Let $N$ be a positive integer. Denote by $A_N$ (resp. $B_N$) the graded $\mathbb{Z}[1/N]$-algebra of holomorphic modular forms of level $\Gamma_1(N)$ (resp. $\Gamma(N)$). $A_N$ and $B_N$ are both finitely presented as $\mathbb{Z}[1/N]$-algebras. What is an explicit presentation for $A_3$, $B_3$ and $A_5$, $B_5$? It is written down anywhere in the literature?
A presentation of $A_1=B_1$ has been given by Deligne (though it was known for a long time). The paper "An explicit structure of the graded ring of modular forms of small level" studies the question over $\mathbb{C}$ but it is not directly helpful to me.
You might want to consult this paper of Rustom:
Generators of graded rings of modular forms
This explains how to find generators for $A_N$ over $Z[1/N]$. Rustom doesn't treat your $B_N$ (the case of $\Gamma(N)$ levels) but I imagine similar techniques should work; it is rather the case of $\Gamma_0(N)$ levels which is hard, because of elliptic points.
Some more recent work is described in Aaron Landesman's answer to an old MathOverflow question of mine; note that my question asked for generators over $\mathbb{C}$, but the answer went beyond this to study generators over other base rings as well, and also addresses finding all relations among the generators.