Rational-cusp inversion of a level-one cusp form
= Rational-cusp inversion of a level-one cusp form
{title2=$f(a/N-1/(N^2\tau))=(N\tau)^kf(\tau-d/N)$}
If $ad\equiv1\pmod N$, 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.