Let $\Gamma$ be a congruence subgroup of $\operatorname{SL}_2(\mathbf Z)$. Let $\mathfrak H^*=\mathfrak H\cup\mathbf Q\cup\{\infty\}$. Shimura in his book *Introduction to Arithmetic Theory of Automorphic Functions, section 6.7,* considers the notion of a model for the compact Riemann surfac $\Gamma\backslash\mathfrak H^*$: let $\varphi$ be a modular function invariant under $\Gamma$ and let $V$ be a projective non-singular algebraic curve isomorphic to $\Gamma\backslash\mathfrak H^*$; then the pair $(\varphi,V)$ is called a model for $\Gamma\backslash\mathfrak H^*$ if $\varphi$ provides an isomorphism between $\Gamma\backslash\mathfrak H^*$ and $V$.

Let $\Gamma'$ be another congruence subgroup and suppose that $\alpha\Gamma\alpha^{-1}\subset \Gamma'$, where $\alpha \in \operatorname{GL}_2^+(\mathbf Q)$. Let $(\varphi',V')$ be a corresponding model. Then, according to Shimura, we have the following commutative diagram $\require{AMScd}$ \begin{CD} \mathfrak H^* @>\alpha>> \mathfrak H^*\\ @V \varphi V V @VV \varphi' V\\ V @>>T> V' \end{CD} Here $T$ is a rational map. For example if $\alpha=1$ and $\Gamma$ is a genus zero congruence subgroup of $\Gamma'=\operatorname{SL}_2(\mathbf Z)$ then we get the expression for the $j$-invariant as a rational function of the uniformizer for $\Gamma$.

What is a concrete example of this, when $\alpha$ is not the identity matrix?