Computing bases of modular forms using the graded algebra structure

Michael Lam; Noah McClelland; Matthew Petty; and John Webb

arXiv:1709.07475·math.NT·September 25, 2017

Computing bases of modular forms using the graded algebra structure

Michael Lam, Noah McClelland, Matthew Petty, and John Webb

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Abstract

We develop a new algorithm to compute a basis for $M_{k} (Γ_{0} (N))$ , the space of weight $k$ holomorphic modular forms on $Γ_{0} (N)$ , in the case when the graded algebra of modular forms over $Γ_{0} (N)$ is generated at weight two. Our tests show that this algorithm significantly outperforms a commonly used algorithm which relies more heavily on modular symbols.

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