There are specific details which I'm a little stuck on in Deligne and Serre's paper on attaching Galois representations to modular forms of weight 1.
In the proof of Lemma 8.3, they use the Chebotarev Density Theorem to conclude $|H_l|\geq (1-\eta)|G_l|$. It might be obvious but I can't see how Chebotarev (in the form given, say, on Wikipedia) gives the result (I understand up till here).
Here, $\eta$ is a positive number, $G_l$ is the image of $\rho_l: Gal(\bar Q/Q) \rightarrow GL_2(F_l)$, and $H_l$ is formed by the Frobenius elements and their conjugates for primes not in a particular set of primes $X_\eta$ which has Dirichlet density $\leq \eta$, and for which the $a_p$, for $p \notin X_\eta$, form a finite set.
Secondly, right after Lemma 8.4, in the paragraph marked 8.5, how do they come up with "there exists R(T) $\in$ Y such that ..." . The polynomial in question is clearly the characteristic polynomial of the Frobenius element at prime p, but why must this be congruent to a polynomial with roots of unity as its roots?
Not sure if it is what you want but
$G$, the image of $\rho_l$ is the Galois group of $Gal(K/\Bbb{Q})$ where $K=\overline{\Bbb{Q}}^{\ker(\rho_l)}$ so Chebotarev is directly telling the density of primes whose Frobenius element is in $C=\{g\rho_l(\sigma)g^{-1},g\in G_l\}$. And hence the ($\lim \inf$ of the) density of primes in your set is $\ge |C|/|G_l| -\eta$
The roots of the characteristic polynomials are roots of unity because $K/\Bbb{Q}$ is a finite extension (the representation has finite image, its kernel has finite index)