Convolution of the Fourier coefficients of Eisentein–Maas series

The convolution is determinate as the sum
$$ N^{s-1}\sum_{n\geqslant 1}\tau_\nu(n)\left(\sigma_{1-2s}(n-N)w_0\left(\sqrt\frac nN\right)+\sigma_{1-2s}(n+N)w_1\left(\sqrt\frac nN\right)\right), $$
where $\tau_\nu(n)=n^{\nu-\frac12}\sigma_{1-2\nu}(n)$ for $n\ne0$ $\sigma_\nu(n)=\sum_{d|n, d>0}d^\nu$ and $w_0$, $w_1$ are arbitrary smooth functions.
The question: how to express this sum as a combination of the $N$'s Fourier coefficients of the eigenfunctions of the automorphic Laplacian? The answer is given in the terms of the biliear form of Hecke's series associated with the eigenfunctions of automorphic Laplacian and the regular cusp forms.
The final identity can give new opportunitys to the problem of moments of the Riemann zeta-functions.