Let $N \geq 1$ be an integer, let $\varepsilon: (\mathbb{Z}/N\mathbb{Z})^\times \to \mathbb{C}^\times$ be a Dirichlet character, and let $f \in S_2(\Gamma_1(N), \varepsilon)$ be a normalized newform with Fourier expansion $a_1(f)=1$. Let $K_f := \mathbb{Q}(a_n(f) : n \geq 1)$ be the coefficient field of $f$. For every prime $\lambda \subset \mathcal{O}_{K_f}$ lying above a rational prime $l$, let $K_{f,\lambda}$ denote the completion of $K_f$ at $\lambda$.

There exists a continuous semisimple representation

\begin{align*} \rho_{f,\lambda}: \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \longrightarrow GL_2(K_{f,\lambda}) \end{align*}

such that for every rational prime $p$ with $p \nmid Nl$, the representation $\rho_{f,\lambda}$ is unramified at $p$, and for any arithmetic Frobenius element $\operatorname{Frob}_p$ at $p$ one has

\begin{align*} \operatorname{tr}\rho_{f,\lambda}(\operatorname{Frob}_p) &= a_p(f), \\ \det\rho_{f,\lambda}(\operatorname{Frob}_p) &= \varepsilon(p)p. \end{align*}

Here $a_p(f)$ and $\varepsilon(p)$ are viewed in $K_{f,\lambda}$ through the natural embedding of the coefficient field $K_f$ into its completion $K_{f,\lambda}$.

Equivalently, for every rational prime $p \nmid Nl$,

\begin{align*} \det\left(XI_2 - \rho_{f,\lambda}(\operatorname{Frob}_p)\right) = X^2 - a_p(f)X + \varepsilon(p)p. \end{align*}