Euler products for congruence subgroups of the Siegel group of genus $2$

In this paper the construction is begun of a theory of Dirichlet series with Euler expansion which correspond to analytic automorphic forms for congruence subgroups of the integral symplectic group of genus $2$. Namely, for an arbitrary positive integer $q$ a connection is revealed between the eigenvalues $\lambda_F(m)$ of an eigenfunction $F\in\mathfrak M_k\bigl(\Gamma_2(q)\bigr)$ of all the Hecke operators $T_k(m)$ ($(m,q)=1$), where $\Gamma_2(q)$ is the principal congruence subgroup of degree $q$ of the group $\Gamma_2=\operatorname{Sp}_2(\mathbf Z)$, and its Fourier coefficients. This connection can be written in the language of Dirichlet series in the form of identities; here an infinite sequence of identities arises, indexed by classes of positive definite integral primitive binary quadratic forms equivalent modulo the principal congruence subgroup of degree $q$ of $\operatorname{SL}_2(\mathbf Z)$.
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