Let $f=\sum_{n\ge 1}a_nq^n\in S_2(\Gamma_0(N))$ be a normalized newform, let $K_f := \mathbb{Q}(a_n : n \ge 1)$ be its Hecke field, let $\lambda\mid l$ be a prime of $K_f$, let $K_{f,\lambda}$ be the completion of $K_f$ at $\lambda$, and let $p\nmid Nl$. Then $\rho_{f,\lambda}: G_{\mathbb{Q}} \to GL_2(K_{f,\lambda})$ is unramified at $p$, and

\begin{align*} \det(1-X\rho_{f,\lambda}(\operatorname{Frob}_p))=1-a_pX+pX^2. \end{align*}

Here $a_p$ means $a_p(f)\in K_f$ viewed in $K_{f,\lambda}$ through the natural completion map $K_f\hookrightarrow K_{f,\lambda}$.