$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Frob{Frob}$In his paper, [Duke paper](https://projecteuclid.org/journals/duke-mathematical-journal/volume-54/issue-1/Sur-les-repr%C3%A9sentations-modulaires-de-degr%C3%A9-2-de-GalQQ/10.1215/S0012-7094-87-05413-5.short), Serre consider continuous, odd Galois representation
$\rho: G\_{\mathbb{Q}}\longrightarrow \GL\_{n}(\overline{\mathbb{F}}\_{p})$ where $p$ is a rational prime.
Roughly, (I don't understand much French except for the help from Google translation) Serre claims (section 1.3) that


$\det(\rho(\Frob\_{l})) = \varepsilon(\Frob\_{l})\omega^{k}(\Frob\_{l})$ for all prime $l\nmid pN$ where $N$ is defined as the level of the representation (with an explicit formula given in the paper) and $\epsilon$ is a Dirichlet character and $k$ is some positive integer.


This seems to be standard since other papers cited it without reproving and I could not find any reference for the proof. In particular, my questions are:


1. Where can I find a proof for this.
2. What exactly is $\varepsilon$, in some paper, there is the claim that $\varepsilon$ is the unique quadratic character mod $p$ ramified only at $p$, and I do not understand where this comes from?
3. How can one find $k$.


For motivation, I think $\det(\rho(\Frob\_{l}))$ is an important invariant to compute since, for example, it appears in the attachment equation that associates these representations with modular forms.


Thanks in advance for any insight.



On the determinant of an odd, continuous Galois representation
$\det (\rho)$ is a one dimensional rep of the absolute Galois group of the rationals, i.e., it is a character. All such characters can be described by class field theory or, more simply, by the Kronecker-Weber theorem. So is a Dirichlet character and, by the hypotheses, its conductor divides $pN$. Factor it as a character of conductor $p$ (that will be $\epsilon$) times a character of conductor $N$. The latter is a power of the cyclotomic character and $k$ is defined to be that power. The bit about quadratic character must be under additional hypotheses.


Edit: I got $p$ and $N$ switched above. The character of conductor $N$ is $\epsilon$. The character of conductor $p$ is a power of the cyclotomic character because $(\mathbb{Z}/p)^\*$ is cyclic.


