= Degeneracy map for cusp forms
{title2=$f(\tau)\mapsto f(D\tau)$}
Let $M\mid N$ and $D\mid N/M$. For a weight-$k$ <cusp form> $f$ on $\Gamma_0(M)$, the function $\tau\mapsto f(D\tau)$ is a weight-$k$ <cusp form> on $\Gamma_0(N)$. Indeed, conjugating a <matrix> $\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\Gamma_0(N)$ by $\operatorname{diag}(D,1)$ gives
$$
\begin{pmatrix}a&Db\\c/D&d\end{pmatrix}\in\Gamma_0(M),
$$
because $MD\mid N$. The same argument applies to $\Gamma_1(M)$ and $\Gamma_1(N)$: the congruences $a,d\equiv1\pmod N$ imply the required congruences modulo $M$ as well. This supplies both common versions of the degeneracy maps used in the <old subspace of cusp forms>.
At a 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 <Fourier expansion of a modular form> then remains holomorphic and has zero constant term. Thus the <cusp form> condition, as well as the weight transformation, is preserved.
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