On the change of root numbers under twisting and applications

Abstract

The purpose of this article is to show how the root number of a modular form changes by twisting in terms of the local Weil-Deligne representation at each prime ideal. As an application, we show how one can for each odd prime p, determine whether a modular form (or a Hilbert modular form) with trivial nebentypus is Steinberg, Principal Series or Supercuspidal at p by analyzing the change of sign under a suitable twist. We also explain the case p = 2, where twisting is not enough in general.