Euler decompositions for theta-series of even quadratic forms

For a generating Dirichlet vector series with coefficients equal to the number of representations of a quadratic form by another one we abtain a decomposition into the product of a finite number of Dirichlet $L$-functions and an infinite number of matrix polynomials. The coefficients of the polynomials are the Eichler–Brandt matrices of the basis double cosets of the local orthogonal Hecke rings. Bibliography: 3 titles.