= Degeneracy map for cusp forms
{title2=$f(\tau)\mapsto f(D\tau)$}
If $M\mid N$ and $D\mid N/M$, conjugating $\Gamma_0(N)$ by $\operatorname{diag}(D,1)$ gives matrices $\begin{pmatrix}a&Db\\c/D&d\end{pmatrix}$ 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 $N$.
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