + 12q^{5} + 8q^{6} + 10q^{7} + q^{8} + 8q^{9} + 4q^{10} + 2q^{11} + 7q^{12} + 3q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(9)ω^{6} & 0 \\ 0 & u(3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 512q^{2} + 24642q^{3} + 262144q^{4}+−1953125q^{5}+12616704q^{6}+−171901114q^{7}+134217728q^{8}+−555033303q^{9}+−1000000000q^{10}+ −11873833248q^{11} + 6459752448q^{12} + 2767074434 ∗ 13q^{13} + −88013370368q^{14} + −48128906250q^{15} + 68719476736q^{16} + −746262181734q^{17} + −284177051136q^{18} + 2682185926820q^{19} + −512000000000q^{20} + −4235987251188q^{21} +−6079402622976q^{22} +15924407922q^{23} +3307393253376q^{24} +3814697265625q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

10. Consider f = q + 8q^{2} + 5q^{3} + 12q^{4} + 5q^{5} + q^{6} + 6q^{7} + 5q^{8} + 9q^{9} + q^{10} + 6q^{11} + 8q^{12} + 9q^{14} + 12q^{15} + q^{16} + 6q^{17} + 7q^{18} + 5q^{19} + 8q^{20} + 4q^{21} + 9q^{22} + 8q^{23} + 12q^{24} + 12q^{25} + ⋯ ∈ S_{20}(Γ_{0}(10); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-6 newform g = q + 4q^{2} + 6q^{3} + 3q^{4} + q^{5} + 11q^{6} + 12q^{7} + 12q^{8} + q^{9} + 4q^{10} + 10q^{11} + 5q^{12} + q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{6} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −512q^{2} + 38628q^{3} + 262144q^{4}+1953125q^{5}+−19777536q^{6}+−144185776q^{7}+−134217728q^{8}+329860917q^{9}+−1000000000q^{10}+ −5156500668q^{11}+10126098432q^{12}+264270446∗13q^{13}+73823117312q^{14}+75445312500q^{15}+68719476736q^{16}+ 366347849874q^{17}+−168888789504q^{18}+−1604379002260q^{19}+512000000000q^{20}+−5569608155328q^{21}+ 2640128342016q^{22} + −13649676405552q^{23} + −5184562397184q^{24} + 3814697265625q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

11. Consider f = q + 5q^{2} + 12q^{4} + 5q^{5} + 2q^{7} + 8q^{8} + 10q^{9} + 12q^{10} + 2q^{11} + 10q^{14} + q^{16} + 4q^{17} + 11q^{18} + 9q^{19} + 8q^{20} + 10q^{22} + 4q^{23} + 12q^{25} + ⋯ ∈ S_{20}(Γ_{0}(10); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-6 newform g = q + 9q^{2} + 3q^{4} + q^{5} + 4q^{7} + q^{8} + 4q^{9} + 9q^{10} + 12q^{11} + 6q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(11)ω^{6} & 0 \\ 0 & u(6)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+512q^{2}+−566542535044659880093∗ 13+O(13^{2}0)q^{3}+262144q^{4}+1953125q^{5}+−609560448527528693570∗13+O(13^{2}0)q^{6}+−7577049053588250539509+ O(13^{2}0)q^{7}+134217728q^{8}+−3174300082259761889658+O(13^{2}0)q^{9}+1000000000q^{10}+356345409146849625349+ O(13^{2}0)q^{11}+−705927796124669533639∗13+O(13^{2}0)q^{12}+−501834182164444892866∗13+O(13^{2}1)q^{13}+ −2436505361501190713204+O(13^{2}0)q^{14}+540946364208310891352∗13+O(13^{2}0)q^{15}+68719476736q^{16}+ −4342088982852629585105+O(13^{2}0)q^{17}+9185242522750664231990+O(13^{2}0)q^{18}+−3380902695112483591158+ O(13^{2}0)q^{19}+512000000000q^{20}+−3802635564120399309531∗13+O(13^{2}1)q^{21}+−7600788265620986209322+ O(13^{2}0)q^{22} + −2197294840581841328961 + O(13^{2}0)q^{23} + −340719893095611885949 ∗ 13 + O(13^{2}0)q^{24} + 3814697265625q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

12. Consider f = q + 5q^{2} + 2q^{3} + 12q^{4} + 5q^{5} + 10q^{6} + 8q^{8} + q^{9} + 12q^{10} + 11q^{12} + 10q^{15} + q^{16} + 7q^{17} + 5q^{18} + 8q^{20} + 3q^{24} + 12q^{25} + ⋯ ∈ S_{20}(Γ_{0}(10); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + 8q^{2} + 2q^{3} + 12q^{4} + 8q^{5} + 3q^{6} + 5q^{8} + q^{9} + 12q^{10} + 11q^{12} + 4q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(10)ω^{1} & 0 \\ 0 & u(4)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+512q^{2}+7365052955580578474933+ O(13^{2}0)q^{3} +262144q^{4} +1953125q^{5} +7924285830857890283098+O(13^{2}0)q^{6} +582849927199102584677∗ 13+O(13^{2}0)q^{7}+134217728q^{8}+3174300082260346702612+O(13^{2}0)q^{9}+1000000000q^{10}+−27411185318715395785∗ 13+O(13^{2}0)q^{11} +9177061349629544481563+O(13^{2}0)q^{12} +501834182166771833294∗13+O(13^{2}1)q^{13} + 187423489349516776516∗13+O(13^{2}0)q^{14}+−7032302734642174400076+O(13^{2}0)q^{15}+68719476736q^{16}+ 4342088983182285364517+O(13^{2}0)q^{17}+−9185242522451239999542+O(13^{2}0)q^{18}+260069438115043107766∗ 13+O(13^{2}0)q^{19}+512000000000q^{20}+3802635564153054324539∗13+O(13^{2}1)q^{21}+584676020572178464850∗ 13+O(13^{2}0)q^{22} +169022680204976725065∗13+O(13^{2}0)q^{23} +4429358614769313176409+O(13^{2}0)q^{24} + 3814697265625q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

13. Consider f = q + 10q^{2} + 2q^{3} + 9q^{4} + q^{5} + 7q^{6} + 10q^{7} + 12q^{8} + 3q^{9} + 10q^{10} + 12q^{11} + 5q^{12} + 9q^{14} + 2q^{15} + 3q^{16} + 11q^{17} + 4q^{18} + 9q^{19} + 9q^{20} + 7q^{21} + 3q^{22} + 2q^{23} + 11q^{24} + q^{25} + ⋯ ∈ S_{22}(Γ_{0}(10); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 4q^{2} + 6q^{3} + 3q^{4} + q^{5} + 11q^{6} + 12q^{7} + 12q^{8} + q^{9} + 4q^{10} + 10q^{11} + 5q^{12} + q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 1024q^{2} + −4333712508483719717250+O(13^{2}0)q^{3}+1048576q^{4}+−9765625q^{5}+9439914634778078215434+O(13^{2}0)q^{6}+ 907652846834922860422+O(13^{2}0)q^{7}+1073741824q^{8}+8315329225325640360664+O(13^{2}0)q^{9}+−10000000000q^{10}+ −5070769192053196570979+O(13^{2}0)q^{11}+−7053975401574821745293+O(13^{2}0)q^{12}+−44193890101988694114∗ 13+O(13^{2}0)q^{13}+−1806709810198163429121+O(13^{2}0)q^{14}+−1421033506555318302411+O(13^{2}0)q^{15}+ 1099511627776q^{16} + 6373611491822027886834 + O(13^{2}0)q^{17} + 673355586857580737088 + O(13^{2}0)q^{18} + −4709838576946527781618+O(13^{2}0)q^{19}+−10240000000000q^{20}+9087230755824462436072+O(13^{2}0)q^{21}+ −4112542120015041889823+O(13^{2}0)q^{22}+−804679443091824327194+O(13^{2}0)q^{23}+−1384576757913680435652+ O(13^{2}0)q^{24} + 95367431640625q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

14. Consider f = q + 3q^{2} + 8q^{3} + 9q^{4} + 12q^{5} + 11q^{6} + 4q^{7} + q^{8} + 11q^{9} + 10q^{10} + 5q^{11} + 7q^{12} + 12q^{14} + 5q^{15} + 3q^{16} + 10q^{17} + 7q^{18} + 5q^{19} + 4q^{20} + 6q^{21} + 2q^{22} + 5q^{23} + 8q^{24} + q^{25} + ⋯ ∈ S_{22}(Γ_{0}(10); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 9q^{2} + 11q^{3} + 3q^{4} + 12q^{5} + 8q^{6} + 10q^{7} + q^{8} + 8q^{9} + 4q^{10} + 2q^{11} + 7q^{12} + 3q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(9)ω^{1} & 0 \\ 0 & u(3)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −1024q^{2} + −2182514257910090808111+O(13^{2}0)q^{3}+1048576q^{4}+9765625q^{5}+−7691125336001346272854+O(13^{2}0)q^{6}+ −64095447778050070906663+O(13^{2}1)q^{7}+−1073741824q^{8}+−24467423368332892936289+O(13^{2}1)q^{9}+ −10000000000q^{10} + 9304507966360211699957 + O(13^{2}0)q^{11} + 7657341264727615738882 + O(13^{2}0)q^{12} + 131883947366894019823∗13+O(13^{2}0)q^{13}+−85426208814531850950946+O(13^{2}1)q^{14}+−5045505011116530194099+ O(13^{2}0)q^{15}+1099511627776q^{16}+6634716121556819924534+O(13^{2}0)q^{17}+101124092754392703614223+ O(13^{2}1)q^{18}+2708938302964255251752+O(13^{2}0)q^{19}+10240000000000q^{20}+3245919033952906222593+ O(13^{2}0)q^{21} + −6329306337576261916667 + O(13^{2}0)q^{22} + 113380493221561808177564 + O(13^{2}1)q^{23} + 7932583944691651609645 + O(13^{2}0)q^{24} + 95367431640625q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

15. Consider f = q + 3q^{2} + 9q^{4} + q^{5} + 12q^{7} + q^{8} + 12q^{9} + 3q^{10} + 4q^{11} + 10q^{14} + 3q^{16} + 3q^{17} + 10q^{18} + 11q^{19} + 9q^{20} + 12q^{22} + q^{23} + q^{25} + ⋯ ∈ S_{22}(Γ_{0}(10); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 9q^{2} + 3q^{4} + q^{5} + 4q^{7} + q^{8} + 4q^{9} + 9q^{10} + 12q^{11} + 6q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(11)ω^{1} & 0 \\ 0 & u(6)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −1024q^{2} + −673231335278085472982 ∗ 13 + O(13^{2}0)q^{3} + 1048576q^{4} + −9765625q^{5} + −637489732451039905976 ∗ 13+O(13^{2}0)q^{6} +8290078361526150441888+O(13^{2}0)q^{7} +−1073741824q^{8} +2714759795177909432810+ O(13^{2}0)q^{9} + 10000000000q^{10} + −1150925507540268283885 + O(13^{2}0)q^{11} + −688883767959547753195 ∗ 13+O(13^{2}0)q^{12}+−2926504386896869073500∗13+O(13^{2}1)q^{13}+6178565168939296650735+O(13^{2}0)q^{14}+ 219423243205291010182∗13+O(13^{2}0)q^{15}+1099511627776q^{16}+2943607868816927282207+O(13^{2}0)q^{17}+ −5189319129582541132494+O(13^{2}0)q^{18}+8305016099064855143127+O(13^{2}0)q^{19}+−10240000000000q^{20}+ 91363551402537480826918∗13+O(13^{2}2)q^{21}+239965678625157492578+O(13^{2}0)q^{22}+−4346575539190584705316+ O(13^{2}0)q^{23}+−690521860763572185311∗13+O(13^{2}0)q^{24}+95367431640625q^{25}+⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

16. Consider f = q + 3q^{2} + 6q^{3} + 9q^{4} + q^{5} + 5q^{6} + q^{8} + 9q^{9} + 3q^{10} + 2q^{12} + 6q^{15} + 3q^{16} + 2q^{17} + q^{18} + 9q^{20} + 6q^{24} + q^{25} + ⋯ ∈ S_{22}(Γ_{0}(10); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-8 newform g = q + 8q^{2} + 2q^{3} + 12q^{4} + 8q^{5} + 3q^{6} + 5q^{8} + q^{9} + 12q^{10} + 11q^{12} + 4q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(10)ω^{8} & 0 \\ 0 & u(4)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −1024q^{2} + 8752007358615111022074+O(13^{2}0)q^{3}+1048576q^{4}+−9765625q^{5}+8287366521863648510296+O(13^{2}0)q^{6}+ −637698335502034080044∗13+O(13^{2}0)q^{7}+−1073741824q^{8}+−2714759795166042694984+O(13^{2}0)q^{9}+ 10000000000q^{10} + 6810210103544032261 ∗ 13^{2} + O(13^{2}0)q^{11} + 8955488983341274600943 + O(13^{2}0)q^{12} + 2926504387042026879056∗13+O(13^{2}1)q^{13}+−475274243741513507563∗13+O(13^{2}0)q^{14}+−2852502160431556569866+ O(13^{2}0)q^{15}+1099511627776q^{16}+−2943607871917341214571+O(13^{2}0)q^{17}+5189319117431001598670+ O(13^{2}0)q^{18}+−638847391683773883899∗13+O(13^{2}0)q^{19}+−10240000000000q^{20}+−91363551387174511296590∗ 13+O(13^{2}2)q^{21}+−1419915007730661266∗13^{2}+O(13^{2}0)q^{22}+25719379772962098256∗13^{2}+O(13^{2}0)q^{23}+ 8976784325960937575251 + O(13^{2}0)q^{24} + 95367431640625q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

17. Consider f = q + 6q^{2} + 2q^{3} + 10q^{4} + 8q^{5} + 12q^{6} + 2q^{7} + 8q^{8} + 8q^{9} + 9q^{10} + 9q^{11} + 7q^{12} + 12q^{14} + 3q^{15} + 9q^{16} + 11q^{17} + 9q^{18} + 2q^{19} + 2q^{20} + 4q^{21} + 2q^{22} + 11q^{23} + 3q^{24} + 12q^{25} + ⋯ ∈ S_{24}(Γ_{0}(10); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-10 newform g = q + 3q^{2} + 5q^{3} + 9q^{4} + 12q^{5} + 2q^{6} + 4q^{7} + q^{8} + 11q^{9} + 10q^{10} + 2q^{11} + 6q^{12} + 4q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(10)ω^{10} & 0 \\ 0 & u(4)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −2048q^{2} + 5550952966004306747499+O(13^{2}0)q^{3}+4194304q^{4}+48828125q^{5}+−3383336998102154474954+O(13^{2}0)q^{6}+ 7905646570715568744903 + O(13^{2}0)q^{7} + −8589934592q^{8} + −2601023980417575960967 + O(13^{2}0)q^{9} + −100000000000q^{10}+3897024043389300743435+O(13^{2}0)q^{11}+−7737605718279430456573+O(13^{2}0)q^{12}+ −604161831359886760334∗13+O(13^{2}0)q^{13}+1464959372956332297108+O(13^{2}0)q^{14}+8549055915068862357700+ O(13^{2}0)q^{15}+17592186044416q^{16}+8290515195125789442685+O(13^{2}0)q^{17}+5507254928571725196136+ O(13^{2}0)q^{18}+−6918691369622940467571+O(13^{2}0)q^{19}+204800000000000q^{20}+−7361388165670397757804+ O(13^{2}0)q^{21}+979544588647841741540+O(13^{2}0)q^{22}+−2954445576960596326467+O(13^{2}0)q^{23}+−3523277214313156898530+ O(13^{2}0)q^{24} + 2384185791015625q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

18. Consider f = q + 7q^{2} + 12q^{3} + 10q^{4} + 5q^{5} + 6q^{6} + 12q^{7} + 5q^{8} + 5q^{9} + 9q^{10} + 11q^{11} + 3q^{12} + 6q^{14} + 8q^{15} + 9q^{16} + 3q^{17} + 9q^{18} + 6q^{19} + 11q^{20} + q^{21} + 12q^{22} + 6q^{23} + 8q^{24} + 12q^{25} + ⋯ ∈ S_{24}(Γ_{0}(10); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-10 newform g = q + 10q^{2} + 4q^{3} + 9q^{4} + q^{5} + q^{6} + 11q^{7} + 12q^{8} + 2q^{9} + 10q^{10} + q^{11} + 10q^{12} + 3q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(9)ω^{10} & 0 \\ 0 & u(3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 2048q^{2} + −1886460467644181469262 + O(13^{2}0)q^{3} + 4194304q^{4} + −48828125q^{5} + −5463391434481362971973 + O(13^{2}0)q^{6} + 2176758444870334437137 + O(13^{2}0)q^{7} + 8589934592q^{8} + −103765800544215856440909 + O(13^{2}1)q^{9} + −100000000000q^{10} + 5546583606965766047122 + O(13^{2}0)q^{11} + 4898005586959502853085 + O(13^{2}0)q^{12} + −690638981629407138888 ∗ 13 + O(13^{2}0)q^{13} + −8165192002542940861659 + O(13^{2}0)q^{14} + 340821258743000408199+O(13^{2}0)q^{15} +17592186044416q^{16} +6778493609192251116940+O(13^{2}0)q^{17} + −36864511386736265186452+O(13^{2}1)q^{18}+−2804025773289980797318+O(13^{2}0)q^{19}+−204800000000000q^{20}+ −3105420900360898742401+O(13^{2}0)q^{21}+−5565110312829199897142+O(13^{2}0)q^{22}+−4203483828858371601018+ O(13^{2}0)q^{23}+−3505431044000260568848+O(13^{2}0)q^{24}+2384185791015625q^{25}+⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

19. Consider f = q+7q^{2}+8q^{3}+10q^{4}+8q^{5}+4q^{6}+2q^{7}+5q^{8}+3q^{9}+4q^{10}+8q^{11}+2q^{12}+q^{14}+12q^{15}+9q^{16}+5q^{17}+ 8q^{18}+10q^{19}+2q^{20}+3q^{21}+4q^{22}+q^{24}+12q^{25}+⋯ ∈ S_{24}(Γ_{0}(10); \overline{F}_{13}). We have ρ_{f}⊗χ^{11} ∼ ρ_{g}, where g is the weight-10 newform g = q+10q^{2}+7q^{3}+9q^{4}+12q^{5}+5q^{6}+4q^{7}+12q^{8}+9q^{9}+3q^{10}+9q^{11}+11q^{12}+5q^{13}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{10} & 0 \\ 0 & u(5)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 2048q^{2} + −1454560930489161042436+O(13^{2}0)q^{3}+4194304q^{4}+48828125q^{5}+4838527014483696982829+O(13^{2}0)q^{6}+ 4711311919966700593638+O(13^{2}0)q^{7}+8589934592q^{8}+7494609184842990611183+O(13^{2}0)q^{9}+100000000000q^{10}+ 1612840231723143230179+O(13^{2}0)q^{11}+7717198949714913218471+O(13^{2}0)q^{12}+−7589894248550297846∗ 13 + O(13^{2}0)q^{13} + −5754785547643299140284 + O(13^{2}0)q^{14} + 5198993518819146782202 + O(13^{2}0)q^{15} + 17592186044416q^{16}+−327586959202312179066+O(13^{2}0)q^{17}+−7051119545241174848424+O(13^{2}0)q^{18}+ 3079035303917499118410+O(13^{2}0)q^{19}+204800000000000q^{20}+−3532688936539010958798+O(13^{2}0)q^{21}+ −3766902260261766944782+O(13^{2}0)q^{22}+10934878264088016985∗13+O(13^{2}0)q^{23}+−7306411684682861653824+ O(13^{2}0)q^{24} + 2384185791015625q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

20. Consider f = q + 7q^{2} + 2q^{3} + 10q^{4} + 8q^{5} + q^{6} + 10q^{7} + 5q^{8} + 8q^{9} + 4q^{10} + 3q^{11} + 7q^{12} + 5q^{14} + 3q^{15} + 9q^{16} + 9q^{17} + 4q^{18} + 3q^{19} + 2q^{20} + 7q^{21} + 8q^{22} + 6q^{23} + 10q^{24} + 12q^{25} + ⋯ ∈ S_{24}(Γ_{0}(10); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 2q^{2} + 5q^{3} + 4q^{4} + 5q^{5} + 10q^{6} + 9q^{7} + 8q^{8} + 11q^{9} + 10q^{10} + 12q^{11} + 7q^{12} + 7q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(7)ω^{10} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 2048q^{2} + −6683726462785577909286 + O(13^{2}0)q^{3} + 4194304q^{4} + 48828125q^{5} + −4697877870687962281008 + O(13^{2}0)q^{6}+2550196361704524399591+O(13^{2}0)q^{7}+8589934592q^{8}+6526340998244414538004+O(13^{2}0)q^{9}+ 100000000000q^{10}+−8463760591694016521676+O(13^{2}0)q^{11}+−4742209079262235471078+O(13^{2}0)q^{12}+ 359575425039164602224∗13+O(13^{2}0)q^{13}+−3562889321353875307907+O(13^{2}0)q^{14}+−4094571937083982093919+ O(13^{2}0)q^{15}+17592186044416q^{16}+3934720913482468337575+O(13^{2}0)q^{17}+5456830663358968355089+ O(13^{2}0)q^{18}+1553116784735532499902+O(13^{2}0)q^{19}+204800000000000q^{20}+8772217145621443989976+ O(13^{2}0)q^{21}+−1254729098056748205936+O(13^{2}0)q^{22}+1974019218740352987092+O(13^{2}0)q^{23}+−507705364969731540433+ O(13^{2}0)q^{24} + 2384185791015625q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

21. Consider f = q + q^{2} + 3q^{3} + q^{4} + 12q^{5} + 3q^{6} + 10q^{7} + q^{8} + 6q^{9} + 12q^{10} + 3q^{11} + 3q^{12} + 10q^{14} + 10q^{15} + q^{16} + 11q^{17} + 6q^{18} + 8q^{19} + 12q^{20} + 4q^{21} + 3q^{22} + 12q^{23} + 3q^{24} + q^{25} + ⋯ ∈ S_{26}(Γ_{0}(10); \overline{F}_{13}). We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-12 newform g = q + 7q^{2} + q^{3} + 10q^{4} + 5q^{5} + 7q^{6} + 7q^{7} + 5q^{8} + 5q^{9} + 9q^{10} + 5q^{11} + 10q^{12} + 7q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{0} & 0 \\ 0 & u(7)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 4096q^{2} + 799388608741227338252+O(13^{2}0)q^{3}+16777216q^{4}+−244140625q^{5}+5441972124569674006420+O(13^{2}0)q^{6}+ 806137608461033785856+O(13^{2}0)q^{7}+68719476736q^{8}+2033256709535406802141+O(13^{2}0)q^{9}+−1000000000000q^{10}+ 1827367493356826101944+O(13^{2}0)q^{11}+−2504685697793011417253+O(13^{2}0)q^{12}+−376471700160638547836∗ 13+O(13^{2}0)q^{13}+−4924052572864715485198+O(13^{2}0)q^{14}+−7224103634058119257634+O(13^{2}0)q^{15}+ 281474976710656q^{16}+4464243490235902003434+O(13^{2}0)q^{17}+4045348859236107374698+O(13^{2}0)q^{18}+ 4590590540809823759789+O(13^{2}0)q^{19}+−4096000000000000q^{20}+−9006511517887337608605+O(13^{2}0)q^{21}+ −3058474513475265324970+O(13^{2}0)q^{22}+5338408674397254920697+O(13^{2}0)q^{23}+3487820275456931884252+ O(13^{2}0)q^{24} + 59604644775390625q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

22. Consider f = q+12q^{2}+8q^{3}+q^{4}+q^{5}+5q^{6}+3q^{7}+12q^{8}+9q^{9}+12q^{10}+10q^{11}+8q^{12}+10q^{14}+8q^{15}+q^{16}+9q^{17}+ 4q^{18}+2q^{19}+q^{20}+11q^{21}+3q^{22}+2q^{23}+5q^{24}+q^{25}+⋯ ∈ S_{26}(Γ_{0}(10); \overline{F}_{13}). We have ρ_{f} ⊗χ^{11} ∼ ρ_{g}, where g is the weight-12 newform g = q+6q^{2}+7q^{3}+10q^{4}+8q^{5}+3q^{6}+6q^{7}+8q^{8}+q^{9}+9q^{10}+8q^{11}+5q^{12}+7q^{13}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{0} & 0 \\ 0 & u(7)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −4096q^{2} + −5440919326801226990857 + O(13^{2}0)q^{3} + 16777216q^{4} + 244140625q^{5} + −6816945357351987163301 + O(13^{2}0)q^{6} + 8860479430041967187125 + O(13^{2}0)q^{7} + −68719476736q^{8} + 1659237739840450032974 + O(13^{2}0)q^{9}+−1000000000000q^{10}+2996334936048666953010+O(13^{2}0)q^{11}+3916398413845045282227+ O(13^{2}0)q^{12} + 301181733465006770603 ∗ 13 + O(13^{2}0)q^{13} + 6957064570429329645910 + O(13^{2}0)q^{14} + 8375208159075572146444+O(13^{2}0)q^{15}+281474976710656q^{16}+658070930564382960296+O(13^{2}0)q^{17}+ 7539249020842864029254+O(13^{2}0)q^{18}+73832689003758461671598+O(13^{2}1)q^{19}+4096000000000000q^{20}+ 4680893961682969186973+O(13^{2}0)q^{21}+4218700517656597936486+O(13^{2}0)q^{22}+1720071554731851270796+ O(13^{2}0)q^{23} + −1378477109910749653748 + O(13^{2}0)q^{24} + 59604644775390625q^{25} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

23. Consider f = q+12q^{2}+z^{1}27q^{3}+q^{4}+12q^{5}+z^{4}3q^{6}+z^{3}5q^{7}+12q^{8}+z^{2}6q^{9}+q^{10}+z^{5}1q^{11}+z^{1}27q^{12}+z^{1}19q^{14}+ z^{4}3q^{15} + q^{16} + z^{2}5q^{17} + z^{1}10q^{18} + z^{1}22q^{19} + 12q^{20} + z^{1}62q^{21} + z^{1}35q^{22} + z^{1}00q^{23} + z^{4}3q^{24} + q^{25} + ⋯ ∈ S_{26}(Γ_{0}(10); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-12 newform g = q + 6q^{2} + z^{7}1q^{3} + 10q^{4} + 5q^{5} + z^{1}41q^{6} + z^{4}9q^{7} + 8q^{8} + z^{8}2q^{9} + 4q^{10} + z^{1}21q^{11} + z^{4}3q^{12} + z^{5}3q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}15)ω^{0} & 0 \\ 0 & u(z^{5}3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −4096q^{2} + aq^{3} + 16777216q^{4} + −244140625q^{5} + −4096 ∗ aq^{6} + 20517 ∗ a + 13690871180q^{7} + −68719476736q^{8} + 545212∗a+28077173721q^{9} +1000000000000q^{10} +−11164578∗a−1146053987940q^{11} +16777216∗aq^{12} + −5577276 ∗ 13 ∗ a − 4227437144782 ∗ 13q^{13} + −84037632 ∗ a − 56077808353280q^{14} + −244140625 ∗ aq^{15} + 281474976710656q^{16}+−386452452∗a+1784976406757874q^{17}+−2233188352∗a−115004103561216q^{18}+ −11796885528∗a+4960632484176428q^{19}+−4096000000000000q^{20}+24876985784∗a+17959879773175788q^{21}+ 45730111488 ∗ a + 4694237134602240q^{22} + 66205713831 ∗ a + 18137026139397780q^{23} + −68719476736 ∗ aq^{24} + 59604644775390625q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −545212x + −875365783164. The slope of f_{0} is 1.

24. Consider f = q + 12q^{2} + z^{1}39q^{3} + q^{4} + 12q^{5} + z^{5}5q^{6} + z^{1}19q^{7} + 12q^{8} + z^{2}q^{9} + q^{10} + z^{1}59q^{11} + z^{1}39q^{12} + z^{3}5q^{14} +z^{5}5q^{15} +q^{16} +z^{1}57q^{17} +z^{8}6q^{18} +z^{7}4q^{19} +12q^{20} +z^{9}0q^{21} +z^{7}5q^{22} +z^{1}24q^{23} +z^{5}5q^{24} +q^{25} +⋯ ∈ S_{26}(Γ_{0}(10); \overline{F}_{13}). Also, z satisfies the equation x^{2} + 12x + 2 = 0. We have ρ_{f} ⊗ χ^{11} ∼ ρ_{g}, where g is the weight-12 newform g = q + 6q^{2} + z^{8}3q^{3} + 10q^{4} + 5q^{5} + z^{1}53q^{6} + z^{1}33q^{7} + 8q^{8} + z^{5}8q^{9} + 4q^{10} + z^{6}1q^{11} + z^{5}5q^{12} + z^{1}7q^{13} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}51)ω^{0} & 0 \\ 0 & u(z^{1}7)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −4096q^{2} + aq^{3} + 16777216q^{4} + −244140625q^{5} + −4096 ∗ aq^{6} + 20517 ∗ a + 13690871180q^{7} + −68719476736q^{8} + 545212∗a+28077173721q^{9} +1000000000000q^{10} +−11164578∗a−1146053987940q^{11} +16777216∗aq^{12} + −5577276 ∗ 13 ∗ a − 4227437144782 ∗ 13q^{13} + −84037632 ∗ a − 56077808353280q^{14} + −244140625 ∗ aq^{15} + 281474976710656q^{16}+−386452452∗a+1784976406757874q^{17}+−2233188352∗a−115004103561216q^{18}+ −11796885528∗a+4960632484176428q^{19}+−4096000000000000q^{20}+24876985784∗a+17959879773175788q^{21}+ 45730111488 ∗ a + 4694237134602240q^{22} + 66205713831 ∗ a + 18137026139397780q^{23} + −68719476736 ∗ aq^{24} + 59604644775390625q^{25} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −545212x + −875365783164. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

25. Consider f = q + 2q2 + 5q3 + 4q4 + 5q5 + 10q6 + 9q7 + 8q8 + 11q9 + 10q10 + 12q11 + 7q12 + 7q13 + 5q14 + 12q15 + 3q16 + q17 + 9q18 + 4q19 + 7q20 + 6q21 + 11q22 + 2q23 + q24 + 12q25 + ⋯ ∈ S16(Γ0(10); \overline{F}13). Since f is ordinary, we do not continue our analysis.

26. Consider f = q + 9q2 + 3q4 + q5 + 4q7 + q8 + 4q9 + 9q10 + 12q11 + 6q13 + 10q14 + 9q16 + q17 + 10q18 + 8q19 + 3q20 + 4q22 + 3q23 + q25 + ⋯ ∈ S18(Γ0(10); \overline{F}13). Since f is ordinary, we do not continue our analysis.

27. Consider f = q + 4q2 + 6q3 + 3q4 + q5 + 11q6 + 12q7 + 12q8 + q9 + 4q10 + 10q11 + 5q12 + q13 + 9q14 + 6q15 + 9q16 + 8q17 + 4q18 + 3q19 + 3q20 + 7q21 + q22 + 6q23 + 7q24 + q25 + ⋯ ∈ S18(Γ0(10); \overline{F}13). Since f is ordinary, we do not continue our analysis.

28. Consider f = q + 9q2 + 11q3 + 3q4 + 12q5 + 8q6 + 10q7 + q8 + 8q9 + 4q10 + 2q11 + 7q12 + 3q13 + 12q14 + 2q15 + 9q16 + 12q17 + 7q18 + 6q19 + 10q20 + 6q21 + 5q22 + 2q23 + 11q24 + q25 + ⋯ ∈ S18(Γ0(10); \overline{F}13). Since f is ordinary, we do not continue our analysis.

29. Consider f = q+8q2 +2q3 +12q4 +8q5 +3q6 +5q8 +q9 +12q10 +11q12 +4q13 +3q15 +q16 +7q17 +8q18 +5q20 +10q24 +12q25 +⋯ ∈ S20(Γ0(10); \overline{F}13). Since f is ordinary, we do not continue our analysis.

30. Consider f = q + 10q2 + 7q3 + 9q4 + 12q5 + 5q6 + 4q7 + 12q8 + 9q9 + 3q10 + 9q11 + 11q12 + 5q13 + q14 + 6q15 + 3q16 + 11q17 + 12q18 + 6q19 + 4q20 + 2q21 + 12q22 + 6q24 + q25 + ⋯ ∈ S22(Γ0(10); \overline{F}13). Since f is ordinary, we do not continue our analysis.

31. Consider f = q + 10q2 + 4q3 + 9q4 + q5 + q6 + 11q7 + 12q8 + 2q9 + 10q10 + q11 + 10q12 + 3q13 + 6q14 + 4q15 + 3q16 + 4q17 + 7q18 + q19 + 9q20 + 5q21 + 10q22 + 11q23 + 9q24 + q25 + ⋯ ∈ S22(Γ0(10); \overline{F}13). Since f is ordinary, we do not continue our analysis.

32. Consider f = q + 3q2 + 5q3 + 9q4 + 12q5 + 2q6 + 4q7 + q8 + 11q9 + 10q10 + 2q11 + 6q12 + 4q13 + 12q14 + 8q15 + 3q16 + 6q17 + 7q18 + 9q19 + 4q20 + 7q21 + 6q22 + 5q23 + 5q24 + q25 + ⋯ ∈ S22(Γ0(10); \overline{F}13). Since f is ordinary, we do not continue our analysis.

33. Consider f = q + 6q2 + 7q3 + 10q4 + 8q5 + 3q6 + 6q7 + 8q8 + q9 + 9q10 + 8q11 + 5q12 + 7q13 + 10q14 + 4q15 + 9q16 + 12q17 + 6q18 + 9q19 + 2q20 + 3q21 + 9q22 + 8q23 + 4q24 + 12q25 + ⋯ ∈ S24(Γ0(10); \overline{F}13). Since f is ordinary, we do not continue our analysis.

34. Consider f = q + 7q2 + q3 + 10q4 + 5q5 + 7q6 + 7q7 + 5q8 + 5q9 + 9q10 + 5q11 + 10q12 + 7q13 + 10q14 + 5q15 + 9q16 + 6q17 + 9q18 + 10q19 + 11q20 + 7q21 + 9q22 + 9q23 + 5q24 + 12q25 + ⋯ ∈ S24(Γ0(10); \overline{F}13). Since f is ordinary, we do not continue our analysis.

35. Consider f = q + 6q2 + z71q3 + 10q4 + 5q5 + z141q6 + z49q7 + 8q8 + z82q9 + 4q10 + z121q11 + z43q12 + z53q13 + z119q14 + z29q15 + 9q16 + z165q17 + z152q18 + z52q19 + 11q20 + z120q21 + z23q22 + z128q23 + z113q24 + 12q25 + ⋯ ∈ S24(Γ0(10); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

36. Consider f = q + 6q2 + z83q3 + 10q4 + 5q5 + z153q6 + z133q7 + 8q8 + z58q9 + 4q10 + z61q11 + z55q12 + z17q13 + z35q14 + z41q15 + 9q16 + z129q17 + z128q18 + z4q19 + 11q20 + z48q21 + z131q22 + z152q23 + z125q24 + 12q25 + ⋯ ∈ S24(Γ0(10); \overline{F}13). Also, z satisfies the equation x2 + 12x + 2 = 0. Since f is ordinary, we do not continue our analysis.

37. Consider f = q + 7q2 + 10q3 + 10q4 + 8q5 + 5q6 + 3q7 + 5q8 + 4q10 + 7q11 + 9q12 + q13 + 8q14 + 2q15 + 9q16 + 11q17 + 12q19 + 2q20 + 4q21 + 10q22 + 5q23 + 11q24 + 12q25 + ⋯ ∈ S24(Γ0(10); \overline{F}13). Since f is ordinary, we do not continue our analysis.

38. Consider f = q + q2 + q4 + q5 + q8 + 10q9 + q10 + q13 + q16 + 2q17 + 10q18 + 5q19 + q20 + 9q23 + q25 + ⋯ ∈ S26(Γ0(10); \overline{F}13). Since f is ordinary, we do not continue our analysis.

39. Consider f = q + q2 + 2q3 + q4 + 12q5 + 2q6 + 9q7 + q8 + q9 + 12q10 + 11q11 + 2q12 + 12q13 + 9q14 + 11q15 + q16 + 2q17 + q18 + 6q19 + 12q20 + 5q21 + 11q22 + 6q23 + 2q24 + q25 + ⋯ ∈ S26(Γ0(10); \overline{F}13). Since f is ordinary, we do not continue our analysis.

40. Consider f = q + 12q2 + 11q3 + q4 + q5 + 2q6 + 9q7 + 12q8 + q9 + 12q10 + 7q11 + 11q12 + q13 + 4q14 + 11q15 + q16 + 7q17 + 12q18 + 2q19 + q20 + 8q21 + 6q22 + 6q23 + 2q24 + q25 + ⋯ ∈ S26(Γ0(10); \overline{F}13). Since f is ordinary, we do not continue our analysis.

17 2 ≤ N ≤ 6, ℓ = 17

17.1 N = 2, ℓ = 17 (18 forms)

1. Consider f = q + 15q^{2} + 15q^{3} + 4q^{4} + 16q^{5} + 4q^{6} + 7q^{7} + 9q^{8} + 11q^{9} + 2q^{10} + 11q^{11} + 9q^{12} + 6q^{13} + 3q^{14} + 2q^{15} + 16q^{16} + 12q^{18} + ⋯ ∈ S_{20}(Γ_{0}(2); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{14} ∼ ρ_{g}, where g is the weight-16 newform g = q + 8q^{2} + 13q^{3} + 13q^{4} + 2q^{5} + 2q^{6} + 5q^{7} + 2q^{8} + 10q^{9} + 16q^{10} + 14q^{11} + 16q^{12} + 11q^{13} + 6q^{14} + 9q^{15} + 16q^{16} + 16q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(16)ω^{1} & 0 \\ 0 & u(16)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−512q^{2}+−13092q^{3}+ 262144q^{4} + 6546750q^{5} + 6703104q^{6} + 96674264q^{7} + −134217728q^{8} + −990861003q^{9} + −3351936000q^{10} + 11799694452q^{11}+−3431989248q^{12}+34401727958q^{13}+−49497223168q^{14}+−85710051000q^{15}+68719476736q^{16}+ −23570447598 ∗ 17q^{17} + 507320833536q^{18} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 4q^{2} + 3q^{3} + 16q^{4} + 11q^{5} + 12q^{6} + 7q^{7} + 13q^{8} + 4q^{9} + 10q^{10} + 5q^{11} + 14q^{12} + 11q^{13} + 11q^{14} + 16q^{15} + q^{16} + 16q^{18} + ⋯ ∈ S_{22}(Γ_{0}(2); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{12} ∼ ρ_{g}, where g is the weight-14 newform g = q + 13q^{2} + 12q^{3} + 16q^{4} + 10q^{5} + 3q^{6} + 6q^{7} + 4q^{8} + 13q^{9} + 11q^{10} + 14q^{11} + 5q^{12} + 11q^{13} + 10q^{14} + q^{15} + q^{16} + 8q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(15)ω^{1} & 0 \\ 0 & u(8)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 1024q^{2} + 59316q^{3} + 1048576q^{4}+4975350q^{5}+60739584q^{6}+1427425832q^{7}+1073741824q^{8}+−6941965347q^{9}+5094758400q^{10}+ −106767894948q^{11} + 62197334016q^{12} + −150150565474q^{13} + 1461684051968q^{14} + 295117860600q^{15} + 1099511627776q^{16}+−659057690574∗17q^{17}+−7108572515328q^{18}+⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q +13q^{2} +16q^{4} +3q^{5} +4q^{8} +12q^{9} +5q^{10} +14q^{11} +4q^{13} +q^{16} +3q^{18} +⋯ ∈ S_{22}(Γ_{0}(2); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{12} ∼ ρ_{g}, where g is the weight-14 newform g = q + 4q^{2} + 16q^{4} + 12q^{5} + 13q^{8} + 5q^{9} + 14q^{10} + 12q^{11} + 4q^{13} + q^{16} + 15q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{1} & 0 \\ 0 & u(15)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−1024q^{2}+4212∗17q^{3}+ 1048576q^{4} + −28693770q^{5} + −4313088 ∗ 17q^{6} + −50188376 ∗ 17q^{7} + −1073741824q^{8} + −5333220387q^{9} + 29382420480q^{10} + 86731179612q^{11} + 4416602112 ∗ 17q^{12} + −895323442786q^{13} + 51392897024 ∗ 17q^{14} + −120858159240 ∗ 17q^{15} + 1099511627776q^{16} + 191621576754 ∗ 17q^{17} + 5461217676288q^{18} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

4. Consider f = q + q^{2} + 2q^{3} + q^{4} + 4q^{5} + 2q^{6} + 6q^{7} + q^{8} + 7q^{9} + 4q^{10} + 10q^{11} + 2q^{12} + 14q^{13} + 6q^{14} + 8q^{15} + q^{16} + 7q^{18} + ⋯ ∈ S_{26}(Γ_{0}(2); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-8 newform g = q + 9q^{2} + 12q^{3} + 13q^{4} + 11q^{5} + 6q^{6} + 13q^{7} + 15q^{8} + 14q^{9} + 14q^{10} + 4q^{11} + 3q^{12} + 5q^{13} + 15q^{14} + 13q^{15} + 16q^{16} + q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{8} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−4096q^{2} +97956q^{3} + 16777216q^{4} +341005350q^{5} +−401227776q^{6} +−40882637368q^{7} +−68719476736q^{8} +−837693231507q^{9} + −1396757913600q^{10}+−14506222377108q^{11}+1643428970496q^{12}+87843989537006q^{13}+167455282659328q^{14}+ 33403520064600q^{15} + 281474976710656q^{16} + −156201521699214 ∗ 17q^{17} + 3431191476252672q^{18} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

5. Consider f = q + 16q^{2} + 3q^{3} + q^{4} + 14q^{5} + 14q^{6} + 16q^{8} + 12q^{9} + 3q^{10} + 14q^{11} + 3q^{12} + 2q^{13} + 8q^{15} + q^{16} + 5q^{18} + ⋯ ∈ S_{26}(Γ_{0}(2); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 16q^{2} + 14q^{3} + q^{4} + 3q^{5} + 3q^{6} + 16q^{8} + 12q^{9} + 14q^{10} + 3q^{11} + 14q^{12} + 2q^{13} + 8q^{15} + q^{16} + 11q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(14)ω^{1} & 0 \\ 0 & u(11)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 4096q^{2} + 1630119222196490464912966 + O(17^{2}0)q^{3} + 16777216q^{4} + 191454872521517712919072 + O(17^{2}0)q^{5} + −563867005136710022321707+O(17^{2}0)q^{6}+670092053825310685724211∗17+O(17^{2}1)q^{7}+68719476736q^{8}+ −4385415491752386589744805+O(17^{2}1)q^{9}+−197503634844944706990081+O(17^{2}0)q^{10}+2020843995094981523233495+ O(17^{2}0)q^{11}+−1115814064143058705602504+O(17^{2}0)q^{12}+−12788397285502986992927536+O(17^{2}1)q^{13}+ 1340852981361116105287581∗17+O(17^{2}1)q^{14}+29315958188393666433903210+O(17^{2}1)q^{15}+281474976710656q^{16}+ −12470086570088639482980 ∗ 17 + O(17^{2}0)q^{17} + 1240963164495077420355140 + O(17^{2}1)q^{18} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

6. Consider f = q + 15q^{2} + 8q^{3} + 4q^{4} + 15q^{5} + q^{6} + 9q^{8} + 6q^{9} + 4q^{10} + 16q^{11} + 15q^{12} + 9q^{13} + q^{15} + 16q^{16} + 5q^{18} + ⋯ ∈ S_{28}(Γ_{0}(2); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-10 newform g = q +16q^{2} +14q^{3} +q^{4} +3q^{5} +3q^{6} +16q^{8} +12q^{9} +14q^{10} +3q^{11} +14q^{12} +2q^{13} +8q^{15} +q^{16} +11q^{17} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(14)ω^{10} & 0 \\ 0 & u(11)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 8192q^{2} + −1016388q^{3} +67108864q^{4} +−3341197410q^{5} +−8326250496q^{6} +−3001256552∗17q^{7} +549755813888q^{8} + −6592552918443q^{9}+−27371089182720q^{10}+−177413845094508q^{11}+−68208644063232q^{12}+−264386643393418q^{13}+ −24586293673984∗17q^{14} +3395952953155080q^{15} +4503599627370496q^{16} +4518346386556818∗17q^{17} + −54006193507885056q^{18} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

7. Consider f = q + 2q^{2} + 11q^{3} + 4q^{4} + 14q^{5} + 5q^{6} + 9q^{7} + 8q^{8} + 12q^{9} + 11q^{10} + 9q^{11} + 10q^{12} + 12q^{13} + q^{14} +q^{15} +16q^{16} +7q^{18} +⋯ ∈ S_{28}(Γ_{0}(2); \overline{F}_{17}). We have ρ_{f} ⊗χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + 9q^{2} + 12q^{3} + 13q^{4} + 11q^{5} + 6q^{6} + 13q^{7} + 15q^{8} + 14q^{9} + 14q^{10} + 4q^{11} + 3q^{12} + 5q^{13} + 15q^{14} + 13q^{15} + 16q^{16} + q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{10} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −8192q^{2} + 3984828q^{3} + 67108864q^{4} + −2851889250q^{5} + −32643710976q^{6} + 368721063704q^{7} + −549755813888q^{8} + 8253256704597q^{9}+23362676736000q^{10}+59896911912852q^{11}+267417280315392q^{12}+1902611126010998q^{13}+ −3020562953863168q^{14}+−11364288136299000q^{15}+4503599627370496q^{16}+−202933185722478∗17q^{17}+ −67610678924058624q^{18} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

8. Consider f = q + 9q^{2} + 2q^{3} + 13q^{4} + 16q^{5} + q^{6} + 8q^{7} + 15q^{8} + 15q^{9} + 8q^{10} + q^{11} + 9q^{12} + 7q^{13} + 4q^{14} + 15q^{15} +16q^{16} +16q^{18} +⋯ ∈ S_{32}(Γ_{0}(2); \overline{F}_{17}). We have ρ_{f} ⊗χ^{15} ∼ ρ_{g}, where g is the weight-14 newform g = q + 13q^{2} + 12q^{3} + 16q^{4} + 10q^{5} + 3q^{6} + 6q^{7} + 4q^{8} + 13q^{9} + 11q^{10} + 14q^{11} + 5q^{12} + 11q^{13} + 10q^{14} + q^{15} + q^{16} + 8q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(15)ω^{14} & 0 \\ 0 & u(8)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 32768q^{2} + −19984212q^{3}+1073741824q^{4}+42951708750q^{5}+−654842658816q^{6}+−16835358997576q^{7}+35184372088832q^{8}+ −218304667023003q^{9} + 1407441592320000q^{10} + −7207832704992348q^{11} + −21457884244082688q^{12} + −270053634881821882q^{13}+−551661043632570368q^{14}+−858356053422255000q^{15}+1152921504606846976q^{16}+ −957381350642703918 ∗ 17q^{17} + −7153407329009762304q^{18} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

9. Consider f = q +8q^{2} +13q^{4} +9q^{5} +2q^{8} +11q^{9} +4q^{10} +13q^{11} +q^{13} +16q^{16} +3q^{18} +⋯ ∈ S_{32}(Γ_{0}(2); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-14 newform g = q + 4q^{2} + 16q^{4} + 12q^{5} + 13q^{8} + 5q^{9} + 14q^{10} + 12q^{11} + 4q^{13} + q^{16} + 15q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{14} & 0 \\ 0 & u(15)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −32768q^{2} + −116871999554412156583879∗17+O(17^{2}0)q^{3}+1073741824q^{4}+504948118188794342368698+O(17^{2}0)q^{5}+ −39665841461525785996835∗17+O(17^{2}0)q^{6}+−15657621942055835994908∗17+O(17^{2}0)q^{7}+−35184372088832q^{8}+ 1526124729998896126630041+O(17^{2}0)q^{9}+−653880348145272040578393+O(17^{2}0)q^{10}+907282171299103144028880+ O(17^{2}0)q^{11}+−66539808890797649681581∗17+O(17^{2}0)q^{12}+−1759790651324801722765794+O(17^{2}0)q^{13}+ 19508816950890759155006∗17+O(17^{2}0)q^{14}+−36120743832813361351747∗17+O(17^{2}0)q^{15}+1152921504606846976q^{16}+ −87891482314211885443524 ∗ 17 + O(17^{2}0)q^{17} + −1751925212095961783884784 + O(17^{2}0)q^{18} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

10. Consider f = q + 16q^{2} + 5q^{3} + q^{4} + 10q^{5} + 12q^{6} + q^{7} + 16q^{8} + 5q^{9} + 7q^{10} + q^{11} + 5q^{12} + 7q^{13} + 16q^{14} + 16q^{15} + q^{16} + 12q^{18} + ⋯ ∈ S_{34}(Γ_{0}(2); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-16 newform g = q + 8q^{2} + 13q^{3} + 13q^{4} + 2q^{5} + 2q^{6} + 5q^{7} + 2q^{8} + 10q^{9} + 16q^{10} + 14q^{11} + 16q^{12} + 11q^{13} + 6q^{14} + 9q^{15} + 16q^{16} + 16q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(16)ω^{0} & 0 \\ 0 & u(16)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −65536q^{2} + −133005564q^{3}+4294967296q^{4}+538799132550q^{5}+8716652642304q^{6}+−33347311051768q^{7}+−281474976710656q^{8}+ 12131419488402573q^{9}+−35310739950796800q^{10}+−85882263625386228q^{11}+−571254547566034944q^{12}+ 1144054008875905166q^{13}+2185449377088667648q^{14}+−71663282507523508200q^{15}+18446744073709551616q^{16}+ −8183157392316801294 ∗ 17q^{17} + −795044707591951024128q^{18} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

11. Consider f = q+2q2 +12q3 +4q4 +13q5 +7q6 +5q7 +8q8 +15q9 +9q10 +13q11 +14q12 +12q13 +10q14 +3q15 +16q16 +13q18 +⋯ ∈ S20(Γ0(2); \overline{F}17). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

12. Consider f = q + 9q2 + 12q3 + 13q4 + 11q5 + 6q6 + 13q7 + 15q8 + 14q9 + 14q10 + 4q11 + 3q12 + 5q13 + 15q14 + 13q15 + 16q16 + q17 + 7q18 + ⋯ ∈ S24(Γ0(2); \overline{F}17). Since f is ordinary, we do not continue our analysis.

