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We obtain a spectral decomposition for number-theoretic convolutions of the form r(n) • r (n • l, Xq), where the function 7" is the number of divisors, X q is the quadratic Dirichlet character of module q, l is a fixed shift, and n is the summation parameter. This is done by using the shortened functional equation for the convolution, obtained by the author (Zap. Nauchn. Semin. POMI, 211, 104-119 (1994)

of the form HiTi, where each T i is contained in Lm(q) and is a linear combination of left classes of integral matrices of identical factors Di = Pi di" Then applying formula (8) successively for each T i (this can be done since the determinant, the level, and the signature of each of the matrices ~ [~] in the right-hand side are the same as for Q), we obtain that the formula (12) is valid with some constant coefficients r independent of n.

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