Let $F$ be an unramified extension of $\mathbf{Q}_p$ of degree $n$, and let $K = F(\alpha)$ where $\alpha$ satisfies $\alpha^{p^n - 1} = -p$.

I'm interested in the local $\varepsilon$-factors attached to finite-order characters of the Galois group of $G_k = \operatorname{Gal}(K(\mu_{p^k}) / F)$, for $k \gg 0$. Specifically, I'd like to know if there's any relation between these $\varepsilon$-factors and the quantities $$ \tau(u, \omega) = \sum_{\gamma \in G_k} \omega(g)^{-1} g(u) $$ for suitable $u \in K(\mu_{p^k})$. If $n = 1$, so $K = F = \mathbf{Q}_p$, then $\tau(u, \omega)$ for $u = \zeta_{p^k}$ is just a Gauss sum, and it's well known that these are the epsilon-factors. Can one say anything useful for $n \ge 2$?