Let $\Gamma_1\le SL_2(\mathbb{Z})$ be a subgroup of finite index (not necessarily congruence).

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 $\Gamma_i$.

1. Is it true that $M_0(\Gamma_1\cap\Gamma_2)$ is generated by $M_0(\Gamma_1)$ and $M_0(\Gamma_2)$?

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

2. Fixing an arbitrary $\Gamma_1$, can we always find a torsion-free $\Gamma_2$ such that $M_0(\Gamma_1\cap\Gamma_2)$ is generated by $M_0(\Gamma_1)$ and $M_0(\Gamma_2)$?

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