A note on quadratic twisting of epsilon factors for modular forms with arbitrary nebentypus

Abstract

In this article, we investigate the variance of local ε-factor for a modular form with arbitrary nebentypus with respect to twisting by a quadratic character. We detect the type of the supercuspidal representation from that. For modular forms with trivial nebentypus, similar results are proved by Pacetti [16]. Our method however is completely different from that of Pacetti and we use representation theory crucially. For ramified principal series (with p N and p odd) and unramified supercuspidal representations of level zero, we relate these numbers with the Morita's p-adic Gamma function.