Abstract

Let $f$ be a normalized holomorphic cuspidal Hecke eigenform of even integral weight $\kappa$ for the full modular group $\Gamma =\mathrm{SL}<!--FUNCTION\; APPLICATION-->(2,\mathbb{Z})$. We investigate the asymptotic behaviour of the higher power moments of a general divisor problem of the coefficients of $\ell$-fold product $L$-functions associated to $f$ on arithmetic progressions. As an application, we also provide quantitative results for the sign changes of the Dirichlet coefficients of the $\ell$-fold product $L$-functions over arithmetic progressions. By analogy, we also consider a similar problem which is supported at certain integral binary quadratic forms.

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