The Manin constant

Let $A$ be a quotient of $J_0(N)$ .

Definition 8.6.1 (Manin constant)   The Manin constant of $A$ is the index in its saturation of $\H ^1(\mathcal{A},\Omega_{\mathcal{A}/\mathbb{Z}})$ in $S_2(\Gamma_0(N);\mathbb{Z})$ under the appropriate natural identifications (see [ARS06]).

Problem 8.6.2   Find (and implement) an algorithm to compute the prime divisors of the Manin constant of $A$ .

This is closely related to understanding $q$ -expansions at cusps other than $\infty$ .

Problem 8.6.3   Give a modular form $f=\sum a_n q^n\in S_2(\Gamma_0(N))$ compute the $q^{1/h}$ -expansion of $f$ at all the cusps.

Remark 8.6.4 (From John Cremona:)   Prob 8.6.3 was dealt with in Delaunay's thesis.

Problem 8.6.5   Implement in SAGE Delaunay's algorithm to compute the $q^{1/h}$ -expansion of a modular form at all cusps.

Problem 8.6.6   Give a modular form $f=\sum a_n q^n\in S_k(\Gamma_1(N))$ compute the $q^{1/h}$ -expansion of $f$ at all the cusps.