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

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