13. Consider f = q + 16q2 + 14q3 + q4 + 3q5 + 3q6 + 16q8 + 12q9 + 14q10 + 3q11 + 14q12 + 2q13 + 8q15 + q16 + 11q17 + 5q18 + ⋯ ∈ S26(Γ0(2); \overline{F}17). Since f is ordinary, we do not continue our analysis.

14. Consider f = q + 4q2 + 16q4 + 12q5 + 13q8 + 5q9 + 14q10 + 12q11 + 4q13 + q16 + 15q17 + 3q18 + ⋯ ∈ S30(Γ0(2); \overline{F}17). Since f is ordinary, we do not continue our analysis.

15. Consider f = q + 13q2 + 12q3 + 16q4 + 10q5 + 3q6 + 6q7 + 4q8 + 13q9 + 11q10 + 14q11 + 5q12 + 11q13 + 10q14 + q15 + q16 + 8q17 + 16q18 + ⋯ ∈ S30(Γ0(2); \overline{F}17). Since f is ordinary, we do not continue our analysis.

16. Consider f = q + 8q2 + 13q3 + 13q4 + 2q5 + 2q6 + 5q7 + 2q8 + 10q9 + 16q10 + 14q11 + 16q12 + 11q13 + 6q14 + 9q15 + 16q16 + 16q17 + 12q18 + ⋯ ∈ S32(Γ0(2); \overline{F}17). Since f is ordinary, we do not continue our analysis.

17. Consider f = q + q2 + 15q3 + q4 + 15q6 + 13q7 + q8 + q9 + 6q11 + 15q12 + 2q13 + 13q14 + q16 + 16q17 + q18 + ⋯ ∈ S34(Γ0(2); \overline{F}17). Since f is ordinary, we do not continue our analysis.

18. Consider f = q + q2 + 4q3 + q4 + 6q5 + 4q6 + 8q7 + q8 + 13q9 + 6q10 + 12q11 + 4q12 + 14q13 + 8q14 + 7q15 + q16 + 13q18 + ⋯ ∈ S34(Γ0(2); \overline{F}17). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

17.2 N = 3, ℓ = 17 (34 forms)

1. Consider f = q + q^{2} + 14q^{3} + 10q^{4} + 7q^{5} + 14q^{6} + 9q^{7} + 2q^{8} + 9q^{9} + 7q^{10} + 12q^{11} + 4q^{12} + 8q^{13} + 9q^{14} + 13q^{15} + 7q^{16} + 9q^{18} + 5q^{19} + 2q^{20} + 7q^{21} + ⋯ ∈ S_{20}(Γ_{0}(3); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{14} ∼ ρ_{g}, where g is the weight-16 newform g = q + 13q^{2} + 11q^{3} + 7q^{4} + 3q^{5} + 7q^{6} + 4q^{7} + 8q^{8} + 2q^{9} + 5q^{10} + 6q^{11} + 9q^{12} + 9q^{13} + q^{14} + 16q^{15} + 7q^{16} + 12q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(10)ω^{1} & 0 \\ 0 & u(12)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−1104q^{2} +19683q^{3} + 694528q^{4}+3516270q^{5}+−21730032q^{6}+−195590584q^{7}+−187944960q^{8}+387420489q^{9}+−3881962080q^{10}+ −2746857948q^{11}+13670394624q^{12}+−44400445258q^{13}+215932004736q^{14}+69210742410q^{15}+−156641460224q^{16}+ −46234265742∗17q^{17}+−427712219856q^{18}+315410465180q^{19}+2442147970560q^{20}+−3849809464872q^{21}+ ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 16q^{2} + 3q^{3} + 10q^{4} + 14q^{5} + 14q^{6} + 15q^{7} + 15q^{8} + 9q^{9} + 3q^{10} + 3q^{11} + 13q^{12} + 3q^{13} + 2q^{14} + 8q^{15} + 7q^{16} + 8q^{18} + 7q^{19} + 4q^{20} + 11q^{21} + ⋯ ∈ S_{20}(Γ_{0}(3); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{14} ∼ ρ_{g}, where g is the weight-16 newform g = q + 4q^{2} + 6q^{3} + 7q^{4} + 6q^{5} + 7q^{6} + q^{7} + 9q^{8} + 2q^{9} + 7q^{10} + 10q^{11} + 8q^{12} + 14q^{13} + 4q^{14} + 2q^{15} + 7q^{16} + 8q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(15)ω^{1} & 0 \\ 0 & u(8)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+1099584231386133112868181+ O(17^{2}0)q^{2} + −19683q^{3} + −295836829973334022701288 + O(17^{2}0)q^{4} + 2010070087794661831379745 + O(17^{2}0)q^{5} + −1084185974934378795881298 + O(17^{2}0)q^{6} + 57454285902788242261250 + O(17^{2}0)q^{7} + −458941204396488314496624+O(17^{2}0)q^{8}+387420489q^{9}+−1262281502444974749065977+O(17^{2}0)q^{10}+ 1496106220934711387273317+O(17^{2}0)q^{11}+−1087281360837855772042529+O(17^{2}0)q^{12}+−669161775266028850572673+ O(17^{2}0)q^{13} +−268455220351074217974878+O(17^{2}0)q^{14} +1083205651789678532064900+O(17^{2}0)q^{15} + −1113259566565944723461774+O(17^{2}0)q^{16}+−1278632570213675971104654∗17+O(17^{2}1)q^{17}+−1246571673025475799218317+ O(17^{2}0)q^{18} +1625952104445755803292097+O(17^{2}0)q^{19} +1773593985545814365647203+O(17^{2}0)q^{20} + −1016378376555811200538672 + O(17^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q + 12q^{2} + 9q^{3} + 10q^{4} + 4q^{5} + 6q^{6} + 14q^{7} + 8q^{8} + 13q^{9} + 14q^{10} + 9q^{11} + 5q^{12} + 12q^{13} + 15q^{14} + 2q^{15} + 14q^{16} + 3q^{18} + 15q^{19} + 6q^{20} + 7q^{21} + ⋯ ∈ S_{22}(Γ_{0}(3); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{12} ∼ ρ_{g}, where g is the weight-14 newform g = q + 5q^{2} + 2q^{3} + 10q^{4} + 16q^{5} + 10q^{6} + 12q^{7} + 9q^{8} + 4q^{9} + 12q^{10} + 15q^{11} + 3q^{12} + 12q^{13} + 9q^{14} + 15q^{15} + 14q^{16} + 11q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(14)ω^{1} & 0 \\ 0 & u(11)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −2844q^{2} + −59049q^{3} + 5991184q^{4} + 3109950q^{5} + 167935356q^{6} + 363303920q^{7} + −11074627008q^{8} + 3486784401q^{9} + −8844697800q^{10} + 14581833156q^{11} + −353773424016q^{12} + 113350790702q^{13} + −1033236348480q^{14} + −183639437550q^{15}+18931815702784q^{16}+−505258211646∗17q^{17}+−9916414836444q^{18}+−29202939273796q^{19}+ 18632282680800q^{20} + −21452733172080q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

4. Consider f = q+z^{7}4q^{2}+8q^{3}+z^{1}77q^{4}+z^{2}27q^{5}+z^{2}54q^{6}+z^{1}92q^{7}+z^{2}6q^{8}+13q^{9}+z^{1}3q^{10}+z^{9}3q^{11}+z^{6}9q^{12}+ z^{1}41q^{13} + z^{2}66q^{14} + z^{1}19q^{15} + z^{1}37q^{16} + z^{1}46q^{18} + z^{3}5q^{19} + z^{1}16q^{20} + z^{8}4q^{21} + ⋯ ∈ S_{22}(Γ_{0}(3); \overline{F}_{17}). Also, z satisfies the equation x^{2} + 16x + 3 = 0. We have ρ_{f} ⊗ χ^{12} ∼ ρ_{g}, where g is the weight-14 newform g = q + z^{2}18q^{2} + 15q^{3} + z^{1}77q^{4} + z^{1}55q^{5} + z^{3}8q^{6} + z^{2}64q^{7} + z^{1}70q^{8} + 4q^{9} + z^{8}5q^{10} + z^{1}65q^{11} + z^{2}85q^{12} +z^{1}41q^{13} +z^{1}94q^{14} +z^{2}63q^{15} +z^{1}37q^{16} +z^{1}83q^{17} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}05)ω^{1} & 0 \\ 0 & u(z^{1}83)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + 59049q^{3} + 666 ∗ a + 367840q^{4} + 13568 ∗ a − 4019706q^{5} + 59049 ∗ aq^{6} + −182016 ∗ a + 400559384q^{7} + −1285756 ∗ a + 1641684672q^{8}+3486784401q^{9}+5016582∗a+33445011456q^{10}+31263232∗a+99523905228q^{11}+39326634∗ a+21720584160q^{12}+−475236864∗a+134019420734q^{13}+279336728∗a−448667983872q^{14}+801176832∗ a−237359619594q^{15} +−611332056∗a−3940794645632q^{16} +98664960∗17∗a−366194521134∗17q^{17} + 3486784401 ∗ aq^{18} + −22384332288 ∗ a + 13434275157716q^{19} + 8331896732 ∗ a + 20795768974656q^{20} + −10747862784 ∗ a + 23652631065816q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −666x + −2464992. The slope of f_{0} is 1.

5. Consider f = q +z^{1}06q^{2} +8q^{3} +z^{1}29q^{4} +z^{1}15q^{5} +z^{2}86q^{6} +z^{9}6q^{7} +z^{1}54q^{8} +13q^{9} +z^{2}21q^{10} +z^{1}41q^{11} + z^{2}1q^{12} +z^{9}3q^{13} +z^{2}02q^{14} +z^{7}q^{15} +z^{2}5q^{16} +z^{1}78q^{18} +z^{1}9q^{19} +z^{2}44q^{20} +z^{2}76q^{21} +⋯ ∈ S_{22}(Γ_{0}(3); \overline{F}_{17}). Also, z satisfies the equation x^{2}+16x+3 = 0. We have ρ_{f} ⊗χ^{12} ∼ ρ_{g}, where g is the weight-14 newform g = q + z^{2}50q^{2} + 15q^{3} + z^{1}29q^{4} + z^{4}3q^{5} + z^{7}0q^{6} + z^{1}68q^{7} + z^{1}0q^{8} + 4q^{9} + z^{5}q^{10} + z^{2}13q^{11} + z^{2}37q^{12} + z^{9}3q^{13} + z^{1}30q^{14} + z^{1}51q^{15} + z^{2}5q^{16} + z^{2}31q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{5}7)ω^{1} & 0 \\ 0 & u(z^{2}31)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + aq^{2} + 59049q^{3} + 666 ∗ a + 367840q^{4} + 13568 ∗ a − 4019706q^{5} + 59049 ∗ aq^{6} + −182016 ∗ a + 400559384q^{7} + −1285756 ∗ a + 1641684672q^{8}+3486784401q^{9}+5016582∗a+33445011456q^{10}+31263232∗a+99523905228q^{11}+39326634∗ a+21720584160q^{12}+−475236864∗a+134019420734q^{13}+279336728∗a−448667983872q^{14}+801176832∗ a−237359619594q^{15} +−611332056∗a−3940794645632q^{16} +98664960∗17∗a−366194521134∗17q^{17} + 3486784401 ∗ aq^{18} + −22384332288 ∗ a + 13434275157716q^{19} + 8331896732 ∗ a + 20795768974656q^{20} + −10747862784 ∗ a + 23652631065816q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −666x + −2464992. The slope of f_{0} is 1.

6. Consider f = q + 11q^{2} + 10q^{3} + 10q^{4} + 4q^{5} + 8q^{6} + 9q^{7} + 11q^{8} + 15q^{9} + 10q^{10} + 2q^{11} + 15q^{12} + 10q^{13} + 14q^{14} + 6q^{15} + 14q^{16} + 12q^{18} + 10q^{19} + 6q^{20} + 5q^{21} + ⋯ ∈ S_{24}(Γ_{0}(3); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{10} ∼ ρ_{g}, where g is the weight-12 newform g = q + 10q^{2} + 12q^{3} + 7q^{4} + 2q^{5} + q^{6} + q^{7} + 7q^{8} + 8q^{9} + 3q^{10} + 13q^{11} + 16q^{12} + 7q^{13} + 10q^{14} + 7q^{15} + 14q^{16} + 10q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(12)ω^{1} & 0 \\ 0 & u(10)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+1809487775611657116861377+ O(17^{2}0)q^{2} + −177147q^{3} + 136150566429465742254479 + O(17^{2}0)q^{4} + 1475487136319254018803826 + O(17^{2}0)q^{5} + 600056015821561171919451 + O(17^{2}0)q^{6} + −17804217483214445823053783 + O(17^{2}1)q^{7} + −501485567361067812848535+O(17^{2}0)q^{8}+31381059609q^{9}+−1246547326781613792773448+O(17^{2}0)q^{10}+ 2016730508814045867001613+O(17^{2}0)q^{11}+−1515224233872495223091079+O(17^{2}0)q^{12}+1027531096930299476663774+ O(17^{2}0)q^{13} +−1849114175003465883020795+O(17^{2}0)q^{14} +730486771792391650399090+O(17^{2}0)q^{15} + 20229865008411040602674438+O(17^{2}1)q^{16}+1451567020705977279436211∗17+O(17^{2}1)q^{17}+1849406125162401398887858+ O(17^{2}0)q^{18} + 1798793038014490707530038 + O(17^{2}1)q^{19} + 641816010620520858857796 + O(17^{2}0)q^{20} + −13976695946213060373131732 + O(17^{2}1)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

7. Consider f = q +5q^{2} +10q^{3} +16q^{4} +13q^{5} +16q^{6} +9q^{7} +q^{8} +15q^{9} +14q^{10} +q^{11} +7q^{12} +15q^{13} +11q^{14} + 11q^{15} + 14q^{16} + 7q^{18} + 10q^{19} + 4q^{20} + 5q^{21} + ⋯ ∈ S_{24}(Γ_{0}(3); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-6 newform g = q + 11q^{2} + 9q^{3} + 4q^{4} + 6q^{5} + 14q^{6} + 11q^{7} + 15q^{8} + 13q^{9} + 15q^{10} + 14q^{11} + 2q^{12} + 9q^{13} + 2q^{14} + 3q^{15} + 3q^{16} + 15q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{6} & 0 \\ 0 & u(15)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−1809487775611657116862619+ O(17^{2}0)q^{2} +−177147q^{3} +−136150566429465748713195+O(17^{2}0)q^{4} +−1475487136319254065612646+ O(17^{2}0)q^{5} + −600056015821560951902877 + O(17^{2}0)q^{6} + 17804217483214445611089879 + O(17^{2}1)q^{7} + 501485567361070384717551+O(17^{2}0)q^{8}+31381059609q^{9}+1246547326782241662563628+O(17^{2}0)q^{10}+ −2016730508812576894635125+O(17^{2}0)q^{11}+1515224233873639365254331+O(17^{2}0)q^{12}+−1027531096919807822399026+ O(17^{2}0)q^{13} +1849114175038374438025211+O(17^{2}0)q^{14} +−730486771784099608362550+O(17^{2}0)q^{15} + −20229865008462013753351158 + O(17^{2}1)q^{16} + −1451567020718382456584495 ∗ 17 + O(17^{2}1)q^{17} + −1849406125201376674922236+O(17^{2}0)q^{18}+−1798793038921873156067982+O(17^{2}1)q^{19}+−641816011213069900616556+ O(17^{2}0)q^{20} + 13976695946250609142833620 + O(17^{2}1)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

8. Consider f = q + 15q^{2} + 13q^{3} + 2q^{4} + 5q^{5} + 8q^{6} + 9q^{7} + 16q^{9} + 7q^{10} + 3q^{11} + 9q^{12} + 16q^{13} + 16q^{14} + 14q^{15} + 13q^{16} + 2q^{18} + 5q^{19} + 10q^{20} + 15q^{21} + ⋯ ∈ S_{26}(Γ_{0}(3); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 15q^{2} + 4q^{3} + 2q^{4} + 12q^{5} + 9q^{6} + 8q^{7} + 16q^{9} + 10q^{10} + 14q^{11} + 8q^{12} + 16q^{13} + q^{14} + 14q^{15} + 13q^{16} + 8q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(15)ω^{1} & 0 \\ 0 & u(8)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−1764669555281112657569613+ O(17^{2}0)q^{2} + −531441q^{3} + −1303692426227224593319769 + O(17^{2}0)q^{4} + 1134817270801757665565442 + O(17^{2}0)q^{5} + 420275629079879805673184 + O(17^{2}0)q^{6} + 1784289998134858659270670 + O(17^{2}0)q^{7} + −36650837876711176819269 ∗ 17 + O(17^{2}0)q^{8} + 282429536481q^{9} + −1714647668969815676990467 + O(17^{2}0)q^{10} +−655863320549338118402183+O(17^{2}0)q^{11} +2014564004129632128033058+O(17^{2}0)q^{12} + −1693998265115593409846128+O(17^{2}0)q^{13}+525587583930945627466326+O(17^{2}1)q^{14}+−1191011130256519112891133+ O(17^{2}0)q^{15}+−228406845365428852710900+O(17^{2}0)q^{16}+75081120712014029821916∗17+O(17^{2}0)q^{17}+ −1863641523055838182595189+O(17^{2}0)q^{18}+−1479703166450609328709405+O(17^{2}0)q^{19}+−1961825766925148793076152+ O(17^{2}0)q^{20} + 1289743190962322665401845 + O(17^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

9. Consider f = q + 12q^{2} + 4q^{3} + 6q^{4} + 12q^{5} + 14q^{6} + 11q^{7} + 14q^{8} + 16q^{9} + 8q^{10} + 10q^{11} + 7q^{12} + 9q^{13} + 13q^{14} +14q^{15} +3q^{16} +5q^{18} +15q^{19} +4q^{20} +10q^{21} +⋯ ∈ S_{26}(Γ_{0}(3); \overline{F}_{17}). We have ρ_{f} ⊗χ^{15} ∼ ρ_{g}, where g is the weight-8 newform g = q + 6q^{2} + 7q^{3} + 10q^{4} + 16q^{5} + 8q^{6} + 4q^{7} + 6q^{8} + 15q^{9} + 11q^{10} + 4q^{11} + 2q^{12} + 2q^{13} + 7q^{14} + 10q^{15} + 14q^{16} + 13q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{8} & 0 \\ 0 & u(13)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−1036939727548302922360101+ O(17^{2}0)q^{2} + 531441q^{3} + −683661026388986385077647 + O(17^{2}0)q^{4} + 140387410757534150046618 + O(17^{2}0)q^{5} + 914910753352522981045650 + O(17^{2}0)q^{6} + −1838711535300333796225993 + O(17^{2}0)q^{7} + −515126309031449965058321+O(17^{2}0)q^{8}+282429536481q^{9}+285153994003318202587249+O(17^{2}0)q^{10}+ −1922100421574615272915373+O(17^{2}0)q^{11}+531303477079740563723669+O(17^{2}0)q^{12}+1226729857761592714572321+ O(17^{2}0)q^{13} +−1127093537943432043120608+O(17^{2}0)q^{14} +530028565217441198526981+O(17^{2}0)q^{15} + −1283840083026547220936677+O(17^{2}0)q^{16}+84126596741267149777860∗17+O(17^{2}0)q^{17}+825569542474424888147616+ O(17^{2}0)q^{18} +1454189115260760060761978+O(17^{2}0)q^{19} +−572970728930424683196091+O(17^{2}0)q^{20} + 524300137816133401384118 + O(17^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

10. Consider f = q + q^{2} + 4q^{3} + 16q^{4} + 4q^{6} + q^{7} + 14q^{8} + 16q^{9} + 16q^{11} + 13q^{12} + 2q^{13} + q^{14} + 16q^{16} + 16q^{18} + 13q^{19} + 4q^{21} + ⋯ ∈ S_{26}(Γ_{0}(3); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + q^{2} + 13q^{3} + 16q^{4} + 13q^{6} + 16q^{7} + 14q^{8} + 16q^{9} + q^{11} + 4q^{12} + 2q^{13} + 16q^{14} + 16q^{16} + 2q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(9)ω^{1} & 0 \\ 0 & u(2)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+1187469094584318741225099+ O(17^{2}0)q^{2} + 531441q^{3} + 1430363211784221889745260 + O(17^{2}0)q^{4} + 43644816301627748181793 ∗ 17 + O(17^{2}0)q^{5} + 295659189760312021643985 + O(17^{2}0)q^{6} + 1810434252382908064876886 + O(17^{2}0)q^{7} + 1924843515115554982348695+O(17^{2}0)q^{8}+282429536481q^{9}+111368707898142702827691∗17+O(17^{2}0)q^{10}+ 1933122873150520438142105+O(17^{2}0)q^{11}+134491489938580727276625+O(17^{2}0)q^{12}+−1148681544945859278353180+ O(17^{2}0)q^{13}+−97616843731742301751982+O(17^{2}0)q^{14}+76182415655736134316806∗17+O(17^{2}0)q^{15}+ −1628929288270122182874300+O(17^{2}0)q^{16}+−85306435529369754912294∗17+O(17^{2}0)q^{17}+−1834946991820194067263876+ O(17^{2}0)q^{18}+−409027557800706036730972+O(17^{2}0)q^{19}+−22342815001101411759147∗17+O(17^{2}0)q^{20}+ 1295930725258960538963193 + O(17^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

11. Consider f = q + 13q^{2} + 12q^{3} + 8q^{4} + 9q^{5} + 3q^{6} + 5q^{7} + 8q^{9} + 15q^{10} + q^{11} + 11q^{12} + 4q^{13} + 14q^{14} + 6q^{15} + 4q^{16} + 2q^{18} + 10q^{19} + 4q^{20} + 9q^{21} + ⋯ ∈ S_{28}(Γ_{0}(3); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-10 newform g = q + 15q^{2} + 4q^{3} + 2q^{4} + 12q^{5} + 9q^{6} + 8q^{7} + 16q^{9} + 10q^{10} + 14q^{11} + 8q^{12} + 16q^{13} + q^{14} + 14q^{15} + 13q^{16} + 8q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(15)ω^{10} & 0 \\ 0 & u(8)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−498840862120232058054261+ O(17^{2}0)q^{2} + 1594323q^{3} + 658165989010551295738341 + O(17^{2}0)q^{4} + 1453121960907295861028584 + O(17^{2}0)q^{5} + −272776877858874563867017 + O(17^{2}0)q^{6} + 362916201877635297763828 + O(17^{2}0)q^{7} + −31443063760598692213768 ∗ 17 + O(17^{2}0)q^{8} + 2541865828329q^{9} + 1181802975223362082237918 + O(17^{2}0)q^{10} +1459625181751197300953558+O(17^{2}0)q^{11} +1524140559014206659282357+O(17^{2}0)q^{12} + 477428980703432512151809+O(17^{2}0)q^{13}+−1191044947261685870377201+O(17^{2}0)q^{14}+−257345933048619476694201+ O(17^{2}0)q^{15}+−2005127986181907365715461+O(17^{2}0)q^{16}+103187330478696446376977∗17+O(17^{2}0)q^{17}+ −1368570270973528570829486+O(17^{2}0)q^{18}+1226172466595712370928270+O(17^{2}0)q^{19}+−430852350620440381691444+ O(17^{2}0)q^{20} + 1343518775478685015622079 + O(17^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

12. Consider f = q + 7q^{2} + 5q^{3} + 7q^{4} + 8q^{5} + q^{6} + 8q^{7} + 10q^{8} + 8q^{9} + 5q^{10} + 9q^{11} + q^{12} + 15q^{13} + 5q^{14} + 6q^{15} + 14q^{16} + 5q^{18} + 13q^{19} + 5q^{20} + 6q^{21} + ⋯ ∈ S_{28}(Γ_{0}(3); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + 6q^{2} + 7q^{3} + 10q^{4} + 16q^{5} + 8q^{6} + 4q^{7} + 6q^{8} + 15q^{9} + 11q^{10} + 4q^{11} + 2q^{12} + 2q^{13} + 7q^{14} + 10q^{15} + 14q^{16} + 13q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{1} & 0 \\ 0 & u(13)ω^{10} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+1078879147100498046268459+ O(17^{2}0)q^{2} + −1594323q^{3} + 388024039005853595520569 + O(17^{2}0)q^{4} + 183857111306788588703878 + O(17^{2}0)q^{5} + −1565595695113399973176633 + O(17^{2}0)q^{6} + 332903030674895106125856 + O(17^{2}0)q^{7} + 13508308556010320594125289+O(17^{2}1)q^{8}+2541865828329q^{9}+−1448946953756872354678940+O(17^{2}0)q^{10}+ −1794974519879740519902477+O(17^{2}0)q^{11}+1333622930729526219566428+O(17^{2}0)q^{12}+−1258049687254691478724355+ O(17^{2}0)q^{13}+−1599097734932922025409803+O(17^{2}0)q^{14}+1004703076417497560185930+O(17^{2}0)q^{15}+ 675149908603050894191270+O(17^{2}0)q^{16}+1795107071007808435761046∗17+O(17^{2}1)q^{17}+1250102047930511856193905+ O(17^{2}0)q^{18}+1578509825228872958667892+O(17^{2}0)q^{19}+−1034545783729012166742082+O(17^{2}0)q^{20}+ 1149282229000561576762304 + O(17^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

13. Consider f = q + 2q^{2} + 5q^{3} + 13q^{4} + 10q^{6} + 10q^{7} + 10q^{8} + 8q^{9} + 11q^{11} + 14q^{12} + 9q^{13} + 3q^{14} + q^{16} + 16q^{18} + 9q^{19} + 16q^{21} + ⋯ ∈ S_{28}(Γ_{0}(3); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-10 newform g = q + q^{2} + 13q^{3} + 16q^{4} + 13q^{6} + 16q^{7} + 14q^{8} + 16q^{9} + q^{11} + 4q^{12} + 2q^{13} + 16q^{14} + 16q^{16} + 2q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(9)ω^{10} & 0 \\ 0 & u(2)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−1078879147100498046246877+ O(17^{2}0)q^{2} + −1594323q^{3} + −388024039005853392916725 + O(17^{2}0)q^{4} + −10815124194517080038234 ∗ 17+O(17^{2}0)q^{5}+1565595695113365564497647+O(17^{2}0)q^{6}+−332903030674525440925952+O(17^{2}0)q^{7}+ −13508308556005891274252465 + O(17^{2}1)q^{8} + 2541865828329q^{9} + 85232173749526040471920 ∗ 17 + O(17^{2}0)q^{10}+1794974519955502855570725+O(17^{2}0)q^{11}+−1333622931052542187944040+O(17^{2}0)q^{12}+ 1258049687151670399546767+O(17^{2}0)q^{13}+1599097743204389711992939+O(17^{2}0)q^{14}+−59100180799555478717390∗ 17 + O(17^{2}0)q^{15} + −675149840375521460490390 + O(17^{2}0)q^{16} + −1795107068967125401693522 ∗ 17 + O(17^{2}1)q^{17}+−1250101993071963549197427+O(17^{2}0)q^{18}+−1578509713608396400025676+O(17^{2}0)q^{19}+ 60855629099196552921146 ∗ 17 + O(17^{2}0)q^{20} + −1149282818366292083307296 + O(17^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

14. Consider f = q + 6q^{2} + 2q^{3} + 4q^{4} + 7q^{5} + 12q^{6} + 7q^{7} + 2q^{8} + 4q^{9} + 8q^{10} + 12q^{11} + 8q^{12} + 9q^{13} + 8q^{14} + 14q^{15} + 3q^{16} + 7q^{18} + 12q^{19} + 11q^{20} + 14q^{21} + ⋯ ∈ S_{30}(Γ_{0}(3); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 11q^{2} + 9q^{3} + 4q^{4} + 6q^{5} + 14q^{6} + 11q^{7} + 15q^{8} + 13q^{9} + 15q^{10} + 14q^{11} + 2q^{12} + 9q^{13} + 2q^{14} + 3q^{15} + 3q^{16} + 15q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{1} & 0 \\ 0 & u(15)ω^{12} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−41756625880499173621179+ O(17^{2}0)q^{2} + 4782969q^{3} + 1195958254061210381257976 + O(17^{2}0)q^{4} + 1406507172053536273255844 + O(17^{2}0)q^{5} + −251576956890632379825710 + O(17^{2}0)q^{6} + 432104670952007023111472 + O(17^{2}0)q^{7} + −1058416697312187358705777+O(17^{2}0)q^{8}+22876792454961q^{9}+1483592276952417433397901+O(17^{2}0)q^{10}+ −182978824082042187167764+O(17^{2}0)q^{11}+311562920876373290072087+O(17^{2}0)q^{12}+1782573321757109280051330+ O(17^{2}0)q^{13}+−1792097706999618935561794+O(17^{2}0)q^{14}+1808670402393378204881596+O(17^{2}0)q^{15}+ −269800250264507421788205+O(17^{2}0)q^{16}+114421886085716684977494∗17+O(17^{2}0)q^{17}+13949695790887278470277+ O(17^{2}0)q^{18}+1116048844506935818035457+O(17^{2}0)q^{19}+−1995640923513901301769324+O(17^{2}0)q^{20}+ 291010226500232791979848 + O(17^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

15. Consider f = q + 3q^{2} + 2q^{3} + 11q^{4} + 10q^{5} + 6q^{6} + 7q^{7} + 5q^{8} + 4q^{9} + 13q^{10} + 7q^{11} + 5q^{12} + 6q^{13} + 4q^{14} + 3q^{15} + 3q^{16} + 12q^{18} + 12q^{19} + 8q^{20} + 14q^{21} + ⋯ ∈ S_{30}(Γ_{0}(3); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-12 newform g = q + 10q^{2} + 12q^{3} + 7q^{4} + 2q^{5} + q^{6} + q^{7} + 7q^{8} + 8q^{9} + 3q^{10} + 13q^{11} + 16q^{12} + 7q^{13} + 10q^{14} + 7q^{15} + 14q^{16} + 10q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(12)ω^{12} & 0 \\ 0 & u(10)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+147803801266069632315553+ O(17^{2}0)q^{2} + 4782969q^{3} + 99943124569845925464412 + O(17^{2}0)q^{4} + −892878413205990061974723 + O(17^{2}0)q^{5} + 460202679743515108935715 + O(17^{2}0)q^{6} + −792543588828618589738538 + O(17^{2}0)q^{7} + −719114756950792703453950+O(17^{2}0)q^{8}+22876792454961q^{9}+−502931698089669576154435+O(17^{2}0)q^{10}+ 823523447359484785862997+O(17^{2}0)q^{11}+−1903006362788667238307190+O(17^{2}0)q^{12}+−1683715987488189306164034+ O(17^{2}0)q^{13}+−704097967053289467655985+O(17^{2}0)q^{14}+−757887711473216546995408+O(17^{2}0)q^{15}+ −1724206512101752995985057+O(17^{2}0)q^{16}+67368342939991404476342∗17+O(17^{2}0)q^{17}+256098121857381231957048+ O(17^{2}0)q^{18} + 1156603596771722211833142 + O(17^{2}0)q^{19} + 410756779555204006239364 + O(17^{2}0)q^{20} + 1280695568511673350294979 + O(17^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

16. Consider f = q + 10q^{2} + 6q^{3} + 6q^{4} + 12q^{5} + 9q^{6} + 16q^{7} + 4q^{8} + 2q^{9} + q^{10} + 12q^{11} + 2q^{12} + 3q^{13} + 7q^{14} + 4q^{15} + 3q^{16} + 3q^{18} + 4q^{19} + 4q^{20} + 11q^{21} + ⋯ ∈ S_{32}(Γ_{0}(3); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-14 newform g = q + 5q^{2} + 2q^{3} + 10q^{4} + 16q^{5} + 10q^{6} + 12q^{7} + 9q^{8} + 4q^{9} + 12q^{10} + 15q^{11} + 3q^{12} + 12q^{13} + 9q^{14} + 15q^{15} + 14q^{16} + 11q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(14)ω^{14} & 0 \\ 0 & u(11)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−1527147078044190977533742+ O(17^{2}0)q^{2}+14348907q^{3}+−959382005613716159140757+O(17^{2}0)q^{4}+−1872337447359805329164212+ O(17^{2}0)q^{5} + 1544316512925673226703651 + O(17^{2}0)q^{6} + 277041826161315286335252 + O(17^{2}0)q^{7} + −396763266531381095565739+O(17^{2}0)q^{8}+205891132094649q^{9}+1616544504040887855558169+O(17^{2}0)q^{10}+ −1912107838086011026998860+O(17^{2}0)q^{11}+1012604907873466735094132+O(17^{2}0)q^{12}+−1538466958665134039568739+ O(17^{2}0)q^{13} + −243165367714294167272998 + O(17^{2}0)q^{14} + 238380945464748279855667 + O(17^{2}0)q^{15} + −1349067725820102977588158+O(17^{2}0)q^{16}+−34447113725164759943397∗17+O(17^{2}0)q^{17}+−435136324013773793112005+ O(17^{2}0)q^{18} +837714112760718225962537+O(17^{2}0)q^{19} +−1572516595467718306106826+O(17^{2}0)q^{20} + −1725531550528293238578142 + O(17^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

17. Consider f = q + z^{1}82q^{2} + 11q^{3} + z^{1}05q^{4} + z^{2}45q^{5} + z^{2}0q^{6} + z^{1}74q^{7} + z^{6}2q^{8} + 2q^{9} + z^{1}39q^{10} + z^{3}q^{11} + z^{2}31q^{12}+z^{2}13q^{13}+z^{6}8q^{14}+z^{8}3q^{15}+z^{2}81q^{16}+z^{1}46q^{18}+z^{1}43q^{19}+z^{6}2q^{20}+z^{1}2q^{21}+⋯ ∈ S_{32}(Γ_{0}(3); \overline{F}_{17}). Also, z satisfies the equation x^{2}+16x+3 = 0. We have ρ_{f} ⊗χ^{15} ∼ ρ_{g}, where g is the weight-14 newform g = q + z^{2}18q^{2} + 15q^{3} + z^{1}77q^{4} + z^{1}55q^{5} + z^{3}8q^{6} + z^{2}64q^{7} + z^{1}70q^{8} + 4q^{9} + z^{8}5q^{10} + z^{1}65q^{11} + z^{2}85q^{12} + z^{1}41q^{13} + z^{1}94q^{14} + z^{2}63q^{15} + z^{1}37q^{16} + z^{1}83q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}05)ω^{14} & 0 \\ 0 & u(z^{1}83)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2} +−14348907q^{3} + (1181748630991128078220631+O(17^{2}0))∗a−138019272764055822528443+O(17^{2}0)q^{4}+−(1125623730094310354160619+ O(17^{2}0))∗a−76346776274125345461539+O(17^{2}0)q^{5}+−14348907∗aq^{6}+(1142610847855095116448918+ O(17^{2}0))∗a−1345096020355181393279755+O(17^{2}0)q^{7} +−(1614031400674441339588285+O(17^{2}0))∗ a − 100960223071720572675637 + O(17^{2}0)q^{8} + 205891132094649q^{9} + −(1226290992152330921377439 + O(17^{2}0))∗a−710025218026609947631258+O(17^{2}0)q^{10}+−(970531699224239726932865+O(17^{2}0))∗a+ 1792951367773677279429543+O(17^{2}0)q^{11}+(1328271614048777489598287+O(17^{2}0))∗a−1099194972493591763476681+ O(17^{2}0)q^{12}+(1738883593340265125992006+O(17^{2}0))∗a+1497848179485714086708313+O(17^{2}0)q^{13}+ (126338698042369198435712+O(17^{2}0))∗a+1398928902543507301823355+O(17^{2}0)q^{14}+−(794165645024379864565420+ O(17^{2}0))∗a−461997588847180244353672+O(17^{2}0)q^{15} +−(18415290268544453768316064+O(17^{2}1))∗ a − 28906573058019856791552084 + O(17^{2}1)q^{16} + (71055441058204435752355 ∗ 17 + O(17^{2}0)) ∗ a + 31199459463129961211615∗17+O(17^{2}0)q^{17}+205891132094649∗aq^{18}+(1570515310662150888206751+ O(17^{2}0))∗a−603459583328178250347080+O(17^{2}0)q^{19}+−(998833513964889380396968+O(17^{2}0))∗a− 434312634174380531675904+O(17^{2}0)q^{20}+−(1628631028909826053757000+O(17^{2}0))∗a+725498101095661009677678+ O(17^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −1181748630991128078220631 + O(17^{2}0)x + 138019272764053675044795 + O(17^{2}0). The slope of f_{0} is 1.

18. Consider f = q+z^{2}14q^{2}+11q^{3}+z^{5}7q^{4}+z^{1}33q^{5}+z^{5}2q^{6}+z^{7}8q^{7}+z^{1}90q^{8}+2q^{9}+z^{5}9q^{10}+z^{5}1q^{11}+z^{1}83q^{12}+ z^{1}65q^{13} + z^{4}q^{14} + z^{2}59q^{15} + z^{1}69q^{16} + z^{1}78q^{18} + z^{1}27q^{19} + z^{1}90q^{20} + z^{2}04q^{21} + ⋯ ∈ S_{32}(Γ_{0}(3); \overline{F}_{17}). Also, z satisfies the equation x^{2} + 16x + 3 = 0. We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-14 newform g = q + z^{2}50q^{2} + 15q^{3} + z^{1}29q^{4} + z^{4}3q^{5} + z^{7}0q^{6} + z^{1}68q^{7} + z^{1}0q^{8} + 4q^{9} + z^{5}q^{10} + z^{2}13q^{11} + z^{2}37q^{12} + z^{9}3q^{13} + z^{1}30q^{14} + z^{1}51q^{15} + z^{2}5q^{16} + z^{2}31q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{5}7)ω^{14} & 0 \\ 0 & u(z^{2}31)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2} +−14348907q^{3} + (1181748630991128078220631+O(17^{2}0))∗a−138019272764055822528443+O(17^{2}0)q^{4}+−(1125623730094310354160619+ O(17^{2}0))∗a−76346776274125345461539+O(17^{2}0)q^{5}+−14348907∗aq^{6}+(1142610847855095116448918+ O(17^{2}0))∗a−1345096020355181393279755+O(17^{2}0)q^{7} +−(1614031400674441339588285+O(17^{2}0))∗ a − 100960223071720572675637 + O(17^{2}0)q^{8} + 205891132094649q^{9} + −(1226290992152330921377439 + O(17^{2}0))∗a−710025218026609947631258+O(17^{2}0)q^{10}+−(970531699224239726932865+O(17^{2}0))∗a+ 1792951367773677279429543+O(17^{2}0)q^{11}+(1328271614048777489598287+O(17^{2}0))∗a−1099194972493591763476681+ O(17^{2}0)q^{12}+(1738883593340265125992006+O(17^{2}0))∗a+1497848179485714086708313+O(17^{2}0)q^{13}+ (126338698042369198435712+O(17^{2}0))∗a+1398928902543507301823355+O(17^{2}0)q^{14}+−(794165645024379864565420+ O(17^{2}0))∗a−461997588847180244353672+O(17^{2}0)q^{15} +−(18415290268544453768316064+O(17^{2}1))∗ a − 28906573058019856791552084 + O(17^{2}1)q^{16} + (71055441058204435752355 ∗ 17 + O(17^{2}0)) ∗ a + 31199459463129961211615∗17+O(17^{2}0)q^{17}+205891132094649∗aq^{18}+(1570515310662150888206751+ O(17^{2}0))∗a−603459583328178250347080+O(17^{2}0)q^{19}+−(998833513964889380396968+O(17^{2}0))∗a− 434312634174380531675904+O(17^{2}0)q^{20}+−(1628631028909826053757000+O(17^{2}0))∗a+725498101095661009677678+ O(17^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −1181748630991128078220631 + O(17^{2}0)x + 138019272764053675044795 + O(17^{2}0). The slope of f_{0} is 1.

