Let $\Gamma_1\le SL_2(\mathbb{Z})$ be a noncongruence subgroup of finite index.

Let $\Gamma_2\le SL_2(\mathbb{Z})$ be another subgroup of finite index.

Let $M_0(\Gamma_i)$ denote the ring of modular functions for $\Gamma_i$, by which I mean holomorphic functions on the upper half plane $\mathcal{H}$ which are invariant under $\Gamma_i$ - ie, regular functions on the affine modular curve $\mathcal{H}/\Gamma_i$. Since all modular curves are defined over a number field, I'm thinking of $M_0(\Gamma_i)$ as a $\overline{\mathbb{Q}}$-algebra.

1. For any $\Gamma_1,\Gamma_2$ as above, is it true that $M_0(\Gamma_1\cap\Gamma_2)$ is generated by $M_0(\Gamma_1)$ and $M_0(\Gamma_2)$? (as $\overline{\mathbb{Q}}$-algebras)

That was a long shot, so if that's false, then

2. Fixing an arbitrary $\Gamma_1$, can we always find a torsion-free congruence subgroup $\Gamma_2$ such that $M_0(\Gamma_1\cap\Gamma_2)$ is generated by $M_0(\Gamma_1)$ and $M_0(\Gamma_2)$? (as $\overline{\mathbb{Q}}$-algebras)

I'd also appreciate pointers to references that specifically address questions about modular functions like this.