Announcements Projections of C ° ° Automorphic Forms

The purpose of this paper is to exhibit an explicit formula which describes the projection operator from the space of C°° automorphic forms to the subspace of holomorphic cusp forms, and to apply it to the zeta functions of Rankin type. Fix a number k > 0 such that 2k G Z. Let TV be a positive integer such that TV = 0 mod(4) if k fi Z, and let x(Z/NZ) —+ C be a Dirichlet character modulo TV. Define r0(iV) = {Q b d) G SL2(Z)\c = 0 mod(A0}