19. Consider f = q + 8q^{2} + q^{3} + 11q^{4} + 13q^{5} + 8q^{6} + 7q^{7} + 4q^{8} + q^{9} + 2q^{10} + 8q^{11} + 11q^{12} + 12q^{13} + 5q^{14} + 13q^{15} + 10q^{16} + 8q^{18} + 12q^{19} + 7q^{20} + 7q^{21} + ⋯ ∈ S_{34}(Γ_{0}(3); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-16 newform g = q + 4q^{2} + 6q^{3} + 7q^{4} + 6q^{5} + 7q^{6} + q^{7} + 9q^{8} + 2q^{9} + 7q^{10} + 10q^{11} + 8q^{12} + 14q^{13} + 4q^{14} + 2q^{15} + 7q^{16} + 8q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(15)ω^{0} & 0 \\ 0 & u(8)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+19849414159963566068725+ O(17^{2}0)q^{2} + 43046721q^{3} + 534286844283380130949320 + O(17^{2}0)q^{4} + −1167613572269368006186308 + O(17^{2}0)q^{5} + 375118035294333326511288 + O(17^{2}0)q^{6} + −599253262563688918418900 + O(17^{2}0)q^{7} + 1331760863295274880044190+O(17^{2}0)q^{8}+1853020188851841q^{9}+−127397375926119859457541+O(17^{2}0)q^{10}+ −691378947800319228900011+O(17^{2}0)q^{11}+−1855736797682015786405634+O(17^{2}0)q^{12}+953255537497169962501948+ O(17^{2}0)q^{13}+−1296982816164155440488268+O(17^{2}0)q^{14}+−426885770076484959892370+O(17^{2}0)q^{15}+ −1607472867433239862335204+O(17^{2}0)q^{16}+−20661404777815467174828∗17+O(17^{2}0)q^{17}+−458599557448993005448053+ O(17^{2}0)q^{18}+−1250257929189255572738059+O(17^{2}0)q^{19}+−573048542390504088307786+O(17^{2}0)q^{20}+ 76106426871483977003352 + O(17^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

20. Consider f = q + 9q^{2} + 16q^{3} + 11q^{4} + 15q^{5} + 8q^{6} + 11q^{7} + 13q^{8} + q^{9} + 16q^{10} + 15q^{11} + 6q^{12} + 15q^{13} + 14q^{14} + 2q^{15} + 10q^{16} + 9q^{18} + 11q^{19} + 12q^{20} + 6q^{21} + ⋯ ∈ S_{34}(Γ_{0}(3); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-16 newform g = q + 13q^{2} + 11q^{3} + 7q^{4} + 3q^{5} + 7q^{6} + 4q^{7} + 8q^{8} + 2q^{9} + 5q^{10} + 6q^{11} + 9q^{12} + 9q^{13} + q^{14} + 16q^{15} + 7q^{16} + 12q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(10)ω^{0} & 0 \\ 0 & u(12)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−137626568599290369593589+ O(17^{2}0)q^{2}+−43046721q^{3}+1163672051649318360067854+O(17^{2}0)q^{4}+−1828741216582296655179825+ O(17^{2}0)q^{5} + −721549460062007391083617 + O(17^{2}0)q^{6} + 1411538042051639939058617 + O(17^{2}0)q^{7} + 1131951653367826599320240+O(17^{2}0)q^{8}+1853020188851841q^{9}+1108677274531242990187294+O(17^{2}0)q^{10}+ −829060279608925907981982+O(17^{2}0)q^{11}+−356972062163967837239983+O(17^{2}0)q^{12}+−1261671733587516234079192+ O(17^{2}0)q^{13}+−1513553515882598274483513+O(17^{2}0)q^{14}+−389128722155821676243874+O(17^{2}0)q^{15}+ 388359084701184594646736+O(17^{2}0)q^{16}+−59513413531618917605518∗17+O(17^{2}0)q^{17}+−1559074300716249362756508+ O(17^{2}0)q^{18}+1728095203031579888061258+O(17^{2}0)q^{19}+−1127335354598668089787365+O(17^{2}0)q^{20}+ 92199581920853204623992 + O(17^{2}0)q^{21} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

21. Consider f = q+6q2 +3q3 +11q4 +13q5 +q6 +5q7 +q8 +9q9 +10q10 +13q11 +16q12 +12q13 +13q14 +5q15 +3q16 +3q18 +6q19 +7q20 +15q21 +⋯ ∈ S20(Γ0(3); \overline{F}17). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

22. Consider f = q + 11q2 + 9q3 + 4q4 + 6q5 + 14q6 + 11q7 + 15q8 + 13q9 + 15q10 + 14q11 + 2q12 + 9q13 + 2q14 + 3q15 + 3q16 + 15q17 + 7q18 + 5q19 + 7q20 + 14q21 + ⋯ ∈ S22(Γ0(3); \overline{F}17). Since f is ordinary, we do not continue our analysis.

23. Consider f = q + 6q2 + 7q3 + 10q4 + 16q5 + 8q6 + 4q7 + 6q8 + 15q9 + 11q10 + 4q11 + 2q12 + 2q13 + 7q14 + 10q15 + 14q16 + 13q17 + 5q18 + 16q19 + 7q20 + 11q21 + ⋯ ∈ S24(Γ0(3); \overline{F}17). Since f is ordinary, we do not continue our analysis.

24. Consider f = q+q2 +13q3 +16q4 +13q6 +16q7 +14q8 +16q9 +q11 +4q12 +2q13 +16q14 +16q16 +2q17 +16q18 +13q19 +4q21 +⋯ ∈ S26(Γ0(3); \overline{F}17). Since f is ordinary, we do not continue our analysis.

25. Consider f = q+15q2 +4q3 +2q4 +12q5 +9q6 +8q7 +16q9 +10q10 +14q11 +8q12 +16q13 +q14 +14q15 +13q16 +8q17 +2q18 +5q19 +7q20 +15q21 +⋯ ∈ S26(Γ0(3); \overline{F}17). Since f is ordinary, we do not continue our analysis.

26. Consider f = q + 10q2 + 12q3 + 7q4 + 2q5 + q6 + q7 + 7q8 + 8q9 + 3q10 + 13q11 + 16q12 + 7q13 + 10q14 + 7q15 + 14q16 + 10q17 + 12q18 + 6q19 + 14q20 + 12q21 + ⋯ ∈ S28(Γ0(3); \overline{F}17). Since f is ordinary, we do not continue our analysis.

27. Consider f = q + z218q2 + 15q3 + z177q4 + z155q5 + z38q6 + z264q7 + z170q8 + 4q9 + z85q10 + z165q11 + z285q12 + z141q13 + z194q14 + z263q15 + z137q16 + z183q17 + z146q18 + z179q19 + z44q20 + z84q21 + ⋯ ∈ S30(Γ0(3); \overline{F}17). Also, z satisfies the equation x2 + 16x + 3 = 0. Since f is ordinary, we do not continue our analysis.

28. Consider f = q + z250q2 + 15q3 + z129q4 + z43q5 + z70q6 + z168q7 + z10q8 + 4q9 + z5q10 + z213q11 + z237q12 + z93q13 + z130q14 + z151q15 + z25q16 + z231q17 + z178q18 + z163q19 + z172q20 + z276q21 + ⋯ ∈ S30(Γ0(3); \overline{F}17). Also, z satisfies the equation x2 + 16x + 3 = 0. Since f is ordinary, we do not continue our analysis.

29. Consider f = q + 5q2 + 2q3 + 10q4 + 16q5 + 10q6 + 12q7 + 9q8 + 4q9 + 12q10 + 15q11 + 3q12 + 12q13 + 9q14 + 15q15 + 14q16 + 11q17 + 3q18 + 2q19 + 7q20 + 7q21 + ⋯ ∈ S30(Γ0(3); \overline{F}17). Since f is ordinary, we do not continue our analysis.

30. Consider f = q+4q2 +6q3 +7q4 +6q5 +7q6 +q7 +9q8 +2q9 +7q10 +10q11 +8q12 +14q13 +4q14 +2q15 +7q16 +8q17 +8q18 +6q19 +8q20 +6q21 +⋯ ∈ S32(Γ0(3); \overline{F}17). Since f is ordinary, we do not continue our analysis.

31. Consider f = q+13q2+11q3+7q4+3q5+7q6+4q7+8q8+2q9+5q10+6q11+9q12+9q13+q14+16q15+7q16+12q17+9q18+14q19+4q20+10q21+⋯ ∈ S32(Γ0(3); \overline{F}17). Also, z satisfies the equation x2 + 16x + 3 = 0. Since f is ordinary, we do not continue our analysis.

32. Consider f = q + q3 + 15q4 + 3q5 + 13q7 + q9 + 14q11 + 15q12 + 16q13 + 3q15 + 4q16 + 16q17 + 16q19 + 11q20 + 13q21 + ⋯ ∈ S34(Γ0(3); \overline{F}17). Since f is ordinary, we do not continue our analysis.

33. Consider f = q + 3q2 + q3 + 7q4 + 6q5 + 3q6 + 8q7 + 15q8 + q9 + q10 + 12q11 + 7q12 + 14q13 + 7q14 + 6q15 + 14q16 + 3q18 + 3q19 + 8q20 + 8q21 + ⋯ ∈ S34(Γ0(3); \overline{F}17). I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

34. Consider f = q+8q2+16q3+11q4+10q5+9q6+4q8+q9+12q10+8q11+6q12+11q13+7q15+10q16+q17+8q18+10q19+8q20+⋯ ∈ S34(Γ0(3); \overline{F}17). Since f is ordinary, we do not continue our analysis.

17.3 N = 4, ℓ = 17 (17 forms)

1. Consider f = q +15q^{3} +9q^{5} +5q^{7} +11q^{9} +7q^{11} +13q^{13} +16q^{15} +4q^{19} +7q^{21} +q^{23} +7q^{25} +15q^{27} +⋯ ∈ S_{20}(Γ_{0}(4); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{14} ∼ ρ_{g}, where g is the weight-16 newform g = q + 13q^{3} + 16q^{5} + 6q^{7} + 10q^{9} + 12q^{11} + 4q^{13} + 4q^{15} + 11q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(14)ω^{1} & 0 \\ 0 & u(11)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −36q^{3} + −196290q^{5} + −35905576q^{7} + −1162260171q^{9} + −12016099980q^{11} + −45529656874q^{13} + 7066440q^{15} + 29209602834∗17q^{17}+1410273986444q^{19}+1292600736q^{21}+−7039745388792q^{23}+−19034956564025q^{25}+ 83682778968q^{27} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q +15q^{3} +3q^{5} +8q^{7} +16q^{9} +2q^{11} +8q^{13} +11q^{15} +7q^{19} +q^{21} +16q^{23} +12q^{25} +12q^{27} +⋯ ∈ S_{22}(Γ_{0}(4); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{12} ∼ ρ_{g}, where g is the weight-14 newform g = q + 9q^{3} + 12q^{5} + 2q^{7} + q^{9} + 9q^{11} + 8q^{13} + 6q^{15} + q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{1} & 0 \\ 0 & u(1)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−1222747516153087907728288+ O(17^{2}0)q^{3} + 26927242356804998346887498 + O(17^{2}1)q^{5} + −661941095775512221607723 + O(17^{2}0)q^{7} + −705141812428086821544991+O(17^{2}0)q^{9}+−1641046322576108099072683+O(17^{2}0)q^{11}+−1137230688856013931827387+ O(17^{2}0)q^{13}+681820478611502628967010+O(17^{2}0)q^{15}+−102699512293818649567368∗17+O(17^{2}0)q^{17}+ −383782217965166857586912+O(17^{2}0)q^{19}+−1584876214099092368820700+O(17^{2}0)q^{21}+−556455887269917245645817+ O(17^{2}0)q^{23}+−17075227400904300742107806+O(17^{2}1)q^{25}+−1663988983675815567852138+O(17^{2}0)q^{27}+ ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q +15q^{3} +15q^{5} +13q^{7} +10q^{9} +7q^{11} +6q^{13} +4q^{15} +6q^{19} +8q^{21} +9q^{23} +11q^{25} +2q^{27} +⋯ ∈ S_{24}(Γ_{0}(4); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-6 newform g = q + 5q^{3} + 3q^{5} + 14q^{7} + 3q^{9} + 13q^{11} + 7q^{13} + 15q^{15} + 16q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(16)ω^{6} & 0 \\ 0 & u(16)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−144330902402074399056751+ O(17^{2}0)q^{3} + 718290570816297564922511 + O(17^{2}0)q^{5} + 856475039799254683086629 + O(17^{2}0)q^{7} + 1679921055972658268319845+O(17^{2}0)q^{9}+−1362195325676858164122032+O(17^{2}0)q^{11}+327373116144457622327086+ O(17^{2}0)q^{13}+−1605991550633559246881436+O(17^{2}0)q^{15}+507604241537257103389671∗17+O(17^{2}1)q^{17}+ 1300121183486264584144841+O(17^{2}0)q^{19}+1610156026852349016313948+O(17^{2}0)q^{21}+122735997540332967355225+ O(17^{2}0)q^{23} +−60182982229896237383841+O(17^{2}0)q^{25} +28810522476503310675170793+O(17^{2}1)q^{27} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

4. Consider f = q +12q^{3} +3q^{5} +2q^{7} +14q^{9} +10q^{11} +15q^{13} +2q^{15} +5q^{19} +7q^{21} +2q^{23} +16q^{25} +2q^{27} +⋯ ∈ S_{24}(Γ_{0}(4); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{10} ∼ ρ_{g}, where g is the weight-12 newform g = q + 11q^{3} + 10q^{5} + 4q^{7} + 12q^{9} + 14q^{11} + 2q^{13} + 8q^{15} + 11q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(14)ω^{1} & 0 \\ 0 & u(11)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+144330902402074399227271+ O(17^{2}0)q^{3} + −718290570816297657188531 + O(17^{2}0)q^{5} + −856475039799254491003189 + O(17^{2}0)q^{7} + −1679921055972649668440027+O(17^{2}0)q^{9}+1362195325675610989426232+O(17^{2}0)q^{11}+−327373116151917922308066+ O(17^{2}0)q^{13}+1605991550527224886789356+O(17^{2}0)q^{15}+−507604241559309888560451∗17+O(17^{2}1)q^{17}+ −1300121184326624863357393+O(17^{2}0)q^{19}+−1610156026451947721234716+O(17^{2}0)q^{21}+−122735991106975044282505+ O(17^{2}0)q^{23} +60183015817137533062991+O(17^{2}0)q^{25} +−28810522460729449584046713+O(17^{2}1)q^{27} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

5. Consider f = q +10q^{3} +3q^{5} +4q^{7} +q^{9} +7q^{11} +13q^{13} +13q^{15} +13q^{19} +6q^{21} +7q^{23} +14q^{25} +6q^{27} +⋯ ∈ S_{26}(Γ_{0}(4); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 7q^{3} + 14q^{5} + 13q^{7} + q^{9} + 10q^{11} + 13q^{13} + 13q^{15} + 6q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(3)ω^{1} & 0 \\ 0 & u(6)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−711470265209058623093979+ O(17^{2}0)q^{3} + 1117251965688628093643184 + O(17^{2}0)q^{5} + 2002532775569047953158953 + O(17^{2}0)q^{7} + −566377435401707999136507+O(17^{2}1)q^{9}+1235137330092096372992322+O(17^{2}0)q^{11}+−263768615577574456023250+ O(17^{2}1)q^{13}+11106086423221115858930545+O(17^{2}1)q^{15}+73399867830850282055128∗17+O(17^{2}0)q^{17}+ −421270683092509446108055880+O(17^{2}2)q^{19}+−14403277468384892440718209+O(17^{2}1)q^{21}+−1660983000587492066175252+ O(17^{2}0)q^{23}+−17124904275956446035034840+O(17^{2}1)q^{25}+−1402623840580821986547138+O(17^{2}0)q^{27}+ ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

6. Consider f = q + 4q^{3} + 2q^{5} + 6q^{7} + 9q^{9} + 8q^{11} + 16q^{13} + 8q^{15} + 9q^{19} + 7q^{21} + 9q^{23} + 10q^{25} + 8q^{27} + ⋯ ∈ S_{28}(Γ_{0}(4); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-10 newform g = q + 7q^{3} + 14q^{5} + 13q^{7} + q^{9} + 10q^{11} + 13q^{13} + 13q^{15} + 6q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(3)ω^{10} & 0 \\ 0 & u(6)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+1276053912854697948640261+ O(17^{2}0)q^{3} + −778421631537567486346256 + O(17^{2}0)q^{5} + −1371030742086365928726583 + O(17^{2}0)q^{7} + −1718072881062479412624817+O(17^{2}0)q^{9}+1640089634315536876102871+O(17^{2}0)q^{11}+1128336425173433586807050+ O(17^{2}0)q^{13}+8290785025720049101408018+O(17^{2}1)q^{15}+101119013574096500322613∗17+O(17^{2}0)q^{17}+ −2025404785410380986040327+O(17^{2}0)q^{19}+1841476217469071619768925+O(17^{2}0)q^{21}+544987947296047984391315+ O(17^{2}0)q^{23}+−1040583522682114715288447+O(17^{2}0)q^{25}+1931215765283152280949348+O(17^{2}0)q^{27}+ ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

7. Consider f = q + 16q^{3} + 16q^{5} + 11q^{7} + 6q^{9} + q^{11} + 9q^{13} + q^{15} + 6q^{19} + 6q^{21} + 10q^{23} + 15q^{25} + 6q^{27} + ⋯ ∈ S_{30}(Γ_{0}(4); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-12 newform g = q + 11q^{3} + 10q^{5} + 4q^{7} + 12q^{9} + 14q^{11} + 2q^{13} + 8q^{15} + 11q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(14)ω^{12} & 0 \\ 0 & u(11)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−508361781394698961908743+ O(17^{2}0)q^{3} + −716023627278416958072988 + O(17^{2}0)q^{5} + 1425910426593735256738198 + O(17^{2}0)q^{7} + 33659392063554492646421+O(17^{2}0)q^{9}+1910612643100910713131145+O(17^{2}0)q^{11}+−1781424677041798706957987+ O(17^{2}0)q^{13}+−589913424902190569737204+O(17^{2}0)q^{15}+−13945148979351436586341∗17+O(17^{2}0)q^{17}+ 1578170777036444865278148+O(17^{2}0)q^{19}+−324074626493115168550678+O(17^{2}0)q^{21}+−1683734858112881481361200+ O(17^{2}0)q^{23}+−401549972782207487252212+O(17^{2}0)q^{25}+−270726655466762218628454+O(17^{2}0)q^{27}+ ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

8. Consider f = q+3q^{3} +12q^{5} +12q^{7} +14q^{9} +16q^{11} +7q^{13} +2q^{15} +14q^{19} +2q^{21} +11q^{23} +5q^{25} +6q^{27} +⋯ ∈ S_{30}(Γ_{0}(4); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 5q^{3} + 3q^{5} + 14q^{7} + 3q^{9} + 13q^{11} + 7q^{13} + 15q^{15} + 16q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(16)ω^{1} & 0 \\ 0 & u(16)ω^{12} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−702193762329398510293818+ O(17^{2}0)q^{3} + −599578859644492641242337 + O(17^{2}0)q^{5} + −217752783173430695124511 + O(17^{2}0)q^{7} + −1954884322338570020203294+O(17^{2}0)q^{9}+−424983670754770160220129+O(17^{2}0)q^{11}+−1414483053212825347482671+ O(17^{2}0)q^{13}+353040933777905739390160+O(17^{2}0)q^{15}+−10362591414896381348878∗17+O(17^{2}0)q^{17}+ 1414718884949989587741804+O(17^{2}0)q^{19}+−1591380390341565914744829+O(17^{2}0)q^{21}+−131830353351002795356447+ O(17^{2}0)q^{23}+277996712659906673763280+O(17^{2}0)q^{25}+543167857310704095747561+O(17^{2}0)q^{27}+⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

9. Consider f = q +10q^{3} +9q^{5} +14q^{7} +9q^{9} +14q^{11} +2q^{13} +5q^{15} +3q^{19} +4q^{21} +7q^{23} +6q^{25} +13q^{27} +⋯ ∈ S_{32}(Γ_{0}(4); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-14 newform g = q + 9q^{3} + 12q^{5} + 2q^{7} + q^{9} + 9q^{11} + 8q^{13} + 6q^{15} + q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(1)ω^{14} & 0 \\ 0 & u(1)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+970442270063727724690945+ O(17^{2}0)q^{3} + −1327296510757648167836176 + O(17^{2}0)q^{5} + 276626578827322618077093 + O(17^{2}0)q^{7} + 1371325187987872575254603+O(17^{2}0)q^{9}+606358036514418400133938+O(17^{2}0)q^{11}+351811593177435427003987+ O(17^{2}0)q^{13}+1901822200876072615356622+O(17^{2}0)q^{15}+−1513445978080418017274∗17+O(17^{2}0)q^{17}+ 1251494747789147653625394+O(17^{2}0)q^{19}+−153375751662595471826543+O(17^{2}0)q^{21}+−1384872623277689414152936+ O(17^{2}0)q^{23}+644118151097387201854206+O(17^{2}0)q^{25}+292774366585619012623914+O(17^{2}0)q^{27}+⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

10. Consider f = q + 5q^{3} + 12q^{5} + 8q^{7} + 5q^{9} + 13q^{11} + q^{13} + 9q^{15} + 2q^{19} + 6q^{21} + 3q^{23} + 3q^{25} + 10q^{27} + ⋯ ∈ S_{34}(Γ_{0}(4); \overline{F}_{17}). Also, z satisfies the equation x^{2} + 16x + 3 = 0. We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-16 newform g = q + 13q^{3} + 16q^{5} + 6q^{7} + 10q^{9} + 12q^{11} + 4q^{13} + 4q^{15} + 11q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(14)ω^{0} & 0 \\ 0 & u(11)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+1685315824699842889936056+ O(17^{2}0)q^{3} + 792292955438408334730960 + O(17^{2}0)q^{5} + 944108204830909082302389 + O(17^{2}0)q^{7} + 1281067744752349350110293+O(17^{2}0)q^{9}+−1219076824306713699880628+O(17^{2}0)q^{11}+1519808825959009448013505+ O(17^{2}0)q^{13} +491502544642709257857044+O(17^{2}0)q^{15} +74296268790736392902482∗17+O(17^{2}0)q^{17} + −454427100080586871962654+O(17^{2}0)q^{19}+1963674126632261505538386+O(17^{2}0)q^{21}+1480102833798779238354940+ O(17^{2}0)q^{23} +1621774410343940684510344+O(17^{2}0)q^{25} +−130814445933570264294648+O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

11. Consider f = q + 5q3 + 3q5 + 14q7 + 3q9 + 13q11 + 7q13 + 15q15 + 16q17 + 3q19 + 2q21 + 10q23 + 12q25 + 7q27 + ⋯ ∈ S22(Γ0(4); \overline{F}17). Since f is ordinary, we do not continue our analysis.

12. Consider f = q + 7q3 + 14q5 + 13q7 + q9 + 10q11 + 13q13 + 13q15 + 6q17 + 13q19 + 6q21 + 10q23 + 14q25 + 11q27 + ⋯ ∈ S26(Γ0(4); \overline{F}17). Since f is ordinary, we do not continue our analysis.

13. Consider f = q + 11q3 + 10q5 + 4q7 + 12q9 + 14q11 + 2q13 + 8q15 + 11q17 + 3q19 + 10q21 + 13q23 + 4q25 + 4q27 + ⋯ ∈ S28(Γ0(4); \overline{F}17). Since f is ordinary, we do not continue our analysis.

14. Consider f = q + 9q3 + 12q5 + 2q7 + q9 + 9q11 + 8q13 + 6q15 + q17 + 10q19 + q21 + 4q23 + 5q25 + 3q27 + ⋯ ∈ S30(Γ0(4); \overline{F}17). Since f is ordinary, we do not continue our analysis.

15. Consider f = q + 13q3 + 16q5 + 6q7 + 10q9 + 12q11 + 4q13 + 4q15 + 11q17 + q19 + 10q21 + 9q23 + 11q25 + q27 + ⋯ ∈ S32(Γ0(4); \overline{F}17). Since f is ordinary, we do not continue our analysis.

16. Consider f = q + z278q3 + z117q5 + z134q7 + z95q9 + z127q11 + z242q13 + z107q15 + 16q17 + z82q19 + z124q21 + z127q23 + 7q25 + 4q27 + ⋯ ∈ S34(Γ0(4); \overline{F}17). Also, z satisfies the equation x2 + 16x + 3 = 0. Since f is ordinary, we do not continue our analysis.

17. Consider f = q + z118q3 + z261q5 + z262q7 + z175q9 + z143q11 + z82q13 + z91q15 + 16q17 + z242q19 + z92q21 + z143q23 + 7q25 + 4q27 + ⋯ ∈ S34(Γ0(4); \overline{F}17). Also, z satisfies the equation x2 + 16x + 3 = 0. Since f is ordinary, we do not continue our analysis.

17.4 N = 5, ℓ = 17 (67 forms)

1. Consider f = q+z^{1}54q^{2}+z^{2}73q^{3}+z^{1}9q^{4}+12q^{5}+z^{1}39q^{6}+z^{2}18q^{7}+z^{3}4q^{8}+z^{2}79q^{9}+z^{1}00q^{10}+z^{2}27q^{11}+ z^{4}q^{12} + z^{2}08q^{13} + z^{8}4q^{14} + z^{2}19q^{15} + z^{2}54q^{16} + z^{1}45q^{18} + z^{9}q^{19} + z^{2}53q^{20} + z^{2}03q^{21} + z^{9}3q^{22} + z^{2}15q^{23} + z^{1}9q^{24}+8q^{25}+z^{7}4q^{26}+z^{1}95q^{27}+⋯ ∈ S_{20}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{2}+16x+3 = 0. We have ρ_{f} ⊗χ^{14} ∼ ρ_{g}, where g is the weight-16 newform g = q+z^{2}26q^{2}+z^{2}37q^{3}+z^{1}63q^{4}+10q^{5}+z^{1}75q^{6}+ z^{1}10q^{7}+z^{2}50q^{8}+z^{2}07q^{9}+z^{2}80q^{10}+z^{2}63q^{11}+z^{1}12q^{12}+z^{6}4q^{13}+z^{4}8q^{14}+z^{3}q^{15}+z^{2}54q^{16}+z^{1}11q^{17}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}77)ω^{1} & 0 \\ 0 & u(z^{1}11)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(250399559683515016856563+ O(17^{2}0))∗a−1417399447917430689156970+O(17^{2}0)q^{3} +−(1668317733990871382092296+O(17^{2}0))∗ a+1616295801892343690938185+O(17^{2}0)q^{4}+1953125q^{5}+(617761799727948381592369+O(17^{2}0))∗a+ 805510712041663026173639+O(17^{2}0)q^{6}+(1765648541307117640059943+O(17^{2}0))∗a+207373424594557014377353+ O(17^{2}0)q^{7} + −(199930550630841367302157 + O(17^{2}0)) ∗ a − 301014044668744071956302 + O(17^{2}0)q^{8} + −(943735203582624203575579 + O(17^{2}0)) ∗ a + 1764204051127044464852599 + O(17^{2}0)q^{9} + 1953125 ∗ aq^{10}+(108855331230527627538257+O(17^{2}0))∗a+455467954676397259509978+O(17^{2}0)q^{11}+−(1035274435414893607082263+ O(17^{2}0)) ∗ a − 1293154191019080817976571 + O(17^{2}0)q^{12} + −(952204510913753091890545 + O(17^{2}0)) ∗ a − 94454857266699060054725 ∗ 17 + O(17^{2}0)q^{13} + −(1005568458533313196280610 + O(17^{2}0)) ∗ a − 1383489117482187848185667+O(17^{2}0)q^{14}+(482150742922959822756242+O(17^{2}0))∗a+1054387075504019188293102+ O(17^{2}0)q^{15}+(1601214088963795732037390+O(17^{2}0))∗a−1386844032500031076853584+O(17^{2}0)q^{16}+ −(106028600028530372698936 ∗ 17 + O(17^{2}0)) ∗ a − 111334787476180341611697 ∗ 17 + O(17^{2}0)q^{17} + −(549259361557811072969395+O(17^{2}0))∗a−1785523223265395194447091+O(17^{2}0)q^{18}+−(977741964290834219950468+ O(17^{2}0))∗a−1233330862295666876010889+O(17^{2}0)q^{19}+−(571623735759473890448866+O(17^{2}0))∗a+ 1020659988175748557426991+O(17^{2}0)q^{20}+−(710042726446020646693925+O(17^{2}0))∗a+627335863947226216900881+ O(17^{2}0)q^{21}+−(924642340649712897725009+O(17^{2}0))∗a+1189489996364715814158435+O(17^{2}0)q^{22}+ (164496474710905055041458+O(17^{2}0))∗a+1474691965668885820549142+O(17^{2}0)q^{23}+−(257928528938763165781998+ O(17^{2}0))∗a−1670414579728352805947660+O(17^{2}0)q^{24}+3814697265625q^{25}+−(229474065987222506289242614+ O(17^{2}2))∗a−530843352365512289337414531+O(17^{2}2)q^{26} +(277055228882508030319541+O(17^{2}0))∗ a − 1774489103110601630348364 + O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 1668317733990871382092296 + O(17^{2}0)x + −1616295801892343691462473 + O(17^{2}0). The slope of f_{0} is 1.

2. Consider f = q + z^{2}6q^{2} + z^{3}3q^{3} + z^{3}5q^{4} + 12q^{5} + z^{5}9q^{6} + z^{2}50q^{7} + z^{2}q^{8} + z^{1}35q^{9} + z^{2}60q^{10} + z^{1}15q^{11} + z^{6}8q^{12}+z^{8}0q^{13}+z^{2}76q^{14}+z^{2}67q^{15}+z^{2}86q^{16}+z^{1}61q^{18}+z^{1}53q^{19}+z^{2}69q^{20}+z^{2}83q^{21}+z^{1}41q^{22}+z^{1}99q^{23}+ z^{3}5q^{24} +8q^{25} +z^{1}06q^{26} +z^{1}47q^{27} +⋯ ∈ S_{20}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{2} +16x+3 = 0. We have ρ_{f} ⊗χ^{14} ∼ ρ_{g}, where g is the weight-16 newform g = q+z^{9}8q^{2}+z^{2}85q^{3}+z^{1}79q^{4}+10q^{5}+z^{9}5q^{6}+ z^{1}42q^{7}+z^{2}18q^{8}+z^{6}3q^{9}+z^{1}52q^{10}+z^{1}51q^{11}+z^{1}76q^{12}+z^{2}24q^{13}+z^{2}40q^{14}+z^{5}1q^{15}+z^{2}86q^{16}+z^{1}59q^{17}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}29)ω^{1} & 0 \\ 0 & u(z^{1}59)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(250399559683515016856563+ O(17^{2}0))∗a−1417399447917430689156970+O(17^{2}0)q^{3} +−(1668317733990871382092296+O(17^{2}0))∗ a+1616295801892343690938185+O(17^{2}0)q^{4}+1953125q^{5}+(617761799727948381592369+O(17^{2}0))∗a+ 805510712041663026173639+O(17^{2}0)q^{6}+(1765648541307117640059943+O(17^{2}0))∗a+207373424594557014377353+ O(17^{2}0)q^{7} + −(199930550630841367302157 + O(17^{2}0)) ∗ a − 301014044668744071956302 + O(17^{2}0)q^{8} + −(943735203582624203575579 + O(17^{2}0)) ∗ a + 1764204051127044464852599 + O(17^{2}0)q^{9} + 1953125 ∗ aq^{10}+(108855331230527627538257+O(17^{2}0))∗a+455467954676397259509978+O(17^{2}0)q^{11}+−(1035274435414893607082263+ O(17^{2}0)) ∗ a − 1293154191019080817976571 + O(17^{2}0)q^{12} + −(952204510913753091890545 + O(17^{2}0)) ∗ a − 94454857266699060054725 ∗ 17 + O(17^{2}0)q^{13} + −(1005568458533313196280610 + O(17^{2}0)) ∗ a − 1383489117482187848185667+O(17^{2}0)q^{14}+(482150742922959822756242+O(17^{2}0))∗a+1054387075504019188293102+ O(17^{2}0)q^{15}+(1601214088963795732037390+O(17^{2}0))∗a−1386844032500031076853584+O(17^{2}0)q^{16}+ −(106028600028530372698936 ∗ 17 + O(17^{2}0)) ∗ a − 111334787476180341611697 ∗ 17 + O(17^{2}0)q^{17} + −(549259361557811072969395+O(17^{2}0))∗a−1785523223265395194447091+O(17^{2}0)q^{18}+−(977741964290834219950468+ O(17^{2}0))∗a−1233330862295666876010889+O(17^{2}0)q^{19}+−(571623735759473890448866+O(17^{2}0))∗a+ 1020659988175748557426991+O(17^{2}0)q^{20}+−(710042726446020646693925+O(17^{2}0))∗a+627335863947226216900881+ O(17^{2}0)q^{21}+−(924642340649712897725009+O(17^{2}0))∗a+1189489996364715814158435+O(17^{2}0)q^{22}+ (164496474710905055041458+O(17^{2}0))∗a+1474691965668885820549142+O(17^{2}0)q^{23}+−(257928528938763165781998+ O(17^{2}0))∗a−1670414579728352805947660+O(17^{2}0)q^{24}+3814697265625q^{25}+−(229474065987222506289242614+ O(17^{2}2))∗a−530843352365512289337414531+O(17^{2}2)q^{26} +(277055228882508030319541+O(17^{2}0))∗ a − 1774489103110601630348364 + O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 1668317733990871382092296 + O(17^{2}0)x + −1616295801892343691462473 + O(17^{2}0). The slope of f_{0} is 1.

3. Consider f = q + z^{1}60q^{2} + z^{1}34q^{3} + z^{1}56q^{4} + 5q^{5} + z^{6}q^{6} + z^{5}2q^{7} + z^{1}04q^{8} + z^{9}1q^{9} + z^{2}50q^{10} + z^{2}83q^{11} + z^{2}q^{12}+z^{1}16q^{13}+z^{2}12q^{14}+z^{2}24q^{15}+z^{6}6q^{16}+z^{2}51q^{18}+z^{2}54q^{19}+z^{2}46q^{20}+z^{1}86q^{21}+z^{1}55q^{22}+z^{2}76q^{23}+ z^{2}38q^{24}+8q^{25}+z^{2}76q^{26}+z^{2}29q^{27}+⋯ ∈ S_{20}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{2}+16x+3 = 0.

We have ρ_{f} ⊗χ^{14} ∼ ρ_{g}, where g is the weight-16 newform g = q +z^{2}32q^{2} +z^{9}8q^{3} +z^{1}2q^{4} +7q^{5} +z^{4}2q^{6} + z^{2}32q^{7} +z^{3}2q^{8} +z^{1}9q^{9} +z^{1}42q^{10} +z^{3}1q^{11} +z^{1}10q^{12} +z^{2}60q^{13} +z^{1}76q^{14} +z^{8}q^{15} +z^{6}6q^{16} +z^{8}8q^{17} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}00)ω^{1} & 0 \\ 0 & u(z^{8}8)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(1617239658273756954012593+ O(17^{2}0))∗a−33665922420166127716600+O(17^{2}0)q^{3} +−(1845887032442664764210017+O(17^{2}0))∗a+ 527232013905825862726809+O(17^{2}0)q^{4} +−1953125q^{5} +(1056553418421739372568530+O(17^{2}0))∗a− 34765950056660672666113+O(17^{2}0)q^{6}+−(709345925451948218066339+O(17^{2}0))∗a−48424167466288836090684∗ 17+O(17^{2}0)q^{7}+−(937385854295767639206603+O(17^{2}0))∗a+117318084745157915260666+O(17^{2}0)q^{8}+ −(771010327519114974129674 + O(17^{2}0)) ∗ a + 362504568580645136546157 + O(17^{2}0)q^{9} + −1953125 ∗ aq^{10} + −(923310011986719955471895 + O(17^{2}0)) ∗ a − 1444687991833736925903422 + O(17^{2}0)q^{11} + (683806733492771360617172+O(17^{2}0))∗a+1023857151766182416843845+O(17^{2}0)q^{12}+−(419860754052961636049632+ O(17^{2}0))∗a−1679079859013818962195584+O(17^{2}0)q^{13}+(1685650237798719024506287+O(17^{2}0))∗a+ 604008023010185547922805+O(17^{2}0)q^{14}+−(670903682042734409774863+O(17^{2}0))∗a−1445201264107643451127579+ O(17^{2}0)q^{15}+(1738515431579143233850328+O(17^{2}0))∗a−1539540118365628828517067+O(17^{2}0)q^{16}+ −(519076973631468998081094 ∗ 17 + O(17^{2}1)) ∗ a + 837814525898476737969193 ∗ 17 + O(17^{2}1)q^{17} + (1579542139363176477439580+O(17^{2}0))∗a−155327777115437356644824+O(17^{2}0)q^{18}+−(1844338468883796576451831+ O(17^{2}0))∗a+1707225660391594651102464+O(17^{2}0)q^{19} +−(1515192469244704053942743+O(17^{2}0))∗ a+220111072664290072415644+O(17^{2}0)q^{20}+(1126765903261274943864129+O(17^{2}0))∗a+1911070886520096605625148+ O(17^{2}0)q^{21}+(1313800044670080446420063+O(17^{2}0))∗a+1562459779089454764380538+O(17^{2}0)q^{22}+ −(936483535158156489305625+O(17^{2}0))∗a+92999311849922072578088+O(17^{2}0)q^{23}+−(6816144112132365662194+ O(17^{2}0))∗a+1569674899576808236266052+O(17^{2}0)q^{24}+3814697265625q^{25}+(1948643193247926744540460+ O(17^{2}0)) ∗ a + 1610322262545238300878998 + O(17^{2}0)q^{26} + −(438124956422217403053033 + O(17^{2}0)) ∗ a + 823946846116189118631543 + O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 1845887032442664764210017 + O(17^{2}0)x + −527232013905825863251097 + O(17^{2}0). The slope of f_{0} is 1.

4. Consider f = q+3q^{2}+10q^{3}+q^{4}+5q^{5}+13q^{6}+12q^{7}+13q^{8}+5q^{9}+15q^{10}+4q^{11}+10q^{12}+5q^{13}+2q^{14}+16q^{15}+ 14q^{16} +15q^{18} +2q^{19} +5q^{20} +q^{21} +12q^{22} +8q^{23} +11q^{24} +8q^{25} +15q^{26} +q^{27} +⋯ ∈ S_{20}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{2} + 16x + 3 = 0. We have ρ_{f} ⊗ χ^{14} ∼ ρ_{g}, where g is the weight-16 newform g = q+5q^{2}+3q^{3}+16q^{4}+7q^{5}+15q^{6}+11q^{7}+q^{8}+3q^{9}+q^{10}+2q^{11}+14q^{12}+12q^{13}+4q^{14}+4q^{15}+14q^{16}+14q^{17}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(11)ω^{1} & 0 \\ 0 & u(14)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−982333413913293296834451+ O(17^{2}0)q^{2} + −1729142472823060718388340 + O(17^{2}0)q^{3} + −209980463097334856262578 + O(17^{2}0)q^{4} + −1953125q^{5} + 615743348263438746819401 + O(17^{2}0)q^{6} + −1451613269604913705014643 + O(17^{2}0)q^{7} + 1808598529630148594793261+O(17^{2}0)q^{8}+445587430657507085783132+O(17^{2}0)q^{9}+1971987654819439573718901+ O(17^{2}0)q^{10} + 1537859007247253618694036 + O(17^{2}0)q^{11} + 918474076941618255180160 + O(17^{2}0)q^{12} + 567293381533229707469821+O(17^{2}0)q^{13}+−184624016730039454801744+O(17^{2}0)q^{14}+1405639047012097302589136+ O(17^{2}0)q^{15}+1540835394962438595036217+O(17^{2}0)q^{16}+−1420163305769624502371554∗17+O(17^{2}1)q^{17}+ −2019908493095942967681545+O(17^{2}0)q^{18}+−1689116592960653014848727+O(17^{2}0)q^{19}+564973582245474824500941+ O(17^{2}0)q^{20}+−1312399243660355217532704+O(17^{2}0)q^{21}+1847613169531716411437821+O(17^{2}0)q^{22}+ 82381731425744908907040+O(17^{2}0)q^{23}+1408518494117543803997301+O(17^{2}0)q^{24}+3814697265625q^{25}+ 1452803721411748029251027 + O(17^{2}0)q^{26} + −241243149405357879505448 + O(17^{2}0)q^{27} + ⋯ is an ℓadic newform that lifts f. The slope of f_{0} is 1.

5. Consider f = q +z^{1}28q^{2} +z^{2}62q^{3} +z^{6}0q^{4} +5q^{5} +z^{1}02q^{6} +z^{2}0q^{7} +z^{4}0q^{8} +z^{1}07q^{9} +z^{2}18q^{10} +z^{2}03q^{11} + z^{3}4q^{12} +z^{2}44q^{13} +z^{1}48q^{14} +z^{6}4q^{15} +z^{2}58q^{16} +z^{2}35q^{18} +z^{2}86q^{19} +z^{1}50q^{20} +z^{2}82q^{21} +z^{4}3q^{22} +z^{8}4q^{23} + z^{1}4q^{24}+8q^{25}+z^{8}4q^{26}+z^{1}49q^{27}+⋯ ∈ S_{20}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{2}+16x+3 = 0. We have ρ_{f} ⊗χ^{14} ∼ ρ_{g}, where g is the weight-16 newform g = q +z^{2}00q^{2} +z^{2}26q^{3} +z^{2}04q^{4} +7q^{5} +z^{1}38q^{6} + z^{2}00q^{7}+z^{2}56q^{8}+z^{3}5q^{9}+z^{1}10q^{10}+z^{2}39q^{11}+z^{1}42q^{12}+z^{1}00q^{13}+z^{1}12q^{14}+z^{1}36q^{15}+z^{2}58q^{16}+z^{5}6q^{17}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}32)ω^{1} & 0 \\ 0 & u(z^{5}6)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(1617239658273756954012593+ O(17^{2}0))∗a−33665922420166127716600+O(17^{2}0)q^{3} +−(1845887032442664764210017+O(17^{2}0))∗a+ 527232013905825862726809+O(17^{2}0)q^{4} +−1953125q^{5} +(1056553418421739372568530+O(17^{2}0))∗a− 34765950056660672666113+O(17^{2}0)q^{6}+−(709345925451948218066339+O(17^{2}0))∗a−48424167466288836090684∗ 17+O(17^{2}0)q^{7}+−(937385854295767639206603+O(17^{2}0))∗a+117318084745157915260666+O(17^{2}0)q^{8}+ −(771010327519114974129674 + O(17^{2}0)) ∗ a + 362504568580645136546157 + O(17^{2}0)q^{9} + −1953125 ∗ aq^{10} + −(923310011986719955471895 + O(17^{2}0)) ∗ a − 1444687991833736925903422 + O(17^{2}0)q^{11} + (683806733492771360617172+O(17^{2}0))∗a+1023857151766182416843845+O(17^{2}0)q^{12}+−(419860754052961636049632+ O(17^{2}0))∗a−1679079859013818962195584+O(17^{2}0)q^{13}+(1685650237798719024506287+O(17^{2}0))∗a+ 604008023010185547922805+O(17^{2}0)q^{14}+−(670903682042734409774863+O(17^{2}0))∗a−1445201264107643451127579+ O(17^{2}0)q^{15}+(1738515431579143233850328+O(17^{2}0))∗a−1539540118365628828517067+O(17^{2}0)q^{16}+ −(519076973631468998081094 ∗ 17 + O(17^{2}1)) ∗ a + 837814525898476737969193 ∗ 17 + O(17^{2}1)q^{17} + (1579542139363176477439580+O(17^{2}0))∗a−155327777115437356644824+O(17^{2}0)q^{18}+−(1844338468883796576451831+ O(17^{2}0))∗a+1707225660391594651102464+O(17^{2}0)q^{19} +−(1515192469244704053942743+O(17^{2}0))∗ a+220111072664290072415644+O(17^{2}0)q^{20}+(1126765903261274943864129+O(17^{2}0))∗a+1911070886520096605625148+ O(17^{2}0)q^{21}+(1313800044670080446420063+O(17^{2}0))∗a+1562459779089454764380538+O(17^{2}0)q^{22}+ −(936483535158156489305625+O(17^{2}0))∗a+92999311849922072578088+O(17^{2}0)q^{23}+−(6816144112132365662194+ O(17^{2}0))∗a+1569674899576808236266052+O(17^{2}0)q^{24}+3814697265625q^{25}+(1948643193247926744540460+ O(17^{2}0)) ∗ a + 1610322262545238300878998 + O(17^{2}0)q^{26} + −(438124956422217403053033 + O(17^{2}0)) ∗ a + 823946846116189118631543 + O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 1845887032442664764210017 + O(17^{2}0)x + −527232013905825863251097 + O(17^{2}0). The slope of f_{0} is 1.

