Local Constants for Heisenberg Representations

We can attach a local constant to every finite dimensional continuous complex representation of a local Galois group of a non-archimedean local field $F/\mathbb{Q}_p$ by Deligne and Langlands. Tate \cite{JT1} gives an explicit formula for computing local constants for linear characters of $F^\times$, but there is no explicit formula of local constant for any arbitrary representation of a local Galois group. In this article we study Heisenberg representations of the absolute Galois group $G_F$ of $F$ and give invariant formulas of local constants for Heisenberg representations of dimension prime to $p$.