Petersson trace formula

In analytic number theory, the Petersson trace formula is a kind of orthogonality relation between coefficients of a holomorphic modular form. It is a specialization of the more general Kuznetsov trace formula.

In its simplest form the Petersson trace formula is as follows. Let ${\displaystyle {\mathcal {F}}}$ be an orthonormal basis of ${\displaystyle S_{k}(\Gamma (1))}$, the space of cusp forms of weight ${\displaystyle k>2}$ on ${\displaystyle SL_{2}(\mathbb {Z} )}$. Then for any positive integers ${\displaystyle m,n}$ we have

${\displaystyle {\frac {\Gamma (k-1)}{(4\pi {\sqrt {mn}})^{k-1}}}\sum _{f\in {\mathcal {F}}}{\bar {\hat {f}}}(m){\hat {f}}(n)=\delta _{mn}+2\pi i^{-k}\sum _{c>0}{\frac {S(m,n;c)}{c}}J_{k-1}\left({\frac {4\pi {\sqrt {mn}}}{c}}\right),}$

where ${\displaystyle \delta }$ is the Kronecker delta function, ${\displaystyle S}$ is the Kloosterman sum and ${\displaystyle J}$ is the Bessel function of the first kind.