# Periodic twists of $GL_3$-automorphic forms

## TL;DR

This paper demonstrates that sums of Hecke eigenvalues of automorphic forms on SL_3(Z) over lengths about q^{3/2} do not correlate with q-periodic functions with bounded Fourier transform, extending previous results to trace functions of small conductor.

## Contribution

It generalizes prior correlation results from Dirichlet characters to trace functions of small conductor for automorphic forms on SL_3(Z).

## Key findings

- Hecke eigenvalue sums of length about q^{3/2} show no correlation with q-periodic functions.
- The results extend earlier work by Munshi and Holowinsky--Nelson.
- Application to trace functions of small conductor modulo primes.

## Abstract

We prove that sums of length about $q^{3/2}$ of Hecke eigenvalues of automorphic forms on $SL_3(\Zz)$ do not correlate with $q$-periodic functions with bounded Fourier transform. This generalizes the earlier results of Munshi and Holowinsky--Nelson, corresponding to multiplicative Dirichlet characters, and applies in particular to trace functions of small conductor modulo primes.