In his paper, Duke paper, 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

$\det(\rho(Frob_{l})) = \epsilon(Frob_{l})\omega^{k}(Frob_{l})$ 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 $\epsilon$, in some paper, there is the claim that $\epsilon$ is the unique quadratic character mod $p$ ramified only at $p$, and I do not understand where this comes from?

3) How can one finds $k$.

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.