  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.
