On M_{k}(Γ_{0}(2)), I found exactly 81 triples (j, n, α) in the box where

0 ≤ j ≤ 14, 1 ≤ n ≤ 6, 3 ≤ α ≤ 200

such that there are d > 0 eigenforms of weight k and slope α in M_{k}(Γ_{0}(2); Z_{2}) for all k in my table, i.e., for k ≤ 5000, with k ≡ −j mod 2^{n}. These triples all satify the following conjecture:

Suppose j is a nonnegative integer and there are d forms of weight k and slope α in M_{k}(Γ_{0}(2); Z_{2}) for all k in your table, i.e., k ≤ 5000, congruent to −j mod 2^{n}. Then for all k in your table congruent to j + 2 mod 2^{n+⌈log_{2}(j+2)⌉} there are d forms of slope α + j + 1.

\begin{matrix}(j, n, α) & d \\ (0, 2, 3) & 1 \\ (0, 3, 3) & 1 \\ (0, 3, 7) & 1 \\ (0, 4, 3) & 1 \\ (0, 4, 7) & 1 \\ (0, 5, 3) & 1 \\ (0, 5, 7) & 1 \\ (0, 6, 3) & 1 \\ (0, 6, 7) & 1 \\ (0, 6, 13) & 1 \\ (0, 6, 15) & 1 \\ (0, 6, 17) & 1 \\ (2, 4, 6) & 2 \\ (2, 5, 6) & 2 \\ (2, 5, 14) & 2 \\ (2, 6, 6) & 2 \\ (2, 6, 14) & 2 \\ (4, 2, 3) & 1 \\ (4, 3, 3) & 1 \\ (4, 4, 3) & 1 \\ (4, 4, 8) & 1 \\ (4, 5, 3) & 1 \\ (4, 5, 8) & 1 \\ (4, 5, 19) & 1 \\ (4, 6, 3) & 1 \\ (4, 6, 8) & 1 \\ (4, 6, 12) & 1 \\ (4, 6, 14) & 1 \\ (4, 6, 19) & 1 \end{matrix}  \begin{matrix}(j, n, α) & d \\ (4, 6, 40) & 1 \\ (6, 3, 4) & 1 \\ (6, 4, 4) & 1 \\ (6, 5, 4) & 1 \\ (6, 6, 4) & 1 \\ (6, 6, 10) & 1 \\ (6, 6, 12) & 1 \\ (6, 6, 14) & 1 \\ (8, 2, 3) & 1 \\ (8, 3, 3) & 1 \\ (8, 3, 7) & 1 \\ (8, 4, 3) & 1 \\ (8, 4, 7) & 1 \\ (8, 4, 11) & 1 \\ (8, 5, 3) & 1 \\ (8, 5, 7) & 1 \\ (8, 5, 11) & 1 \\ (8, 5, 16) & 1 \\ (8, 6, 3) & 1 \\ (8, 6, 7) & 1 \\ (8, 6, 11) & 1 \\ (8, 6, 16) & 1 \\ (8, 6, 39) & 1 \\ (10, 4, 5) & 1 \\ (10, 5, 5) & 1 \\ (10, 5, 16) & 1 \\ (10, 6, 5) & 1 \\ (10, 6, 9) & 1 \\ (10, 6, 11) & 1 \end{matrix}  \begin{matrix}(j, n, α) & d \\ (10, 6, 16) & 1 \\ (10, 6, 37) & 1 \\ (12, 2, 3) & 1 \\ (12, 3, 3) & 1 \\ (12, 4, 3) & 1 \\ (12, 4, 9) & 2 \\ (12, 5, 3) & 1 \\ (12, 5, 9) & 2 \\ (12, 5, 16) & 1 \\ (12, 6, 3) & 1 \\ (12, 6, 9) & 2 \\ (12, 6, 16) & 1 \\ (12, 6, 35) & 1 \\ (14, 3, 4) & 1 \\ (14, 4, 4) & 1 \\ (14, 4, 8) & 1 \\ (14, 5, 4) & 1 \\ (14, 5, 8) & 1 \\ (14, 5, 13) & 1 \\ (14, 6, 4) & 1 \\ (14, 6, 8) & 1 \\ (14, 6, 13) & 1 \\ (14, 6, 36) & 1 \end{matrix}