Computing actions on cusp forms

Abstract

For positive integers $k$ and $N$, we describe how to compute the natural action of $S{L}_2(\mathbb{Z})$ on the space of cusp forms $S_k(\Gamma (N))$, where a cusp form is given by sufficiently many terms of its $q$-expansion. This will reduce to computing the action of the Atkin--Lehner operator on $S_k(\Gamma )$ for a congruence subgroup ${\Gamma }_1(N)\subseteq \Gamma \subseteq {\Gamma }_0(N)$. Our motivating application of such fundamental computations is to compute explicit models of some modular curves $X_G$.

No substantial change; minor corrections made

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• $\ell$-adic images of Galois for elliptic curves over $\mathbb{Q}$