6. Consider f = q +z^{1}86q^{2} +z^{2}06q^{3} +z^{1}60q^{4} +8q^{5} +z^{1}04q^{6} +z^{2}59q^{7} +z^{5}1q^{8} +z^{9}6q^{9} +z^{7}8q^{10} +z^{2}86q^{11} + z^{7}8q^{12} + z^{1}90q^{13} + z^{1}57q^{14} + z^{9}8q^{15} + z^{3}1q^{16} + z^{2}82q^{18} + z^{6}8q^{19} + z^{5}2q^{20} + z^{1}77q^{21} + z^{1}84q^{22} + z^{2}8q^{23} + z^{2}57q^{24} +13q^{25} +z^{8}8q^{26} +z^{5}5q^{27} +⋯ ∈ S_{22}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{2} +16x+3 = 0. We have ρ_{f} ⊗χ^{12} ∼ ρ_{g}, where g is the weight-14 newform g = q+z^{4}2q^{2}+z^{1}34q^{3}+z^{1}60q^{4}+15q^{5}+z^{1}76q^{6}+ z^{4}3q^{7} +z^{1}95q^{8} +z^{2}40q^{9} +z^{1}50q^{10} +z^{7}0q^{11} +z^{6}q^{12} +z^{1}90q^{13} +z^{8}5q^{14} +z^{2}42q^{15} +z^{3}1q^{16} +z^{2}22q^{17} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{6}6)ω^{1} & 0 \\ 0 & u(z^{2}22)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(186273203567523576990785+ O(17^{2}0))∗a−434825027740575180721934+O(17^{2}0)q^{3} +−(110026137702734874193325+O(17^{2}0))∗a+ 936256412369042056388237+O(17^{2}0)q^{4} +−9765625q^{5} +(1461289641290438787649288+O(17^{2}0))∗a− 472499187126158498552202+O(17^{2}0)q^{6}+−(337589458239747598907988+O(17^{2}0))∗a+264011167539250350017180+ O(17^{2}0)q^{7} +−(1958037973324455866228916+O(17^{2}0))∗a−970528220616276027784178+O(17^{2}0)q^{8} + −(120585768710920892283083+O(17^{2}0))∗a−64420151897847049169726∗17+O(17^{2}0)q^{9}+−9765625∗ aq^{10} + −(375964881657093226864400 + O(17^{2}0)) ∗ a + 1082097952066579200739787 + O(17^{2}0)q^{11} + (1272330583544252961696563+O(17^{2}0))∗a−38326823489288779213003∗17+O(17^{2}0)q^{12}+(1018492709974032123089525+ O(17^{2}0))∗a+1861203328425809050457617+O(17^{2}0)q^{13} +−(1794298543207498197608200+O(17^{2}0))∗ a−1316179126132599202720470+O(17^{2}0)q^{14}+−(1496370370274401603788444+O(17^{2}0))∗a+738243824194824787182743+ O(17^{2}0)q^{15}+−(436110150161316557886180152+O(17^{2}2))∗a−492952938294632116810531870+O(17^{2}2)q^{16}+ (91545488010883577099669∗17+O(17^{2}0))∗a−63287641130609383574903∗17+O(17^{2}0)q^{17}+(375076509952914122553321+ O(17^{2}0))∗a−790711997955147139347377+O(17^{2}0)q^{18}+(5168087043650076011604270+O(17^{2}1))∗a− 26246933199469001759863293+O(17^{2}1)q^{19}+(951333631565329310991952+O(17^{2}0))∗a+1670774538373684563949333+ O(17^{2}0)q^{20}+−(366572723935533932844038+O(17^{2}0))∗a+1126093408646406916419089+O(17^{2}0)q^{21}+ −(328294014410168996658421+O(17^{2}0))∗a−1548550997020942544021259+O(17^{2}0)q^{22}+(764492808226580447302235+ O(17^{2}0))∗a+1176608433049850487996409+O(17^{2}0)q^{23} +−(1279532254013164676384953+O(17^{2}0))∗ a + 465611718544649628063744 + O(17^{2}0)q^{24} + 95367431640625q^{25} + (1485592771912603433647839 + O(17^{2}0)) ∗ a − 924473590614702560669195 + O(17^{2}0)q^{26} + (4607465199805893337853871 + O(17^{2}1)) ∗ a − 20129375134376376097841300 + O(17^{2}1)q^{27} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 110026137702734874193325 + O(17^{2}0)x + −936256412369042058485389 + O(17^{2}0). The slope of f_{0} is 1.

7. Consider f = q + z^{2}82q^{2} + z^{4}6q^{3} + z^{1}28q^{4} + 8q^{5} + z^{4}0q^{6} + z^{8}3q^{7} + z^{3}q^{8} + z^{1}92q^{9} + z^{1}74q^{10} + z^{2}54q^{11} + z^{1}74q^{12} +z^{6}2q^{13} +z^{7}7q^{14} +z^{2}26q^{15} +z^{2}39q^{16} +z^{1}86q^{18} +z^{4}q^{19} +z^{2}0q^{20} +z^{1}29q^{21} +z^{2}48q^{22} +z^{1}88q^{23} + z^{4}9q^{24}+13q^{25}+z^{5}6q^{26}+z^{7}1q^{27}+⋯ ∈ S_{22}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{2}+16x+3 = 0. We have ρ_{f} ⊗χ^{12} ∼ ρ_{g}, where g is the weight-14 newform g = q+z^{1}38q^{2}+z^{2}62q^{3}+z^{1}28q^{4}+15q^{5}+z^{1}12q^{6}+ z^{1}55q^{7} +z^{1}47q^{8} +z^{4}8q^{9} +z^{2}46q^{10} +z^{3}8q^{11} +z^{1}02q^{12} +z^{6}2q^{13} +z^{5}q^{14} +z^{8}2q^{15} +z^{2}39q^{16} +z^{3}0q^{17} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}58)ω^{1} & 0 \\ 0 & u(z^{3}0)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(186273203567523576990785+ O(17^{2}0))∗a−434825027740575180721934+O(17^{2}0)q^{3} +−(110026137702734874193325+O(17^{2}0))∗a+ 936256412369042056388237+O(17^{2}0)q^{4} +−9765625q^{5} +(1461289641290438787649288+O(17^{2}0))∗a− 472499187126158498552202+O(17^{2}0)q^{6}+−(337589458239747598907988+O(17^{2}0))∗a+264011167539250350017180+ O(17^{2}0)q^{7} +−(1958037973324455866228916+O(17^{2}0))∗a−970528220616276027784178+O(17^{2}0)q^{8} + −(120585768710920892283083+O(17^{2}0))∗a−64420151897847049169726∗17+O(17^{2}0)q^{9}+−9765625∗ aq^{10} + −(375964881657093226864400 + O(17^{2}0)) ∗ a + 1082097952066579200739787 + O(17^{2}0)q^{11} + (1272330583544252961696563+O(17^{2}0))∗a−38326823489288779213003∗17+O(17^{2}0)q^{12}+(1018492709974032123089525+ O(17^{2}0))∗a+1861203328425809050457617+O(17^{2}0)q^{13} +−(1794298543207498197608200+O(17^{2}0))∗ a−1316179126132599202720470+O(17^{2}0)q^{14}+−(1496370370274401603788444+O(17^{2}0))∗a+738243824194824787182743+ O(17^{2}0)q^{15}+−(436110150161316557886180152+O(17^{2}2))∗a−492952938294632116810531870+O(17^{2}2)q^{16}+ (91545488010883577099669∗17+O(17^{2}0))∗a−63287641130609383574903∗17+O(17^{2}0)q^{17}+(375076509952914122553321+ O(17^{2}0))∗a−790711997955147139347377+O(17^{2}0)q^{18}+(5168087043650076011604270+O(17^{2}1))∗a− 26246933199469001759863293+O(17^{2}1)q^{19}+(951333631565329310991952+O(17^{2}0))∗a+1670774538373684563949333+ O(17^{2}0)q^{20}+−(366572723935533932844038+O(17^{2}0))∗a+1126093408646406916419089+O(17^{2}0)q^{21}+ −(328294014410168996658421+O(17^{2}0))∗a−1548550997020942544021259+O(17^{2}0)q^{22}+(764492808226580447302235+ O(17^{2}0))∗a+1176608433049850487996409+O(17^{2}0)q^{23} +−(1279532254013164676384953+O(17^{2}0))∗ a + 465611718544649628063744 + O(17^{2}0)q^{24} + 95367431640625q^{25} + (1485592771912603433647839 + O(17^{2}0)) ∗ a − 924473590614702560669195 + O(17^{2}0)q^{26} + (4607465199805893337853871 + O(17^{2}1)) ∗ a − 20129375134376376097841300 + O(17^{2}1)q^{27} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 110026137702734874193325 + O(17^{2}0)x + −936256412369042058485389 + O(17^{2}0). The slope of f_{0} is 1.

8. Consider f = q +z^{3}891q^{2} +z^{1}696q^{3} +z^{4}653q^{4} +9q^{5} +z^{6}75q^{6} +z^{2}965q^{7} +z^{6}63q^{8} +z^{2}078q^{9} +z^{4}505q^{10} + z^{4}557q^{11} + z^{1}437q^{12} + z^{4}468q^{13} + z^{1}944q^{14} + z^{2}310q^{15} + z^{1}983q^{16} + z^{1}057q^{18} + z^{4}336q^{19} + z^{3}55q^{20} + z^{4}661q^{21} + z^{3}536q^{22} + z^{7}69q^{23} + z^{2}359q^{24} + 13q^{25} + z^{3}447q^{26} + z^{8}44q^{27} + ⋯ ∈ S_{22}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{3} + x + 14 = 0. We have ρ_{f} ⊗ χ^{12} ∼ ρ_{g}, where g is the weight-14 newform g = q + z^{1}435q^{2} + z^{4}68q^{3} + z^{4}653q^{4} + 2q^{5} + z^{1}903q^{6} + z^{4}193q^{7} + z^{3}119q^{8} + z^{4}534q^{9} + z^{8}21q^{10} + z^{8}73q^{11} + z^{2}09q^{12} + z^{4}468q^{13} + z^{7}16q^{14} + z^{4}766q^{15} + z^{1}983q^{16} + z^{2}247q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}665)ω^{1} & 0 \\ 0 & u(z^{2}247)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(833582029566494312677720+ O(17^{2}0))∗a^{2} −(1742014582096321598832977+O(17^{2}0))∗a−36110027972914803094334+O(17^{2}0)q^{3} + a^{2}−2097152q^{4}+9765625q^{5}+(1410352658708518461669315+O(17^{2}0))∗a^{2}+(81821667492026207161863+ O(17^{2}0))∗a−139778426582116329937290+O(17^{2}0)q^{6}+(1662393154057433747792467+O(17^{2}0))∗a^{2}− (748069141633582612697737+O(17^{2}0))∗a−406533678321147724264890+O(17^{2}0)q^{7}+−(1526487700155323457001062+ O(17^{2}0)) ∗ a^{2} − (59295991787488211671791 + O(17^{2}0)) ∗ a + 119870152901000023300585 + O(17^{2}0)q^{8} + −(997865391926632711730819+O(17^{2}0))∗a^{2}+(1728181238070987130607668+O(17^{2}0))∗a−241458415411354295059614+ O(17^{2}0)q^{9}+9765625∗aq^{10}+(868845365714653824646584+O(17^{2}0))∗a^{2}+(96454370530524892290841+ O(17^{2}0))∗a+1917117096199609531208012+O(17^{2}0)q^{11}+(1770648604888380082579440+O(17^{2}0))∗a^{2}+ (195759972223420104720897+O(17^{2}0))∗a−1078024051192504720834270+O(17^{2}0)q^{12}+(1769370291472858728608802+ O(17^{2}0))∗a^{2}+(266940548076317363901666+O(17^{2}0))∗a+145988595391434212251387+O(17^{2}0)q^{13}+ −(1883634245854313047476687+O(17^{2}0))∗a^{2}+(116393989062643530686753+O(17^{2}0))∗a−593350237557540523451473+ O(17^{2}0)q^{14}+−(1275771947286071595053651+O(17^{2}0))∗a^{2}−(478677907280388267228274+O(17^{2}0))∗ a+110306187153510591843616+O(17^{2}0)q^{15}+(7395294929735712359465∗17+O(17^{2}0))∗a^{2}−(1853617022392653461551768+ O(17^{2}0))∗a−701756029772462021744968+O(17^{2}0)q^{16}+−(1938027221574071341462654∗17+O(17^{2}1))∗ a^{2} + (74669488362443535712279 ∗ 17 + O(17^{2}1)) ∗ a + 1970680495137177739533023 ∗ 17 + O(17^{2}1)q^{17} + −(595762586195589871917342+O(17^{2}0))∗a^{2}−(1116624646241293770335123+O(17^{2}0))∗a−191437728182320265352635+ O(17^{2}0)q^{18}+(22367305016309876988996∗17+O(17^{2}0))∗a^{2}−(446376371592565286706282+O(17^{2}0))∗ a+1839425491342111341939189+O(17^{2}0)q^{19}+9765625∗a^{2}−20480000000000q^{20}+(288805852805444248691674+ O(17^{2}0))∗a^{2}+(341071781421548097975468+O(17^{2}0))∗a+148846977667231751821624+O(17^{2}0)q^{21}+ −(718741810038686574308001+O(17^{2}0))∗a^{2}−(352958576179122003756517+O(17^{2}0))∗a−1138312368294593500192219+ O(17^{2}0)q^{22}+(276518344958886107256747+O(17^{2}0))∗a^{2}−(101694839331243639121941+O(17^{2}0))∗a− 1669426247618798553169082+O(17^{2}0)q^{23}+(1241203463280647750326662+O(17^{2}0))∗a^{2}−(858396397738064082880934+ O(17^{2}0))∗a+711317925085956156057698+O(17^{2}0)q^{24}+95367431640625q^{25}+(995984952021009396607575+ O(17^{2}0))∗a^{2} +(615242954805837227756106+O(17^{2}0))∗a−95896592414595495417770+O(17^{2}0)q^{26} + (7787706506406865596881∗17+O(17^{2}0))∗a^{2}+(262997351304824749924833+O(17^{2}0))∗a−1501653424194860163600053+ O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 1526487700155323457001062 + O(17^{2}0)x^{2} + 59295991787488207477487 + O(17^{2}0)x + −119870152901000023300585 + O(17^{2}0). The slope of f_{0} is 1.

9. Consider f = q + z^{4}563q^{2} + z^{3}856q^{3} + z^{3}741q^{4} + 9q^{5} + z^{3}507q^{6} + z^{2}197q^{7} + z^{3}9q^{8} + z^{1}278q^{9} + z^{2}65q^{10} + z^{5}57q^{11} + z^{2}685q^{12} + z^{4}308q^{13} + z^{1}848q^{14} + z^{4}470q^{15} + z^{3}295q^{16} + z^{9}29q^{18} + z^{5}44q^{19} + z^{4}355q^{20} + z^{1}141q^{21} + z^{2}08q^{22} + z^{1}201q^{23} + z^{3}895q^{24} + 13q^{25} + z^{3}959q^{26} + z^{3}228q^{27} + ⋯ ∈ S_{22}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{3} + x + 14 = 0. We have ρ_{f} ⊗ χ^{12} ∼ ρ_{g}, where g is the weight-14 newform g = q + z^{2}107q^{2} + z^{2}628q^{3} + z^{3}741q^{4} + 2q^{5} + z^{4}735q^{6} + z^{3}425q^{7} + z^{2}495q^{8} + z^{3}734q^{9} + z^{1}493q^{10} + z^{1}785q^{11} + z^{1}457q^{12} + z^{4}308q^{13} + z^{6}20q^{14} + z^{2}014q^{15} + z^{3}295q^{16} + z^{9}99q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{3}913)ω^{1} & 0 \\ 0 & u(z^{9}99)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(833582029566494312677720+ O(17^{2}0))∗a^{2} −(1742014582096321598832977+O(17^{2}0))∗a−36110027972914803094334+O(17^{2}0)q^{3} + a^{2}−2097152q^{4}+9765625q^{5}+(1410352658708518461669315+O(17^{2}0))∗a^{2}+(81821667492026207161863+ O(17^{2}0))∗a−139778426582116329937290+O(17^{2}0)q^{6}+(1662393154057433747792467+O(17^{2}0))∗a^{2}− (748069141633582612697737+O(17^{2}0))∗a−406533678321147724264890+O(17^{2}0)q^{7}+−(1526487700155323457001062+ O(17^{2}0)) ∗ a^{2} − (59295991787488211671791 + O(17^{2}0)) ∗ a + 119870152901000023300585 + O(17^{2}0)q^{8} + −(997865391926632711730819+O(17^{2}0))∗a^{2}+(1728181238070987130607668+O(17^{2}0))∗a−241458415411354295059614+ O(17^{2}0)q^{9}+9765625∗aq^{10}+(868845365714653824646584+O(17^{2}0))∗a^{2}+(96454370530524892290841+ O(17^{2}0))∗a+1917117096199609531208012+O(17^{2}0)q^{11}+(1770648604888380082579440+O(17^{2}0))∗a^{2}+ (195759972223420104720897+O(17^{2}0))∗a−1078024051192504720834270+O(17^{2}0)q^{12}+(1769370291472858728608802+ O(17^{2}0))∗a^{2}+(266940548076317363901666+O(17^{2}0))∗a+145988595391434212251387+O(17^{2}0)q^{13}+ −(1883634245854313047476687+O(17^{2}0))∗a^{2}+(116393989062643530686753+O(17^{2}0))∗a−593350237557540523451473+ O(17^{2}0)q^{14}+−(1275771947286071595053651+O(17^{2}0))∗a^{2}−(478677907280388267228274+O(17^{2}0))∗ a+110306187153510591843616+O(17^{2}0)q^{15}+(7395294929735712359465∗17+O(17^{2}0))∗a^{2}−(1853617022392653461551768+ O(17^{2}0))∗a−701756029772462021744968+O(17^{2}0)q^{16}+−(1938027221574071341462654∗17+O(17^{2}1))∗ a^{2} + (74669488362443535712279 ∗ 17 + O(17^{2}1)) ∗ a + 1970680495137177739533023 ∗ 17 + O(17^{2}1)q^{17} + −(595762586195589871917342+O(17^{2}0))∗a^{2}−(1116624646241293770335123+O(17^{2}0))∗a−191437728182320265352635+ O(17^{2}0)q^{18}+(22367305016309876988996∗17+O(17^{2}0))∗a^{2}−(446376371592565286706282+O(17^{2}0))∗ a+1839425491342111341939189+O(17^{2}0)q^{19}+9765625∗a^{2}−20480000000000q^{20}+(288805852805444248691674+ O(17^{2}0))∗a^{2}+(341071781421548097975468+O(17^{2}0))∗a+148846977667231751821624+O(17^{2}0)q^{21}+ −(718741810038686574308001+O(17^{2}0))∗a^{2}−(352958576179122003756517+O(17^{2}0))∗a−1138312368294593500192219+ O(17^{2}0)q^{22}+(276518344958886107256747+O(17^{2}0))∗a^{2}−(101694839331243639121941+O(17^{2}0))∗a− 1669426247618798553169082+O(17^{2}0)q^{23}+(1241203463280647750326662+O(17^{2}0))∗a^{2}−(858396397738064082880934+ O(17^{2}0))∗a+711317925085956156057698+O(17^{2}0)q^{24}+95367431640625q^{25}+(995984952021009396607575+ O(17^{2}0))∗a^{2} +(615242954805837227756106+O(17^{2}0))∗a−95896592414595495417770+O(17^{2}0)q^{26} + (7787706506406865596881∗17+O(17^{2}0))∗a^{2}+(262997351304824749924833+O(17^{2}0))∗a−1501653424194860163600053+ O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 1526487700155323457001062 + O(17^{2}0)x^{2} + 59295991787488207477487 + O(17^{2}0)x + −119870152901000023300585 + O(17^{2}0). The slope of f_{0} is 1.

10. Consider f = q + 9q^{2} + 6q^{3} + 15q^{4} + 9q^{5} + 3q^{6} + 8q^{7} + 14q^{9} + 13q^{10} + 12q^{11} + 5q^{12} + 16q^{13} + 4q^{14} + 3q^{15} + 13q^{16} + 7q^{18} + 13q^{19} + 16q^{20} + 14q^{21} + 6q^{22} + 8q^{23} + 13q^{25} + 8q^{26} + 3q^{27} + ⋯ ∈ S_{22}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{3} + x + 14 = 0. We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-4 newform g = q+13q^{2}+2q^{3}+8q^{4}+12q^{5}+9q^{6}+6q^{7}+11q^{9}+3q^{10}+15q^{11}+16q^{12}+13q^{13}+10q^{14}+7q^{15}+4q^{16}+9q^{17}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{4} & 0 \\ 0 & u(9)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+1526487700155323457003972+ O(17^{2}0)q^{2} + −1450832132095731381348419 + O(17^{2}0)q^{3} + −66424022018018893468150 + O(17^{2}0)q^{4} + 9765625q^{5} + −79584432307192974907430 + O(17^{2}0)q^{6} + −1583327354601960835660356 + O(17^{2}0)q^{7} + 80052288446290138964998∗17+O(17^{2}0)q^{8}+1929391542770114388867259+O(17^{2}0)q^{9}+1483477645626483074589822+ O(17^{2}0)q^{10} + 328300551006931251056873 + O(17^{2}0)q^{11} + 1076465074816746051479363 + O(17^{2}0)q^{12} + −1801226958812998770409619+O(17^{2}0)q^{13}+−2000862254596870212482745+O(17^{2}0)q^{14}+−2011365934028789134665184+ O(17^{2}0)q^{15}+2011514845897569769179429+O(17^{2}0)q^{16}+−900275558023468892393557∗17+O(17^{2}1)q^{17}+ 1360965653552052971899817+O(17^{2}0)q^{18}+−995932419663563308811581+O(17^{2}0)q^{19}+−436361729944086994816145+ O(17^{2}0)q^{20} + 1404041249372710470936576 + O(17^{2}0)q^{21} + 926310187804422186502366 + O(17^{2}0)q^{22} + −1451552907900118763945478+O(17^{2}0)q^{23}+−12992447881200386924058∗17+O(17^{2}0)q^{24}+95367431640625q^{25}+ −626011632019047788232147+O(17^{2}0)q^{26}+271170908334796335664293+O(17^{2}0)q^{27}+⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

11. Consider f = q +z^{2}291q^{2} +z^{4}272q^{3} +z^{5}09q^{4} +9q^{5} +z^{1}651q^{6} +z^{1}285q^{7} +z^{1}447q^{8} +z^{9}42q^{9} +z^{2}905q^{10} + z^{3}789q^{11} + z^{4}781q^{12} + z^{2}276q^{13} + z^{3}576q^{14} + z^{4}886q^{15} + z^{4}239q^{16} + z^{3}233q^{18} + z^{3}2q^{19} + z^{1}123q^{20} + z^{6}45q^{21} + z^{1}168q^{22} + z^{3}249q^{23} + z^{8}07q^{24} + 13q^{25} + z^{4}567q^{26} + z^{4}524q^{27} + ⋯ ∈ S_{22}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{3} + x + 14 = 0. We have ρ_{f} ⊗ χ^{12} ∼ ρ_{g}, where g is the weight-14 newform g = q + z^{4}747q^{2} + z^{3}044q^{3} + z^{5}09q^{4} + 2q^{5} + z^{2}879q^{6} + z^{2}513q^{7} + z^{3}903q^{8} + z^{3}398q^{9} + z^{4}133q^{10} + z^{1}05q^{11} + z^{3}553q^{12} + z^{2}276q^{13} + z^{2}348q^{14} + z^{2}430q^{15} + z^{4}239q^{16} + z^{3}815q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}097)ω^{1} & 0 \\ 0 & u(z^{3}815)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(833582029566494312677720+ O(17^{2}0))∗a^{2} −(1742014582096321598832977+O(17^{2}0))∗a−36110027972914803094334+O(17^{2}0)q^{3} + a^{2}−2097152q^{4}+9765625q^{5}+(1410352658708518461669315+O(17^{2}0))∗a^{2}+(81821667492026207161863+ O(17^{2}0))∗a−139778426582116329937290+O(17^{2}0)q^{6}+(1662393154057433747792467+O(17^{2}0))∗a^{2}− (748069141633582612697737+O(17^{2}0))∗a−406533678321147724264890+O(17^{2}0)q^{7}+−(1526487700155323457001062+ O(17^{2}0)) ∗ a^{2} − (59295991787488211671791 + O(17^{2}0)) ∗ a + 119870152901000023300585 + O(17^{2}0)q^{8} + −(997865391926632711730819+O(17^{2}0))∗a^{2}+(1728181238070987130607668+O(17^{2}0))∗a−241458415411354295059614+ O(17^{2}0)q^{9}+9765625∗aq^{10}+(868845365714653824646584+O(17^{2}0))∗a^{2}+(96454370530524892290841+ O(17^{2}0))∗a+1917117096199609531208012+O(17^{2}0)q^{11}+(1770648604888380082579440+O(17^{2}0))∗a^{2}+ (195759972223420104720897+O(17^{2}0))∗a−1078024051192504720834270+O(17^{2}0)q^{12}+(1769370291472858728608802+ O(17^{2}0))∗a^{2}+(266940548076317363901666+O(17^{2}0))∗a+145988595391434212251387+O(17^{2}0)q^{13}+ −(1883634245854313047476687+O(17^{2}0))∗a^{2}+(116393989062643530686753+O(17^{2}0))∗a−593350237557540523451473+ O(17^{2}0)q^{14}+−(1275771947286071595053651+O(17^{2}0))∗a^{2}−(478677907280388267228274+O(17^{2}0))∗ a+110306187153510591843616+O(17^{2}0)q^{15}+(7395294929735712359465∗17+O(17^{2}0))∗a^{2}−(1853617022392653461551768+ O(17^{2}0))∗a−701756029772462021744968+O(17^{2}0)q^{16}+−(1938027221574071341462654∗17+O(17^{2}1))∗ a^{2} + (74669488362443535712279 ∗ 17 + O(17^{2}1)) ∗ a + 1970680495137177739533023 ∗ 17 + O(17^{2}1)q^{17} + −(595762586195589871917342+O(17^{2}0))∗a^{2}−(1116624646241293770335123+O(17^{2}0))∗a−191437728182320265352635+ O(17^{2}0)q^{18}+(22367305016309876988996∗17+O(17^{2}0))∗a^{2}−(446376371592565286706282+O(17^{2}0))∗ a+1839425491342111341939189+O(17^{2}0)q^{19}+9765625∗a^{2}−20480000000000q^{20}+(288805852805444248691674+ O(17^{2}0))∗a^{2}+(341071781421548097975468+O(17^{2}0))∗a+148846977667231751821624+O(17^{2}0)q^{21}+ −(718741810038686574308001+O(17^{2}0))∗a^{2}−(352958576179122003756517+O(17^{2}0))∗a−1138312368294593500192219+ O(17^{2}0)q^{22}+(276518344958886107256747+O(17^{2}0))∗a^{2}−(101694839331243639121941+O(17^{2}0))∗a− 1669426247618798553169082+O(17^{2}0)q^{23}+(1241203463280647750326662+O(17^{2}0))∗a^{2}−(858396397738064082880934+ O(17^{2}0))∗a+711317925085956156057698+O(17^{2}0)q^{24}+95367431640625q^{25}+(995984952021009396607575+ O(17^{2}0))∗a^{2} +(615242954805837227756106+O(17^{2}0))∗a−95896592414595495417770+O(17^{2}0)q^{26} + (7787706506406865596881∗17+O(17^{2}0))∗a^{2}+(262997351304824749924833+O(17^{2}0))∗a−1501653424194860163600053+ O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + 1526487700155323457001062 + O(17^{2}0)x^{2} + 59295991787488207477487 + O(17^{2}0)x + −119870152901000023300585 + O(17^{2}0). The slope of f_{0} is 1.

12. Consider f = q + 3q^{3} + 8q^{4} + 11q^{5} + 11q^{7} + 15q^{9} + 15q^{11} + 7q^{12} + 8q^{13} + 16q^{15} + 13q^{16} + 12q^{19} + 3q^{20} + 16q^{21} + 13q^{23} + 2q^{25} + 12q^{27} + ⋯ ∈ S_{24}(Γ_{0}(5); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{10} ∼ ρ_{g}, where g is the weight-12 newform g = q + 7q^{3} + 9q^{4} + 14q^{5} + 5q^{7} + 8q^{9} + 4q^{11} + 12q^{12} + 9q^{13} + 13q^{15} + 13q^{16} + 6q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(3)ω^{1} & 0 \\ 0 & u(6)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+12775391882152734228393∗ 17+O(17^{2}0)q^{2}+−937457193063410966001789+O(17^{2}0)q^{3}+−8292151762848225901955038+O(17^{2}1)q^{4}+ 48828125q^{5}+−15014016557182872157382∗17+O(17^{2}0)q^{6}+−84738778111342006324334+O(17^{2}0)q^{7}+ −74742365378102924701571∗17+O(17^{2}0)q^{8}+1634191199203478201707229+O(17^{2}0)q^{9}+113368871992546267070793∗ 17+O(17^{2}0)q^{10}+475092980835738206590288+O(17^{2}0)q^{11}+−761870049783018685612611+O(17^{2}0)q^{12}+ 272111258143319923965788+O(17^{2}0)q^{13}+−10744914702109300442235∗17+O(17^{2}0)q^{14}+−1042160055460238278310511+ O(17^{2}0)q^{15}+122859689855024918667602+O(17^{2}0)q^{16}+−62014911015168916181094∗17+O(17^{2}0)q^{17}+ −42347077931426059050171∗17+O(17^{2}0)q^{18}+−16258678478640495956408+O(17^{2}0)q^{19}+−20804062796237592630905728+ O(17^{2}1)q^{20} +565423561038012698178394+O(17^{2}0)q^{21} +92173535724760156746060∗17+O(17^{2}0)q^{22} + 612226698901253505981055+O(17^{2}0)q^{23}+61344036674205953103010∗17+O(17^{2}0)q^{24}+2384185791015625q^{25}+ −97230122977775800015780 ∗ 17 + O(17^{2}0)q^{26} + −1045629135293605277463392 + O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

13. Consider f = q + 8q^{2} + 10q^{3} + 4q^{4} + 6q^{5} + 12q^{6} + 15q^{7} + 11q^{8} + 4q^{9} + 14q^{10} + 7q^{11} + 6q^{12} + 8q^{13} + q^{14} + 9q^{15}+q^{16}+15q^{18}+14q^{19}+7q^{20}+14q^{21}+5q^{22}+6q^{23}+8q^{24}+2q^{25}+13q^{26}+15q^{27}+⋯ ∈ S_{24}(Γ_{0}(5); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{10} ∼ ρ_{g}, where g is the weight-12 newform g = q + 15q^{2} + 12q^{3} + 13q^{4} + 3q^{5} + 10q^{6} + 13q^{7} + 7q^{8} + q^{9} + 11q^{10} + 3q^{11} + 3q^{12} + 9q^{13} + 8q^{14} + 2q^{15} + q^{16} + 8q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(15)ω^{1} & 0 \\ 0 & u(8)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+155622332426566814048379+ O(17^{2}0)q^{2} + 1782837064066288951668005 + O(17^{2}0)q^{3} + −942641202623435313809419 + O(17^{2}0)q^{4} + −48828125q^{5} + 543229804967107528293971 + O(17^{2}0)q^{6} + 256718911119690829977329 + O(17^{2}0)q^{7} + −1750643619163279139961831+O(17^{2}0)q^{8}+1245815621458798111075680+O(17^{2}0)q^{9}+448162369315317461072689+ O(17^{2}0)q^{10}+−729685643436266435463178+O(17^{2}0)q^{11}+−305808656151868425378320+O(17^{2}0)q^{12}+ −1598076314553166620977115+O(17^{2}0)q^{13}+1491135311481599389292787+O(17^{2}0)q^{14}+−1609364673198767412099824+ O(17^{2}0)q^{15}+−1143138364010211605110670+O(17^{2}0)q^{16}+−35297068327481788247497∗17+O(17^{2}0)q^{17}+ −961659424226718233776362+O(17^{2}0)q^{18}+1348168839929533979032035+O(17^{2}0)q^{19}+−1951510704793807504609221+ O(17^{2}0)q^{20} + 922377464392475635003293 + O(17^{2}0)q^{21} + 267441829698675481836560 + O(17^{2}0)q^{22} + −752728493784455152728595+O(17^{2}0)q^{23}+1861662786112740786813063+O(17^{2}0)q^{24}+2384185791015625q^{25}+ 1157034161342437571282064+O(17^{2}0)q^{26}+1364207246759348652135189+O(17^{2}0)q^{27}+⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

14. Consider f = q +4q^{2} +5q^{3} +7q^{4} +6q^{5} +3q^{6} +q^{7} +9q^{8} +14q^{9} +7q^{10} +4q^{11} +q^{12} +12q^{13} +4q^{14} +13q^{15} + 7q^{16} + 5q^{18} + 12q^{19} + 8q^{20} + 5q^{21} + 16q^{22} + 6q^{23} + 11q^{24} + 2q^{25} + 14q^{26} + 15q^{27} + ⋯ ∈ S_{24}(Γ_{0}(5); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-6 newform g = q + 2q^{2} + 13q^{3} + 6q^{4} + 8q^{5} + 9q^{6} + 5q^{7} + 16q^{8} + 11q^{9} + 16q^{10} + 5q^{11} + 10q^{12} + 14q^{13} + 10q^{14} + 2q^{15} + 10q^{16} + 5q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(7)ω^{6} & 0 \\ 0 & u(5)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(587427422373205749935235+ O(17^{2}0))∗a+23772875552033314480348∗17+O(17^{2}0)q^{3}+(1240606626846018068523411+O(17^{2}0))∗a− 1432997100225198264150287+O(17^{2}0)q^{4}+−48828125q^{5}+(668484446547446892718773+O(17^{2}0))∗a− 1344389323283294255248219+O(17^{2}0)q^{6}+−(691313563333689135434713+O(17^{2}0))∗a+1365844795723999342979382+ O(17^{2}0)q^{7} + (1210342042946692355113526 + O(17^{2}0)) ∗ a − 148255019539204764364025 + O(17^{2}0)q^{8} + (1427190266372239115501643 + O(17^{2}0)) ∗ a + 493206879229482919617262 + O(17^{2}0)q^{9} + −48828125 ∗ aq^{10} + −(1506712734930847940170135 + O(17^{2}0)) ∗ a − 520805314960478309959159 + O(17^{2}0)q^{11} + −(25098327225515734969188+O(17^{2}0))∗a+1968828314323009501843537+O(17^{2}0)q^{12}+−(1820994960747280413096251+ O(17^{2}0))∗a+1994746671690575480785311+O(17^{2}0)q^{13} +−(1224715988879121162513646+O(17^{2}0))∗ a+845928544781108823185162+O(17^{2}0)q^{14}+(277373210967346065641843+O(17^{2}0))∗a−46083949180638945773043∗ 17+O(17^{2}0)q^{15}+(329721640698934822363031+O(17^{2}0))∗a−573976214716741860206∗17+O(17^{2}0)q^{16}+ −(32430622915240244101522 ∗ 17 + O(17^{2}0)) ∗ a + 16976592034870216194395 ∗ 17 + O(17^{2}0)q^{17} + (374056123731492872611920+O(17^{2}0))∗a−1290293259925377192688311+O(17^{2}0)q^{18}+(1122657734390069605862517+ O(17^{2}0))∗a−356067325618301488984784+O(17^{2}0)q^{19}+−(145136086783816131173590+O(17^{2}0))∗a+ 1754778670645130625309690+O(17^{2}0)q^{20}+−(1511714595805881056107187+O(17^{2}0))∗a−409029821674258735148458+ O(17^{2}0)q^{21}+(12562734077963440465864215+O(17^{2}1))∗a−19083647500021499571259277+O(17^{2}1)q^{22}+ (223941540782542433289918+O(17^{2}0))∗a−96335251614387928705676∗17+O(17^{2}0)q^{23}+(1079702807405594027335338+ O(17^{2}0))∗a+1986323175628466766532926+O(17^{2}0)q^{24}+2384185791015625q^{25}+(1548690689031866979366305+ O(17^{2}0))∗a−1554227838232719234086673+O(17^{2}0)q^{26} +−(1276748070972387134933232+O(17^{2}0))∗ a + 73989092262878452095094 + O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −1240606626846018068523411 + O(17^{2}0)x + 1432997100225198255761679 + O(17^{2}0). The slope of f_{0} is 1.

15. Consider f = q + 6q^{2} + 9q^{3} + 13q^{5} + 3q^{6} + q^{7} + 5q^{8} + 16q^{9} + 10q^{10} + 5q^{11} + 5q^{13} + 6q^{14} + 15q^{15} + 13q^{16} + 11q^{18} + 4q^{19} + 9q^{21} + 13q^{22} + 6q^{23} + 11q^{24} + 16q^{25} + 13q^{26} + q^{27} + ⋯ ∈ S_{26}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{2} + 16x + 3 = 0. We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-8 newform g = q + 3q^{2} + 3q^{3} + 6q^{5} + 9q^{6} + 5q^{7} + 7q^{8} + 15q^{9} + q^{10} + 2q^{11} + 3q^{13} + 15q^{14} + q^{15} + 4q^{16} + 3q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{8} & 0 \\ 0 & u(3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+1958424543004790655903631+ O(17^{2}0)q^{2} +−1743477954173833808897463+O(17^{2}0)q^{3} +61432030969640928304948∗17+O(17^{2}0)q^{4} + −244140625q^{5}+−1786029708766539714985344+O(17^{2}0)q^{6}+−774606465622810530311543+O(17^{2}0)q^{7}+ −496459642099665311795724+O(17^{2}0)q^{8}+−1491409003605123868720375+O(17^{2}0)q^{9}+792737286563123560964669+ O(17^{2}0)q^{10} +160640617166987961953436+O(17^{2}0)q^{11} +69695746923125718674234∗17+O(17^{2}0)q^{12} + 1035436771019833282121367+O(17^{2}0)q^{13}+−71727830468463093201220+O(17^{2}0)q^{14}+−1978167920032890219291709+ O(17^{2}0)q^{15}+895172512491473571269555+O(17^{2}0)q^{16}+115559288140431734999913∗17+O(17^{2}0)q^{17}+ −1963068378313892436821597+O(17^{2}0)q^{18}+1993373315768476452058323+O(17^{2}0)q^{19}+−102177847875722552812644∗ 17+O(17^{2}0)q^{20}+−1708798935431019707236739+O(17^{2}0)q^{21}+969615065582402460402664+O(17^{2}0)q^{22}+ −1616287680497482106748874+O(17^{2}0)q^{23}+−1649911886022320953722279+O(17^{2}0)q^{24}+59604644775390625q^{25}+ 1316839932232267722047983+O(17^{2}0)q^{26}+1611616170932381800816524+O(17^{2}0)q^{27}+⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

16. Consider f = q +9q^{2} +12q^{3} +11q^{4} +13q^{5} +6q^{6} +8q^{7} +13q^{8} +11q^{9} +15q^{10} +2q^{11} +13q^{12} +6q^{13} +4q^{14} + 3q^{15}+10q^{16}+14q^{18}+6q^{19}+7q^{20}+11q^{21}+q^{22}+12q^{23}+3q^{24}+16q^{25}+3q^{26}+15q^{27}+⋯ ∈ S_{26}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{2} +16x+3 = 0. We have ρ_{f} ⊗χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + 9q^{2} + 5q^{3} + 11q^{4} + 4q^{5} + 11q^{6} + 9q^{7} + 13q^{8} + 11q^{9} + 2q^{10} + 15q^{11} + 4q^{12} + 6q^{13} + 13q^{14} + 3q^{15} + 10q^{16} + 6q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(3)ω^{1} & 0 \\ 0 & u(6)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+1798285489273704686914506+ O(17^{2}0)q^{2} + −587203369435627062905775 + O(17^{2}0)q^{3} + 1684992636323611245843638 + O(17^{2}0)q^{4} + −244140625q^{5}+−1015299182471886464694271+O(17^{2}0)q^{6}+−1074815832478417557002719+O(17^{2}0)q^{7}+ −1842530399630430203882938+O(17^{2}0)q^{8}+−517116189538697107180269+O(17^{2}0)q^{9}+−1679552870873681225179048+ O(17^{2}0)q^{10} +−1466648749932152648002170+O(17^{2}0)q^{11} +899032205118976250125274+O(17^{2}0)q^{12} + −1801809010900298679917026+O(17^{2}0)q^{13}+−1364640658145264783873852+O(17^{2}0)q^{14}+−1320577531796203082873856+ O(17^{2}0)q^{15}+−1153364436581377107471790+O(17^{2}0)q^{16}+−30348388277982883261106∗17+O(17^{2}0)q^{17}+ 1408810476506119234205757+O(17^{2}0)q^{18}+126899854497245916362643+O(17^{2}0)q^{19}+−701034044608084371674512+ O(17^{2}0)q^{20} +−426087490917274880130414+O(17^{2}0)q^{21} +−37600078867820406081002+O(17^{2}0)q^{22} + 1409427294183161589409616+O(17^{2}0)q^{23}+910086008389936499736192+O(17^{2}0)q^{24}+59604644775390625q^{25}+ 205867635707322522491028+O(17^{2}0)q^{26} +1421918710315487166254760+O(17^{2}0)q^{27} +⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

17. Consider f = q + z^{2}54q^{2} + z^{1}57q^{3} + z^{6}5q^{4} + 4q^{5} + z^{1}23q^{6} + z^{1}89q^{7} + zq^{8} + z^{6}5q^{9} + z^{1}82q^{10} + z^{4}4q^{11} + z^{2}22q^{12} + z^{2}02q^{13} + z^{1}55q^{14} + z^{8}5q^{15} + z^{2}2q^{16} + z^{3}1q^{18} + z^{4}9q^{19} + z^{2}81q^{20} + z^{5}8q^{21} + z^{1}0q^{22} + z^{2}10q^{23} + z^{1}58q^{24}+16q^{25}+z^{1}68q^{26}+z^{1}81q^{27}+⋯ ∈ S_{26}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{2}+16x+3 = 0. We have ρ_{f} ⊗χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q+z^{2}54q^{2}+z^{1}3q^{3}+z^{6}5q^{4}+13q^{5}+z^{2}67q^{6}+ z^{4}5q^{7} + zq^{8} + z^{6}5q^{9} + z^{3}8q^{10} + z^{1}88q^{11} + z^{7}8q^{12} + z^{2}02q^{13} + z^{1}1q^{14} + z^{8}5q^{15} + z^{2}2q^{16} + z^{5}3q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}35)ω^{1} & 0 \\ 0 & u(z^{5}3)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(552872681228445313879200+ O(17^{2}0))∗a+571154033604853445648338+O(17^{2}0)q^{3} +(1287778682659478026147624+O(17^{2}0))∗a+ 1639352480729082476246681+O(17^{2}0)q^{4} +244140625q^{5} +(744817814130256655905915+O(17^{2}0))∗a+ 1084469833797034528676436+O(17^{2}0)q^{6}+−(396252811293356223248695+O(17^{2}0))∗a−1235646123506262081735278+ O(17^{2}0)q^{7} + (378087339577689203448352 + O(17^{2}0)) ∗ a − 714159483646659019861257 + O(17^{2}0)q^{8} + (1100132761060402197060925 + O(17^{2}0)) ∗ a − 1734797861992367215553940 + O(17^{2}0)q^{9} + 244140625 ∗ aq^{10} + −(120256396766007306568027 + O(17^{2}0)) ∗ a + 841774733398631340545758 + O(17^{2}0)q^{11} + (810120793035431061035272+O(17^{2}0))∗a+99845629602357671836442+O(17^{2}0)q^{12}+−(161440241870113460139740+ O(17^{2}0))∗a+1186775462344216774967487+O(17^{2}0)q^{13} +−(2000863595222289065283584+O(17^{2}0))∗ a+1175998097059797042850655+O(17^{2}0)q^{14}+−(1121439636008193770301433+O(17^{2}0))∗a+848391581188895188760912+ O(17^{2}0)q^{15}+−(1002439290197462600795867+O(17^{2}0))∗a+1981848689119577465313478+O(17^{2}0)q^{16}+ (101855075189473146773512 ∗ 17 + O(17^{2}0)) ∗ a − 15522736120267085792576 ∗ 17 + O(17^{2}0)q^{17} + (933668420082525473115824+O(17^{2}0))∗a+1342280394677092063166624+O(17^{2}0)q^{18}+−(1160389426274792199060328+ O(17^{2}0))∗a−1556130282820477504738450+O(17^{2}0)q^{19} +−(1539396275787318808655777+O(17^{2}0))∗ a−723735678174778329455584+O(17^{2}0)q^{20}+−(1224190660770212856493538+O(17^{2}0))∗a+1853234727676277948923426+ O(17^{2}0)q^{21}+−(1636479742858747207841032+O(17^{2}0))∗a−117655352318147150587990+O(17^{2}0)q^{22}+ −(14747682404265625018643+O(17^{2}0))∗a−1196408250443918296625019+O(17^{2}0)q^{23}+−(123862727866441636627403+ O(17^{2}0))∗a−1291555332114531248062789+O(17^{2}0)q^{24}+59604644775390625q^{25}+−(1429060666158755063124932+ O(17^{2}0)) ∗ a − 1606546510818278280248168 + O(17^{2}0)q^{26} + −(101348293038478177508583 + O(17^{2}0)) ∗ a + 1237231142855204039748889 + O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −1287778682659478026147624 + O(17^{2}0)x + −1639352480729082509801113 + O(17^{2}0). The slope of f_{0} is 1.

