# A constraint for twist equivalence of cusp forms on GL$(n)$

## TL;DR

This paper establishes a divisibility constraint on characters relating two cuspidal automorphic representations of GL(n), generalizing a question by Kaisa Matomäki and providing bounds involving conductors over number fields.

## Contribution

It generalizes a question of Kaisa Matomäki by proving a divisibility condition on conductors for characters relating cuspidal automorphic representations of GL(n).

## Key findings

- For two cuspidal automorphic representations, the conductor of the character satisfies Q^n divides N_1 N_2.
- If local components are discrete series at ramified places, then Q^n divides the least common multiple of N_1 and N_2.
- The result extends previous bounds and applies over general number fields.

## Abstract

This Note answers, and generalizes, a question of Kaisa Matom\"aki. We show that give two cuspidal automorphic representations $\pi_1$ and $\pi_2$ of $GL_n$ over a number field $F$ of respective conductors $N_1,$ $N_2,$ every character $\chi$ such that $\pi_1\otimes\chi\simeq\pi_2$ of conductor $Q,$ satisfies the bound: $Q^n\mid N_1N_2.$ If at every finite place $v,$ $\pi_{1,v}$ is a discrete series whenever it is ramified, then $Q^n$ divides the least common multiple $[N_1, N_2].$