Computing Classical Modular Forms for Arbitrary Congruence Subgroups

Abstract

In this paper, we prove the existence of an efficient algorithm for the computation of $q$-expansions of modular forms of weight $k$ and level $\Gamma$, where $\Gamma \subseteq SL_{2}({\mathbb{Z}})$ is an arbitrary congruence subgroup. We also discuss some practical aspects and provide the necessary theoretical background.