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

TL;DR

This paper proves an improved exponent of distribution for convolutions of Hecke eigenvalues of holomorphic cusp forms in arithmetic progressions with certain moduli.

Contribution

It establishes a new, larger exponent of distribution for these convolutions when the modulus is square-free with small prime factors.

Findings

Exponent of distribution is at least 1/2 + 1/70 for certain moduli.

Results apply to convolutions of Hecke eigenvalues with the trivial function.

Distribution results hold for square-free moduli with small prime factors.

Abstract

Let $(\lambda_f(n))_{n\geqslant1}$ be the Hecke eigenvalues of a holomorphic cusp form $f$. We prove that the exponent of distribution of $\lambda_f*1$ 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.