[proofplan] The proof is an application of [Deligne's construction of the $\lambda$-adic representation attached to a newform](/page/Deligne%20Galois%20Representation), with $K_{f,\lambda}$ the completion of the Hecke field at $\lambda$. That construction realizes $\rho_{f,\lambda}$ on the $\lambda$-adic Tate module of the $f$-isotypic quotient of the Jacobian of the modular curve $X_0(N)$, so good reduction away from $Nl$ gives unramifiedness at such primes $p$. With the arithmetic Frobenius normalization used in the theorem, the [Eichler-Shimura relation](/page/Eichler-Shimura%20Relation) gives trace $a_p$ and determinant $p$ on $\rho_{f,\lambda}(\operatorname{Frob}_p)$, which is exactly the displayed characteristic polynomial. [/proofplan]

[step:Recall the Galois representation attached to the newform]Let $K_f := \mathbb{Q}(a_n : n \ge 1)$ be the [Hecke field](/page/Hecke%20Field) of $f$, let $K_{f,\lambda}$ denote the completion of $K_f$ at the prime $\lambda \mid l$, and let \begin{align*} \rho_{f,\lambda}: G_{\mathbb{Q}} &\to GL_2(K_{f,\lambda}) \end{align*} be the [$\lambda$-adic Galois representation attached to $f$](/page/Deligne%20Galois%20Representation). Here $G_{\mathbb{Q}} := \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, and $X_0(N)$ denotes the modular curve over $\mathbb{Q}$ classifying elliptic curves equipped with a cyclic subgroup of order $N$. By [Deligne's construction of the Galois representation attached to a normalized newform](/page/Deligne%20Galois%20Representation), $\rho_{f,\lambda}$ is obtained from the $f$-isotypic summand of the $\lambda$-adic Tate module of the Jacobian $J_0(N)$ of $X_0(N)$, and for every prime $p \nmid Nl$ its Frobenius characteristic polynomial is governed by the [Eichler-Shimura relation](/page/Eichler-Shimura%20Relation) at $p$.[/step]

[guided]We first name all objects that the theorem uses implicitly. The Hecke field is \begin{align*} K_f := \mathbb{Q}(a_n : n \ge 1), \end{align*} and $K_{f,\lambda}$ is its completion at the prime $\lambda \mid l$. The representation in the statement is a continuous homomorphism \begin{align*} \rho_{f,\lambda}: G_{\mathbb{Q}} &\to GL_2(K_{f,\lambda}), \end{align*} where $G_{\mathbb{Q}} := \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. The curve $X_0(N)$ is the modular curve over $\mathbb{Q}$ classifying elliptic curves with a cyclic subgroup of order $N$, and $J_0(N)$ is its Jacobian. For a normalized weight two newform $f$, Deligne's construction realizes $\rho_{f,\lambda}$ on the two-dimensional $f$-isotypic part of the $\lambda$-adic Tate module of $J_0(N)$. This is exactly the construction needed here because it connects two operations: the arithmetic action of Frobenius on Tate modules and the analytic Hecke eigenvalue $a_p$ of the newform.[/guided]

[step:Use good reduction away from $Nl$ to obtain unramifiedness at $p$]Since $p \nmid N$, the standard integral model of $X_0(N)$ is smooth at $p$, and hence the Jacobian $J_0(N)$ has good reduction at $p$. Since also $p \ne l$, the $\lambda$-adic Tate module is an $l$-adic representation at a prime different from $p$. The [Neron-Ogg-Shafarevich criterion](/page/Neron-Ogg-Shafarevich%20Criterion), applied to the abelian variety $J_0(N)$ with good reduction at $p$, implies that the inertia subgroup $I_p \subset G_{\mathbb{Q}}$ acts as the identity on the $\lambda$-adic Tate module of $J_0(N)$. Passing to the $f$-isotypic quotient preserves identity action of inertia, so $I_p$ acts as the identity on the two-dimensional representation space of $\rho_{f,\lambda}$. Hence $\rho_{f,\lambda}$ is [unramified at $p$](/page/Unramified%20Galois%20Representation).[/step]

[guided]The hypothesis $p \nmid Nl$ has two independent roles. First, $p \nmid N$ is the good-reduction condition: the standard integral model of the modular curve $X_0(N)$ is smooth at $p$, and therefore its Jacobian $J_0(N)$ has good reduction at $p$. Second, $p \ne l$ means that the $\lambda$-adic Tate module is an $l$-adic Tate module at a prime different from $p$. The [Neron-Ogg-Shafarevich criterion](/page/Neron-Ogg-Shafarevich%20Criterion) applies in exactly this setting: for an abelian variety with good reduction at $p$, its $l$-adic Tate module is unramified at $p$ for every $l \ne p$. Applying this to $J_0(N)$ shows that the inertia subgroup $I_p \subset G_{\mathbb{Q}}$ acts as the identity on the full Tate module. The representation $\rho_{f,\lambda}$ is obtained from the $f$-isotypic quotient of that module, and a quotient of a representation on which $I_p$ acts as the identity is again a representation on which $I_p$ acts as the identity. Therefore $\rho_{f,\lambda}$ is [unramified at $p$](/page/Unramified%20Galois%20Representation).[/guided]

