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 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 {displaystyle {mathcal {F}}} be an orthonormal basis of S k ( Γ ( 1 ) ) {displaystyle S_{k}(Gamma (1))} , the space of cusp forms of weight k > 2 {displaystyle k>2} on S L 2 ( Z ) {displaystyle SL_{2}(mathbb {Z} )} . Then for any positive integers m , n {displaystyle m,n} we have where δ {displaystyle delta } is the Kronecker delta function, S {displaystyle S} is the Kloosterman sum and J {displaystyle J} is the Bessel function of the first kind.