# An explicit conductor formula for ${\rm GL}_n \times {\rm GL}_1$

## TL;DR

This paper derives an explicit formula for the conductor of irreducible admissible representations of GL_n over non-archimedean local fields when twisted by characters, enabling precise quantification of character twists with fixed conductor.

## Contribution

It provides the first explicit conductor formula for GL_n representations twisted by characters, advancing understanding of local representation theory.

## Key findings

- Explicit conductor formula for GL_n representations twisted by characters
- Quantification of character twists with fixed conductor
- Enhanced tools for local representation analysis

## Abstract

We prove an explicit formula for the conductor of an irreducible, admissible representation of ${\rm GL}_n(F)$ twisted by a character of $F^{\times}$ where the field $F$ is local and non-archimedean. As a consequence, we quantify the number of character twists of such a representation of fixed conductor.