Let $F$ be a non-Archimedean local field. For a multiplicative character $\chi$ of $F^\times$, let $a(\chi)$ be the conductor of $\chi$. For a nontrivial additive character $\phi$ of $F$, we denote the level of $\phi$ by $n(\phi)$.
The following Theorem of Deligne states that (see Theorem 4.16 from this paper):
Given $\chi \in \widehat{F^\times}$, there exists a $c$ of valuation $-(a(\chi)+n(\phi))$ such that $\chi(1+x)=\phi(cx)$ for all $x\in\mathfrak{p}_{{F}}^r$ with $r \geq \left[\frac{a(\chi)+1}{2}\right]$. If $\beta \in \widehat{F^\times}$ satisfies that $2a(\beta) \leq a(\chi)$, then
$$\varepsilon(\alpha\beta, \phi)= \beta^{-1}(c)\varepsilon(\alpha, \phi).$$
My question is, when we say $c$ of valuation $-(a(\chi)+n(\phi))$, do we mean $c=p^n$ with $n=-(a(\chi)+n(\phi))$ (which means we are taking the unit to be $1$) or $c=p^nu$ with $n=-(a(\chi)+n(\phi))$ and $u$ is a $p$-adic unit.
The same confusion appears in many situation while reading other notes/textbooks related to epsilon factor.