18. Consider f = q + z^{2}86q^{2} + z^{7}7q^{3} + z^{2}41q^{4} + 4q^{5} + z^{7}5q^{6} + z^{4}5q^{7} + z^{1}7q^{8} + z^{2}41q^{9} + z^{2}14q^{10} + z^{1}72q^{11} + z^{3}0q^{12} +z^{2}66q^{13} +z^{4}3q^{14} +z^{5}q^{15} +z^{8}6q^{16} +z^{2}39q^{18} +z^{2}57q^{19} +z^{1}69q^{20} +z^{1}22q^{21} +z^{1}70q^{22} +z^{1}14q^{23} + z^{9}4q^{24}+16q^{25}+z^{2}64q^{26}+z^{1}97q^{27}+⋯ ∈ S_{26}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{2}+16x+3 = 0. We have ρ_{f} ⊗χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q+z^{2}86q^{2}+z^{2}21q^{3}+z^{2}41q^{4}+13q^{5}+z^{2}19q^{6}+ z^{1}89q^{7} +z^{1}7q^{8} +z^{2}41q^{9} +z^{7}0q^{10} +z^{2}8q^{11} +z^{1}74q^{12} +z^{2}66q^{13} +z^{1}87q^{14} +z^{5}q^{15} +z^{8}6q^{16} +z^{3}7q^{17} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}51)ω^{1} & 0 \\ 0 & u(z^{3}7)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(552872681228445313879200+ O(17^{2}0))∗a+571154033604853445648338+O(17^{2}0)q^{3} +(1287778682659478026147624+O(17^{2}0))∗a+ 1639352480729082476246681+O(17^{2}0)q^{4} +244140625q^{5} +(744817814130256655905915+O(17^{2}0))∗a+ 1084469833797034528676436+O(17^{2}0)q^{6}+−(396252811293356223248695+O(17^{2}0))∗a−1235646123506262081735278+ O(17^{2}0)q^{7} + (378087339577689203448352 + O(17^{2}0)) ∗ a − 714159483646659019861257 + O(17^{2}0)q^{8} + (1100132761060402197060925 + O(17^{2}0)) ∗ a − 1734797861992367215553940 + O(17^{2}0)q^{9} + 244140625 ∗ aq^{10} + −(120256396766007306568027 + O(17^{2}0)) ∗ a + 841774733398631340545758 + O(17^{2}0)q^{11} + (810120793035431061035272+O(17^{2}0))∗a+99845629602357671836442+O(17^{2}0)q^{12}+−(161440241870113460139740+ O(17^{2}0))∗a+1186775462344216774967487+O(17^{2}0)q^{13} +−(2000863595222289065283584+O(17^{2}0))∗ a+1175998097059797042850655+O(17^{2}0)q^{14}+−(1121439636008193770301433+O(17^{2}0))∗a+848391581188895188760912+ O(17^{2}0)q^{15}+−(1002439290197462600795867+O(17^{2}0))∗a+1981848689119577465313478+O(17^{2}0)q^{16}+ (101855075189473146773512 ∗ 17 + O(17^{2}0)) ∗ a − 15522736120267085792576 ∗ 17 + O(17^{2}0)q^{17} + (933668420082525473115824+O(17^{2}0))∗a+1342280394677092063166624+O(17^{2}0)q^{18}+−(1160389426274792199060328+ O(17^{2}0))∗a−1556130282820477504738450+O(17^{2}0)q^{19} +−(1539396275787318808655777+O(17^{2}0))∗ a−723735678174778329455584+O(17^{2}0)q^{20}+−(1224190660770212856493538+O(17^{2}0))∗a+1853234727676277948923426+ O(17^{2}0)q^{21}+−(1636479742858747207841032+O(17^{2}0))∗a−117655352318147150587990+O(17^{2}0)q^{22}+ −(14747682404265625018643+O(17^{2}0))∗a−1196408250443918296625019+O(17^{2}0)q^{23}+−(123862727866441636627403+ O(17^{2}0))∗a−1291555332114531248062789+O(17^{2}0)q^{24}+59604644775390625q^{25}+−(1429060666158755063124932+ O(17^{2}0)) ∗ a − 1606546510818278280248168 + O(17^{2}0)q^{26} + −(101348293038478177508583 + O(17^{2}0)) ∗ a + 1237231142855204039748889 + O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −1287778682659478026147624 + O(17^{2}0)x + −1639352480729082509801113 + O(17^{2}0). The slope of f_{0} is 1.

19. Consider f = q + 10q^{2} + 14q^{3} + 13q^{4} + 4q^{5} + 4q^{6} + 2q^{7} + 8q^{8} + 12q^{9} + 6q^{10} + 12q^{12} + 4q^{13} + 3q^{14} + 5q^{15} + 3q^{16} + q^{18} + 4q^{19} + q^{20} + 11q^{21} + 16q^{23} + 10q^{24} + 16q^{25} + 6q^{26} + 6q^{27} + ⋯ ∈ S_{26}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{2} + 16x + 3 = 0. We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-8 newform g = q+5q^{2}+16q^{3}+16q^{4}+11q^{5}+12q^{6}+10q^{7}+q^{8}+7q^{9}+4q^{10}+q^{12}+16q^{13}+16q^{14}+6q^{15}+14q^{16}+2q^{17}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(9)ω^{8} & 0 \\ 0 & u(2)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−1960614826264891096713919+ O(17^{2}0)q^{2} + 1182241792039471593005988 + O(17^{2}0)q^{3} + 794376716812316641332910 + O(17^{2}0)q^{4} + 244140625q^{5}+−131414100551079704526321+O(17^{2}0)q^{6}+−391904205974278479801799+O(17^{2}0)q^{7}+ 806103465672700472492165+O(17^{2}0)q^{8}+−1295728529284973921039169+O(17^{2}0)q^{9}+−1340906644854730756740160+ O(17^{2}0)q^{10}+50308428731694791961308∗17+O(17^{2}0)q^{11}+−1000619851752511089404598+O(17^{2}0)q^{12}+ 1723219589694632233136915+O(17^{2}0)q^{13}+1694236859326458396342791+O(17^{2}0)q^{14}+1303546065738829998577383+ O(17^{2}0)q^{15}+1783484838169381981313698+O(17^{2}0)q^{16}+−96922231064358263426567∗17+O(17^{2}0)q^{17}+ 1191019600474742108745624+O(17^{2}0)q^{18}+−1923799102014386308640226+O(17^{2}0)q^{19}+243646619261963367713302+ O(17^{2}0)q^{20} +403739423343834899755261+O(17^{2}0)q^{21} +21953823726204578871496∗17+O(17^{2}0)q^{22} + −968704220056646052208872+O(17^{2}0)q^{23}+1777380266690794783885959+O(17^{2}0)q^{24}+59604644775390625q^{25}+ −1088726651317586430135068 + O(17^{2}0)q^{26} + 445854854798640677514892 + O(17^{2}0)q^{27} + ⋯ is an ℓadic newform that lifts f. The slope of f_{0} is 1.

20. Consider f = q +13q^{2} +12q^{3} +14q^{4} +4q^{5} +3q^{6} +12q^{7} +3q^{8} +11q^{9} +q^{10} +4q^{11} +15q^{12} +14q^{13} +3q^{14} + 14q^{15}+11q^{16}+7q^{18}+13q^{19}+5q^{20}+8q^{21}+q^{22}+16q^{23}+2q^{24}+16q^{25}+12q^{26}+15q^{27}+⋯ ∈ S_{26}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{2} +16x+3 = 0. We have ρ_{f} ⊗χ^{15} ∼ ρ_{g}, where g is the weight-8 newform g = q + 15q^{2} + 4q^{3} + 12q^{4} + 11q^{5} + 9q^{6} + 9q^{7} + 11q^{8} + 5q^{9} + 12q^{10} + 5q^{11} + 14q^{12} + 5q^{13} + 16q^{14} + 10q^{15} + 6q^{16} + 11q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(14)ω^{8} & 0 \\ 0 & u(11)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+1569774345434466488306891+ O(17^{2}0)q^{2} + −167012742469246246359768 + O(17^{2}0)q^{3} + −626713992365046312297409 + O(17^{2}0)q^{4} + 244140625q^{5} + 1763663801470938598817093 + O(17^{2}0)q^{6} + 223224898846360375582926 + O(17^{2}0)q^{7} + 2024957258758332689181529+O(17^{2}0)q^{8}+1847315454409024163276138+O(17^{2}0)q^{9}+−766367810433478958089537+ O(17^{2}0)q^{10} +1492257419638100702312535+O(17^{2}0)q^{11} +1505447856919809582667319+O(17^{2}0)q^{12} + 1394327563313602637619453+O(17^{2}0)q^{13}+−1030372037601504870271324+O(17^{2}0)q^{14}+1340893468549297402934348+ O(17^{2}0)q^{15}+−1316420583514707018117930+O(17^{2}0)q^{16}+12430026758843424343731∗17+O(17^{2}0)q^{17}+ 1788275230958053902431739+O(17^{2}0)q^{18}+1370135671901661832320311+O(17^{2}0)q^{19}+1570772300176679708196031+ O(17^{2}0)q^{20} +1191982483642830967715310+O(17^{2}0)q^{21} +−146512383785238123434127+O(17^{2}0)q^{22} + 404631743562708943579336+O(17^{2}0)q^{23}+571767810948041178700553+O(17^{2}0)q^{24}+59604644775390625q^{25}+ −149677887207978122996313 + O(17^{2}0)q^{26} + −1210537710962935084302565 + O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

21. Consider f = q + 12q^{2} + 7q^{3} + 3q^{5} + 16q^{6} + 10q^{7} + 6q^{8} + 8q^{9} + 2q^{10} + 13q^{11} + 14q^{13} + q^{14} + 4q^{15} + 4q^{16} + 11q^{18} + 8q^{19} + 2q^{21} + 3q^{22} + 15q^{23} + 8q^{24} + 9q^{25} + 15q^{26} + 7q^{27} + ⋯ ∈ S_{28}(Γ_{0}(5); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + 3q^{2} + 3q^{3} + 6q^{5} + 9q^{6} + 5q^{7} + 7q^{8} + 15q^{9} + q^{10} + 2q^{11} + 3q^{13} + 15q^{14} + q^{15} + 4q^{16} + 3q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(6)ω^{1} & 0 \\ 0 & u(3)ω^{10} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+188016018946329831569988+ O(17^{2}0)q^{2} + 747812238660666653750969 + O(17^{2}0)q^{3} + 111983530002098451959905 ∗ 17 + O(17^{2}0)q^{4} + 1220703125q^{5} +1257034206303197080588722+O(17^{2}0)q^{6} +1164128749802670904900938+O(17^{2}0)q^{7} + 316311628239604741896790+O(17^{2}0)q^{8}+1076858119381525095406025+O(17^{2}0)q^{9}+1762966100581239285533370+ O(17^{2}0)q^{10}+1664012886918245807416948+O(17^{2}0)q^{11}+63662246089796628716954∗17+O(17^{2}0)q^{12}+ −1276769522049621781420515+O(17^{2}0)q^{13}+1393282712225553414808623+O(17^{2}0)q^{14}+−1159562253728613616882946+ O(17^{2}0)q^{15}+−1608124953115722051273681+O(17^{2}0)q^{16}+−31609260669689076644137∗17+O(17^{2}0)q^{17}+ 429448990683058141257606+O(17^{2}0)q^{18}+−1867446125283817792895504+O(17^{2}0)q^{19}+119350796914846096539292∗ 17+O(17^{2}0)q^{20}+596618317893967272628348+O(17^{2}0)q^{21}+1446061735020266996187035+O(17^{2}0)q^{22}+ 1122444127708437327808588+O(17^{2}0)q^{23}+−1653795851374261207304332+O(17^{2}0)q^{24}+1490116119384765625q^{25}+ 243338253514083754817919 + O(17^{2}0)q^{26} + 134809324335971578420029 + O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

22. Consider f = q + q^{2} + 15q^{3} + 10q^{4} + 3q^{5} + 15q^{6} + 12q^{7} + 2q^{8} + 14q^{9} + 3q^{10} + 12q^{11} + 14q^{12} + 10q^{13} + 12q^{14} + 11q^{15} + 7q^{16} + 14q^{18} + 12q^{19} + 13q^{20} + 10q^{21} + 12q^{22} + 13q^{23} + 13q^{24} + 9q^{25} + 10q^{26} + 3q^{27} + ⋯ ∈ S_{28}(Γ_{0}(5); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-10 newform g = q + 9q^{2} + 5q^{3} + 11q^{4} + 4q^{5} + 11q^{6} + 9q^{7} + 13q^{8} + 11q^{9} + 2q^{10} + 15q^{11} + 4q^{12} + 6q^{13} + 13q^{14} + 3q^{15} + 10q^{16} + 6q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(3)ω^{10} & 0 \\ 0 & u(6)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+1624886012353228748919323+ O(17^{2}0)q^{2} + 1407513661836793432887596 + O(17^{2}0)q^{3} + −950065427396313353420151 + O(17^{2}0)q^{4} + 1220703125q^{5} +−458813091854518546424651+O(17^{2}0)q^{6} +472402167541232173943750+O(17^{2}0)q^{7} + 790222075512592117102293+O(17^{2}0)q^{8}+−1804607889578714083078530+O(17^{2}0)q^{9}+1579500291385303008545375+ O(17^{2}0)q^{10}+−1212273232676555923667280+O(17^{2}0)q^{11}+−1033100682980206117303157+O(17^{2}0)q^{12}+ −1533744279354443949526412+O(17^{2}0)q^{13}+672628856467415458055432+O(17^{2}0)q^{14}+−1178209388597011888797115+ O(17^{2}0)q^{15}+946971118282231874040664+O(17^{2}0)q^{16}+−15651309192536047128087∗17+O(17^{2}0)q^{17}+ −949420344198297033677988+O(17^{2}0)q^{18}+1863801618902814142572952+O(17^{2}0)q^{19}+−1649520169220688973078701+ O(17^{2}0)q^{20}+1788603161811806829944943+O(17^{2}0)q^{21}+−1860390304936008957939784+O(17^{2}0)q^{22}+ 1305051255121062343706835+O(17^{2}0)q^{23}+1173464142061612815595289+O(17^{2}0)q^{24}+1490116119384765625q^{25}+ −1442853753394903160407991 + O(17^{2}0)q^{26} + 1572329764292690615133803 + O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

23. Consider f = q +z^{2}18q^{2} +z^{3}1q^{3} +z^{2}81q^{4} +14q^{5} +z^{2}49q^{6} +z^{2}43q^{7} +z^{1}81q^{8} +z^{1}01q^{9} +z^{9}2q^{10} +z^{2}6q^{11} + z^{2}4q^{12} +z^{2}74q^{13} +z^{1}73q^{14} +z^{1}93q^{15} +z^{1}66q^{16} +z^{3}1q^{18} +z^{1}3q^{19} +z^{1}55q^{20} +z^{2}74q^{21} +z^{2}44q^{22} +z^{4}8q^{23} + z^{2}12q^{24} +9q^{25} +z^{2}04q^{26} +z^{9}1q^{27} +⋯ ∈ S_{28}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{2} +16x+3 = 0. We have ρ_{f} ⊗χ^{15} ∼ ρ_{g}, where g is the weight-10 newform g = q+z^{2}54q^{2}+z^{1}3q^{3}+z^{6}5q^{4}+13q^{5}+z^{2}67q^{6}+ z^{4}5q^{7} + zq^{8} + z^{6}5q^{9} + z^{3}8q^{10} + z^{1}88q^{11} + z^{7}8q^{12} + z^{2}02q^{13} + z^{1}1q^{14} + z^{8}5q^{15} + z^{2}2q^{16} + z^{5}3q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}35)ω^{10} & 0 \\ 0 & u(z^{5}3)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(513745639460272990291303+ O(17^{2}0)) ∗ a − 960055887336129390899552 + O(17^{2}0)q^{3} + (814114344927358890581802 + O(17^{2}0)) ∗ a − 1158007072521574626289086+O(17^{2}0)q^{4}+−1220703125q^{5}+(1452034829088452946804454+O(17^{2}0))∗ a−938942130841747238466733+O(17^{2}0)q^{6}+(1985200627752591788116615+O(17^{2}0))∗a+187299676395870479747822+ O(17^{2}0)q^{7} +(1643861449026332522400582+O(17^{2}0))∗a+1433265377621805004969412+O(17^{2}0)q^{8} + −(981740030213170496770131+O(17^{2}0))∗a+1257759881546368596573581+O(17^{2}0)q^{9}+−1220703125∗ aq^{10} + −(184990031482950986652021 + O(17^{2}0)) ∗ a − 1382032021251658760935533 + O(17^{2}0)q^{11} + −(584674275708104227605709+O(17^{2}0))∗a−1655847535974884276992766+O(17^{2}0)q^{12}+(1996416540643190116071927+ O(17^{2}0)) ∗ a − 980618274850491164467102 + O(17^{2}0)q^{13} + (11824542586310152095418 + O(17^{2}0)) ∗ a − 1399505803471799656669067+O(17^{2}0)q^{14}+−(1415805999690738459843354+O(17^{2}0))∗a−1487124406810876258354637+ O(17^{2}0)q^{15} +(528364656806157049906524+O(17^{2}0))∗a+1030565878413360809996836+O(17^{2}0)q^{16} + −(27198808235538640045763 ∗ 17 + O(17^{2}0)) ∗ a + 31805385120845980877961 ∗ 17 + O(17^{2}0)q^{17} + (1304825400115979844408942+O(17^{2}0))∗a+262948895281931275929061+O(17^{2}0)q^{18}+(1832006258261017560746179+ O(17^{2}0))∗a−215363575343727255510718+O(17^{2}0)q^{19}+−(1637663151021468392224159+O(17^{2}0))∗a− 1298839357574815344012072+O(17^{2}0)q^{20}+(111945834431718966566112+O(17^{2}0))∗a+1099392062493849624303167+ O(17^{2}0)q^{21}+−(1927287298199419050885626+O(17^{2}0))∗a−270151904932727488339721+O(17^{2}0)q^{22}+ (1043715007447892025881975+O(17^{2}0))∗a+1324345704708519718865426+O(17^{2}0)q^{23}+−(1927678871060120621110646+ O(17^{2}0))∗a−1947957860658188741912040+O(17^{2}0)q^{24}+1490116119384765625q^{25}+−(204236800894001976966410+ O(17^{2}0)) ∗ a − 612862235842903654965567 + O(17^{2}0)q^{26} + −(1716490685183586645082262 + O(17^{2}0)) ∗ a − 756493559278310593138678 + O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −814114344927358890581802 + O(17^{2}0)x + 1158007072521574492071358 + O(17^{2}0). The slope of f_{0} is 1.

24. Consider f = q + z^{2}50q^{2} + z^{2}39q^{3} + z^{1}69q^{4} + 14q^{5} + z^{2}01q^{6} + z^{9}9q^{7} + z^{1}97q^{8} + z^{2}77q^{9} + z^{1}24q^{10} + z^{1}54q^{11} + z^{1}20q^{12} + z^{5}0q^{13} + z^{6}1q^{14} + z^{1}13q^{15} + z^{2}30q^{16} + z^{2}39q^{18} + z^{2}21q^{19} + z^{4}3q^{20} + z^{5}0q^{21} + z^{1}16q^{22} + z^{2}40q^{23} + z^{1}48q^{24} + 9q^{25} + z^{1}2q^{26} + z^{1}07q^{27} + ⋯ ∈ S_{28}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{2} + 16x + 3 = 0. We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-10 newform g = q + z^{2}86q^{2} + z^{2}21q^{3} + z^{2}41q^{4} + 13q^{5} + z^{2}19q^{6} + z^{1}89q^{7} + z^{1}7q^{8} + z^{2}41q^{9} + z^{7}0q^{10} + z^{2}8q^{11} + z^{1}74q^{12} + z^{2}66q^{13} + z^{1}87q^{14} + z^{5}q^{15} + z^{8}6q^{16} + z^{3}7q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}51)ω^{10} & 0 \\ 0 & u(z^{3}7)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(513745639460272990291303+ O(17^{2}0)) ∗ a − 960055887336129390899552 + O(17^{2}0)q^{3} + (814114344927358890581802 + O(17^{2}0)) ∗ a − 1158007072521574626289086+O(17^{2}0)q^{4}+−1220703125q^{5}+(1452034829088452946804454+O(17^{2}0))∗ a−938942130841747238466733+O(17^{2}0)q^{6}+(1985200627752591788116615+O(17^{2}0))∗a+187299676395870479747822+ O(17^{2}0)q^{7} +(1643861449026332522400582+O(17^{2}0))∗a+1433265377621805004969412+O(17^{2}0)q^{8} + −(981740030213170496770131+O(17^{2}0))∗a+1257759881546368596573581+O(17^{2}0)q^{9}+−1220703125∗ aq^{10} + −(184990031482950986652021 + O(17^{2}0)) ∗ a − 1382032021251658760935533 + O(17^{2}0)q^{11} + −(584674275708104227605709+O(17^{2}0))∗a−1655847535974884276992766+O(17^{2}0)q^{12}+(1996416540643190116071927+ O(17^{2}0)) ∗ a − 980618274850491164467102 + O(17^{2}0)q^{13} + (11824542586310152095418 + O(17^{2}0)) ∗ a − 1399505803471799656669067+O(17^{2}0)q^{14}+−(1415805999690738459843354+O(17^{2}0))∗a−1487124406810876258354637+ O(17^{2}0)q^{15} +(528364656806157049906524+O(17^{2}0))∗a+1030565878413360809996836+O(17^{2}0)q^{16} + −(27198808235538640045763 ∗ 17 + O(17^{2}0)) ∗ a + 31805385120845980877961 ∗ 17 + O(17^{2}0)q^{17} + (1304825400115979844408942+O(17^{2}0))∗a+262948895281931275929061+O(17^{2}0)q^{18}+(1832006258261017560746179+ O(17^{2}0))∗a−215363575343727255510718+O(17^{2}0)q^{19}+−(1637663151021468392224159+O(17^{2}0))∗a− 1298839357574815344012072+O(17^{2}0)q^{20}+(111945834431718966566112+O(17^{2}0))∗a+1099392062493849624303167+ O(17^{2}0)q^{21}+−(1927287298199419050885626+O(17^{2}0))∗a−270151904932727488339721+O(17^{2}0)q^{22}+ (1043715007447892025881975+O(17^{2}0))∗a+1324345704708519718865426+O(17^{2}0)q^{23}+−(1927678871060120621110646+ O(17^{2}0))∗a−1947957860658188741912040+O(17^{2}0)q^{24}+1490116119384765625q^{25}+−(204236800894001976966410+ O(17^{2}0)) ∗ a − 612862235842903654965567 + O(17^{2}0)q^{26} + −(1716490685183586645082262 + O(17^{2}0)) ∗ a − 756493559278310593138678 + O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −814114344927358890581802 + O(17^{2}0)x + 1158007072521574492071358 + O(17^{2}0). The slope of f_{0} is 1.

25. Consider f = q + 3q^{2} + 9q^{3} + q^{4} + 14q^{5} + 10q^{6} + 3q^{7} + 13q^{8} + 6q^{9} + 8q^{10} + 9q^{12} + q^{13} + 9q^{14} + 7q^{15} + 14q^{16} + q^{18} + 8q^{19} + 14q^{20} + 10q^{21} + 6q^{23} + 15q^{24} + 9q^{25} + 3q^{26} + 8q^{27} + ⋯ ∈ S_{28}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{2} + 16x + 3 = 0. We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q+5q^{2}+16q^{3}+16q^{4}+11q^{5}+12q^{6}+10q^{7}+q^{8}+7q^{9}+4q^{10}+q^{12}+16q^{13}+16q^{14}+6q^{15}+14q^{16}+2q^{17}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(9)ω^{1} & 0 \\ 0 & u(2)ω^{10} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−1702395884833057978430973+ O(17^{2}0)q^{2} + −163982665143703222759232 + O(17^{2}0)q^{3} + −1135405160867682038613991 + O(17^{2}0)q^{4} + −1220703125q^{5}+−1500846003831261254169388+O(17^{2}0)q^{6}+1637520932383089498588209+O(17^{2}0)q^{7}+ 1160519886979751784889560+O(17^{2}0)q^{8}+−2028979910902212328668882+O(17^{2}0)q^{9}+−1603355184514612216316680+ O(17^{2}0)q^{10}+35272439757282282642556∗17+O(17^{2}0)q^{11}+−1270124287038402825467872+O(17^{2}0)q^{12}+ 31930209546600752481054+O(17^{2}0)q^{13}+−1803008332796920998107777+O(17^{2}0)q^{14}+1053052161889592501113754+ O(17^{2}0)q^{15}+−1485822443267913250883449+O(17^{2}0)q^{16}+−94469871823864265498685∗17+O(17^{2}0)q^{17}+ −32268283526231585683729+O(17^{2}0)q^{18}+−1551821294674611042057374+O(17^{2}0)q^{19}+437183200740091526079800+ O(17^{2}0)q^{20}+1659700248164775330010372+O(17^{2}0)q^{21}+−50776001850998762823476∗17+O(17^{2}0)q^{22}+ 1049339073469590161522897+O(17^{2}0)q^{23}+−610117145532696139555819+O(17^{2}0)q^{24}+1490116119384765625q^{25}+ −1914087054389696650966590 + O(17^{2}0)q^{26} + −1991138983945847518363809 + O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

26. Consider f = q + 9q^{2} + 15q^{3} + 5q^{4} + 14q^{5} + 16q^{6} + q^{7} + 7q^{8} + 14q^{9} + 7q^{10} + 7q^{11} + 7q^{12} + 12q^{13} + 9q^{14} + 6q^{15} +6q^{16} +7q^{18} +9q^{19} +2q^{20} +15q^{21} +12q^{22} +6q^{23} +3q^{24} +9q^{25} +6q^{26} +3q^{27} +⋯ ∈ S_{28}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{2} + 16x + 3 = 0. We have ρ_{f} ⊗ χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + 15q^{2} + 4q^{3} + 12q^{4} + 11q^{5} + 9q^{6} + 9q^{7} + 11q^{8} + 5q^{9} + 12q^{10} + 5q^{11} + 14q^{12} + 5q^{13} + 16q^{14} + 10q^{15} + 6q^{16} + 11q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(14)ω^{1} & 0 \\ 0 & u(11)ω^{10} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−709591279240712004996081+ O(17^{2}0)q^{2} + −586309122935307685694084 + O(17^{2}0)q^{3} + 1901331812679539932739798 + O(17^{2}0)q^{4} + −1220703125q^{5}+−474078780596136949883646+O(17^{2}0)q^{6}+1255200669446926991938985+O(17^{2}0)q^{7}+ −418400268285146414453147+O(17^{2}0)q^{8}+2009848463775703238218035+O(17^{2}0)q^{9}+75322929441079992005425+ O(17^{2}0)q^{10} + −897279244710055415327782 + O(17^{2}0)q^{11} + 919370103840628507055140 + O(17^{2}0)q^{12} + 712440466123861106750661+O(17^{2}0)q^{13}+−2016029784053921101570311+O(17^{2}0)q^{14}+−122604958090016036189464+ O(17^{2}0)q^{15}+1003515194957622418757992+O(17^{2}0)q^{16}+103815733913609041683559∗17+O(17^{2}0)q^{17}+ 1090137674667914260750890+O(17^{2}0)q^{18}+115994956959805938091437+O(17^{2}0)q^{19}+694565949278976877017510+ O(17^{2}0)q^{20} + −702191778607356781572669 + O(17^{2}0)q^{21} + 685534580748698341998536 + O(17^{2}0)q^{22} + 1378672205940298726955952+O(17^{2}0)q^{23}+−1703709694966266131145583+O(17^{2}0)q^{24}+1490116119384765625q^{25}+ −889306520256646053755956 + O(17^{2}0)q^{26} + −765851826818686233990055 + O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

27. Consider f = q + 4q^{3} + 2q^{4} + 2q^{5} + q^{7} + 4q^{9} + 10q^{11} + 8q^{12} + 15q^{13} + 8q^{15} + 4q^{16} + 11q^{19} + 4q^{20} + 4q^{21} + 14q^{23} + 4q^{25} + 2q^{27} + ⋯ ∈ S_{30}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{3} + x + 14 = 0. We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-12 newform g = q + 7q^{3} + 9q^{4} + 14q^{5} + 5q^{7} + 8q^{9} + 4q^{11} + 12q^{12} + 9q^{13} + 13q^{15} + 13q^{16} + 6q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(3)ω^{12} & 0 \\ 0 & u(6)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−41291870264471669467382∗ 17+O(17^{2}0)q^{2}+1840655966151185781376310+O(17^{2}0)q^{3}+−16174467088173692479656360+O(17^{2}1)q^{4}+ −6103515625q^{5}+58995004340342164693829∗17+O(17^{2}0)q^{6}+−703805673903317021989065+O(17^{2}0)q^{7}+ 19302377614439935451890∗17+O(17^{2}0)q^{8}+1573650869184117266755470+O(17^{2}0)q^{9}+10995374253148640026902∗ 17+O(17^{2}0)q^{10}+−1400064989626644148525886+O(17^{2}0)q^{11}+1259459479625206866115562+O(17^{2}0)q^{12}+ 2006779930754789765721728+O(17^{2}0)q^{13}+5397689053050566264510∗17+O(17^{2}0)q^{14}+237971848304449344801159+ O(17^{2}0)q^{15}+1921638411524219749606437+O(17^{2}0)q^{16}+−22772904418819065599847∗17+O(17^{2}0)q^{17}+ −9488387942122623545807∗17+O(17^{2}0)q^{18}+−288771028459016846504444+O(17^{2}0)q^{19}+−14124282794845923578412255+ O(17^{2}1)q^{20}+−1229254137659585010751588+O(17^{2}0)q^{21}+−57638845681953040162796∗17+O(17^{2}0)q^{22}+ 21893497195755165891774+O(17^{2}0)q^{23}+−48309182276033819998619∗17+O(17^{2}0)q^{24}+37252902984619140625q^{25}+ 81151856008051999955991 ∗ 17 + O(17^{2}0)q^{26} + 1856642364543293254159329 + O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

28. Consider f = q + 15q^{2} + q^{3} + 6q^{4} + 15q^{5} + 15q^{6} + 14q^{7} + q^{8} + 6q^{9} + 4q^{10} + 14q^{11} + 6q^{12} + 14q^{13} + 6q^{14} + 15q^{15}+10q^{16}+5q^{18}+11q^{19}+5q^{20}+14q^{21}+6q^{22}+13q^{23}+q^{24}+4q^{25}+6q^{26}+11q^{27}+⋯ ∈ S_{30}(Γ_{0}(5); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q + 2q^{2} + 13q^{3} + 6q^{4} + 8q^{5} + 9q^{6} + 5q^{7} + 16q^{8} + 11q^{9} + 16q^{10} + 5q^{11} + 10q^{12} + 14q^{13} + 10q^{14} + 2q^{15} + 10q^{16} + 5q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(7)ω^{1} & 0 \\ 0 & u(5)ω^{12} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+−(1303791718492555110573272+ O(17^{2}0))∗a+44579345013815318634004∗17+O(17^{2}0)q^{3}+−(241262051544956541389384+O(17^{2}0))∗a− 1375822160579054951939983+O(17^{2}0)q^{4}+6103515625q^{5}+−(1564192734288245891215722+O(17^{2}0))∗ a−502881446480875444651697+O(17^{2}0)q^{6}+(324479364226703625364140+O(17^{2}0))∗a−927414111054932934996975+ O(17^{2}0)q^{7} + −(1764935734889902331728720 + O(17^{2}0)) ∗ a + 72670737360722429333343 + O(17^{2}0)q^{8} + (225687345461093157624697 + O(17^{2}0)) ∗ a + 332295184313858892993225 + O(17^{2}0)q^{9} + 6103515625 ∗ aq^{10}+(1709029428414096632888374+O(17^{2}0))∗a−711936664738639655823542+O(17^{2}0)q^{11}+−(1254120727569414333824226+ O(17^{2}0))∗a−1186356583535298795864442+O(17^{2}0)q^{12}+−(950192817785893255174677+O(17^{2}0))∗a+ 829345723898500395905619+O(17^{2}0)q^{13}+−(113484976279054687868228+O(17^{2}0))∗a−818253309635498717866204+ O(17^{2}0)q^{14}+−(675961367205077550646987+O(17^{2}0))∗a+22749806565809068871112∗17+O(17^{2}0)q^{15}+ (1187009652623117427573001+O(17^{2}0))∗a+108944007589949355458756∗17+O(17^{2}0)q^{16}+−(48168521595414975821928∗ 17 + O(17^{2}0)) ∗ a − 15373900386999221321950 ∗ 17 + O(17^{2}0)q^{17} + −(1876911058770831380465298 + O(17^{2}0))∗a+1570917281350852333960855+O(17^{2}0)q^{18} +−(1771139050356655233882464+O(17^{2}0))∗ a−567609072795055226831278+O(17^{2}0)q^{19}+(1100315442407685139780817+O(17^{2}0))∗a−38159409229530083096051+ O(17^{2}0)q^{20}+(1449156393210383789908331+O(17^{2}0))∗a−1096108372834404161581803+O(17^{2}0)q^{21}+ (149188146748083909051291+O(17^{2}0))∗a+1037024199224940528635493+O(17^{2}0)q^{22}+(2028314218415512759860207+ O(17^{2}0))∗a−1765718950529322426067∗17+O(17^{2}0)q^{23}+−(1411474280160103127460508+O(17^{2}0))∗ a+692170908825958001319971+O(17^{2}0)q^{24}+37252902984619140625q^{25}+−(684373666592712314207610+ O(17^{2}0)) ∗ a + 1317148263387232611491132 + O(17^{2}0)q^{26} + (1858836526082782488176976 + O(17^{2}0)) ∗ a − 45550602145838098720922 + O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 241262051544956541389384 + O(17^{2}0)x + 1375822160579054415069071 + O(17^{2}0). The slope of f_{0} is 1.

29. Consider f = q+13q^{2}+2q^{3}+q^{4}+15q^{5}+9q^{6}+6q^{7}+5q^{8}+9q^{9}+8q^{10}+16q^{11}+2q^{12}+15q^{13}+10q^{14}+13q^{15}+ 16q^{16} +15q^{18} +10q^{19} +15q^{20} +12q^{21} +4q^{22} +13q^{23} +10q^{24} +4q^{25} +8q^{26} +11q^{27} +⋯ ∈ S_{30}(Γ_{0}(5); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-12 newform g = q + 15q^{2} + 12q^{3} + 13q^{4} + 3q^{5} + 10q^{6} + 13q^{7} + 7q^{8} + q^{9} + 11q^{10} + 3q^{11} + 3q^{12} + 9q^{13} + 8q^{14} + 2q^{15} + q^{16} + 8q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(15)ω^{12} & 0 \\ 0 & u(8)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−450399263827747796185917+ O(17^{2}0)q^{2} + 1735612209329867751822665 + O(17^{2}0)q^{3} + 255405606689263760247408 + O(17^{2}0)q^{4} + 6103515625q^{5} + 1402525132997241667855908 + O(17^{2}0)q^{6} + 754528010068377133145752 + O(17^{2}0)q^{7} + 295929730454237132786021+O(17^{2}0)q^{8}+926153863228013090169085+O(17^{2}0)q^{9}+1679845819254925802440988+ O(17^{2}0)q^{10} +−157404191717544532582682+O(17^{2}0)q^{11} +1560438869499238932480652+O(17^{2}0)q^{12} + 419794754384625943913749+O(17^{2}0)q^{13}+−1691377705575175578852780+O(17^{2}0)q^{14}+−327889346784340389994172+ O(17^{2}0)q^{15}+848620854356418124070274+O(17^{2}0)q^{16}+−15505015229470731865813∗17+O(17^{2}0)q^{17}+ 1474820057096838010710619+O(17^{2}0)q^{18}+550336625037902601509387+O(17^{2}0)q^{19}+−281744097355984754442496+ O(17^{2}0)q^{20} + 386478697336586825950227 + O(17^{2}0)q^{21} + 1497866197879428685986351 + O(17^{2}0)q^{22} + 1540004463949689738833404+O(17^{2}0)q^{23}+−1113827370619596400509289+O(17^{2}0)q^{24}+37252902984619140625q^{25}+ −1219749644465125665078161 + O(17^{2}0)q^{26} + 1273898361161354886477007 + O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

30. Consider f = q + z^{6}q^{2} + z^{1}52q^{3} + z^{8}8q^{4} + 7q^{5} + z^{1}58q^{6} + z^{2}41q^{7} + z^{8}7q^{8} + z^{2}76q^{9} + z^{2}04q^{10} + z^{1}96q^{11} + z^{2}40q^{12}+z^{2}62q^{13}+z^{2}47q^{14}+z^{6}2q^{15}+z^{1}75q^{16}+z^{2}82q^{18}+z^{1}76q^{19}+z^{2}86q^{20}+z^{1}05q^{21}+z^{2}02q^{22}+z^{8}2q^{23}+ z^{2}39q^{24}+15q^{25}+z^{2}68q^{26}+z^{1}81q^{27}+⋯ ∈ S_{32}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{2}+16x+3 = 0. We have ρ_{f} ⊗χ^{15} ∼ ρ_{g}, where g is the weight-14 newform g = q+z^{4}2q^{2}+z^{1}34q^{3}+z^{1}60q^{4}+15q^{5}+z^{1}76q^{6}+ z^{4}3q^{7} +z^{1}95q^{8} +z^{2}40q^{9} +z^{1}50q^{10} +z^{7}0q^{11} +z^{6}q^{12} +z^{1}90q^{13} +z^{8}5q^{14} +z^{2}42q^{15} +z^{3}1q^{16} +z^{2}22q^{17} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{6}6)ω^{14} & 0 \\ 0 & u(z^{2}22)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(1713047268128909497077884+ O(17^{2}0))∗a+405615723784218178074114+O(17^{2}0)q^{3} +(1878386572888313988202987+O(17^{2}0))∗a− 355694520189278367082899+O(17^{2}0)q^{4}+30517578125q^{5}+−(1626800617721159059976635+O(17^{2}0))∗ a+1522966468227442873024025+O(17^{2}0)q^{6}+−(188266226940180103922745+O(17^{2}0))∗a−1477559131845870548046255+ O(17^{2}0)q^{7} + (40899516379191388320917 + O(17^{2}0)) ∗ a + 570107733680533376227080 + O(17^{2}0)q^{8} + −(543655560309608583193995+O(17^{2}0))∗a+25358819025619995320540∗17+O(17^{2}0)q^{9}+30517578125∗ aq^{10} + −(396850165213001994464719 + O(17^{2}0)) ∗ a − 1192371919720826005695607 + O(17^{2}0)q^{11} + −(435460930019375733886829+O(17^{2}0))∗a−70732269013106188339494∗17+O(17^{2}0)q^{12}+(632006856367042666799038+ O(17^{2}0))∗a−1144501353099093255153779+O(17^{2}0)q^{13} +−(1065218880950680494234375+O(17^{2}0))∗ a−1628604420523901443036888+O(17^{2}0)q^{14}+−(1155099101199219042354425+O(17^{2}0))∗a+1068779065713777921357185+ O(17^{2}0)q^{15}+−(1108194616588419834707233+O(17^{2}0))∗a+1856158132859875026050779+O(17^{2}0)q^{16}+ −(48699765790780039685028 ∗ 17 + O(17^{2}0)) ∗ a + 1536906235185157031390 ∗ 17^{2} + O(17^{2}0)q^{17} + (1841642121497827236589158+O(17^{2}0))∗a+1339869299523989365553042+O(17^{2}0)q^{18}+−(1251871667485070840085223+ O(17^{2}0))∗a+1267656494542775156735705+O(17^{2}0)q^{19} +−(1806994098991227100554268+O(17^{2}0))∗ a+1005903477170617956912607+O(17^{2}0)q^{20}+(1625938290655544325357527+O(17^{2}0))∗a−1193614692726189259003809+ O(17^{2}0)q^{21}+(1021929889063978043969167+O(17^{2}0))∗a+1837471988149375161553940+O(17^{2}0)q^{22}+ (860250838926636417943192+O(17^{2}0))∗a+1299087180797516180357138+O(17^{2}0)q^{23}+−(1609148898294454152625976+ O(17^{2}0))∗a+1988579586202788337210714+O(17^{2}0)q^{24}+931322574615478515625q^{25}+(832537121287145537584024+ O(17^{2}0)) ∗ a − 290736588006162140443157 + O(17^{2}0)q^{26} + −(448047322903804203686639 + O(17^{2}0)) ∗ a − 810380577968495306299062 + O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −1878386572888313988202987 + O(17^{2}0)x + 355694520189276219599251 + O(17^{2}0). The slope of f_{0} is 1.

