Computing bases of modular forms using the graded algebra structure

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.