Rational-cusp inversion of a level-one cusp form ($f(a/N-1/(N^2\tau ))=(N\tau {)}^{k}f(\tau -d/N)$)

If $ad\equiv 1(\mathrm{mod}N)$, the determinant-one matrix with rows $(a,(ad-1)/N)$ and $(N,d)$ sends $\tau -d/N$ to $a/N-1/(N^2\tau )$. The transformation of the cusp form gives the displayed identity. It turns the small-height portion of an additive-twist Mellin transform into exponentially decaying values at the inverse rational cusp.