[proofplan] We apply Deligne's construction theorem for the $\lambda$-adic Galois representation attached to a normalized cuspidal Hecke eigenform. That theorem gives both unramifiedness away from $N\ell$ and the Frobenius trace and determinant formulas at each good prime $p \nmid N\ell$. Once those two scalar invariants are known, the stated characteristic polynomial follows from the elementary formula for the characteristic polynomial of a two-dimensional linear operator. [/proofplan]

[step:Apply Deligne's theorem at a prime away from $N\ell$]Fix a prime number $p$ with $p \nmid N\ell$. Let $G_{\mathbb{Q}} := \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ denote the absolute [Galois group](/page/Galois%20Group), let $G_{\mathbb{Q}_p} \subset G_{\mathbb{Q}}$ be a decomposition subgroup at $p$, and let $I_p \trianglelefteq G_{\mathbb{Q}_p}$ be its inertia subgroup. Let $V_{f,\lambda}$ be the two-dimensional $K_{f,\lambda}$-[vector space](/page/Vector%20Space) on which \begin{align*} \rho_{f,\lambda}: G_{\mathbb{Q}} &\to GL(V_{f,\lambda}) \end{align*} acts. We use Deligne's modular Galois representation theorem in the following form: for a normalized cuspidal Hecke eigenform $f$ of weight $k$, level $N$, nebentypus $\chi$, and coefficient field $K_f$, and for every finite place $\lambda \mid \ell$ of $K_f$, there is a continuous two-dimensional $\lambda$-adic representation $\rho_{f,\lambda}$ such that, for each prime $p \nmid N\ell$, the restriction $\rho_{f,\lambda}|_{G_{\mathbb{Q}_p}}$ is unramified and arithmetic Frobenius satisfies \begin{align*} \operatorname{tr}(\rho_{f,\lambda}(\operatorname{Frob}_p)) &= a_p,\\ \det(\rho_{f,\lambda}(\operatorname{Frob}_p)) &= \chi(p)p^{k-1}. \end{align*} The hypotheses are exactly the ambient hypotheses on $f$ and $\lambda$, together with the good-prime condition $p \nmid N\ell$. Therefore $\rho_{f,\lambda}(\sigma)=\operatorname{id}_{V_{f,\lambda}}$ for every $\sigma \in I_p$. Hence $\rho_{f,\lambda}(\operatorname{Frob}_p)$ is defined up to conjugacy, and its trace and determinant are well-defined elements of $K_{f,\lambda}$. This proves the asserted unramifiedness, trace formula, and determinant formula for the chosen prime $p$.[/step]

[guided]We first isolate the local Galois-theoretic objects at the prime $p$. Define $G_{\mathbb{Q}} := \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. Choose a decomposition subgroup $G_{\mathbb{Q}_p} \subset G_{\mathbb{Q}}$ at $p$, and let $I_p \trianglelefteq G_{\mathbb{Q}_p}$ denote the inertia subgroup. Let $V_{f,\lambda}$ be the two-dimensional vector space over the completion $K_{f,\lambda}$ on which the representation acts, so that \begin{align*} \rho_{f,\lambda}: G_{\mathbb{Q}} &\to GL(V_{f,\lambda}) \end{align*} is a continuous $\lambda$-adic representation. We now invoke Deligne's modular Galois representation theorem. Its hypotheses are exactly the ambient hypotheses on $f$: $f$ is a normalized cuspidal Hecke eigenform of weight $k$, level $N$, nebentypus $\chi$, and coefficient field $K_f$, and $\lambda$ is a finite place of $K_f$ above $\ell$. The local condition required for the good-prime conclusion is $p \nmid N\ell$, which is the hypothesis fixed at the start of this step. The theorem has two relevant conclusions. First, it says that $\rho_{f,\lambda}$ is unramified at $p$, i.e. the inertia subgroup acts as the identity: \begin{align*} \rho_{f,\lambda}(\sigma) = \operatorname{id}_{V_{f,\lambda}} \quad \text{for every } \sigma \in I_p. \end{align*} This is why the conjugacy class of the arithmetic Frobenius element $\operatorname{Frob}_p$ has a well-defined image for purposes of trace and determinant: changing the lift by inertia does not change the resulting conjugacy class under $\rho_{f,\lambda}$. Second, Deligne's theorem identifies the Frobenius invariants with the Hecke data of $f$: \begin{align*} \operatorname{tr}(\rho_{f,\lambda}(\operatorname{Frob}_p)) &= a_p,\\ \det(\rho_{f,\lambda}(\operatorname{Frob}_p)) &= \chi(p)p^{k-1}. \end{align*} Thus the unramifiedness, trace identity, and determinant identity all follow directly from the good-prime part of Deligne's construction theorem.[/guided]

[step:Convert the Frobenius trace and determinant into the characteristic polynomial]Define the $K_{f,\lambda}$-linear automorphism \begin{align*} T_p: V_{f,\lambda} &\to V_{f,\lambda},\\ v &\mapsto \rho_{f,\lambda}(\operatorname{Frob}_p)(v). \end{align*} Since $\dim_{K_{f,\lambda}} V_{f,\lambda}=2$, the characteristic polynomial of $T_p$ is \begin{align*} \det(X\operatorname{id}_{V_{f,\lambda}} - T_p) = X^2 - \operatorname{tr}(T_p)X + \det(T_p). \end{align*} Substituting the trace and determinant formulas from the previous step gives \begin{align*} \det(X\operatorname{id}_{V_{f,\lambda}} - T_p) = X^2 - a_pX + \chi(p)p^{k-1}. \end{align*} Because $p \nmid N\ell$ was arbitrary, the conclusion holds for every prime $p \nmid N\ell$.[/step]

[guided]We now translate the two scalar identities into the characteristic polynomial. Define the $K_{f,\lambda}$-linear automorphism \begin{align*} T_p: V_{f,\lambda} &\to V_{f,\lambda},\\ v &\mapsto \rho_{f,\lambda}(\operatorname{Frob}_p)(v). \end{align*} This is an automorphism because $\rho_{f,\lambda}(\operatorname{Frob}_p)$ lies in $GL(V_{f,\lambda})$. The vector space $V_{f,\lambda}$ has dimension $2$ over $K_{f,\lambda}$. For any two-dimensional linear automorphism $T: V \to V$ over a field, the characteristic polynomial is \begin{align*} \det(X\operatorname{id}_V - T) = X^2 - \operatorname{tr}(T)X + \det(T). \end{align*} Applying this elementary formula to $T = T_p$ and $V = V_{f,\lambda}$ gives \begin{align*} \det(X\operatorname{id}_{V_{f,\lambda}} - T_p) = X^2 - \operatorname{tr}(T_p)X + \det(T_p). \end{align*} By the previous step, \begin{align*} \operatorname{tr}(T_p) &= a_p,\\ \det(T_p) &= \chi(p)p^{k-1}. \end{align*} Substituting these two identities into the characteristic polynomial formula yields \begin{align*} \det(X\operatorname{id}_{V_{f,\lambda}} - T_p) = X^2 - a_pX + \chi(p)p^{k-1}. \end{align*} Since the prime $p$ was only assumed to satisfy $p \nmid N\ell$, the same argument applies to every prime away from $N\ell$.[/guided]