Let $F$ be a totally real field, and let $\pi$ be a regular algebraic cuspidal automorphic representation of $GL_2(\mathbb A_F)$ associated with a Hilbert modular eigenform. Let $E_\pi \subset \mathbb C$ be the number field generated by the Hecke eigenvalues of $\pi$ at finite unramified places. For each finite place $\lambda$ of $E_\pi$ above a rational prime $\ell$, after replacing $E_{\pi,\lambda}$ by a finite extension $K_\lambda$ if necessary, there is a continuous semisimple representation

\begin{align*} \rho_{\pi,\lambda}: G_F \to GL_2(K_\lambda) \end{align*}

which is unramified at every finite place $v \nmid \ell$ at which $\pi_v$ is unramified, and for such $v$ satisfies

\begin{align*} \det\left(1 - X\rho_{\pi,\lambda}(\operatorname{Frob}_v)\right) = 1 - a_v(\pi)X + \omega_\pi(\varpi_v)q_v X^2, \end{align*}

where $q_v$ is the cardinality of the residue field of $F_v$, $a_v(\pi)$ is the standard Hecke eigenvalue of $\pi_v$, $\omega_\pi$ is the central character of $\pi$, $\varpi_v$ is a uniformizer of $F_v$, and $\operatorname{Frob}_v$ denotes geometric Frobenius.