31. Consider f = q+z^{1}02q^{2}+z^{2}80q^{3}+z^{5}6q^{4}+7q^{5}+z^{9}4q^{6}+z^{6}5q^{7}+z^{3}9q^{8}+z^{8}4q^{9}+z^{1}2q^{10}+z^{1}64q^{11}+z^{4}8q^{12}+ z^{1}34q^{13}+z^{1}67q^{14}+z^{1}90q^{15}+z^{9}5q^{16}+z^{1}86q^{18}+z^{1}12q^{19}+z^{2}54q^{20}+z^{5}7q^{21}+z^{2}66q^{22}+z^{2}42q^{23}+z^{3}1q^{24}+ 15q^{25} + z^{2}36q^{26} + z^{1}97q^{27} + ⋯ ∈ S_{32}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{2} + 16x + 3 = 0. We have ρ_{f} ⊗χ^{15} ∼ ρ_{g}, where g is the weight-14 newform g = q+z^{1}38q^{2}+z^{2}62q^{3}+z^{1}28q^{4}+15q^{5}+z^{1}12q^{6}+ z^{1}55q^{7} +z^{1}47q^{8} +z^{4}8q^{9} +z^{2}46q^{10} +z^{3}8q^{11} +z^{1}02q^{12} +z^{6}2q^{13} +z^{5}q^{14} +z^{8}2q^{15} +z^{2}39q^{16} +z^{3}0q^{17} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}58)ω^{14} & 0 \\ 0 & u(z^{3}0)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(1713047268128909497077884+ O(17^{2}0))∗a+405615723784218178074114+O(17^{2}0)q^{3} +(1878386572888313988202987+O(17^{2}0))∗a− 355694520189278367082899+O(17^{2}0)q^{4}+30517578125q^{5}+−(1626800617721159059976635+O(17^{2}0))∗ a+1522966468227442873024025+O(17^{2}0)q^{6}+−(188266226940180103922745+O(17^{2}0))∗a−1477559131845870548046255+ O(17^{2}0)q^{7} + (40899516379191388320917 + O(17^{2}0)) ∗ a + 570107733680533376227080 + O(17^{2}0)q^{8} + −(543655560309608583193995+O(17^{2}0))∗a+25358819025619995320540∗17+O(17^{2}0)q^{9}+30517578125∗ aq^{10} + −(396850165213001994464719 + O(17^{2}0)) ∗ a − 1192371919720826005695607 + O(17^{2}0)q^{11} + −(435460930019375733886829+O(17^{2}0))∗a−70732269013106188339494∗17+O(17^{2}0)q^{12}+(632006856367042666799038+ O(17^{2}0))∗a−1144501353099093255153779+O(17^{2}0)q^{13} +−(1065218880950680494234375+O(17^{2}0))∗ a−1628604420523901443036888+O(17^{2}0)q^{14}+−(1155099101199219042354425+O(17^{2}0))∗a+1068779065713777921357185+ O(17^{2}0)q^{15}+−(1108194616588419834707233+O(17^{2}0))∗a+1856158132859875026050779+O(17^{2}0)q^{16}+ −(48699765790780039685028 ∗ 17 + O(17^{2}0)) ∗ a + 1536906235185157031390 ∗ 17^{2} + O(17^{2}0)q^{17} + (1841642121497827236589158+O(17^{2}0))∗a+1339869299523989365553042+O(17^{2}0)q^{18}+−(1251871667485070840085223+ O(17^{2}0))∗a+1267656494542775156735705+O(17^{2}0)q^{19} +−(1806994098991227100554268+O(17^{2}0))∗ a+1005903477170617956912607+O(17^{2}0)q^{20}+(1625938290655544325357527+O(17^{2}0))∗a−1193614692726189259003809+ O(17^{2}0)q^{21}+(1021929889063978043969167+O(17^{2}0))∗a+1837471988149375161553940+O(17^{2}0)q^{22}+ (860250838926636417943192+O(17^{2}0))∗a+1299087180797516180357138+O(17^{2}0)q^{23}+−(1609148898294454152625976+ O(17^{2}0))∗a+1988579586202788337210714+O(17^{2}0)q^{24}+931322574615478515625q^{25}+(832537121287145537584024+ O(17^{2}0)) ∗ a − 290736588006162140443157 + O(17^{2}0)q^{26} + −(448047322903804203686639 + O(17^{2}0)) ∗ a − 810380577968495306299062 + O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −1878386572888313988202987 + O(17^{2}0)x + 355694520189276219599251 + O(17^{2}0). The slope of f_{0} is 1.

32. Consider f = q +16q^{2} +4q^{3} +9q^{4} +10q^{5} +13q^{6} +14q^{7} +10q^{9} +7q^{10} +16q^{11} +2q^{12} +4q^{13} +3q^{14} +6q^{15} + 4q^{16}+7q^{18}+8q^{19}+5q^{20}+5q^{21}+q^{22}+12q^{23}+15q^{25}+13q^{26}+16q^{27}+⋯ ∈ S_{32}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{6} + 2x^{4} + 10x^{2} + 3x + 3 = 0. We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q+13q^{2}+2q^{3}+8q^{4}+12q^{5}+9q^{6}+6q^{7}+11q^{9}+3q^{10}+15q^{11}+16q^{12}+13q^{13}+10q^{14}+7q^{15}+4q^{16}+9q^{17}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(2)ω^{1} & 0 \\ 0 & u(9)ω^{14} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+867876285717285179640057+ O(17^{2}0)q^{2} + −384292937355616474213624 + O(17^{2}0)q^{3} + 1642954137430482588373948 + O(17^{2}0)q^{4} + −30517578125q^{5}+1684851121335933711778946+O(17^{2}0)q^{6}+−2011920667572647031488216+O(17^{2}0)q^{7}+ 76708567230431496167297∗17+O(17^{2}0)q^{8}+1452319161311190169399780+O(17^{2}0)q^{9}+−486437045024751393076756+ O(17^{2}0)q^{10}+−1765534669889169308356052+O(17^{2}0)q^{11}+−1939732782083608625091238+O(17^{2}0)q^{12}+ 1651947112189451020079408+O(17^{2}0)q^{13}+−272977162508119915696924+O(17^{2}0)q^{14}+−918785749631703074404721+ O(17^{2}0)q^{15}+−265916389969215845638070+O(17^{2}0)q^{16}+43857855445925106423438∗17+O(17^{2}0)q^{17}+ 785481416156452197436468+O(17^{2}0)q^{18}+−1314343772662680994988783+O(17^{2}0)q^{19}+−473657492087471817176609+ O(17^{2}0)q^{20} +−68027553056500326401731+O(17^{2}0)q^{21} +−276797243970884419176276+O(17^{2}0)q^{22} + −1706254808624923168911396+O(17^{2}0)q^{23}+−93179123369654835714409∗17+O(17^{2}0)q^{24}+931322574615478515625q^{25}+ −396249617084011529807439 + O(17^{2}0)q^{26} + −305826431725026352658034 + O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

33. Consider f = q+3∗z^{5}+9∗z^{4}+10∗z^{3}+6∗z^{2}+13∗z+4q^{2}+3∗z^{5}+10∗z^{4}+11∗z^{3}+8∗z^{2}+9q^{3}+3∗z^{5}+z^{4}+2∗ z^{3}+7∗z^{2}+15∗z+4q^{4}+10q^{5}+8∗z^{5}+10∗z^{4}+7∗z^{3}+5∗z^{2}+7∗z+1q^{6}+13∗z^{5}+15∗z^{4}+8∗z^{3}+12∗z^{2}+9q^{7}+ 5∗z^{5}+4∗z^{4}+5∗z^{2}+6∗z+12q^{8}+4∗z^{5}+8∗z^{4}+15∗z^{3}+7∗z+16q^{9}+13∗z^{5}+5∗z^{4}+15∗z^{3}+9∗z^{2}+11∗z+ 6q^{10}+10∗z^{5}+16∗z^{4}+8∗z^{3}+9∗z^{2}+10∗z+14q^{11}+7∗z^{5}+z^{4}+9∗z^{3}+8∗z^{2}+7∗z+6q^{12}+7∗z^{5}+6∗z^{4}+14∗ z^{3}+z^{2}+10∗z+10q^{13}+7∗z^{5}+8∗z^{3}+6∗z^{2}+3∗z+9q^{14}+13∗z^{5}+15∗z^{4}+8∗z^{3}+12∗z^{2}+5q^{15}+2∗z^{5}+6∗z^{4}+ z^{3}+4∗z^{2}+3∗z+2q^{16}+8∗z^{5}+9∗z^{4}+6∗z^{3}+3∗z^{2}+3∗z+1q^{18}+9∗z^{5}+10∗z^{4}+13∗z^{3}+z^{2}+5∗z+1q^{19}+13∗ z^{5}+10∗z^{4}+3∗z^{3}+2∗z^{2}+14∗z+6q^{20}+z^{5}+z^{4}+7∗z^{3}+15∗z^{2}+2∗z+1q^{21}+16∗z^{5}+9∗z^{4}+3∗z^{3}+5∗z^{2}+4∗ z+1q^{22}+8∗z^{5}+z^{4}+15∗z^{3}+4∗z^{2}+5∗z+10q^{23}+6∗z^{5}+13∗z^{4}+15∗z^{3}+2∗z^{2}+6∗z+2q^{24}+15q^{25}+6∗z^{5}+ 10∗z^{4}+12∗z^{3}+13∗z^{2}+11∗z+3q^{26}+10∗z^{5}+13∗z^{4}+5∗z^{3}+3∗z^{2}+15∗z+9q^{27}+⋯ ∈ S_{32}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{6} + 2x^{4} + 10x^{2} + 3x + 3 = 0. We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-14 newform g = q +z^{1}435q^{2} +z^{4}68q^{3} +z^{4}653q^{4} +2q^{5} +z^{1}903q^{6} +z^{4}193q^{7} +z^{3}119q^{8} +z^{4}534q^{9} + z^{8}21q^{10} + z^{8}73q^{11} + z^{2}09q^{12} + z^{4}468q^{13} + z^{7}16q^{14} + z^{4}766q^{15} + z^{1}983q^{16} + z^{2}247q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}665)ω^{14} & 0 \\ 0 & u(z^{2}247)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(419234900148323865745694+ O(17^{2}0))∗a^{2} −(372434947545598918662717+O(17^{2}0))∗a+857492562604471570428151+O(17^{2}0)q^{3} + a^{2}−2147483648q^{4}+−30517578125q^{5}+−(1455027625932150412639515+O(17^{2}0))∗a^{2}−(1475984087553079378329575+ O(17^{2}0))∗a+196736261836984789702076+O(17^{2}0)q^{6}+−(1938572546373109577312938+O(17^{2}0))∗a^{2}− (295100274169664936930166+O(17^{2}0))∗a−1865605761234303697092114+O(17^{2}0)q^{7}+(1615830589018350487850406+ O(17^{2}0))∗a^{2}−(1939537599790432880165717+O(17^{2}0))∗a−669093301881960133693900+O(17^{2}0)q^{8}+ (1542368590887774579256914+O(17^{2}0))∗a^{2}−(1940810148480347422938024+O(17^{2}0))∗a−1396932979045386533198404+ O(17^{2}0)q^{9}+−30517578125∗aq^{10}+−(598702446941720269873050+O(17^{2}0))∗a^{2}+(604501668165561442911340+ O(17^{2}0))∗a−231062705185243977173040+O(17^{2}0)q^{11}+(1220571166099670651144157+O(17^{2}0))∗a^{2}+ (301638926910361315334096+O(17^{2}0))∗a+1456200795406697612733646+O(17^{2}0)q^{12}+−(1295314737659843723783846+ O(17^{2}0))∗a^{2}−(46779592140558865862470+O(17^{2}0))∗a−1778576524138258627723963+O(17^{2}0)q^{13}+ −(1354621502516895037485649+O(17^{2}0))∗a^{2}+(359700466586952579584732+O(17^{2}0))∗a−1168506104902611575668546+ O(17^{2}0)q^{14} + (378912462760303365408846 + O(17^{2}0)) ∗ a^{2} + (980746124830342490725404 + O(17^{2}0)) ∗ a − 1694924629827436942286804 + O(17^{2}0)q^{15} + −(37519981203521401013667 ∗ 17 + O(17^{2}0)) ∗ a^{2} − (489791253731283580924997+O(17^{2}0))∗a+1607038909924447707352852+O(17^{2}0)q^{16}+−(40880901899992074170218∗ 17 + O(17^{2}0)) ∗ a^{2} + (115637794785285474424272 ∗ 17 + O(17^{2}0)) ∗ a + 6298949388027553449003 ∗ 17 + O(17^{2}0)q^{17}+(434480204731588004468712+O(17^{2}0))∗a^{2}+(846188819066643008628839+O(17^{2}0))∗a+ 1558666447156561814684596+O(17^{2}0)q^{18}+(143619108782315312793∗17+O(17^{2}0))∗a^{2}−(48299673546037922248287+ O(17^{2}0))∗a+1453559517001287466231144+O(17^{2}0)q^{19}+−30517578125∗a^{2}+65536000000000000000q^{20}+ (1696172169357499609854934+O(17^{2}0))∗a^{2}−(568461264584615280836651+O(17^{2}0))∗a+833657329710643299315738+ O(17^{2}0)q^{21}+−(1939280005914423574393816+O(17^{2}0))∗a^{2}+(1911854259754473848921368+O(17^{2}0))∗ a + 1568362163813179717791348 + O(17^{2}0)q^{22} + −(1151213848548791421402225 + O(17^{2}0)) ∗ a^{2} − (238522337163522929219072+O(17^{2}0))∗a+262818453066577755758436+O(17^{2}0)q^{23}+(1182717415444855659031878+ O(17^{2}0))∗a^{2}+(546732885519048425656536+O(17^{2}0))∗a−834321743762906190502888+O(17^{2}0)q^{24}+ 931322574615478515625q^{25}+(1994113985552644826846884+O(17^{2}0))∗a^{2}+(1018436351482017887437259+ O(17^{2}0)) ∗ a + 599547681978580175742085 + O(17^{2}0)q^{26} + (80767494105542283787645 ∗ 17 + O(17^{2}0)) ∗ a^{2} + (402062343158666521242487 + O(17^{2}0)) ∗ a − 418100930513993195061369 + O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + −1615830589018350487850406+O(17^{2}0)x^{2}+1939537599790428585198421+O(17^{2}0)x+669093301881960133693900+ O(17^{2}0). The slope of f_{0} is 1.

34. Consider f = q+3∗z^{5}+2∗z^{4}+3∗z^{3}+9∗z^{2}+2∗z+14q^{2}+14∗z^{5}+16∗z^{4}+15∗z^{3}+10∗z^{2}+2∗z+16q^{3}+9∗z^{4}+ 9∗z^{3}+z^{2}+2∗z+4q^{4}+10q^{5}+7∗z^{5}+2∗z^{4}+10∗z^{3}+10∗z^{2}+11∗z+13q^{6}+4∗z^{5}+7∗z^{4}+14∗z^{3}+15∗z^{2}+3∗z+ 11q^{7}+3∗z^{5}+8∗z^{4}+9∗z^{3}+4∗z^{2}+9∗z+5q^{8}+11∗z^{5}+9∗z^{4}+7∗z^{3}+8∗z^{2}+14∗z+13q^{9}+13∗z^{5}+3∗z^{4}+13∗z^{3}+ 5∗z^{2}+3∗z+4q^{10}+9∗z^{5}+9∗z^{4}+12∗z^{3}+16∗z^{2}+z+10q^{11}+8∗z^{5}+8∗z^{4}+5∗z^{3}+z^{2}+16∗z+10q^{12}+12∗z^{5}+ 10∗z^{4}+14∗z^{3}+6∗z^{2}+16∗z+16q^{13}+14∗z^{5}+11∗z^{4}+10∗z^{3}+16∗z+16q^{14}+4∗z^{5}+7∗z^{4}+14∗z^{3}+15∗z^{2}+3∗ z+7q^{15}+2∗z^{5}+7∗z^{4}+2∗z^{3}+6∗z^{2}+7∗z+3q^{16}+13∗z^{5}+5∗z^{4}+15∗z^{3}+9∗z^{2}+11∗z+16q^{18}+9∗z^{5}+6∗z^{4}+9∗ z^{3}+10∗z^{2}+6∗z+14q^{19}+5∗z^{4}+5∗z^{3}+10∗z^{2}+3∗z+6q^{20}+13∗z^{5}+z^{4}+11∗z^{3}+z^{2}+12∗z+16q^{21}+12∗z^{5}+ 3∗z^{4}+7∗z^{3}+9∗z^{2}+5∗z+7q^{22}+10∗z^{5}+12∗z^{4}+4∗z^{3}+z^{2}+11∗z+15q^{23}+2∗z^{5}+2∗z^{4}+14∗z^{3}+13∗z^{2}+4∗ z+3q^{24}+15q^{25}+3∗z^{5}+11∗z^{4}+12∗z^{3}+10∗z^{2}+4∗z+13q^{26}+10∗z^{5}+z^{4}+10∗z^{3}+13∗z^{2}+z+14q^{27}+⋯ ∈ S_{32}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{6}+2x^{4}+10x^{2}+3x+3 = 0. We have ρ_{f} ⊗χ^{15} ∼ ρ_{g}, where g is the weight-14 newform g = q + z^{2}107q^{2} + z^{2}628q^{3} + z^{3}741q^{4} + 2q^{5} + z^{4}735q^{6} + z^{3}425q^{7} + z^{2}495q^{8} + z^{3}734q^{9} + z^{1}493q^{10} + z^{1}785q^{11} + z^{1}457q^{12} + z^{4}308q^{13} + z^{6}20q^{14} + z^{2}014q^{15} + z^{3}295q^{16} + z^{9}99q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{3}913)ω^{14} & 0 \\ 0 & u(z^{9}99)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(419234900148323865745694+ O(17^{2}0))∗a^{2} −(372434947545598918662717+O(17^{2}0))∗a+857492562604471570428151+O(17^{2}0)q^{3} + a^{2}−2147483648q^{4}+−30517578125q^{5}+−(1455027625932150412639515+O(17^{2}0))∗a^{2}−(1475984087553079378329575+ O(17^{2}0))∗a+196736261836984789702076+O(17^{2}0)q^{6}+−(1938572546373109577312938+O(17^{2}0))∗a^{2}− (295100274169664936930166+O(17^{2}0))∗a−1865605761234303697092114+O(17^{2}0)q^{7}+(1615830589018350487850406+ O(17^{2}0))∗a^{2}−(1939537599790432880165717+O(17^{2}0))∗a−669093301881960133693900+O(17^{2}0)q^{8}+ (1542368590887774579256914+O(17^{2}0))∗a^{2}−(1940810148480347422938024+O(17^{2}0))∗a−1396932979045386533198404+ O(17^{2}0)q^{9}+−30517578125∗aq^{10}+−(598702446941720269873050+O(17^{2}0))∗a^{2}+(604501668165561442911340+ O(17^{2}0))∗a−231062705185243977173040+O(17^{2}0)q^{11}+(1220571166099670651144157+O(17^{2}0))∗a^{2}+ (301638926910361315334096+O(17^{2}0))∗a+1456200795406697612733646+O(17^{2}0)q^{12}+−(1295314737659843723783846+ O(17^{2}0))∗a^{2}−(46779592140558865862470+O(17^{2}0))∗a−1778576524138258627723963+O(17^{2}0)q^{13}+ −(1354621502516895037485649+O(17^{2}0))∗a^{2}+(359700466586952579584732+O(17^{2}0))∗a−1168506104902611575668546+ O(17^{2}0)q^{14} + (378912462760303365408846 + O(17^{2}0)) ∗ a^{2} + (980746124830342490725404 + O(17^{2}0)) ∗ a − 1694924629827436942286804 + O(17^{2}0)q^{15} + −(37519981203521401013667 ∗ 17 + O(17^{2}0)) ∗ a^{2} − (489791253731283580924997+O(17^{2}0))∗a+1607038909924447707352852+O(17^{2}0)q^{16}+−(40880901899992074170218∗ 17 + O(17^{2}0)) ∗ a^{2} + (115637794785285474424272 ∗ 17 + O(17^{2}0)) ∗ a + 6298949388027553449003 ∗ 17 + O(17^{2}0)q^{17}+(434480204731588004468712+O(17^{2}0))∗a^{2}+(846188819066643008628839+O(17^{2}0))∗a+ 1558666447156561814684596+O(17^{2}0)q^{18}+(143619108782315312793∗17+O(17^{2}0))∗a^{2}−(48299673546037922248287+ O(17^{2}0))∗a+1453559517001287466231144+O(17^{2}0)q^{19}+−30517578125∗a^{2}+65536000000000000000q^{20}+ (1696172169357499609854934+O(17^{2}0))∗a^{2}−(568461264584615280836651+O(17^{2}0))∗a+833657329710643299315738+ O(17^{2}0)q^{21}+−(1939280005914423574393816+O(17^{2}0))∗a^{2}+(1911854259754473848921368+O(17^{2}0))∗ a + 1568362163813179717791348 + O(17^{2}0)q^{22} + −(1151213848548791421402225 + O(17^{2}0)) ∗ a^{2} − (238522337163522929219072+O(17^{2}0))∗a+262818453066577755758436+O(17^{2}0)q^{23}+(1182717415444855659031878+ O(17^{2}0))∗a^{2}+(546732885519048425656536+O(17^{2}0))∗a−834321743762906190502888+O(17^{2}0)q^{24}+ 931322574615478515625q^{25}+(1994113985552644826846884+O(17^{2}0))∗a^{2}+(1018436351482017887437259+ O(17^{2}0)) ∗ a + 599547681978580175742085 + O(17^{2}0)q^{26} + (80767494105542283787645 ∗ 17 + O(17^{2}0)) ∗ a^{2} + (402062343158666521242487 + O(17^{2}0)) ∗ a − 418100930513993195061369 + O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + −1615830589018350487850406+O(17^{2}0)x^{2}+1939537599790428585198421+O(17^{2}0)x+669093301881960133693900+ O(17^{2}0). The slope of f_{0} is 1.

35. Consider f = q+11∗z^{5}+6∗z^{4}+4∗z^{3}+2∗z^{2}+2∗z+11q^{2}+8∗z^{4}+8∗z^{3}+16∗z^{2}+15∗z+16q^{3}+14∗z^{5}+7∗ z^{4}+6∗z^{3}+9∗z^{2}+11q^{4}+10q^{5}+2∗z^{5}+5∗z^{4}+2∗z^{2}+16∗z+14q^{6}+12∗z^{4}+12∗z^{3}+7∗z^{2}+14∗z+11q^{7}+9∗ z^{5}+5∗z^{4}+8∗z^{3}+8∗z^{2}+2∗z+1q^{8}+2∗z^{5}+12∗z^{3}+9∗z^{2}+13∗z+14q^{9}+8∗z^{5}+9∗z^{4}+6∗z^{3}+3∗z^{2}+3∗z+ 8q^{10}+15∗z^{5}+9∗z^{4}+14∗z^{3}+9∗z^{2}+6∗z+9q^{11}+2∗z^{5}+8∗z^{4}+3∗z^{3}+8∗z^{2}+11∗z+11q^{12}+15∗z^{5}+z^{4}+6∗ z^{3}+10∗z^{2}+8∗z+15q^{13}+13∗z^{5}+6∗z^{4}+16∗z^{3}+11∗z^{2}+15∗z+14q^{14}+12∗z^{4}+12∗z^{3}+7∗z^{2}+14∗z+7q^{15}+ 13∗z^{5}+4∗z^{4}+14∗z^{3}+7∗z^{2}+7∗z+1q^{16}+13∗z^{5}+3∗z^{4}+13∗z^{3}+5∗z^{2}+3∗z+14q^{18}+16∗z^{5}+z^{4}+12∗z^{3}+6∗ z^{2}+6∗z+5q^{19}+4∗z^{5}+2∗z^{4}+9∗z^{3}+5∗z^{2}+8q^{20}+3∗z^{5}+15∗z^{4}+16∗z^{3}+z^{2}+3∗z+9q^{21}+6∗z^{5}+5∗z^{4}+7∗ z^{3}+3∗z^{2}+8∗z+10q^{22}+16∗z^{5}+4∗z^{4}+15∗z^{3}+12∗z^{2}+z+6q^{23}+9∗z^{5}+2∗z^{4}+5∗z^{3}+2∗z^{2}+7∗z+16q^{24}+ 15q^{25}+8∗z^{5}+13∗z^{4}+10∗z^{3}+11∗z^{2}+2∗zq^{26}+14∗z^{5}+3∗z^{4}+2∗z^{3}+z^{2}+z+4q^{27}+⋯ ∈ S_{32}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{6} + 2x^{4} + 10x^{2} + 3x + 3 = 0. We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-14 newform g = q +z^{4}747q^{2} +z^{3}044q^{3} +z^{5}09q^{4} +2q^{5} +z^{2}879q^{6} +z^{2}513q^{7} +z^{3}903q^{8} +z^{3}398q^{9} + z^{4}133q^{10} +z^{1}05q^{11} +z^{3}553q^{12} +z^{2}276q^{13} +z^{2}348q^{14} +z^{2}430q^{15} +z^{4}239q^{16} +z^{3}815q^{17} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}097)ω^{14} & 0 \\ 0 & u(z^{3}815)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(419234900148323865745694+ O(17^{2}0))∗a^{2} −(372434947545598918662717+O(17^{2}0))∗a+857492562604471570428151+O(17^{2}0)q^{3} + a^{2}−2147483648q^{4}+−30517578125q^{5}+−(1455027625932150412639515+O(17^{2}0))∗a^{2}−(1475984087553079378329575+ O(17^{2}0))∗a+196736261836984789702076+O(17^{2}0)q^{6}+−(1938572546373109577312938+O(17^{2}0))∗a^{2}− (295100274169664936930166+O(17^{2}0))∗a−1865605761234303697092114+O(17^{2}0)q^{7}+(1615830589018350487850406+ O(17^{2}0))∗a^{2}−(1939537599790432880165717+O(17^{2}0))∗a−669093301881960133693900+O(17^{2}0)q^{8}+ (1542368590887774579256914+O(17^{2}0))∗a^{2}−(1940810148480347422938024+O(17^{2}0))∗a−1396932979045386533198404+ O(17^{2}0)q^{9}+−30517578125∗aq^{10}+−(598702446941720269873050+O(17^{2}0))∗a^{2}+(604501668165561442911340+ O(17^{2}0))∗a−231062705185243977173040+O(17^{2}0)q^{11}+(1220571166099670651144157+O(17^{2}0))∗a^{2}+ (301638926910361315334096+O(17^{2}0))∗a+1456200795406697612733646+O(17^{2}0)q^{12}+−(1295314737659843723783846+ O(17^{2}0))∗a^{2}−(46779592140558865862470+O(17^{2}0))∗a−1778576524138258627723963+O(17^{2}0)q^{13}+ −(1354621502516895037485649+O(17^{2}0))∗a^{2}+(359700466586952579584732+O(17^{2}0))∗a−1168506104902611575668546+ O(17^{2}0)q^{14} + (378912462760303365408846 + O(17^{2}0)) ∗ a^{2} + (980746124830342490725404 + O(17^{2}0)) ∗ a − 1694924629827436942286804 + O(17^{2}0)q^{15} + −(37519981203521401013667 ∗ 17 + O(17^{2}0)) ∗ a^{2} − (489791253731283580924997+O(17^{2}0))∗a+1607038909924447707352852+O(17^{2}0)q^{16}+−(40880901899992074170218∗ 17 + O(17^{2}0)) ∗ a^{2} + (115637794785285474424272 ∗ 17 + O(17^{2}0)) ∗ a + 6298949388027553449003 ∗ 17 + O(17^{2}0)q^{17}+(434480204731588004468712+O(17^{2}0))∗a^{2}+(846188819066643008628839+O(17^{2}0))∗a+ 1558666447156561814684596+O(17^{2}0)q^{18}+(143619108782315312793∗17+O(17^{2}0))∗a^{2}−(48299673546037922248287+ O(17^{2}0))∗a+1453559517001287466231144+O(17^{2}0)q^{19}+−30517578125∗a^{2}+65536000000000000000q^{20}+ (1696172169357499609854934+O(17^{2}0))∗a^{2}−(568461264584615280836651+O(17^{2}0))∗a+833657329710643299315738+ O(17^{2}0)q^{21}+−(1939280005914423574393816+O(17^{2}0))∗a^{2}+(1911854259754473848921368+O(17^{2}0))∗ a + 1568362163813179717791348 + O(17^{2}0)q^{22} + −(1151213848548791421402225 + O(17^{2}0)) ∗ a^{2} − (238522337163522929219072+O(17^{2}0))∗a+262818453066577755758436+O(17^{2}0)q^{23}+(1182717415444855659031878+ O(17^{2}0))∗a^{2}+(546732885519048425656536+O(17^{2}0))∗a−834321743762906190502888+O(17^{2}0)q^{24}+ 931322574615478515625q^{25}+(1994113985552644826846884+O(17^{2}0))∗a^{2}+(1018436351482017887437259+ O(17^{2}0)) ∗ a + 599547681978580175742085 + O(17^{2}0)q^{26} + (80767494105542283787645 ∗ 17 + O(17^{2}0)) ∗ a^{2} + (402062343158666521242487 + O(17^{2}0)) ∗ a − 418100930513993195061369 + O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{3} + −1615830589018350487850406+O(17^{2}0)x^{2}+1939537599790428585198421+O(17^{2}0)x+669093301881960133693900+ O(17^{2}0). The slope of f_{0} is 1.

36. Consider f = q +z^{1}90q^{2} +z^{2}55q^{3} +z^{9}1q^{4} +16q^{5} +z^{1}57q^{6} +z^{2}0q^{7} +z^{1}42q^{8} +z^{2}43q^{9} +z^{4}6q^{10} +z^{1}01q^{11} + z^{5}8q^{12} +z^{1}36q^{13} +z^{2}10q^{14} +z^{1}11q^{15} +z^{1}10q^{16} +z^{1}45q^{18} +z^{4}5q^{19} +z^{2}35q^{20} +z^{2}75q^{21} +z^{3}q^{22} +z^{2}33q^{23} + z^{1}09q^{24}+q^{25}+z^{3}8q^{26}+z^{1}41q^{27}+⋯ ∈ S_{34}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{2}+16x+3 = 0. We have ρ_{f} ⊗χ^{15} ∼ ρ_{g}, where g is the weight-16 newform g = q+z^{2}26q^{2}+z^{2}37q^{3}+z^{1}63q^{4}+10q^{5}+z^{1}75q^{6}+ z^{1}10q^{7}+z^{2}50q^{8}+z^{2}07q^{9}+z^{2}80q^{10}+z^{2}63q^{11}+z^{1}12q^{12}+z^{6}4q^{13}+z^{4}8q^{14}+z^{3}q^{15}+z^{2}54q^{16}+z^{1}11q^{17}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}77)ω^{0} & 0 \\ 0 & u(z^{1}11)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(1458393643319997754061460+ O(17^{2}0))∗a+1723668887195374355892764+O(17^{2}0)q^{3}+(1138128332623662180558778+O(17^{2}0))∗a+ 942164111436591968985898+O(17^{2}0)q^{4}+−152587890625q^{5}+(382833920844806827025814+O(17^{2}0))∗ a+1234721310452452876707397+O(17^{2}0)q^{6}+(1802630623014962423659884+O(17^{2}0))∗a+157752462232979951788553+ O(17^{2}0)q^{7} + (1846483028899772089708636 + O(17^{2}0)) ∗ a + 548422534485550730525205 + O(17^{2}0)q^{8} + −(734536704986020174307624+O(17^{2}0))∗a+534292118397575343904559+O(17^{2}0)q^{9}+−152587890625∗ aq^{10} + −(1428163953694801295354379 + O(17^{2}0)) ∗ a + 1237372402675889063268786 + O(17^{2}0)q^{11} + −(1426181625023343448840805+O(17^{2}0))∗a+23249751078754655003741+O(17^{2}0)q^{12}+(1673150564697082902175720+ O(17^{2}0))∗a+100354188266680249801582∗17+O(17^{2}0)q^{13}+−(797433895479524247798328+O(17^{2}0))∗ a−674527328600335853171759+O(17^{2}0)q^{14}+(1724307088644227149785205+O(17^{2}0))∗a−1669791154776154125428800+ O(17^{2}0)q^{15} + (273487894310104746640979 + O(17^{2}0)) ∗ a − 725906665858185052115155 + O(17^{2}0)q^{16} + (49707249799014459209109∗17+O(17^{2}0))∗a−90234241116668315482254∗17+O(17^{2}0)q^{17}+−(775404819961617934768165+ O(17^{2}0))∗a−576836244456819119075569+O(17^{2}0)q^{18}+(1439565759431516814245577+O(17^{2}0))∗a− 185330458513479225520898+O(17^{2}0)q^{19}+(201390405233136218994958+O(17^{2}0))∗a+1766125415054611562961593+ O(17^{2}0)q^{20}+(1619868794094222828235872+O(17^{2}0))∗a+1805351244614501892636316+O(17^{2}0)q^{21}+ (967181061720558267464114+O(17^{2}0))∗a−1245361751826794258059137+O(17^{2}0)q^{22}+(1467650909033699029924933+ O(17^{2}0))∗a−253579713659214695516863+O(17^{2}0)q^{23} +(463393839888019231084388+O(17^{2}0))∗a+ 1290475444635666316676929+O(17^{2}0)q^{24}+23283064365386962890625q^{25}+(1370069987031612280826477+ O(17^{2}0)) ∗ a + 551314173824756130729312 + O(17^{2}0)q^{26} + (1463882582213283229724251 + O(17^{2}0)) ∗ a − 1957684915531955435092705 + O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −1138128332623662180558778 + O(17^{2}0)x + −942164111436600558920490 + O(17^{2}0). The slope of f_{0} is 1.

37. Consider f = q + z^{6}2q^{2} + z^{1}5q^{3} + z^{1}07q^{4} + 16q^{5} + z^{7}7q^{6} + z^{5}2q^{7} + z^{1}10q^{8} + z^{9}9q^{9} + z^{2}06q^{10} + z^{2}77q^{11} + z^{1}22q^{12} +z^{8}q^{13} +z^{1}14q^{14} +z^{1}59q^{15} +z^{1}42q^{16} +z^{1}61q^{18} +z^{1}89q^{19} +z^{2}51q^{20} +z^{6}7q^{21} +z^{5}1q^{22} +z^{2}17q^{23} + z^{1}25q^{24}+q^{25}+z^{7}0q^{26}+z^{9}3q^{27}+⋯ ∈ S_{34}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{2}+16x+3 = 0. We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-16 newform g = q + z^{9}8q^{2} + z^{2}85q^{3} + z^{1}79q^{4} + 10q^{5} + z^{9}5q^{6} + z^{1}42q^{7}+z^{2}18q^{8}+z^{6}3q^{9}+z^{1}52q^{10}+z^{1}51q^{11}+z^{1}76q^{12}+z^{2}24q^{13}+z^{2}40q^{14}+z^{5}1q^{15}+z^{2}86q^{16}+z^{1}59q^{17}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{1}29)ω^{0} & 0 \\ 0 & u(z^{1}59)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(1458393643319997754061460+ O(17^{2}0))∗a+1723668887195374355892764+O(17^{2}0)q^{3}+(1138128332623662180558778+O(17^{2}0))∗a+ 942164111436591968985898+O(17^{2}0)q^{4}+−152587890625q^{5}+(382833920844806827025814+O(17^{2}0))∗ a+1234721310452452876707397+O(17^{2}0)q^{6}+(1802630623014962423659884+O(17^{2}0))∗a+157752462232979951788553+ O(17^{2}0)q^{7} + (1846483028899772089708636 + O(17^{2}0)) ∗ a + 548422534485550730525205 + O(17^{2}0)q^{8} + −(734536704986020174307624+O(17^{2}0))∗a+534292118397575343904559+O(17^{2}0)q^{9}+−152587890625∗ aq^{10} + −(1428163953694801295354379 + O(17^{2}0)) ∗ a + 1237372402675889063268786 + O(17^{2}0)q^{11} + −(1426181625023343448840805+O(17^{2}0))∗a+23249751078754655003741+O(17^{2}0)q^{12}+(1673150564697082902175720+ O(17^{2}0))∗a+100354188266680249801582∗17+O(17^{2}0)q^{13}+−(797433895479524247798328+O(17^{2}0))∗ a−674527328600335853171759+O(17^{2}0)q^{14}+(1724307088644227149785205+O(17^{2}0))∗a−1669791154776154125428800+ O(17^{2}0)q^{15} + (273487894310104746640979 + O(17^{2}0)) ∗ a − 725906665858185052115155 + O(17^{2}0)q^{16} + (49707249799014459209109∗17+O(17^{2}0))∗a−90234241116668315482254∗17+O(17^{2}0)q^{17}+−(775404819961617934768165+ O(17^{2}0))∗a−576836244456819119075569+O(17^{2}0)q^{18}+(1439565759431516814245577+O(17^{2}0))∗a− 185330458513479225520898+O(17^{2}0)q^{19}+(201390405233136218994958+O(17^{2}0))∗a+1766125415054611562961593+ O(17^{2}0)q^{20}+(1619868794094222828235872+O(17^{2}0))∗a+1805351244614501892636316+O(17^{2}0)q^{21}+ (967181061720558267464114+O(17^{2}0))∗a−1245361751826794258059137+O(17^{2}0)q^{22}+(1467650909033699029924933+ O(17^{2}0))∗a−253579713659214695516863+O(17^{2}0)q^{23} +(463393839888019231084388+O(17^{2}0))∗a+ 1290475444635666316676929+O(17^{2}0)q^{24}+23283064365386962890625q^{25}+(1370069987031612280826477+ O(17^{2}0)) ∗ a + 551314173824756130729312 + O(17^{2}0)q^{26} + (1463882582213283229724251 + O(17^{2}0)) ∗ a − 1957684915531955435092705 + O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + −1138128332623662180558778 + O(17^{2}0)x + −942164111436600558920490 + O(17^{2}0). The slope of f_{0} is 1.

38. Consider f = q +z^{1}64q^{2} +z^{2}44q^{3} +z^{1}32q^{4} +q^{5} +z^{1}20q^{6} +z^{1}10q^{7} +z^{1}48q^{8} +z^{7}1q^{9} +z^{1}64q^{10} +z^{7}7q^{11} + z^{8}8q^{12}+z^{1}72q^{13}+z^{2}74q^{14}+z^{2}44q^{15}+z^{1}14q^{16}+z^{2}35q^{18}+z^{3}4q^{19}+z^{1}32q^{20}+z^{6}6q^{21}+z^{2}41q^{22}+z^{1}02q^{23}+ z^{1}04q^{24}+q^{25}+z^{4}8q^{26}+z^{9}5q^{27}+⋯ ∈ S_{34}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{2}+16x+3 = 0. We have ρ_{f} ⊗χ^{15} ∼ ρ_{g}, where g is the weight-16 newform g = q +z^{2}00q^{2} +z^{2}26q^{3} +z^{2}04q^{4} +7q^{5} +z^{1}38q^{6} + z^{2}00q^{7}+z^{2}56q^{8}+z^{3}5q^{9}+z^{1}10q^{10}+z^{2}39q^{11}+z^{1}42q^{12}+z^{1}00q^{13}+z^{1}12q^{14}+z^{1}36q^{15}+z^{2}58q^{16}+z^{5}6q^{17}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}32)ω^{0} & 0 \\ 0 & u(z^{5}6)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(1092736988502271747125792+ O(17^{2}0)) ∗ a − 309561535934071187296097 + O(17^{2}0)q^{3} + −(1691940963217720593054005 + O(17^{2}0)) ∗ a − 1540469484260492901154827 + O(17^{2}0)q^{4} + 152587890625q^{5} + −(442026959176797122636868 + O(17^{2}0))∗a−546350613214388478080163+O(17^{2}0)q^{6} +−(644038705183790578649648+O(17^{2}0))∗a− 49221449077578088835052∗17+O(17^{2}0)q^{7}+(811343784170104892528698+O(17^{2}0))∗a+382201958739049383038455+ O(17^{2}0)q^{8} + (1302148397952240357104573 + O(17^{2}0)) ∗ a + 909728771852170797191419 + O(17^{2}0)q^{9} + 152587890625 ∗ aq^{10} + −(1304908091910304039927235 + O(17^{2}0)) ∗ a + 140270700745281649300764 + O(17^{2}0)q^{11}+−(1748794345863764139307025+O(17^{2}0))∗a+524234205719852196816621+O(17^{2}0)q^{12}+ (477943123355392223358490+O(17^{2}0))∗a+1500354254673242110292992+O(17^{2}0)q^{13}+−(2005691655746194234739882+ O(17^{2}0))∗a−691506877484549276904255+O(17^{2}0)q^{14}+(1907399434770212156334535+O(17^{2}0))∗a+ 1895840010270349930831534+O(17^{2}0)q^{15}+(148218829914383332294578+O(17^{2}0))∗a+851059944307980581543219+ O(17^{2}0)q^{16}+−(93299577822237332878009∗17+O(17^{2}0))∗a+304682598806985239374∗17^{2}+O(17^{2}0)q^{17}+ (592267028192306902041159+O(17^{2}0))∗a+223257841235412038407907+O(17^{2}0)q^{18}+−(1808318419398901228778976+ O(17^{2}0))∗a−127931753974960884851825+O(17^{2}0)q^{19} +(728438652065181781795779+O(17^{2}0))∗a− 10530178573912132246389+O(17^{2}0)q^{20}+(550200499797393965368786+O(17^{2}0))∗a−1107519803607064925906549+ O(17^{2}0)q^{21}+(1556761794030186485502104+O(17^{2}0))∗a−1874319642645671384788893+O(17^{2}0)q^{22}+ −(382186432163275032849280+O(17^{2}0))∗a+1778727484311205522225381+O(17^{2}0)q^{23}+−(811516204137779551340291+ O(17^{2}0))∗a−1687366896866504846166669+O(17^{2}0)q^{24}+23283064365386962890625q^{25}+−(845377796602842219718071+ O(17^{2}0)) ∗ a − 1486490938671653762078257 + O(17^{2}0)q^{26} + −(279060187511533855983067 + O(17^{2}0)) ∗ a − 1613214370974746640790068 + O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 1691940963217720593054005 + O(17^{2}0)x + 1540469484260484311220235 + O(17^{2}0). The slope of f_{0} is 1.

39. Consider f = q +z^{1}96q^{2} +z^{1}16q^{3} +z^{2}28q^{4} +q^{5} +z^{2}4q^{6} +z^{1}42q^{7} +z^{2}12q^{8} +z^{5}5q^{9} +z^{1}96q^{10} +z^{1}57q^{11} + z^{5}6q^{12} + z^{4}4q^{13} + z^{5}0q^{14} + z^{1}16q^{15} + z^{2}10q^{16} + z^{2}51q^{18} + z^{2}q^{19} + z^{2}28q^{20} + z^{2}58q^{21} + z^{6}5q^{22} + z^{6}q^{23} + z^{4}0q^{24} + q^{25} + z^{2}40q^{26} + z^{1}75q^{27} + ⋯ ∈ S_{34}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{2} + 16x + 3 = 0. We have ρ_{f} ⊗χ^{15} ∼ ρ_{g}, where g is the weight-16 newform g = q +z^{2}32q^{2} +z^{9}8q^{3} +z^{1}2q^{4} +7q^{5} +z^{4}2q^{6} + z^{2}32q^{7} +z^{3}2q^{8} +z^{1}9q^{9} +z^{1}42q^{10} +z^{3}1q^{11} +z^{1}10q^{12} +z^{2}60q^{13} +z^{1}76q^{14} +z^{8}q^{15} +z^{6}6q^{16} +z^{8}8q^{17} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(z^{2}00)ω^{0} & 0 \\ 0 & u(z^{8}8)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+aq^{2}+(1092736988502271747125792+ O(17^{2}0)) ∗ a − 309561535934071187296097 + O(17^{2}0)q^{3} + −(1691940963217720593054005 + O(17^{2}0)) ∗ a − 1540469484260492901154827 + O(17^{2}0)q^{4} + 152587890625q^{5} + −(442026959176797122636868 + O(17^{2}0))∗a−546350613214388478080163+O(17^{2}0)q^{6} +−(644038705183790578649648+O(17^{2}0))∗a− 49221449077578088835052∗17+O(17^{2}0)q^{7}+(811343784170104892528698+O(17^{2}0))∗a+382201958739049383038455+ O(17^{2}0)q^{8} + (1302148397952240357104573 + O(17^{2}0)) ∗ a + 909728771852170797191419 + O(17^{2}0)q^{9} + 152587890625 ∗ aq^{10} + −(1304908091910304039927235 + O(17^{2}0)) ∗ a + 140270700745281649300764 + O(17^{2}0)q^{11}+−(1748794345863764139307025+O(17^{2}0))∗a+524234205719852196816621+O(17^{2}0)q^{12}+ (477943123355392223358490+O(17^{2}0))∗a+1500354254673242110292992+O(17^{2}0)q^{13}+−(2005691655746194234739882+ O(17^{2}0))∗a−691506877484549276904255+O(17^{2}0)q^{14}+(1907399434770212156334535+O(17^{2}0))∗a+ 1895840010270349930831534+O(17^{2}0)q^{15}+(148218829914383332294578+O(17^{2}0))∗a+851059944307980581543219+ O(17^{2}0)q^{16}+−(93299577822237332878009∗17+O(17^{2}0))∗a+304682598806985239374∗17^{2}+O(17^{2}0)q^{17}+ (592267028192306902041159+O(17^{2}0))∗a+223257841235412038407907+O(17^{2}0)q^{18}+−(1808318419398901228778976+ O(17^{2}0))∗a−127931753974960884851825+O(17^{2}0)q^{19} +(728438652065181781795779+O(17^{2}0))∗a− 10530178573912132246389+O(17^{2}0)q^{20}+(550200499797393965368786+O(17^{2}0))∗a−1107519803607064925906549+ O(17^{2}0)q^{21}+(1556761794030186485502104+O(17^{2}0))∗a−1874319642645671384788893+O(17^{2}0)q^{22}+ −(382186432163275032849280+O(17^{2}0))∗a+1778727484311205522225381+O(17^{2}0)q^{23}+−(811516204137779551340291+ O(17^{2}0))∗a−1687366896866504846166669+O(17^{2}0)q^{24}+23283064365386962890625q^{25}+−(845377796602842219718071+ O(17^{2}0)) ∗ a − 1486490938671653762078257 + O(17^{2}0)q^{26} + −(279060187511533855983067 + O(17^{2}0)) ∗ a − 1613214370974746640790068 + O(17^{2}0)q^{27} + ⋯ is an ℓ-adic newform that lifts f. Here the coefficients of f_{0} are written in terms of a root a of x^{2} + 1691940963217720593054005 + O(17^{2}0)x + 1540469484260484311220235 + O(17^{2}0). The slope of f_{0} is 1.

