  We study graded rings of modular forms over congruence subgroups, with
coefficients in a subring $A$ of $\mathbb{C}$, and specifically the highest
weight needed to generate these rings as $A$-algebras. In particular, we
determine upper bounds, independent of $N$, for the highest needed weight that
generates the $\mathbb{C}$-algebras of modular forms over $\Gamma(N)$,
$\Gamma_1(N)$ and $\Gamma_0(N)$ with some conditions on $N$. For $N \geq 5$, we
prove that the $\mathbb{Z}[1/N]$-algebra of modular forms over $\Gamma_1(N)$
with coefficients in $\mathbb{Z}[1/N]$ is generated in weight at most 3. We
give an algorithm that computes the generators, and supply some computations
that allow us to state two conjectures concerning the situation over
$\Gamma_0(N)$.
