  In this article, we investigate the variance of local $\varepsilon$-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. Our method however is completely different from that of Pacetti and we
use representation theory crucially.
  For ramified principal series (with $\infdiv{p}{N}$ and $p$ odd) and
unramified supercuspidal representations of level zero, we relate these numbers
with the Morita's $p$-adic Gamma function.
