Computing bases of modular forms using the graded algebra structure

Abstract

We develop a new algorithm to compute a basis for $M_k(\hat{I}{“}_0(N))$, the space of weight $k$ holomorphic modular forms on $\hat{I}{“}_0(N)$, in the case when the graded algebra of modular forms over $\hat{I}{“}_0(N)$ is generated at weight two. Our tests show that this algorithm significantly outperforms a commonly used algorithm which relies more heavily on modular symbols.