Let $\Gamma(N) \leq \Gamma_{1}(N) \leq \Gamma_{0}(N)$ be the usual congruence subgroups of the modular group $SL_{2}(\mathbb{Z})$, with all containments normal. We have, e.g., the quotient group $\Gamma_{0}(N)/\Gamma_{1}(N)$, which acts on the moduli space $$S_{1}(N) := \{[E, Q] \mid E \text{ complex elliptic curve and } Q \text{ a point of order } N\}/\sim$$ where $\sim$ is equivalence on curves defined by isomorphisms which preserve the order-$N$ points. This action according to Diamond & Shurman works out to be $\Gamma_{1}(N)\gamma: [E, Q] \mapsto [E, dQ]$, where $\gamma := \big(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\big) \in \Gamma_{0}(N)$.
Where does this action come from? The text makes it seem like a natural thing that should just pop out, so I feel like an idiot for not getting it! Does this come from the usual linear fractional action? If so, it's not immediately obvious to me.