[step:Apply Eichler-Shimura to compute the Frobenius characteristic polynomial]Because $p \nmid Nl$, [arithmetic Frobenius](/page/Frobenius%20Element) at $p$ is defined up to conjugacy on the unramified representation $\rho_{f,\lambda}$. Let $V_{f,\lambda}$ denote the two-dimensional $K_{f,\lambda}$-[vector space](/page/Vector%20Space) underlying $\rho_{f,\lambda}$, and let $\operatorname{Id}_{V_{f,\lambda}}: V_{f,\lambda} \to V_{f,\lambda}$ be its identity map. With Deligne's arithmetic Frobenius normalization, the [Eichler-Shimura relation](/page/Eichler-Shimura%20Relation) on the $f$-isotypic summand gives \begin{align*} \operatorname{tr}\left(\rho_{f,\lambda}(\operatorname{Frob}_p)\right) &= a_p, & \det\left(\rho_{f,\lambda}(\operatorname{Frob}_p)\right) &= p, \end{align*} because $f$ has weight $2$ and nebentypus equal to $1$. For an endomorphism of a two-dimensional vector space, the characteristic polynomial is determined by trace and determinant, so \begin{align*} \det\left(T\operatorname{Id}_{V_{f,\lambda}} - \rho_{f,\lambda}(\operatorname{Frob}_p)\right) = T^2 - a_pT + p. \end{align*} Substituting $T = X^{-1}$ and multiplying by $X^2$ gives \begin{align*} \det\left(1 - X\rho_{f,\lambda}(\operatorname{Frob}_p)\right) = 1 - a_pX + pX^2. \end{align*} This is the required identity.[/step]

[guided]Once unramifiedness is known, the conjugacy class of $\rho_{f,\lambda}(\operatorname{Frob}_p)$ is defined, so its trace, determinant, and characteristic polynomial are meaningful. We must also fix the Frobenius convention: $\operatorname{Frob}_p$ in the theorem is arithmetic Frobenius. With Deligne's normalization of the [$\lambda$-adic Galois representation attached to $f$](/page/Deligne%20Galois%20Representation), the [Eichler-Shimura relation](/page/Eichler-Shimura%20Relation) gives the trace and determinant formulas \begin{align*} \operatorname{tr}\left(\rho_{f,\lambda}(\operatorname{Frob}_p)\right) &= a_p, & \det\left(\rho_{f,\lambda}(\operatorname{Frob}_p)\right) &= p. \end{align*} Here $a_p$ enters because $f$ is an eigenvector for the Hecke operator $T_p$ with eigenvalue $a_p$, and the determinant is $p$ because the form has weight $2$ and nebentypus equal to $1$. Let $V_{f,\lambda}$ denote the two-dimensional $K_{f,\lambda}$-vector space underlying $\rho_{f,\lambda}$, and let $\operatorname{Id}_{V_{f,\lambda}}: V_{f,\lambda} \to V_{f,\lambda}$ be its identity map. For any endomorphism $A: V_{f,\lambda} \to V_{f,\lambda}$ of a two-dimensional vector space, \begin{align*} \det(T\operatorname{Id}_{V_{f,\lambda}} - A) = T^2 - \operatorname{tr}(A)T + \det(A). \end{align*} Applying this with $A = \rho_{f,\lambda}(\operatorname{Frob}_p)$ gives \begin{align*} \det\left(T\operatorname{Id}_{V_{f,\lambda}} - \rho_{f,\lambda}(\operatorname{Frob}_p)\right) = T^2 - a_pT + p. \end{align*} The theorem writes the same polynomial in the reciprocal variable. Indeed, \begin{align*} \det\left(1 - X\rho_{f,\lambda}(\operatorname{Frob}_p)\right) &= X^2\det\left(X^{-1}\operatorname{Id}_{V_{f,\lambda}} - \rho_{f,\lambda}(\operatorname{Frob}_p)\right) \\ &= X^2\left(X^{-2} - a_pX^{-1} + p\right) \\ &= 1 - a_pX + pX^2. \end{align*} This proves both asserted conclusions: unramifiedness at $p$ and the Frobenius characteristic polynomial formula.[/guided]