40. Consider f = q+10q^{2}+9q^{3}+13q^{4}+q^{5}+5q^{6}+9q^{7}+8q^{8}+10q^{9}+10q^{10}+5q^{11}+15q^{12}+3q^{13}+5q^{14}+9q^{15}+ 3q^{16}+15q^{18}+q^{19}+13q^{20}+13q^{21}+16q^{22}+7q^{23}+4q^{24}+q^{25}+13q^{26}+12q^{27}+⋯ ∈ S_{34}(Γ_{0}(5); \overline{F}_{17}). Also, z satisfies the equation x^{2} + 16x + 3 = 0. We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-16 newform g = q+5q^{2}+3q^{3}+16q^{4}+7q^{5}+15q^{6}+11q^{7}+q^{8}+3q^{9}+q^{10}+2q^{11}+14q^{12}+12q^{13}+4q^{14}+4q^{15}+14q^{16}+14q^{17}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(11)ω^{0} & 0 \\ 0 & u(14)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−1799015977979176355881323+ O(17^{2}0)q^{2} + 1893631338225180887475427 + O(17^{2}0)q^{3} + −1165202286014341868300667 + O(17^{2}0)q^{4} + 152587890625q^{5}+−1605596090715953157063470+O(17^{2}0)q^{6}+−1610107675030187404024897+O(17^{2}0)q^{7}+ −406457884418919486709081+O(17^{2}0)q^{8}+595898546048220623822863+O(17^{2}0)q^{9}+1041656208498946851803771+ O(17^{2}0)q^{10} +421194231368952308269008+O(17^{2}0)q^{11} +−1611051948618821532325075+O(17^{2}0)q^{12} + −82383877767583701719018+O(17^{2}0)q^{13}+1430492106445746382258450+O(17^{2}0)q^{14}+231497775741699397083942+ O(17^{2}0)q^{15}+−941798814496078133092092+O(17^{2}0)q^{16}+−109816771698978682467238∗17+O(17^{2}0)q^{17}+ 34864839998490201846293+O(17^{2}0)q^{18}+−1694414431744716326049150+O(17^{2}0)q^{19}+−74176238106440743297385+ O(17^{2}0)q^{20}+−1477800742866236770162779+O(17^{2}0)q^{21}+−887644428368214235057846+O(17^{2}0)q^{22}+ −610198052179408493162265+O(17^{2}0)q^{23}+−1608560732078781864118384+O(17^{2}0)q^{24}+23283064365386962890625q^{25}+ 857487062662255361776500+O(17^{2}0)q^{26}+−235895962955204907436256+O(17^{2}0)q^{27}+⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

41. Consider f = q + 13q2 + 2q3 + 8q4 + 12q5 + 9q6 + 6q7 + 11q9 + 3q10 + 15q11 + 16q12 + 13q13 + 10q14 + 7q15 + 4q16 + 9q17 + 7q18 + 15q19 + 11q20 + 12q21 + 8q22 + 7q23 + 8q25 + 16q26 + 2q27 + ⋯ ∈ S20(Γ0(5); \overline{F}17). Also, z satisfies the equation x2 + 16x + 3 = 0. Since f is ordinary, we do not continue our analysis.

42. Consider f = q + 6q2 + 12q3 + 11q4 + 5q5 + 4q6 + 5q7 + q8 + 15q9 + 13q10 + 13q11 + 13q12 + 12q13 + 13q14 + 9q15 + 3q16 + 5q18 + 6q19 + 4q20 + 9q21 + 10q22 + 8q23 + 12q24 + 8q25 + 4q26 + 9q27 + ⋯ ∈ S20(Γ0(5); \overline{F}17). Also, z satisfies the equation x2 + 16x + 3 = 0. I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

43. Consider f = q + 2q2 + 13q3 + 6q4 + 8q5 + 9q6 + 5q7 + 16q8 + 11q9 + 16q10 + 5q11 + 10q12 + 14q13 + 10q14 + 2q15 + 10q16 + 5q17 + 5q18 + 6q19 + 14q20 + 14q21 + 10q22 + q23 + 4q24 + 13q25 + 11q26 + 10q27 + ⋯ ∈ S22(Γ0(5); \overline{F}17). Also, z satisfies the equation x2 + 16x + 3 = 0. Since f is ordinary, we do not continue our analysis.

44. Consider f = q + 15q2 + 4q3 + 12q4 + 11q5 + 9q6 + 9q7 + 11q8 + 5q9 + 12q10 + 5q11 + 14q12 + 5q13 + 16q14 + 10q15 + 6q16 + 11q17 + 7q18 + 15q19 + 13q20 + 2q21 + 7q22 + 14q23 + 10q24 + 2q25 + 7q26 + 10q27 + ⋯ ∈ S24(Γ0(5); \overline{F}17). Since f is ordinary, we do not continue our analysis.

45. Consider f = q + 5q2 + 16q3 + 16q4 + 11q5 + 12q6 + 10q7 + q8 + 7q9 + 4q10 + q12 + 16q13 + 16q14 + 6q15 + 14q16 + 2q17 + q18 + 2q19 + 6q20 + 7q21 + 14q23 + 16q24 + 2q25 + 12q26 + 4q27 + ⋯ ∈ S24(Γ0(5); \overline{F}17). Since f is ordinary, we do not continue our analysis.

46. Consider f = q + 3q2 + 3q3 + 6q5 + 9q6 + 5q7 + 7q8 + 15q9 + q10 + 2q11 + 3q13 + 15q14 + q15 + 4q16 + 3q17 + 11q18 + 2q19 + 15q21 + 6q22 + q23 + 4q24 + 2q25 + 9q26 + 12q27 + ⋯ ∈ S24(Γ0(5); \overline{F}17). Since f is ordinary, we do not continue our analysis.

47. Consider f = q + z254q2 + z13q3 + z65q4 + 13q5 + z267q6 + z45q7 + zq8 + z65q9 + z38q10 + z188q11 + z78q12 + z202q13 + z11q14 + z85q15 + z22q16 + z53q17 + z31q18 + z49q19 + z137q20 + z58q21 + z154q22 + z66q23 + z14q24 + 16q25 + z168q26 + z37q27 + ⋯ ∈ S26(Γ0(5); \overline{F}17). Also, z satisfies the equation x2 + 16x + 3 = 0. Since f is ordinary, we do not continue our analysis.

48. Consider f = q + z286q2 + z221q3 + z241q4 + 13q5 + z219q6 + z189q7 + z17q8 + z241q9 + z70q10 + z28q11 + z174q12 + z266q13 + z187q14 + z5q15 + z86q16 + z37q17 + z239q18 + z257q19 + z25q20 + z122q21 + z26q22 + z258q23 + z238q24 + 16q25 + z264q26 + z53q27 + ⋯ ∈ S26(Γ0(5); \overline{F}17). Also, z satisfies the equation x2 + 16x + 3 = 0. Since f is ordinary, we do not continue our analysis.

49. Consider f = q + 9q2 + 5q3 + 11q4 + 4q5 + 11q6 + 9q7 + 13q8 + 11q9 + 2q10 + 15q11 + 4q12 + 6q13 + 13q14 + 3q15 + 10q16 + 6q17 + 14q18 + 6q19 + 10q20 + 11q21 + 16q22 + 5q23 + 14q24 + 16q25 + 3q26 + 2q27 + ⋯ ∈ S26(Γ0(5); \overline{F}17). Also, z satisfies the equation x2 + 16x + 3 = 0. Since f is ordinary, we do not continue our analysis.

50. Consider f = q + 16q2 + 6q3 + 10q4 + 3q5 + 11q6 + 2q7 + 15q8 + 12q9 + 14q10 + 9q11 + 9q12 + 5q13 + 15q14 + q15 + 7q16 + 7q17 + 5q18 + 14q19 + 13q20 + 12q21 + 8q22 + 5q23 + 5q24 + 9q25 + 12q26 + 13q27 + ⋯ ∈ S28(Γ0(5); \overline{F}17). Since f is ordinary, we do not continue our analysis.

51. Consider f = q + 15q2 + 12q3 + 13q4 + 3q5 + 10q6 + 13q7 + 7q8 + q9 + 11q10 + 3q11 + 3q12 + 9q13 + 8q14 + 2q15 + q16 + 8q17 + 15q18 + 5q19 + 5q20 + 3q21 + 11q22 + 5q23 + 16q24 + 9q25 + 16q26 + 13q27 + ⋯ ∈ S28(Γ0(5); \overline{F}17). Since f is ordinary, we do not continue our analysis.

52. Consider f = q+7q3+9q4+14q5+5q7+8q9+4q11+12q12+9q13+13q15+13q16+6q17+14q19+7q20+q21+8q23+9q25+7q27+⋯ ∈ S28(Γ0(5); \overline{F}17). Also, z satisfies the equation x2 + 16x + 3 = 0. Since f is ordinary, we do not continue our analysis.

53. Consider f = q+z4747q2 +z3044q3 +z509q4 +2q5 +z2879q6 +z2513q7 +z3903q8 +z3398q9 +z4133q10 +z105q11 +z3553q12 +z2276q13 +z2348q14 + z2430q15 +z4239q16 +z3815q17 +z3233q18 +z2488q19 +z4807q20 +z645q21 +z4852q22 +z4477q23 +z2035q24 +4q25 +z2111q26 +z840q27 +⋯ ∈ S30(Γ0(5); \overline{F}17). Also, z satisfies the equation x3 + x + 14 = 0. Since f is ordinary, we do not continue our analysis.

54. Consider f = q +z1435q2 +z468q3 +z4653q4 +2q5 +z1903q6 +z4193q7 +z3119q8 +z4534q9 +z821q10 +z873q11 +z209q12 +z4468q13 +z716q14 + z4766q15 +z1983q16 +z2247q17 +z1057q18 +z1880q19 +z4039q20 +z4661q21 +z2308q22 +z1997q23 +z3587q24 +4q25 +z991q26 +z2072q27 +⋯ ∈ S30(Γ0(5); \overline{F}17). Also, z satisfies the equation x3 + x + 14 = 0. Since f is ordinary, we do not continue our analysis.

55. Consider f = q+z2107q2+z2628q3+z3741q4+2q5+z4735q6+z3425q7+z2495q8+z3734q9+z1493q10+z1785q11+z1457q12+z4308q13+z620q14+ z2014q15 +z3295q16 +z999q17 +z929q18 +z3000q19 +z3127q20 +z1141q21 +z3892q22 +z2429q23 +z211q24 +4q25 +z1503q26 +z4456q27 +⋯ ∈ S30(Γ0(5); \overline{F}17). Also, z satisfies the equation x3 + x + 14 = 0. Since f is ordinary, we do not continue our analysis.

56. Consider f = q + z42q2 + z134q3 + z160q4 + 15q5 + z176q6 + z43q7 + z195q8 + z240q9 + z150q10 + z70q11 + z6q12 + z190q13 + z85q14 + z242q15 + z31q16 + z222q17 + z282q18 + z212q19 + z268q20 + z177q21 + z112q22 + z100q23 + z41q24 + 4q25 + z232q26 + z127q27 + ⋯ ∈ S30(Γ0(5); \overline{F}17). Also, z satisfies the equation x2 + 16x + 3 = 0. Since f is ordinary, we do not continue our analysis.

57. Consider f = q+z200q2 +z226q3 +z204q4 +7q5 +z138q6 +z200q7 +z256q8 +z35q9 +z110q10 +z239q11 +z142q12 +z100q13 +z112q14 +z136q15 + z258q16 + z56q17 + z235q18 + z70q19 + z114q20 + z138q21 + z151q22 + z120q23 + z194q24 + 15q25 + z12q26 + z41q27 + ⋯ ∈ S32(Γ0(5); \overline{F}17). Also, z satisfies the equation x2 + 16x + 3 = 0. Since f is ordinary, we do not continue our analysis.

58. Consider f = q + z232q2 + z98q3 + z12q4 + 7q5 + z42q6 + z232q7 + z32q8 + z19q9 + z142q10 + z31q11 + z110q12 + z260q13 + z176q14 + z8q15 + z66q16 + z88q17 + z251q18 + z38q19 + z210q20 + z42q21 + z263q22 + z24q23 + z130q24 + 15q25 + z204q26 + z121q27 + ⋯ ∈ S32(Γ0(5); \overline{F}17). Also, z satisfies the equation x2 + 16x + 3 = 0. Since f is ordinary, we do not continue our analysis.

59. Consider f = q + 5q2 + 3q3 + 16q4 + 7q5 + 15q6 + 11q7 + q8 + 3q9 + q10 + 2q11 + 14q12 + 12q13 + 4q14 + 4q15 + 14q16 + 14q17 + 15q18 + 9q19 + 10q20 + 16q21 + 10q22 + 4q23 + 3q24 + 15q25 + 9q26 + 8q27 + ⋯ ∈ S32(Γ0(5); \overline{F}17). Also, z satisfies the equation x2 + 16x + 3 = 0. Since f is ordinary, we do not continue our analysis.

60. Consider f = q+15∗z5 +7∗z3 +16∗z2 +15q2 +13∗z5 +14∗z3 +15∗z2 +3q3 +8∗z5 +6∗z3 +4∗z2 +7q4 +10q5 +2∗z5 +10∗z3 +z2 +1q6 +11∗z5 +4∗z3 + 14∗z2 +5q7 +5∗z5 +8∗z3 +11∗z2 +7q8 +10∗z5 +16∗z3 +5∗z2q9 +14∗z5 +2∗z3 +7∗z2 +14q10 +5∗z5 +8∗z3 +11∗z2 +16q11 +13∗z5 +14∗z3 +15∗z2 + 10q12 +12∗z5 +9∗z3 +6∗z2 +12q13 +2∗z5 +10∗z3 +z2 +9q14 +11∗z5 +4∗z3 +14∗z2 +13q15 +6∗z5 +13∗z3 +3∗z2 +12q16 +12∗z5 +9∗z3 +6∗z2 + 8q17 +14∗z5 +2∗z3 +7∗z2 +8q18 +4∗z5 +3∗z3 +2∗z2q19 +12∗z5 +9∗z3 +6∗z2 +2q20 +13∗z5 +14∗z3 +15∗z2 +2q21 +9∗z5 +11∗z3 +13∗z2 +6q22 + 15∗z5 +7∗z3 +16∗z2 +6q23 +4∗z5 +3∗z3 +2∗z2 +12q24 +15q25 +3∗z5 +15∗z3 +10∗z2 +6q26 +3∗z5 +15∗z3 +10∗z2 +15q27 +⋯ ∈ S32(Γ0(5); \overline{F}17). Also, z satisfies the equation x6 + 2x4 + 10x2 + 3x + 3 = 0. Since f is ordinary, we do not continue our analysis.

61. Consider f = q +2∗z5 +10∗z3 +z2 +15q2 +4∗z5 +3∗z3 +2∗z2 +3q3 +9∗z5 +11∗z3 +13∗z2 +7q4 +10q5 +15∗z5 +7∗z3 +16∗z2 +1q6 +6∗z5 +13∗ z3 +3∗z2 +5q7 +12∗z5 +9∗z3 +6∗z2 +7q8 +7∗z5 +z3 +12∗z2q9 +3∗z5 +15∗z3 +10∗z2 +14q10 +12∗z5 +9∗z3 +6∗z2 +16q11 +4∗z5 +3∗z3 +2∗z2 + 10q12 +5∗z5 +8∗z3 +11∗z2 +12q13 +15∗z5 +7∗z3 +16∗z2 +9q14 +6∗z5 +13∗z3 +3∗z2 +13q15 +11∗z5 +4∗z3 +14∗z2 +12q16 +5∗z5 +8∗z3 +11∗z2 + 8q17 +3∗z5 +15∗z3 +10∗z2 +8q18 +13∗z5 +14∗z3 +15∗z2q19 +5∗z5 +8∗z3 +11∗z2 +2q20 +4∗z5 +3∗z3 +2∗z2 +2q21 +8∗z5 +6∗z3 +4∗z2 +6q22 + 2∗z5 +10∗z3 +z2 +6q23 +13∗z5 +14∗z3 +15∗z2 +12q24 +15q25 +14∗z5 +2∗z3 +7∗z2 +6q26 +14∗z5 +2∗z3 +7∗z2 +15q27 +⋯ ∈ S32(Γ0(5); \overline{F}17). Also, z satisfies the equation x6 + 2x4 + 10x2 + 3x + 3 = 0. Since f is ordinary, we do not continue our analysis.

62. Consider f = q + q2 + 2q3 + 16q4 + 16q5 + 2q6 + 15q7 + 14q8 + q9 + 16q10 + 2q11 + 15q12 + 2q13 + 15q14 + 15q15 + 16q16 + q17 + q18 + q20 + 13q21 + 2q22 + 6q23 + 11q24 + q25 + 2q26 + 13q27 + ⋯ ∈ S34(Γ0(5); \overline{F}17). Also, z satisfies the equation x2 + 16x + 3 = 0. Since f is ordinary, we do not continue our analysis.

63. Consider f = q + 10q2 + 4q3 + 13q4 + 16q5 + 6q6 + 9q7 + 8q8 + 13q9 + 7q10 + 7q11 + q12 + 12q13 + 5q14 + 13q15 + 3q16 + 16q17 + 11q18 + 12q19 + 4q20 + 2q21 + 2q22 + 4q23 + 15q24 + q25 + q26 + 6q27 + ⋯ ∈ S34(Γ0(5); \overline{F}17). Also, z satisfies the equation x2 + 16x + 3 = 0. Since f is ordinary, we do not continue our analysis.

64. Consider f = q + 5q2 + 9q3 + 6q4 + 16q5 + 11q6 + 4q7 + 3q8 + 10q9 + 12q10 + 2q11 + 3q12 + 5q13 + 3q14 + 8q15 + 3q16 + 16q17 + 16q18 + 5q19 + 11q20 + 2q21 + 10q22 + 9q23 + 10q24 + q25 + 8q26 + 12q27 + ⋯ ∈ S34(Γ0(5); \overline{F}17). Also, z satisfies the equation x2 + 16x + 3 = 0. Since f is ordinary, we do not continue our analysis.

65. Consider f = q + z153q2 + z278q3 + q4 + q5 + z143q6 + z134q7 + z9q8 + z95q9 + z153q10 + z287q11 + z278q12 + 13q13 + z287q14 + z278q15 + 12q16 + 16q17 + z248q18 + z82q19 + q20 + z124q21 + z152q22 + z152q23 + z287q24 + q25 + z225q26 + 4q27 + ⋯ ∈ S34(Γ0(5); \overline{F}17). Also, z satisfies the equation x2 + 16x + 3 = 0. Since f is ordinary, we do not continue our analysis.

66. Consider f = q + z9q2 + z118q3 + q4 + q5 + z127q6 + z262q7 + z153q8 + z175q9 + z9q10 + z271q11 + z118q12 + 13q13 + z271q14 + z118q15 + 12q16 + 16q17 + z184q18 + z242q19 + q20 + z92q21 + z280q22 + z280q23 + z271q24 + q25 + z81q26 + 4q27 + ⋯ ∈ S34(Γ0(5); \overline{F}17). Also, z satisfies the equation x2 + 16x + 3 = 0. Since f is ordinary, we do not continue our analysis.

67. Consider f = q + 3q2 + 4q3 + 7q4 + q5 + 12q6 + 8q7 + 15q8 + 13q9 + 3q10 + 12q11 + 11q12 + 14q13 + 7q14 + 4q15 + 14q16 + 5q18 + 3q19 + 7q20 + 15q21 + 2q22 + 7q23 + 9q24 + q25 + 8q26 + 6q27 + ⋯ ∈ S34(Γ0(5); \overline{F}17). Also, z satisfies the equation x2 + 16x + 3 = 0. I didn’t find a twist of ρf in low weight, so ρf must be reducible and we do not consider it further.

17.5 N = 6, ℓ = 17 (34 forms)

1. Consider f = q + 2q^{2} + 14q^{3} + 4q^{4} + 5q^{5} + 11q^{6} + 12q^{7} + 8q^{8} + 9q^{9} + 10q^{10} + 3q^{11} + 5q^{12} + 2q^{13} + 7q^{14} + 2q^{15} + 16q^{16} + q^{18} + 3q^{19} + 3q^{20} + 15q^{21} + 6q^{22} + 14q^{23} + 10q^{24} + 2q^{25} + 4q^{26} + 7q^{27} + 14q^{28} + 16q^{29} + 4q^{30} + 11q^{31} + 15q^{32} + 8q^{33} + ⋯ ∈ S_{20}(Γ_{0}(6); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{14} ∼ ρ_{g}, where g is the weight-16 newform g = q + 9q^{2} + 11q^{3} + 13q^{4} + 7q^{5} + 14q^{6} + 11q^{7} + 15q^{8} + 2q^{9} + 12q^{10} + 10q^{11} + 7q^{12} + 15q^{13} + 14q^{14} + 9q^{15} + 16q^{16} + 5q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(7)ω^{1} & 0 \\ 0 & u(5)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 512q^{2} + 19683q^{3} + 262144q^{4} + 1953390q^{5} + 10077696q^{6} + 40488776q^{7} + 134217728q^{8} + 387420489q^{9} + 1000135680q^{10} + 1916860452q^{11}+5159780352q^{12}+3132480182q^{13}+20730253312q^{14}+38448575370q^{15}+68719476736q^{16}+ 35744703858 ∗ 17q^{17} + 198359290368q^{18} + 2507511106460q^{19} + 512069468160q^{20} + 796940578008q^{21} + 981432551424q^{22}+−13588841327928q^{23}+2641807540224q^{24}+−15257753836025q^{25}+1603829853184q^{26}+ 7625597484987q^{27}+10613889695744q^{28}+−95129927516010q^{29}+19685670589440q^{30}+−128131172993488q^{31}+ 35184372088832q^{32} + 37729564276716q^{33} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

2. Consider f = q + 15q^{2} + 3q^{3} + 4q^{4} + 12q^{5} + 11q^{6} + 2q^{7} + 9q^{8} + 9q^{9} + 10q^{10} + 10q^{11} + 12q^{12} + 16q^{13} + 13q^{14} + 2q^{15} + 16q^{16} + 16q^{18} + 2q^{19} + 14q^{20} + 6q^{21} + 14q^{22} + 2q^{23} + 10q^{24} + 2q^{25} + 2q^{26} + 10q^{27} + 8q^{28} + 9q^{29} + 13q^{30} + 11q^{31} + 2q^{32} + 13q^{33} + ⋯ ∈ S_{20}(Γ_{0}(6); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{14} ∼ ρ_{g}, where g is the weight-16 newform g = q + 8q^{2} + 6q^{3} + 13q^{4} + 10q^{5} + 14q^{6} + 16q^{7} + 2q^{8} + 2q^{9} + 12q^{10} + 5q^{11} + 10q^{12} + q^{13} + 9q^{14} + 9q^{15} + 16q^{16} + 4q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(13)ω^{1} & 0 \\ 0 & u(4)ω^{2} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−512q^{2}+−19683q^{3}+ 262144q^{4}+−3732474q^{5}+10077696q^{6}+−149672656q^{7}+−134217728q^{8}+387420489q^{9}+1911026688q^{10}+ −7459672308q^{11}+−5159780352q^{12}+59238459878q^{13}+76632399872q^{14}+73466285742q^{15}+68719476736q^{16}+ 30771201762∗17q^{17} +−198359290368q^{18} +969502037780q^{19} +−978445664256q^{20} +2946006888048q^{21} + 3819352221696q^{22}+−1368374071512q^{23}+2641807540224q^{24}+−5142124167449q^{25}+−30330091457536q^{26}+ −7625597484987q^{27}+−39235788734464q^{28}+−98642915804130q^{29}+−37614738299904q^{30}+194951985476072q^{31}+ −35184372088832q^{32} + 146828730038364q^{33} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

3. Consider f = q + 4q^{2} + 9q^{3} + 16q^{4} + 4q^{5} + 2q^{6} + 14q^{7} + 13q^{8} + 13q^{9} + 16q^{10} + 9q^{11} + 8q^{12} + 12q^{13} + 5q^{14} + 2q^{15} + q^{16} + q^{18} + 15q^{19} + 13q^{20} + 7q^{21} + 2q^{22} + 15q^{23} + 15q^{24} + 2q^{25} + 14q^{26} + 15q^{27} + 3q^{28} + 13q^{29} + 8q^{30} + 10q^{31} + 4q^{32} + 13q^{33} + ⋯ ∈ S_{22}(Γ_{0}(6); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{12} ∼ ρ_{g}, where g is the weight-14 newform g = q + 13q^{2} + 2q^{3} + 16q^{4} + 16q^{5} + 9q^{6} + 12q^{7} + 4q^{8} + 4q^{9} + 4q^{10} + 15q^{11} + 15q^{12} + 12q^{13} + 3q^{14} + 15q^{15} + q^{16} + 11q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(14)ω^{1} & 0 \\ 0 & u(11)ω^{4} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 1024q^{2} + −59049q^{3} +1048576q^{4} +12954174q^{5} +−60466176q^{6} +−479513104q^{7} +1073741824q^{8} +3486784401q^{9} + 13265074176q^{10}+115657781700q^{11}+−61917364224q^{12}+295658246702q^{13}+−491021418496q^{14}+−764931020526q^{15}+ 1099511627776q^{16}+389822554818∗17q^{17}+3570467226624q^{18}+28576184164796q^{19}+13583435956224q^{20}+ 28314769278096q^{21}+118433568460800q^{22}+335385196791000q^{23}+−63403380965376q^{24}+−309026534180849q^{25}+ 302754044622848q^{26}+−205891132094649q^{27}+−502805932539904q^{28}+−699224214482106q^{29}+−783289365018624q^{30}+ −3484957262657992q^{31} + 1125899906842624q^{32} + −6829476351603300q^{33} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

4. Consider f = q + 13q^{2} + 8q^{3} + 16q^{4} + 13q^{5} + 2q^{6} + 7q^{7} + 4q^{8} + 13q^{9} + 16q^{10} + 13q^{11} + 9q^{12} + q^{13} + 6q^{14} + 2q^{15} + q^{16} + 16q^{18} + 6q^{19} + 4q^{20} + 5q^{21} + 16q^{22} + 5q^{23} + 15q^{24} + 2q^{25} + 13q^{26} + 2q^{27} + 10q^{28} + 3q^{29} + 9q^{30} + 9q^{31} + 13q^{32} + 2q^{33} + ⋯ ∈ S_{22}(Γ_{0}(6); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-4 newform g = q + 15q^{2} + 14q^{3} + 4q^{4} + 6q^{5} + 6q^{6} + q^{7} + 9q^{8} + 9q^{9} + 5q^{10} + 12q^{11} + 5q^{12} + 4q^{13} + 15q^{14} + 16q^{15} + 16q^{16} + 10q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(12)ω^{4} & 0 \\ 0 & u(10)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −1024q^{2} + 59049q^{3} + 1048576q^{4} + 26444550q^{5} + −60466176q^{6} + 166115864q^{7} + −1073741824q^{8} + 3486784401q^{9} + −27079219200q^{10} + −104878761780q^{11} + 61917364224q^{12} + 335591325758q^{13} + −170102644736q^{14} + 1561524232950q^{15}+1099511627776q^{16}+858596750802∗17q^{17}+−3570467226624q^{18}+3569529974996q^{19}+ 27729120460800q^{20}+9808975653336q^{21}+107395852062720q^{22}+222369240588600q^{23}+−63403380965376q^{24}+ 222477066499375q^{25}+−343645517576192q^{26}+205891132094649q^{27}+174185108209664q^{28}+2194109701319454q^{29}+ −1599000814540800q^{30} + −8723627187590032q^{31} + −1125899906842624q^{32} + −6192986004347220q^{33} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

5. Consider f = q + 9q^{2} + 7q^{3} + 13q^{4} + 12q^{6} + 3q^{7} + 15q^{8} + 15q^{9} + 6q^{12} + 13q^{13} + 10q^{14} + 16q^{16} + 16q^{18} + 12q^{19} + 4q^{21} + 4q^{23} + 3q^{24} + 7q^{25} + 15q^{26} + 3q^{27} + 5q^{28} + 16q^{31} + 8q^{32} + ⋯ ∈ S_{24}(Γ_{0}(6); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{10} ∼ ρ_{g}, where g is the weight-12 newform g = q + 2q^{2} + 5q^{3} + 4q^{4} + 10q^{6} + 6q^{7} + 8q^{8} + 8q^{9} + 3q^{12} + 4q^{13} + 12q^{14} + 16q^{16} + 12q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(10)ω^{1} & 0 \\ 0 & u(12)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−2048q^{2}+177147q^{3}+ 4194304q^{4} + −2087250 ∗ 17q^{5} + −362797056q^{6} + −2385847912q^{7} + −8589934592q^{8} + 31381059609q^{9} + 4274688000∗17q^{10} +1480399140∗17^{2}q^{11} +743008370688q^{12} +4303510800614q^{13} +4886216523776q^{14} + −369750075750 ∗ 17q^{15} + 17592186044416q^{16} + −12445098770286 ∗ 17q^{17} + −64268410079232q^{18} + −303299666491876q^{19} + −8754561024000 ∗ 17q^{20} + −422645800067064q^{21} + −3031857438720 ∗ 17^{2}q^{22} + −4084826356392600q^{23}+−1521681143169024q^{24}+−10661867924515625q^{25}+−8813590119657472q^{26}+ 5559060566555523q^{27} + −10006971440693248q^{28} + −4513206603894762 ∗ 17q^{29} + 757248155136000 ∗ 17q^{30} + −95662499637633472q^{31} + −36028797018963968q^{32} + 262248266453580 ∗ 17^{2}q^{33} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

6. Consider f = q + 9q^{2} + 10q^{3} + 13q^{4} + 6q^{5} + 5q^{6} + 5q^{7} + 15q^{8} + 15q^{9} + 3q^{10} + 15q^{11} + 11q^{12} + 7q^{13} + 11q^{14} + 9q^{15} + 16q^{16} + 16q^{18} + 15q^{19} + 10q^{20} + 16q^{21} + 16q^{22} + 13q^{23} + 14q^{24} + 9q^{25} + 12q^{26} + 14q^{27} + 14q^{28} + 9q^{29} + 13q^{30} + 4q^{31} + 8q^{32} + 14q^{33} + ⋯ ∈ S_{24}(Γ_{0}(6); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{10} ∼ ρ_{g}, where g is the weight-12 newform g = q + 2q^{2} + 12q^{3} + 4q^{4} + 3q^{5} + 7q^{6} + 10q^{7} + 8q^{8} + 8q^{9} + 6q^{10} + 4q^{11} + 14q^{12} + 10q^{13} + 3q^{14} + 2q^{15} + 16q^{16} + 16q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(16)ω^{1} & 0 \\ 0 & u(16)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −2048q^{2} + −177147q^{3}+4194304q^{4}+196251270q^{5}+362797056q^{6}+−8131131904q^{7}+−8589934592q^{8}+31381059609q^{9}+ −401922600960q^{10}+413280901452q^{11}+−743008370688q^{12}+−5925134073898q^{13}+16652558139392q^{14}+ −34765323726690q^{15}+17592186044416q^{16}+2637120072258∗17q^{17}+−64268410079232q^{18}+869707616627540q^{19}+ 823137486766080q^{20}+1440405623397888q^{21}+−846399286173696q^{22}+−3916239181103928q^{23}+1521681143169024q^{24}+ 26593632021534775q^{25} + 12134674583343104q^{26} + −5559060566555523q^{27} + −34104439069474816q^{28} + 99766495798865310q^{29} + 71199382992261120q^{30} + 56862945627758072q^{31} + −36028797018963968q^{32} + −73211471849517444q^{33} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

7. Consider f = q +8q^{2} +7q^{3} +13q^{4} +10q^{5} +5q^{6} +8q^{7} +2q^{8} +15q^{9} +12q^{10} +3q^{11} +6q^{12} +14q^{13} +13q^{14} + 2q^{15} +16q^{16} +q^{18} +8q^{19} +11q^{20} +5q^{21} +7q^{22} +13q^{23} +14q^{24} +5q^{25} +10q^{26} +3q^{27} +2q^{28} +10q^{29} +16q^{30} + 15q^{31} +9q^{32} +4q^{33} +⋯ ∈ S_{24}(Γ_{0}(6); \overline{F}_{17}). We have ρ_{f} ⊗χ^{15} ∼ ρ_{g}, where g is the weight-6 newform g = q+4q^{2}+8q^{3}+16q^{4}+2q^{5}+15q^{6}+6q^{7}+13q^{8}+13q^{9}+8q^{10}+8q^{11}+9q^{12}+5q^{13}+7q^{14}+16q^{15}+q^{16}+11q^{17}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(14)ω^{6} & 0 \\ 0 & u(11)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +2048q^{2} +177147q^{3} + 4194304q^{4} +877889730901524823555422+O(17^{2}0)q^{5} +362797056q^{6} +−1626086239300254779044211+ O(17^{2}0)q^{7}+8589934592q^{8}+31381059609q^{9}+1527887148095783739996614+O(17^{2}0)q^{10}+717576534152689204996319+ O(17^{2}0)q^{11}+743008370688q^{12}+−567979410483915134036121+O(17^{2}0)q^{13}+−1619096042559891635632909+ O(17^{2}0)q^{14}+1781616049702921197480370+O(17^{2}0)q^{15}+17592186044416q^{16}+1696064698633020094018107∗ 17+O(17^{2}1)q^{17}+64268410079232q^{18}+−509581774641027937715437+O(17^{2}0)q^{19}+−345303818465742736167298+ O(17^{2}0)q^{20} +166144231116754391026459+O(17^{2}0)q^{21} +−1655027261713761276918250+O(17^{2}0)q^{22} + 588608918864419975862110 + O(17^{2}1)q^{23} + 1521681143169024q^{24} + −1945834124116988357344493 + O(17^{2}0)q^{25}+−851650369852453099117922+O(17^{2}0)q^{26}+5559060566555523q^{27}+504132661761108503508784+ O(17^{2}0)q^{28}+−672900741868321375361039+O(17^{2}0)q^{29}+−930133377937512676839938+O(17^{2}0)q^{30}+ 5299696334725867258295575 + O(17^{2}1)q^{31} + 36028797018963968q^{32} + −435410169691185671952584 + O(17^{2}0)q^{33} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

8. Consider f = q + 8q^{2} + 7q^{3} + 13q^{4} + q^{5} + 5q^{6} + 12q^{7} + 2q^{8} + 15q^{9} + 8q^{10} + 2q^{11} + 6q^{12} + 8q^{13} + 11q^{14} + 7q^{15} + 16q^{16} + q^{18} + 13q^{19} + 13q^{20} + 16q^{21} + 16q^{22} + 13q^{23} + 14q^{24} + 8q^{25} + 13q^{26} + 3q^{27} + 3q^{28} + q^{29} + 5q^{30} + 15q^{31} + 9q^{32} + 14q^{33} + ⋯ ∈ S_{24}(Γ_{0}(6); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{10} ∼ ρ_{g}, where g is the weight-12 newform g = q + 15q^{2} + 5q^{3} + 4q^{4} + 9q^{5} + 7q^{6} + 7q^{7} + 9q^{8} + 8q^{9} + 16q^{10} + 13q^{11} + 3q^{12} + 9q^{13} + 3q^{14} + 11q^{15} + 16q^{16} + 13q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{1} & 0 \\ 0 & u(13)ω^{6} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q +2048q^{2} +177147q^{3} + 4194304q^{4} +−877889730901524798307266+O(17^{2}0)q^{5} +362797056q^{6} +1626086239300260543506979+ O(17^{2}0)q^{7}+8589934592q^{8}+31381059609q^{9}+−1527887148095732031773126+O(17^{2}0)q^{10}+−717576534151672083526295+ O(17^{2}0)q^{11}+743008370688q^{12}+567979410486670395120997+O(17^{2}0)q^{13}+1619096042571697255381773+ O(17^{2}0)q^{14}+−1781616049698448562389438+O(17^{2}0)q^{15}+17592186044416q^{16}+−1696064698629590400855063∗ 17+O(17^{2}1)q^{17}+64268410079232q^{18}+509581774222688987146277+O(17^{2}0)q^{19}+345303818571641177870722+ O(17^{2}0)q^{20} +−166144230095597105063563+O(17^{2}0)q^{21} +1655027263796826047527402+O(17^{2}0)q^{22} + −588608903769646438117774 + O(17^{2}1)q^{23} + 1521681143169024q^{24} + 1945834170285324708199579 + O(17^{2}0)q^{25}+851650375495227800943970+O(17^{2}0)q^{26}+5559060566555523q^{27}+−504132637583199257835312+ O(17^{2}0)q^{28} + 672900672064039028850299 + O(17^{2}0)q^{29} + 930133387097469343068674 + O(17^{2}0)q^{30} + −5299696336715355274064151 + O(17^{2}1)q^{31} + 36028797018963968q^{32} + 435410349871202722294112 + O(17^{2}0)q^{33} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

9. Consider f = q+q^{2}+4q^{3}+q^{4}+9q^{5}+4q^{6}+8q^{7}+q^{8}+16q^{9}+9q^{10}+4q^{12}+11q^{13}+8q^{14}+2q^{15}+q^{16}+16q^{18}+ 9q^{20} +15q^{21} +10q^{23} +4q^{24} +q^{25} +11q^{26} +13q^{27} +8q^{28} +7q^{29} +2q^{30} +6q^{31} +q^{32} +⋯ ∈ S_{26}(Γ_{0}(6); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{8} ∼ ρ_{g}, where g is the weight-10 newform g = q + q^{2} + 13q^{3} + q^{4} + 8q^{5} + 13q^{6} + 9q^{7} + q^{8} +16q^{9} +8q^{10} +13q^{12} +11q^{13} +9q^{14} +2q^{15} +q^{16} +15q^{17} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{1} & 0 \\ 0 & u(15)ω^{8} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+−4096q^{2}+531441q^{3}+ 16777216q^{4} + 590425734q^{5} + −2176782336q^{6} + 57857417576q^{7} + −68719476736q^{8} + 282429536481q^{9} + −2418383806464q^{10}+558486249420∗17q^{11}+8916100448256q^{12}+−134968021061458q^{13}+−236983982391296q^{14}+ 313776442502694q^{15} + 281474976710656q^{16} + −148594942164942 ∗ 17q^{17} + −1156831381426176q^{18} + 674632874838868∗17q^{19}+9905700071276544q^{20}+30747803854007016q^{21}+−2287559677624320∗17q^{22}+ 113342630802000600q^{23}+−36520347436056576q^{24}+50579323492485631q^{25}+552829014267731968q^{26}+ 150094635296999121q^{27}+970686391874748416q^{28}+1081348899350530974q^{29}+−1285228308491034624q^{30}+ 4649090467326833408q^{31} + −1152921504606846976q^{32} + 296802490878014220 ∗ 17q^{33} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

10. Consider f = q + 16q^{2} + 13q^{3} + q^{4} + 8q^{5} + 4q^{6} + q^{7} + 16q^{8} + 16q^{9} + 9q^{10} + 4q^{11} + 13q^{12} + 10q^{13} + 16q^{14} + 2q^{15} + q^{16} + q^{18} + 12q^{19} + 8q^{20} + 13q^{21} + 13q^{22} + 12q^{23} + 4q^{24} + q^{25} + 7q^{26} + 4q^{27} + q^{28} + 4q^{29} + 15q^{30} + 11q^{31} + 16q^{32} + q^{33} + ⋯ ∈ S_{26}(Γ_{0}(6); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-8 newform g = q + 8q^{2} + 10q^{3} + 13q^{4} + 5q^{5} + 12q^{6} + 5q^{7} + 2q^{8} + 15q^{9} + 6q^{10} + 5q^{11} + 11q^{12} + 6q^{13} + 6q^{14} + 16q^{15} + 16q^{16} + 7q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{8} & 0 \\ 0 & u(7)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+4096q^{2}+−531441q^{3}+ 16777216q^{4} + −799327650q^{5} + −2176782336q^{6} + 7962409664q^{7} + 68719476736q^{8} + 282429536481q^{9} + −3274046054400q^{10}+−1760169438780q^{11}+−8916100448256q^{12}+164197771122398q^{13}+32614029983744q^{14}+ 424795485643650q^{15}+281474976710656q^{16}+−139852104891678∗17q^{17}+1156831381426176q^{18}+10628455411412156q^{19}+ −13410492638822400q^{20} +−4231550954245824q^{21} +−7209654021242880q^{22} +76636270038478200q^{23} + −36520347436056576q^{24}+340901468177569375q^{25}+672554070517342208q^{26}+−150094635296999121q^{27}+ 133587066813415424q^{28}+2409286328905759014q^{29}+1739962309196390400q^{30}+5300292736548162248q^{31}+ 1152921504606846976q^{32} + 935426206714681980q^{33} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

11. Consider f = q + 15q^{2} + 12q^{3} + 4q^{4} + 11q^{5} + 10q^{6} + 10q^{7} + 9q^{8} + 8q^{9} + 12q^{10} + 7q^{11} + 14q^{12} + 11q^{13} + 14q^{14} + 13q^{15} + 16q^{16} + q^{18} + 7q^{19} + 10q^{20} + q^{21} + 3q^{22} + 13q^{23} + 6q^{24} + 8q^{25} + 12q^{26} + 11q^{27} + 6q^{28} + 3q^{29} +8q^{30} +16q^{31} +2q^{32} +16q^{33} +⋯ ∈ S_{28}(Γ_{0}(6); \overline{F}_{17}). We have ρ_{f} ⊗χ^{6} ∼ ρ_{g}, where g is the weight-8 newform g = q + 8q^{2} + 10q^{3} + 13q^{4} + 5q^{5} + 12q^{6} + 5q^{7} + 2q^{8} + 15q^{9} + 6q^{10} + 5q^{11} + 11q^{12} + 6q^{13} + 6q^{14} + 16q^{15} + 16q^{16} + 7q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(5)ω^{1} & 0 \\ 0 & u(7)ω^{10} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q+8192q^{2}+1594323q^{3}+ 67108864q^{4} + 1220703150q^{5} + 13060694016q^{6} + 96889207016q^{7} + 549755813888q^{8} + 2541865828329q^{9} + 10000000204800q^{10}+34495064342052q^{11}+106993205379072q^{12}+300892562137622q^{13}+793716383875072q^{14}+ 1946195108217450q^{15} + 4503599627370496q^{16} + 670971194843058 ∗ 17q^{17} + 20822964865671168q^{18} + 62694436994411420q^{19} + 81920001677721600q^{20} + 154472691197370168q^{21} + 282583567090089984q^{22} + −894750379460289528q^{23}+876488338465357824q^{24}+−5960464416503905625q^{25}+2464911869031399424q^{26}+ 4052555153018976267q^{27}+6502124616704589824q^{28}+104977877616797619030q^{29}+15943230326517350400q^{30}+ 242604665669598327632q^{31} + 36893488147419103232q^{32} + 54996274467013370796q^{33} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

