The Petersson conjecture for the zeroth weight. I

In the present work, a known result by Eichler–Deligne concerning the Petersson conjecture for finite-dimensional classical spaces is proved for infinite-dimensional Hilbert spaces of weight 0. In this work, the techniques of spectral decompositions of convolutions are used. The work is subdivided into two parts. In this (first) part, an explicit representation of an eigenvalue of the Hecke operator in terms of spectral components of the convolution is obtained. On the basis of this representation, the Petersson conjecture will be proved in the second part.