Computing bases of modular forms using the graded algebra structure

Abstract

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