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We study graded rings of modular forms over congruence subgroups, withcoefficients in a subring $A$ of $\mathbb{C}$, and specifically the highestweight needed to generate these rings as $A$-algebras. In particular, wedetermine upper bounds, independent of $N$, for the highest needed weight thatgenerates the $\mathbb{C}$-algebras of modular forms over $\Gamma(N)$,$\Gamma_1(N)$ and $\Gamma_0(N)$ with some conditions on $N$. For $N \geq 5$, weprove 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. Wegive an algorithm that computes the generators, and supply some computationsthat allow us to state two conjectures concerning the situation over$\Gamma_0(N)$.

Generators of graded rings of modular forms