Theta-transforms and even zeta functions of Siegel modular forms (Q2773334)

The author considers a Siegel modular form \(F\) of weight \(k\) and genus 2 with respect to \(\Gamma^2_0(q)\), \(q\in \mathbb{N}\) square free, and its theta transform NEWLINE\[NEWLINED_F(s,\psi,Q)=\sum_{M:SL_2(\mathbb{Z})\setminus \Delta} \psi(\text{det }M)\alpha_F(MQM^{tr})(\text{det }M)^{-s-k+1},NEWLINE\]NEWLINE where \(\alpha_F\) denote the Fourier coefficients of \(F\), \(Q\) is positive definite, \(\psi\) is a Dirichlet character and \(\Delta = \{M\in\mathbb{Z}^{2\times 2};\text{det }M > 0\}\). The author shows that the list of Hecke operators, which are diagonalizable on the space of new forms as introduced by the author in [St. Petersb. Math. J. 11, 931-987 (2000); translation from Algebra Anal. 11, No.~6, 1-68 (1999; Zbl 1023.11021)], includes all the operators sufficient for the Euler factorization of the corresponding theta transform. Therefore one has to extend the techniques of Euler factorization in order to include also prime divisors of the level \(q\).