Let $f$ be a modular function (that is, a meromorphic modular form of weight 0) holomorphic on $\mathcal{H}$ which is invariant under $\Gamma\le SL_2(\mathbb{Z})$ (not necessarily congruence!), and not invariant under any larger group.
Is $\mathbb{C}(j)[f]$ precisely the function field $\mathbb{C}(\Gamma)$ of the modular curve $\mathcal{H}/\Gamma$?
Certainly if $\mathbb{C}(j)[f]$ is strictly smaller than $\mathbb{C}(\Gamma)$ then it couldn't be the function field of any modular curve. Is this possible?