Computing bases of modular forms using the graded algebra structure

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.