  Let $F/\bbQ_p$ be a non-Archimedean local field, and $G_F$ be the absolute
Galois group of $F$. Let $\rho_1$ and $\rho_2$ be two finite-dimensional
complex representations of $G_F$. Let $\psi$ be a nontrivial additive character
of $F$. Then, the question is:
  What is the twisting formula for the root number
$W(\rho_1\otimes\rho_2,\psi)$?}
  In general, the answer to this question is not yet known. However, if one of
$\rho_i \quad(i=1,2)$ is one-dimensional with ``sufficiently'' large conductor,
then in [13], Deligne gave a twisting formula for
$W(\rho_1\otimes\rho_2,\psi)$. Later, in [12], Deligne and Henniart gave a
general twisting formula for a {\it zero}-dimensional virtual representation
twisted by a finite-dimensional representation of $G_F$. In this paper, we
first extend Deligne's twisting formula for U-isotropic Heisenberg
representation of dimension prime $p$, then we further extend
Deligne-Henniart's result.
  Finally, we provide two very important applications of our twisting formula:
--
  (i) invariant formula for the local root numbers for U-isotropic Heisenberg
representations, and (ii) a converse theorem on the Galois side.
