= Solution
Choose $b=(ad-1)/N\in\mathbb Z$, and take the <matrix>
$$
\gamma=\begin{pmatrix}a&b\\N&d\end{pmatrix}\in SL_2(\mathbb Z).
$$
At $w=\tau-d/N$ its denominator is $Nw+d=N\tau$, while its numerator is $aw+b=a\tau-1/N$. Consequently
$$
\gamma w=\frac aN-\frac1{N^2\tau}.
$$
Applying the weight-$k$ transformation law of the <cusp form> at $w\in\mathbb H$ proves the <rational-cusp inversion of a level-one cusp form>:
$$
\boxed{f\!\left(\frac aN-\frac1{N^2\tau}\right)=(N\tau)^kf\!\left(\tau-\frac dN\right).}
$$
Back to article page
