Let $f\in S_k(\Gamma_1(N),\varepsilon)$ be a normalized cuspidal eigenform with $k\ge 2$, and let $\lambda\mid \ell$ be a finite place of its coefficient field $K_f$. There is a continuous semisimple representation
for every $p\nmid N\ell$, where $\operatorname{Frob}_p$ denotes an arithmetic Frobenius element at $p$, and $a_p(f)$ and $\varepsilon(p)$ are viewed in $K_{f,\lambda}$ through the natural completion map from the coefficient field $K_f$.