  Let $F/\mathbb{Q}_p$ 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 question is:
  What is the twisting formula for the root number
$W(\rho_1\otimes\rho_2,\psi)$?
  In general, answer of this question is not known yet. But if one of $\rho_i
(i=1,2)$ is one-dimensional with "sufficiently" large conductor, then Deligne
gave a twisting formula for $W(\rho_1\otimes\rho_2,\psi)$. Later Deligne and
Henniart give a general twisting formula for a zero dimensional virtual
representation twisted by a finite dimensional representation of $G_F$. In this
paper, first we extend Deligne's twisting formula for Heisenberg representation
of dimension prime $p$, then we further extend Deligne-Henniart's result.
  Finally, we give two very important applications of our twisting formula --
invariant formula of local root numbers for U-isotropic Heisenberg
representations and a converse theorem in the Galois side.
