I am interested in ring-theoretic properties of rings of modular forms. Consider the ring $R$ of integral modular forms for some level, say $\Gamma_1(n)$ -- and to be gentle, let's invert $n$. Algebro-geometrically, this can be defined as the sections of powers of the line bundle $\lambda$ on the compactified or uncompactified moduli stack $\mathcal{M}_1(n)$ of elliptic curves with $\Gamma_1(n)$-level structure; if we consider the compactified moduli, we get holomorphic modular forms, if we consider the uncompactified, we get meromorphic modular forms.
If we consider meromorphic modular forms, then $R$ is certainly known to be regular for $n\geq 2$; indeed, for $n\geq 2$ the $\mathbb{G}_m$-torsor over the moduli stack $\mathcal{M}_1(n)$ associated to $\bigoplus_{i\in \mathbb{Z}} \lambda^{\otimes i}$ is representable by an affine scheme and known to be regular as $\mathcal{M}_1(n)$ is regular.
My question is about the holomorphic case:
For which $n$ is the ring of integral holomorphic modular forms $R$ of level $\Gamma_1(n)$ regular? Is $R$ always normal?