12. Consider f = q+2q^{2}+5q^{3}+4q^{4}+6q^{5}+10q^{6}+12q^{7}+8q^{8}+8q^{9}+12q^{10}+3q^{12}+7q^{13}+7q^{14}+13q^{15}+16q^{16}+ 16q^{18}+7q^{20}+9q^{21}+8q^{23}+6q^{24}+8q^{25}+14q^{26}+6q^{27}+14q^{28}+q^{29}+9q^{30}+q^{31}+15q^{32}+⋯ ∈ S_{28}(Γ_{0}(6); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-10 newform g = q + q^{2} + 13q^{3} + q^{4} + 8q^{5} + 13q^{6} + 9q^{7} + q^{8} +16q^{9} +8q^{10} +13q^{12} +11q^{13} +9q^{14} +2q^{15} +q^{16} +15q^{17} +· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(8)ω^{10} & 0 \\ 0 & u(15)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −8192q^{2} + −1594323q^{3}+67108864q^{4}+1650577510170207343733772+O(17^{2}0)q^{5}+13060694016q^{6}+−847096058492987981534016+ O(17^{2}0)q^{7}+−549755813888q^{8}+2541865828329q^{9}+166926602135222163066303+O(17^{2}0)q^{10}+−1159747449637474905076897∗ 17+O(17^{2}1)q^{11}+−106993205379072q^{12}+384386165649681344561119+O(17^{2}0)q^{13}+1767900027151248987126165+ O(17^{2}0)q^{14}+−430025452102781323543265+O(17^{2}0)q^{15}+4503599627370496q^{16}+96853179981456838006683∗ 17+O(17^{2}0)q^{17}+−20822964865671168q^{18}+112541610236547192569842∗17+O(17^{2}0)q^{19}+−1880972058155592312216240+ O(17^{2}0)q^{20}+632835727728489204978868+O(17^{2}0)q^{21}+−1521921311830134985002914∗17+O(17^{2}1)q^{22}+ −26708337278709575259043 + O(17^{2}0)q^{23} + 876488338465357824q^{24} + 1555177377609427572235484 + O(17^{2}0)q^{25}+887871149679130216553927+O(17^{2}0)q^{26}+−4052555153018976267q^{27}+−1780520537730805220639317+ O(17^{2}0)q^{28} +1052210413075528770638189+O(17^{2}0)q^{29} +−920125937460774455761187+O(17^{2}0)q^{30} + −1476653711854784259060689+O(17^{2}0)q^{31}+−36893488147419103232q^{32}+−1916843131920834649916017∗ 17 + O(17^{2}1)q^{33} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

13. Consider f = q + 4q^{2} + 15q^{3} + 16q^{4} + 9q^{6} + 8q^{7} + 13q^{8} + 4q^{9} + 2q^{12} + q^{13} + 15q^{14} + q^{16} + 16q^{18} + 11q^{19} + q^{21} + 3q^{23} + 8q^{24} + 14q^{25} + 4q^{26} + 9q^{27} + 9q^{28} + 7q^{31} + 4q^{32} + ⋯ ∈ S_{30}(Γ_{0}(6); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-12 newform g = q + 2q^{2} + 5q^{3} + 4q^{4} + 10q^{6} + 6q^{7} + 8q^{8} + 8q^{9} + 3q^{12} + 4q^{13} + 12q^{14} + 16q^{16} + 12q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(10)ω^{12} & 0 \\ 0 & u(12)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −16384q^{2} + −4782969q^{3}+268435456q^{4}+114798750∗17q^{5}+78364164096q^{6}+−1589770990192q^{7}+−4398046511104q^{8}+ 22876792454961q^{9} + −1880862720000 ∗ 17q^{10} + 17868869804676 ∗ 17q^{11} + −1283918464548864q^{12} + 17487095945024654q^{13} +26046807903305728q^{14} +−549078862488750∗17q^{15} +72057594037927936q^{16} + −3109691254400766∗17q^{17}+−374813367582081024q^{18}+3266460704183826620q^{19}+30816054804480000∗ 17q^{20}+7603825363187640048q^{21}+−292763562879811584∗17q^{22}+44207920862534261784q^{23}+21035720123168587776q^{24}+ −182455855305644140625q^{25}+−286508579963283931136q^{26}+−109418989131512359209q^{27}+−426750900687761047552q^{28}+ −27472403590423422810 ∗ 17q^{29} + 8996108083015680000 ∗ 17q^{30} + −3337262440659652380328q^{31} + −1180591620717411303424q^{32} +−85466250340801363044∗17q^{33} +⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

14. Consider f = q + 4q^{2} + 2q^{3} + 16q^{4} + 15q^{5} + 8q^{6} + 2q^{7} + 13q^{8} + 4q^{9} + 9q^{10} + 10q^{11} + 15q^{12} + 11q^{13} + 8q^{14} + 13q^{15} + q^{16} + 16q^{18} + q^{19} + 2q^{20} + 4q^{21} + 6q^{22} + 14q^{23} + 9q^{24} + q^{25} + 10q^{26} + 8q^{27} + 15q^{28} + 3q^{29} + q^{30} + 6q^{31} + 4q^{32} + 3q^{33} + ⋯ ∈ S_{30}(Γ_{0}(6); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-12 newform g = q + 2q^{2} + 12q^{3} + 4q^{4} + 3q^{5} + 7q^{6} + 10q^{7} + 8q^{8} + 8q^{9} + 6q^{10} + 4q^{11} + 14q^{12} + 10q^{13} + 3q^{14} + 2q^{15} + 16q^{16} + 16q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(16)ω^{12} & 0 \\ 0 & u(16)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −16384q^{2} + 4782969q^{3}+268435456q^{4}+−13482080250q^{5}+−78364164096q^{6}+−517740139336q^{7}+−4398046511104q^{8}+ 22876792454961q^{9}+220890402816000q^{10}+826182077337420q^{11}+1283918464548864q^{12}+−7296887698350562q^{13}+ 8482654442881024q^{14}+−64484371891262250q^{15}+72057594037927936q^{16}+−12560650878715758∗17q^{17}+ −374813367582081024q^{18}+−4289316893885018284q^{19}+−3619068359737344000q^{20}+−2476335036499768584q^{21}+ −13536167155096289280q^{22}+89147061874258260600q^{23}+−21035720123168587776q^{24}+−4498027055655640625q^{25}+ 119552208049775607808q^{26}+109418989131512359209q^{27}+−138979810392162697216q^{28}+1247194296870130527774q^{29}+ 1056511949066440704000q^{30}+3232514869817179952528q^{31}+−1180591620717411303424q^{32}+3951603264260482399980q^{33}+ ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

15. Consider f = q + 13q^{2} + 15q^{3} + 16q^{4} + 11q^{5} + 8q^{6} + 15q^{7} + 4q^{8} + 4q^{9} + 7q^{10} + 7q^{11} + 2q^{12} + 15q^{13} + 8q^{14} + 12q^{15} + q^{16} + q^{18} + 2q^{19} + 6q^{20} + 4q^{21} + 6q^{22} + 14q^{23} + 9q^{24} + 16q^{25} + 8q^{26} + 9q^{27} + 2q^{28} + 6q^{29} + 3q^{30} + 14q^{31} + 13q^{32} + 3q^{33} + ⋯ ∈ S_{30}(Γ_{0}(6); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-12 newform g = q + 15q^{2} + 5q^{3} + 4q^{4} + 9q^{5} + 7q^{6} + 7q^{7} + 9q^{8} + 8q^{9} + 16q^{10} + 13q^{11} + 3q^{12} + 9q^{13} + 3q^{14} + 11q^{15} + 16q^{16} + 13q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(4)ω^{12} & 0 \\ 0 & u(13)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 16384q^{2} + −4782969q^{3}+268435456q^{4}+−1211694870968076841690535+O(17^{2}0)q^{5}+−78364164096q^{6}+−1161042335311548132392045+ O(17^{2}0)q^{7}+4398046511104q^{8}+22876792454961q^{9}+1361655532420797674095445+O(17^{2}0)q^{10}+310195155983654774024289+ O(17^{2}0)q^{11}+−1283918464548864q^{12}+443920158693777469061541+O(17^{2}0)q^{13}+−1914638633765196271772600+ O(17^{2}0)q^{14}+−1503257773999352931290762+O(17^{2}0)q^{15}+72057594037927936q^{16}+−915471582176009394528889∗ 17+O(17^{2}1)q^{17}+374813367582081024q^{18}+−1174332971269897482499459+O(17^{2}0)q^{19}+798052093823516917382991+ O(17^{2}0)q^{20} +1887307322982095261129639+O(17^{2}0)q^{21} +1948177326734164611949726+O(17^{2}0)q^{22} + 6974748390253565571770146+O(17^{2}1)q^{23}+−21035720123168587776q^{24}+495516378706112896707723+ O(17^{2}0)q^{25}+−1786337860304761994578046+O(17^{2}0)q^{26}+−109418989131512359209q^{27}+−1701379103010988826721882+ O(17^{2}0)q^{28} +1318612235886616719890247+O(17^{2}0)q^{29} +−133044921108940514142548+O(17^{2}0)q^{30} + −678879268689695762635811+O(17^{2}1)q^{31}+1180591620717411303424q^{32}+1988439524353587569714211+ O(17^{2}0)q^{33} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

16. Consider f = q+13q^{2}+15q^{3}+16q^{4}+8q^{5}+8q^{6}+10q^{7}+4q^{8}+4q^{9}+2q^{10}+2q^{11}+2q^{12}+5q^{13}+11q^{14}+q^{15}+ q^{16} +q^{18} +13q^{19} +9q^{20} +14q^{21} +9q^{22} +14q^{23} +9q^{24} +10q^{25} +14q^{26} +9q^{27} +7q^{28} +9q^{29} +13q^{30} +14q^{31} + 13q^{32} + 13q^{33} + ⋯ ∈ S_{30}(Γ_{0}(6); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{4} ∼ ρ_{g}, where g is the weight-6 newform g = q+4q^{2}+8q^{3}+16q^{4}+2q^{5}+15q^{6}+6q^{7}+13q^{8}+13q^{9}+8q^{10}+8q^{11}+9q^{12}+5q^{13}+7q^{14}+16q^{15}+q^{16}+11q^{17}+· · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(14)ω^{1} & 0 \\ 0 & u(11)ω^{12} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 16384q^{2} + −4782969q^{3}+268435456q^{4}+1211694870968070283743779+O(17^{2}0)q^{5}+−78364164096q^{6}+1161042335310439769245197+ O(17^{2}0)q^{7} + 4398046511104q^{8} + 22876792454961q^{9} + −1361655532528243073745749 + O(17^{2}0)q^{10} + −310195155648315375886809+O(17^{2}0)q^{11} +−1283918464548864q^{12} +−443920176288884383507337+ O(17^{2}0)q^{13} +1914638615605774473814968+O(17^{2}0)q^{14} +1503257805365808968889326+O(17^{2}0)q^{15} + 72057594037927936q^{16} + 915471675663649626525565 ∗ 17 + O(17^{2}1)q^{17} + 374813367582081024q^{18} + 1174339774291094902813691+O(17^{2}0)q^{19}+−798053854208944787963727+O(17^{2}0)q^{20}+−1887302021715523144697927+ O(17^{2}0)q^{21}+−1948171832533465527477406+O(17^{2}0)q^{22}+−6974760900115750157468146+O(17^{2}1)q^{23}+ −21035720123168587776q^{24}+−495282015559817604635437+O(17^{2}0)q^{25}+1786049582073075714656382+ O(17^{2}0)q^{26}+−109418989131512359209q^{27}+1701081579044251088879194+O(17^{2}0)q^{28}+−1317505881764824713569979+ O(17^{2}0)q^{29} + 133558829124660529015124 + O(17^{2}0)q^{30} + 686649756493180651304787 + O(17^{2}1)q^{31} + 1180591620717411303424q^{32} + −1990043442299357794292331 + O(17^{2}0)q^{33} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

17. Consider f = q + 8q^{2} + 11q^{3} + 13q^{4} + 5q^{5} + 3q^{6} + 8q^{7} + 2q^{8} + 2q^{9} + 6q^{10} + 6q^{11} + 7q^{12} + 13q^{13} + 13q^{14} + 4q^{15} + 16q^{16} + 16q^{18} + 5q^{19} + 14q^{20} + 3q^{21} + 14q^{22} + 16q^{23} + 5q^{24} + q^{25} + 2q^{26} + 5q^{27} + 2q^{28} + 8q^{29} + 15q^{30} + 11q^{31} + 9q^{32} + 15q^{33} + ⋯ ∈ S_{32}(Γ_{0}(6); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{2} ∼ ρ_{g}, where g is the weight-4 newform g = q + 15q^{2} + 14q^{3} + 4q^{4} + 6q^{5} + 6q^{6} + q^{7} + 9q^{8} + 9q^{9} + 5q^{10} + 12q^{11} + 5q^{12} + 4q^{13} + 15q^{14} + 16q^{15} + 16q^{16} + 10q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(12)ω^{1} & 0 \\ 0 & u(10)ω^{14} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −32768q^{2} + −14348907q^{3}+1073741824q^{4}+21436113030q^{5}+470184984576q^{6}+14604456549536q^{7}+−35184372088832q^{8}+ 205891132094649q^{9}+−702418551767040q^{10}+10293018436866252q^{11}+−15407021574586368q^{12}+−73352418613693258q^{13}+ −478558832215195648q^{14}+−307584792308958210q^{15}+1152921504606846976q^{16}+−858779793185279742∗ 17q^{17}+−6746640616477458432q^{18}+47146370538017638100q^{19}+23016851104302366720q^{20}+−209557988814832957152q^{21}+ −337281628139233345536q^{22}+−727167447999538420728q^{23}+504857282956046106624q^{24}+−4197105931242456797225q^{25}+ 2403612053133500678144q^{26}+−2954312706550833698643q^{27}+15681415814027530993664q^{28}+31343797618972430901150q^{29}+ 10078938474379942625280q^{30}+150548795843298856679192q^{31}+−37778931862957161709568q^{32}+−147693564299879221386564q^{33}+ ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

18. Consider f = q +9q^{2} +6q^{3} +13q^{4} +12q^{5} +3q^{6} +16q^{7} +15q^{8} +2q^{9} +6q^{10} +12q^{11} +10q^{12} +3q^{13} +8q^{14} + 4q^{15} + 16q^{16} + q^{18} + 4q^{19} + 3q^{20} + 11q^{21} + 6q^{22} + 14q^{23} + 5q^{24} + q^{25} + 10q^{26} + 12q^{27} + 4q^{28} + 12q^{29} + 2q^{30} + 16q^{31} + 8q^{32} + 4q^{33} + ⋯ ∈ S_{32}(Γ_{0}(6); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-14 newform g = q + 13q^{2} + 2q^{3} + 16q^{4} + 16q^{5} + 9q^{6} + 12q^{7} + 4q^{8} + 4q^{9} + 4q^{10} + 15q^{11} + 15q^{12} + 12q^{13} + 3q^{14} + 15q^{15} + q^{16} + 11q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(14)ω^{14} & 0 \\ 0 & u(11)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 32768q^{2} + 14348907q^{3}+1073741824q^{4}+−1933717300441037070415108+O(17^{2}0)q^{5}+470184984576q^{6}+−1197616116772386572218944+ O(17^{2}0)q^{7} + 35184372088832q^{8} + 205891132094649q^{9} + 1383360190400473401102247 + O(17^{2}0)q^{10} + −1804812742979906746820347+O(17^{2}0)q^{11} +15407021574586368q^{12} +−374340697935710735614301+ O(17^{2}0)q^{13} + 733548191397077839502264 + O(17^{2}0)q^{14} + 1637590843319985626028099 + O(17^{2}0)q^{15} + 1152921504606846976q^{16} + 61363331964836925923143 ∗ 17 + O(17^{2}0)q^{17} + 6746640616477458432q^{18} + 917574280980318906922411+O(17^{2}0)q^{19}+1573840702336064973373743+O(17^{2}0)q^{20}+−1506127089411859424107584+ O(17^{2}0)q^{21}+−1472763836756506343434345+O(17^{2}0)q^{22}+−1089348882373602522132000+O(17^{2}0)q^{23}+ 504857282956046106624q^{24}+509991166437884589194324+O(17^{2}0)q^{25}+−545604694995512001383350+ O(17^{2}0)q^{26}+2954312706550833698643q^{27}+1042596785702747327118438+O(17^{2}0)q^{28}+−606260581905175178348458+ O(17^{2}0)q^{29} + 529491941388980420410029 + O(17^{2}0)q^{30} + −807144035331431673414147 + O(17^{2}0)q^{31} + 37778931862957161709568q^{32} + 1302848634898036743886024 + O(17^{2}0)q^{33} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

19. Consider f = q+q^{2}+16q^{3}+q^{4}+q^{5}+16q^{6}+9q^{7}+q^{8}+q^{9}+q^{10}+8q^{11}+16q^{12}+8q^{13}+9q^{14}+16q^{15}+q^{16}+q^{18}+ 10q^{19}+q^{20}+8q^{21}+8q^{22}+8q^{23}+16q^{24}+13q^{25}+8q^{26}+16q^{27}+9q^{28}+7q^{29}+16q^{30}+2q^{31}+q^{32}+9q^{33}+⋯ ∈ S_{34}(Γ_{0}(6); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-16 newform g = q + 9q^{2} + 11q^{3} + 13q^{4} + 7q^{5} + 14q^{6} + 11q^{7} + 15q^{8} + 2q^{9} + 12q^{10} + 10q^{11} + 7q^{12} + 15q^{13} + 14q^{14} + 9q^{15} + 16q^{16} + 5q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(7)ω^{0} & 0 \\ 0 & u(5)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + 65536q^{2} + −43046721q^{3}+4294967296q^{4}+397843662750q^{5}+−2821109907456q^{6}+−19838876991136q^{7}+281474976710656q^{8}+ 1853020188851841q^{9} + 26073082281984000q^{10} + 151075290395738820q^{11} + −184884258895036416q^{12} + −1189064824676669122q^{13}+−1300160642491088896q^{14}+−17125865152017342750q^{15}+18446744073709551616q^{16}+ −3178183189090953438∗17q^{17}+121439531096594251776q^{18}+−901406666568223456324q^{19}+1708725520432103424000q^{20}+ 853998602790750865056q^{21}+9900870231375139307520q^{22}+50700139861678042543800q^{23}+−12116574790945106558976q^{24}+ 41864258163400923109375q^{25}+−77926552350010187579392q^{26}+−79766443076872509863361q^{27}+−85207327866296001888256q^{28}+ 1888095603534789166837734q^{29}+−1122360698602608574464000q^{30}+−1019461206982537879015192q^{31}+ 1208925819614629174706176q^{32} + −6503295875659348573409220q^{33} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1.

20. Consider f = q + 16q^{2} + q^{3} + q^{4} + 16q^{5} + 16q^{6} + 10q^{7} + 16q^{8} + q^{9} + q^{10} + 4q^{11} + q^{12} + 13q^{13} + 7q^{14} + 16q^{15} + q^{16} + 16q^{18} + q^{19} + 16q^{20} + 10q^{21} + 13q^{22} + 6q^{23} + 16q^{24} + 13q^{25} + 4q^{26} + q^{27} + 10q^{28} + 5q^{29} + q^{30} + 2q^{31} + 16q^{32} + 4q^{33} + ⋯ ∈ S_{34}(Γ_{0}(6); \overline{F}_{17}). We have ρ_{f} ⊗ χ^{15} ∼ ρ_{g}, where g is the weight-16 newform g = q + 8q^{2} + 6q^{3} + 13q^{4} + 10q^{5} + 14q^{6} + 16q^{7} + 2q^{8} + 2q^{9} + 12q^{10} + 5q^{11} + 10q^{12} + q^{13} + 9q^{14} + 9q^{15} + 16q^{16} + 4q^{17} + · · · . We deduce that the semisimplification of ρ_{f}|D_{ℓ} is

(\begin{matrix}u(13)ω^{0} & 0 \\ 0 & u(4)ω^{1} \end{matrix}) ,

where u is unramified and ω is the mod-ℓ cyclotomic character. The form f_{0} = q + −65536q^{2} + 43046721q^{3}+4294967296q^{4}+471441446100200521953906+O(17^{2}0)q^{5}+−2821109907456q^{6}+840386396165860459113249+ O(17^{2}0)q^{7} +−281474976710656q^{8} +1853020188851841q^{9} +−99458287895091474212814+O(17^{2}0)q^{10} + −1918789322004931472510528+O(17^{2}0)q^{11}+184884258895036416q^{12}+−518074548696731579908880+ O(17^{2}0)q^{13} +−1163067644575797381791313+O(17^{2}0)q^{14} +430670413085045804136906+O(17^{2}0)q^{15} + 18446744073709551616q^{16}+−7415225338120684655635∗17+O(17^{2}0)q^{17}+−121439531096594251776q^{18}+ −1646716533356262071421384+O(17^{2}0)q^{19}+−928820769991471921189700+O(17^{2}0)q^{20}+536583769278727435344691+ O(17^{2}0)q^{21}+−1606946167352433177973533+O(17^{2}0)q^{22}+1745939902653953954760502+O(17^{2}0)q^{23}+ −12116574790945106558976q^{24}+1448451040097479295266695+O(17^{2}0)q^{25}+−55547744820031234615074+ O(17^{2}0)q^{26}+79766443076872509863361q^{27}+−1858876755765444566537987+O(17^{2}0)q^{28}+−1496737344732823323427498+ O(17^{2}0)q^{29} + 1670927225829348162847329 + O(17^{2}0)q^{30} + 873737148831321065047161 + O(17^{2}0)q^{31} + −1208925819614629174706176q^{32} + 1679437722972831556029165 + O(17^{2}0)q^{33} + ⋯ is an ℓ-adic newform that lifts f. The slope of f_{0} is 1. —————————————————————————————————- (Uninteresting cases below.)

21. Consider f = q + 15q2 + 14q3 + 4q4 + 6q5 + 6q6 + q7 + 9q8 + 9q9 + 5q10 + 12q11 + 5q12 + 4q13 + 15q14 + 16q15 + 16q16 + 10q17 + 16q18 + 3q19 + 7q20 + 14q21 + 10q22 + 15q23 + 7q24 + 13q25 + 9q26 + 7q27 + 4q28 + 13q29 + 2q30 + 14q31 + 2q32 + 15q33 + ⋯ ∈ S20(Γ0(6); \overline{F}17). Since f is ordinary, we do not continue our analysis.

22. Consider f = q + 4q2 + 8q3 + 16q4 + 2q5 + 15q6 + 6q7 + 13q8 + 13q9 + 8q10 + 8q11 + 9q12 + 5q13 + 7q14 + 16q15 + q16 + 11q17 + q18 + 4q19 + 15q20 + 14q21 + 15q22 + 5q23 + 2q24 + 7q25 + 3q26 + 2q27 + 11q28 + 15q29 + 13q30 + 12q31 + 4q32 + 13q33 + ⋯ ∈ S22(Γ0(6); \overline{F}17). Since f is ordinary, we do not continue our analysis.

23. Consider f = q + 8q2 + 10q3 + 13q4 + 5q5 + 12q6 + 5q7 + 2q8 + 15q9 + 6q10 + 5q11 + 11q12 + 6q13 + 6q14 + 16q15 + 16q16 + 7q17 + q18 + 6q19 + 14q20 + 16q21 + 6q22 + 2q23 + 3q24 + 15q25 + 14q26 + 14q27 + 14q28 + 6q29 + 9q30 + 2q31 + 9q32 + 16q33 + ⋯ ∈ S24(Γ0(6); \overline{F}17). Since f is ordinary, we do not continue our analysis.

24. Consider f = q + q2 + 13q3 + q4 + 8q5 + 13q6 + 9q7 + q8 + 16q9 + 8q10 + 13q12 + 11q13 + 9q14 + 2q15 + q16 + 15q17 + 16q18 + 8q20 + 15q21 + 7q23 + 13q24 + q25 + 11q26 + 4q27 + 9q28 + 10q29 + 2q30 + 11q31 + q32 + ⋯ ∈ S26(Γ0(6); \overline{F}17). Since f is ordinary, we do not continue our analysis.

25. Consider f = q + 2q2 + 12q3 + 4q4 + 3q5 + 7q6 + 10q7 + 8q8 + 8q9 + 6q10 + 4q11 + 14q12 + 10q13 + 3q14 + 2q15 + 16q16 + 16q17 + 16q18 + 9q19 + 12q20 + q21 + 8q22 + 8q23 + 11q24 + 15q25 + 3q26 + 11q27 + 6q28 + 13q29 + 4q30 + 15q31 + 15q32 + 14q33 + ⋯ ∈ S28(Γ0(6); \overline{F}17). Since f is ordinary, we do not continue our analysis.

26. Consider f = q + 15q2 + 5q3 + 4q4 + 9q5 + 7q6 + 7q7 + 9q8 + 8q9 + 16q10 + 13q11 + 3q12 + 9q13 + 3q14 + 11q15 + 16q16 + 13q17 + q18 + q19 + 2q20 + q21 + 8q22 + 8q23 + 11q24 + 2q25 + 16q26 + 6q27 + 11q28 + 9q29 + 12q30 + q31 + 2q32 + 14q33 + ⋯ ∈ S28(Γ0(6); \overline{F}17). Since f is ordinary, we do not continue our analysis.

27. Consider f = q + 2q2 + 5q3 + 4q4 + 10q6 + 6q7 + 8q8 + 8q9 + 3q12 + 4q13 + 12q14 + 16q16 + 12q17 + 16q18 + 14q19 + 13q21 + 9q23 + 6q24 + 6q25 + 8q26 + 6q27 + 7q28 + 9q31 + 15q32 + ⋯ ∈ S28(Γ0(6); \overline{F}17). Since f is ordinary, we do not continue our analysis.

28. Consider f = q + 13q2 + 2q3 + 16q4 + 16q5 + 9q6 + 12q7 + 4q8 + 4q9 + 4q10 + 15q11 + 15q12 + 12q13 + 3q14 + 15q15 + q16 + 11q17 + q18 + 2q19 + q20 + 7q21 + 8q22 + 8q23 + 8q24 + 15q25 + 3q26 + 8q27 + 5q28 + q29 + 8q30 + 6q31 + 13q32 + 13q33 + ⋯ ∈ S30(Γ0(6); \overline{F}17). Since f is ordinary, we do not continue our analysis.

29. Consider f = q + 8q2 + 6q3 + 13q4 + 10q5 + 14q6 + 16q7 + 2q8 + 2q9 + 12q10 + 5q11 + 10q12 + q13 + 9q14 + 9q15 + 16q16 + 4q17 + 16q18 + 9q19 + 11q20 + 11q21 + 6q22 + q23 + 12q24 + 8q25 + 8q26 + 12q27 + 4q28 + 16q29 + 4q30 + 5q31 + 9q32 + 13q33 + ⋯ ∈ S32(Γ0(6); \overline{F}17). Since f is ordinary, we do not continue our analysis.

30. Consider f = q + 9q2 + 11q3 + 13q4 + 7q5 + 14q6 + 11q7 + 15q8 + 2q9 + 12q10 + 10q11 + 7q12 + 15q13 + 14q14 + 9q15 + 16q16 + 5q17 + q18 + 5q19 + 6q20 + 2q21 + 5q22 + 7q23 + 12q24 + 8q25 + 16q26 + 5q27 + 7q28 + 2q29 + 13q30 + 5q31 + 8q32 + 8q33 + ⋯ ∈ S32(Γ0(6); \overline{F}17). Since f is ordinary, we do not continue our analysis.

31. Consider f = q + 9q2 + 6q3 + 13q4 + 8q5 + 3q6 + 6q7 + 15q8 + 2q9 + 4q10 + 15q11 + 10q12 + 5q13 + 3q14 + 14q15 + 16q16 + q17 + q18 + 10q19 + 2q20 + 2q21 + 16q22 + 4q23 + 5q24 + 6q25 + 11q26 + 12q27 + 10q28 + 11q29 + 7q30 + 12q31 + 8q32 + 5q33 + ⋯ ∈ S32(Γ0(6); \overline{F}17). Since f is ordinary, we do not continue our analysis.

32. Consider f = q + q2 + q3 + q4 + 15q5 + q6 + q8 + q9 + 15q10 + 13q11 + q12 + 15q13 + 15q15 + q16 + q17 + q18 + 4q19 + 15q20 + 13q22 + q24 + 16q25 + 15q26 + q27 + 7q29 + 15q30 + 8q31 + q32 + 13q33 + ⋯ ∈ S34(Γ0(6); \overline{F}17). Since f is ordinary, we do not continue our analysis.

33. Consider f = q + 16q2 + 16q3 + q4 + 13q5 + q6 + 15q7 + 16q8 + q9 + 4q10 + 16q12 + 11q13 + 2q14 + 4q15 + q16 + 16q17 + 16q18 + 4q19 + 13q20 + 2q21 + 6q23 + q24 + 11q25 + 6q26 + 16q27 + 15q28 + 13q29 + 13q30 + 11q31 + 16q32 + ⋯ ∈ S34(Γ0(6); \overline{F}17). Since f is ordinary, we do not continue our analysis.

34. Consider f = q + 16q2 + q3 + q4 + 16q6 + 2q7 + 16q8 + q9 + q12 + 2q13 + 15q14 + q16 + 16q17 + 16q18 + 13q19 + 2q21 + 11q23 + 16q24 + 12q25 + 15q26 + q27 + 2q28 + 7q31 + 16q32 + ⋯ ∈ S34(Γ0(6); \overline{F}17). Since f is ordinary, we do not continue our analysis.

18 Source code

/*

***************************************************************

breuil.m --- 03-24-01

Savitt and Stein

AIM: Analyze every form of level N <= 20 and weight ell <= 7.

Let ell be a prime number.
Definition: "low weight" means weight between 2 and ell+1, inclusive,
and "medium weight" means weight between ell+2 and 2*ell, inclusive.

****************************************************************/

declare verbose Breuil, 2;    // since mine isn't in the kernel yet.

function AllFormsOfWeight(N, k, ell)
// The newforms of weight k on Gamma_1(N).
local ans, S, ND, f_mod_ell, N_mod_ell, n;

assert Type(N) eq RngIntElt;
assert Type(k) eq RngIntElt;
assert Type(ell) eq RngIntElt;

ans := [];
S := CuspForms(N,k);
ND := NewformDecomposition(S);
for f in ND do
   f_mod_ell := BaseExtend(f,GF(ell));
   N_mod_ell := Newforms(f_mod_ell);
   n := LCM([Degree(fmod) : fmod in N_mod_ell]);
   F<z> := GF(ell^n);
   Append(~ans, BaseExtend(f,F));
end for;
return ans;
end function;
intrinsic FormsOfLowAndMediumWeight(N::RngIntElt, ell:RngIntElt) -> .
{The newforms on Gamma_1(N) of weight between 2 and 2*ell.}
requirege N, 1 ;
the_ans := [* [] *];
for k in [2..2*ell] do
   vprint Breuil: "Computing forms of weight ", k;
   Append(~the_ans, AllFormsOfWeight(N,k,ell));
end for;
return the_ans;
end intrinsic;

/* Reduce a polynomial defined over Zell modulo ell. */
function Reduce_modulo_ell(poly)
local ell, R;
ell  := Prime(BaseRing(Parent(poly)));
R<x> := PolynomialRing(GF(ell));
return &+[(Integers()!Coefficient(poly,i))*x^i : i in [0..Degree(poly)]];
end function;

function QellLift(f,prec)
local ell, f0, f0ell, possible, qexp_f, R, a, c, b;
assert Type(f) eq ModFrmElt;
assert Type(prec) eq RngIntElt;
ell := Characteristic(BaseRing(Parent(f)));
// f0 is the q-expansion of a newform defined over Q (i.e., a Galois conjugacy class

// over Qbar!) that has a reduction to f.
f0  := Lift(f,prec);
// f0ell is the collection of newforms defined over Qell (conjugacy classes over Qellbar)
// that arise from f0.
f0ell := QpBarNewforms(f0, ell, 1);

if #f0ell eq 1 then
   return f0ell[1];
end if;
// Determine which of the f0ell's reduces to f.
possible := [true : i in [1..#f0ell]];
qexp_f := qExpansion(f,prec);
R<x> := PolynomialRing(pAdicField(ell));
for n in [2..prec-1] do
   a := MinimalPolynomial(Coefficient(qexp_f,n));
   for i in [j : j in [1..#f0ell] | possible[j]] do
c := Coefficient(f0ell[i],n);
if Type(c) eq FldLocElt then
 b := x - c;
else
 b := MinimalPolynomial(c);
end if;
b := Reduce_modulo_ell(b);
if (b mod a) ne 0 then
 possible[i] := false;
end if;
   end for;
   r := [i : i in [1..#f0ell] | possible[i]];
   if #r eq 1 then
return f0ell[r[1]];
   end if;
end for;
r := [i : i in [1..#f0ell] | possible[i]];
if #r ge 1 then
   if #r gt 1 then
error "WARNING -- Lift not uniquely determined to given precision!";
   end if;
   return f0ell[r[1]];   // there is a lift, but it is not uniquely determined, perhaps because of precision!
end if;
error "Couldn't find a lift, so terminating, because this indicates there is a bug somewhere.";
end function;

function FindInLowWeight(forms, f, prec)
// Find a form g of low weight such that f tensor chi^i = g, away from Fourier coefficients
// that involves ell or the level, for some integer i.

ell := Characteristic(BaseRing(Parent(f)));
N := Level(f);
qexp_f := qExpansion(f,prec);
P := [p : p in [2..prec-1] |IsPrime(p) and (N*ell) mod p ne 0];
if ell gt 2 then
   chi := DirichletGroup(ell, GF(ell)).1;
else
   chi := DirichletGroup(ell, GF(ell))!1;
end if;
ap_chi := [[Coefficient(qexp_f,p)*Evaluate(chi^i,p) : p in P] : i in [1..ell-1]];
for k in [2..ell+1] do
   F := forms[k];
   for new_space in F do
NF := Newforms(new_space);
for g in NF do
 qexp_g := qExpansion(g,prec);
 ap_g := [Coefficient(qexp_g,p) : p in P];
 // Now compare ap_g with the aplists for each twist of f.
 for i in [1..ell-1] do
    if ap_g eq ap_chi[i] then
// return g and i where f tensor chi^i = g.
return g, i, k;
    end if;
 end for;
end for;
   end for;
end for;
return "eisenstein", 0, 0;
end function;

function ComputeLocalRepresentation(forms, f, prec)
////////////////////////////////////////////////////////////////////////////////////////////////
// The output is as follows, encoded in a list:
// * <ell, k, and N>
// * the q-expansion of f up to q^(prec) (nothing further, if f is ordinary),
// * Let ftilde be a lift of f to an ell-adic modular form:
//   > ell-adic approximation to a_ell(ftilde) where ftilde is *a* lift of f,
//   > the slope: ord_ell(a_ell(ftilde))
// * data that describes the local representation:
//   red:  [<unr(a), k_1>, <unr(a^(-1)), k_2>],
//   irreducible: k_3, where the rep is Induction of omega_2^(k_3).
////////////////////////////////////////////////////////////////////////////////////////////////

ell := Characteristic(BaseRing(Parent(f)));
vprint Breuil : "Computing local representation at", ell, " of ", qExpansion(f,2*ell+1);
N := Level(f);
k := Weight(f);

ans := [* <"'ell,k,N,f'", ell, k, N, qExpansion(f,prec)> *];
if Coefficient(qExpansion(f,ell+1),ell) ne 0 then
   Append(~ans, "ordinary");
   return ans;
end if;
// Find a twist of f having low weight.
g, i, wt_g := FindInLowWeight(forms, f,prec);
if Type(g) eq MonStgElt and g eq "eisenstein" then
   Append(~ans, "eisenstein");
   return ans;
end if;
qexp_g := qExpansion(g, ell+1);
Append(~ans, <"'wt(g), i, g'", wt_g, i, qexp_g>);

// Compute lifts and their valuations
f0 := QellLift(f, prec);
a_ell := Coefficient(f0,ell);
if Type(a_ell) eq FldLocElt then
   v_ell := Valuation(a_ell)/1;
else
   minpoly := MinimalPolynomial(a_ell);
   v_ell := Valuation(Coefficient(minpoly,0))/Degree(minpoly);
end if;
num := 0; // don't bother with it for now!
Append(~ans, <"'v(a_ell(f0)), num, f0'", v_ell, num, f0>);
////////////////////////////////////////////////////////////////////////////////////////////////
// Let rho be the semisimplification of rho_g|D_ell.
//    If g is supersingular, then rho is induced from omega_2^(wt(g)-1).
//    If g is ordinary, then rho = unr(a_ell(g)^(-1))*omega^(wt(g)-1) x unr(a_ell(g))
//
// Let delta be the semisimplication of rho_f|D_ell. Then
// delta = rho \tensor chi^(-i) is as follows:
//    In the supersingular case for g, delta is induced from omega_2^(wt(g)-1-(ell+1)*i).
//    In the ordinary case for g, delta is
//         unr(a_ell(g)^(-1))*omega^(wt(g)-1-i)  x  unr(a_ell(g))*omega^(-i).
////////////////////////////////////////////////////////////////////////////////////////////////
a_ell_of_g := Coefficient(qexp_g,ell);
g_is_supersingular := a_ell_of_g eq 0;
wt_g := Weight(g);
if g_is_supersingular then
   Append(~ans, <"rho (irreducible)", IntegerRing(ell^2-1)!(wt_g - 1 - (ell+1)*i)>);
else
   Append(~ans, <"rho (split)", [<a_ell_of_g^(-1), IntegerRing(ell-1)!(wt_g-1-i)>, <a_ell_of_g, IntegerRing(ell-1)!(-i)>]>);
end if;

return ans;
end function;

intrinsic ComputeSupersingularRepresentations(N::RngIntElt, ell::RngIntElt, prec::RngIntElt) -> .
{}
forms := FormsOfLowAndMediumWeight(N,ell);
return ComputeSupersingularRepresentations(forms, prec);
end intrinsic;
intrinsic ComputeSupersingularRepresentations(forms::List, prec::RngIntElt) -> .
{forms is the output of FormsOfLowAndMediumWeight.  For each newform f of medium weight, i.e., weight greater than
ell+1 and le 2*ell, determine the semisimplification of rho_f and lifts of f to characteristic 0 eigenforms.}
ell := #forms div 2;
require IsPrime(ell) and ell*2 eq #forms : "There is something wrong with argument 1.";
requirege prec, ell+1;
ans := [];
for k in [(ell+1)+1..2*ell] do
   F := forms[k];
   for new_space in F do
NF := Newforms(new_space);
for f in NF do
 Append(~ans,ComputeLocalRepresentation(forms,f,prec));
end for;
   end for;
end for;
return ans;
end intrinsic;

function TexIfy(f)
assert Type(f) eq RngSerPowElt;
ans := "q + ";
if Type(Coefficient(f,0)) eq RngUPolResElt then
  R<a> := Parent(Coefficient(f,0));
end if;
for n in [2..AbsolutePrecision(f)-1] do
   if Coefficient(f,n) ne 0 then
c := Coefficient(f,n);
if c eq 1 then
 c := "";
end if;
ans *:= Sprintf("%oq^{%o} + ",c, n);
   end if;
end for;
return ans cat " \\cdots ";
end function;

function TexIfyPoly(f)
assert Type(f) eq RngUPolElt;
ans := Sprintf("x^{%o}", Degree(f));
for n in [1..Degree(f)] do
   i := Degree(f)-n;
   if Coefficient(f,i) ne 0 then
c := Coefficient(f,i);
if c eq 1 and i gt 0 then
     c := "";
end if;
if i ge 2 then
 ans *:= Sprintf(" + %ox^{%o}", c, i);
elif i eq 1 then
 ans *:= Sprintf(" + %ox", c);
elif i eq 0 then
 ans *:= Sprintf(" + %o", c);
end if;
   end if;
end for;
return ans;
end function;
procedure LaTeX_ssrep(x)
assert Type(x) eq List;
ell := x[1][2];
k   := x[1][3];
N   := x[1][4];
f   := x[1][5];
R<q> := Parent(f);
 printf "\\\\item  Consider \n     $f = %o \\\\in S_{%o}(\\\\Gamma_0(%o);\\\\overline{\\\\mathbf{F}}_{%o}).$\n", TexIfy(f+O(q^(2*ell))), k, N, ell;
if Type(Coefficient(f,0)) ne FldFinElt then
   printf "Here the coefficients of $f$ are written in terms of a root~$a$ of $%o$.\n", TexIfyPoly(Modulus(BaseRing(Parent(f))));
   K := BaseRing(Parent(Coefficient(f,0)));
   if not IsPrime(#K) then
printf "Also,~$z$ satisfies the equation $%o=0$.\n", TexIfyPoly(DefiningPolynomial(K));
   end if;
else
   K := Parent(Coefficient(f,0));
   if not IsPrime(#K) then
printf "Also,~$z$ satisfies the equation $%o=0$.\n", TexIfyPoly(DefiningPolynomial(K));
   end if;
end if;

if x[2] cmpeq "ordinary" then
   print "Since~$f$ is ordinary, we do not continue our analysis.";
   return;
end if;
if x[2] cmpeq "eisenstein" then
   print "I didn't find a twist of $\\rho_f$ in low weight, so $\\rho_f$ must be reducible and ";

   print "we do not consider it further.";
   return;
end if;
wt_g := x[2][2];
i := x[2][3];
g := x[2][4];
R<q> := Parent(g);
printf "We have $\\\\rho_f \\\\otimes \\\\chi^{%o} \\\\sim \\\\rho_g$, where $g$ is the\n\
weight-$%o$ newform\n         $ g = %o.$\n", i, wt_g, TexIfy(g+O(q^(2*ell)));

printf "We deduce that the semisimplification of $\\\\rho_f|D_\\\\ell$";
if x[4][1] eq "rho (split)" then
   printf " is  \n   \
   $$\\\\left(\\\\begin{matrix}u(%o)\\\\omega^{%o}&0\\\\\\\\0&u(%o)\\\\omega^{%o}\\\\\end{matrix}\\\\right),$$\n",
   x[4][2][1][1], x[4][2][1][2], x[4][2][2][1], x[4][2][2][2];
   printf "where $u$ is unramified and $\\\\omega$ is the mod-$\\\\ell$ cyclotomic character. ";
else
   printf " is irreducible; it is the induction of $\\\\omega_2^{%o}$. ", x[4][2];
end if;
f0 := x[3][4];
R<q> := Parent(f0);
printf "The form \n     $f_0 = %o$\n is an $\\\\ell$-adic newform that lifts~$f$.\n", TexIfy(f0+O(q^(2*ell)));
if Type(Coefficient(f0,0)) ne FldLocElt then
   printf "Here the coefficients of $f_0$ are written in terms of a root~$a$ of $%o$.\n", TexIfyPoly(Modulus(BaseRing(Parent(f0))));
end if;
printf "The slope of $f_0$ is $%o$.\n", x[3][2];
/*   if x[3][3] ne 0 then
   printf "Moreover, the inverse of the unit part of $a_\\\\ell$ is \n   $$\\\\ell^{\\\\nu_\\\\ell}/a_\\\\ell = %o.$$\n", x[3][3];
end if;
*/
end procedure;

intrinsic TypesetSupersingularRepComp(ssrepcomp::SeqEnum, N::RngIntElt, ell::RngIntElt)
{Print out a LaTeX version the output of ComputeSupersingularRepresentations.}
if #ssrepcomp eq 0 then
   cnt := "no forms";
elif #ssrepcomp eq 1 then
   cnt := "1 form";
else
   cnt := Sprintf("%o forms", #ssrepcomp);
end if;
printf "\n\n\\\\subsection{$\\\\mathbf{N = %o, \\\\quad \\\\ell = %o}$ \\\\quad (%o)}\n\n", N, ell, cnt;
if #ssrepcomp eq 0 then
   print "There are no forms to analyze.";
else
   print "\\begin{enumerate}";

// divide into interesting and boaring batches.
interesting := [x : x in ssrepcomp | #x eq 4];
uninteresting := [x : x in ssrepcomp | #x ne 4];
for x in interesting do
LaTeX_ssrep(x);
print "\n\n";
end for;
if #interesting eq 0 then
print "\\item[-]";
end if;
print " ----------------------------------------------------------------------------------------------------\\\\";
print "(Uninteresting cases below.)";
print "\\begin{tiny}";
for x in uninteresting do
LaTeX_ssrep(x);
print "\n\n";
end for;
print "\\end{tiny}";
print "\\end{enumerate}";
end if;
end intrinsic;
function idxG0(n)   // index of Gamma_0(N) in SL_2(Z).
return
&*[Integers()| t[1]^t[2] + t[1]^(t[2]-1) : t in Factorization(n)];
end function;
intrinsic Go(Nstart::RngIntElt, Nstop::RngIntElt, ell::RngIntElt)
{}
printf "\n\n\\\\setcounter{section}{%o}\n\\\\section{$\\\\mathbf{%o \\\\leq N \\\\leq %o,  \\\\quad \\\\ell = %o}$}\n\n",
        ell-1, Nstart, Nstop, ell;
for N in [m : m in [Nstart..Nstop] | m mod ell ne 0] do
prec := idxG0(N)*Ceiling(ell/6)+10;  // this *should* be enough, but I haven't proved it.
data := ComputeSupersingularRepresentations(N,ell,prec);
TypesetSupersingularRepComp(data,N,ell);
end for;
end intrinsic;
/*

*/