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 F be an orthonormal basis of S k ( Γ ( 1 ) ) , the space of cusp forms of weight k > 2 on S L 2 ( Z ) . Then for any positive integers m , n we have

where δ is the Kronecker delta function, S is the Kloosterman sum and J is the Bessel function of the first kind.