Fourier expansion of a modular form ($f(\tau )={\sum }_{n\ge 0}{a}_n{q}^n$)

Translation invariance at a cusp gives a Fourier expansion in its local parameter $q$. Holomorphy at the cusp excludes negative powers; vanishing of the constant term defines a cusp form.

A cusp form is a modular form that vanishes at every cusp. At the cusp at infinity for the full modular group it has an expansion ${\sum }_{n\ge 1}{a}_n{q}^n$.

For even $k\ge 12$,The dimension is zero for odd $k$ and for $k<12$.

The modular discriminant is the normalized weight-twelve level-one cusp form

For every real $s$, the rapidly convergent integralis the analytic continuation of $(2\pi {)}^{-s}\Gamma (s)L(\Delta ,s)$. The product for $\Delta$ makes the integrand positive, so $L(\Delta ,s)>0$ for real $s>0$.

For weight-$k$ cusp forms on $\Gamma$, the Petersson inner product isCusp decay makes the integral convergent.

The Rankin–Selberg method represents Dirichlet series built from automorphic forms as integrals against Eisenstein series and studies them by unfolding those integrals.

For cusp forms $f=\sum a_n{q}^n$ and $g=\sum b_n{q}^n$, their Rankin–Selberg convolution in the elementary normalization is $L(f,g,s)={\sum }_{n\ge 1}{a}_n\overline{b_n}{n}^{-s}$.

If $f$ and $g$ have respective weights $k$ and $l$, and $r=k-l>2$, unfolding the weight-$r$ Eisenstein series gives