The general Langlands reciprocity principle predicts that algebraic automorphic representations should correspond to compatible systems of Galois representations with matching local $L$-factors at almost all places. In the classical case proved here, let $f \in S_k(\Gamma_0(N), \varepsilon)$ be a holomorphic normalized cuspidal Hecke eigenform of weight $k \ge 2$, level $N \in \mathbb N$, and nebentypus Dirichlet character $\varepsilon$, and let $E_f \subset \mathbb C$ be the number field generated by the Hecke eigenvalues of $f$. For every finite place $\lambda$ of $E_f$, with completion $E_{f,\lambda}$ and residue characteristic $\ell$, there is a continuous semisimple representation

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

such that for every rational prime $p \nmid N\ell$, the representation $\rho_{f,\lambda}$ is unramified at $p$ and, for arithmetic Frobenius $\operatorname{Frob}_p$, satisfies

\begin{align*} \det\left(1 - T\rho_{f,\lambda}(\operatorname{Frob}_p)\right) &= 1 - a_p(f)T + \varepsilon(p)p^{k-1}T^2. \end{align*}

The coefficients $a_p(f)$ and $\varepsilon(p)$ are regarded as elements of $E_{f,\lambda}$ through the natural embedding $E_f\hookrightarrow E_{f,\lambda}$.

Consequently $(\rho_{f,\lambda})_\lambda$ is a weakly compatible system: for each prime $p \nmid N$, the above Frobenius polynomial lies in $E_f[T]$ and is independent of $\lambda$ for all finite places $\lambda \nmid p$.