On the exponent of distribution for convolutions of $\mathrm{GL}(2)$ coefficients to smooth moduli

Abstract

Let $({\lambda }_f(n){)}_{n⩾1}$ be the Hecke eigenvalues of a holomorphic cusp form $f$. We prove that the exponent of distribution of ${\lambda }_f\ast 1$ in arithmetic progressions is as large as $\frac{1}{2}+\frac{1}{70}$ when the modulus $q$ is square-free and has only sufficiently small prime factors.

14 pages. arXiv admin note: substantial text overlap with arXiv:2511.07945