Let $\Gamma$ be a finite-index subgroup of $\operatorname{SL}_2(\mathbb{Z})$. I've seen it stated (in a comment in the code of a computer program) that the graded ring $$ M(\Gamma, \mathbb{C}) = \bigoplus_{k \ge 0} M_k(\Gamma, \mathbb{C}),$$ where $M_k(\Gamma, \mathbb{C})$ is the space of modular forms of weight $k$ and level $\Gamma$, is always generated as a $\mathbb{C}$-algebra by forms of weight $\le 12$.

Why is this true? Moreover, can one improve on the bound of 12? (For the subgroups $\Gamma_0(N)$, weight $\le 6$ always seems to be sufficient.)