Degeneracy map for cusp forms ($f(\tau )\mapsto f(D\tau )$)

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 $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in {\Gamma }_0(N)$ by $\mathrm{diag}(D,1)$ givesbecause $MD\mid N$. The same argument applies to ${\Gamma }_1(M)$ and ${\Gamma }_1(N)$: the congruences $a,d\equiv 1(\mathrm{mod}N)$ 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.

For prime $p$, the two cusps of ${\Gamma }_0(p)$ have widths one and $p$. At infinity the modular discriminant product gives the first pair of orders. At zero use $S\tau =-1/\tau$ and the parameter $q_0=e^{2\pi i\tau /p}$. The weight-twelve transforms are $\Delta {|}_{12}S=\Delta (\tau )$ and $\Delta (p\tau ){|}_{12}S=p^{-12}\Delta (\tau /p)$, giving orders $p$ and one in $q_0$.