 Degeneracy map for cusp forms

If $M\mid N$ and $D\mid N/M$, conjugating ${\Gamma }_0(N)$ by $\mathrm{diag}(D,1)$ gives matrices $\left(\begin{array}{cc}a& Db\\ c/D& d\end{array}\right)$ in ${\Gamma }_0(M)$. This proves the weight transformation of $f(D\tau )$. At any rational modular cusp, factor the rational change of variable into an integral cusp representative followed by an upper-triangular rational matrix of positive slope. The original cusp expansion then stays holomorphic and has zero constant term. Thus this map takes cusp forms of level $M$ to those of level $N$.