Let $N \in \mathbb{N}$, let $k \geq 2$, and let $f \in S_k(\Gamma_1(N), \varepsilon)$ be a normalized cuspidal Hecke eigenform with Fourier expansion $\sum_{n=1}^{\infty} a_n(f)q^n$. Let $E_f$ be the number field generated by the Hecke eigenvalues of $f$, let $\lambda$ be a finite place of $E_f$ lying above the rational prime $\ell$, and let

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

be Deligne's $\lambda$-adic Galois representation attached to $f$. If $p$ is a rational prime such that $p \nmid N\ell$, and $\operatorname{Frob}_p$ denotes an arithmetic Frobenius element at $p$, then

\begin{align*} \det\left(I_2 - X\rho_{f,\lambda}(\operatorname{Frob}_p)\right) =1-a_p(f)X+\varepsilon(p)p^{k-1}X^2. \end{align*}

The coefficients are viewed in $E_{f,\lambda}$ through the natural embedding $E_f\hookrightarrow E_{f,\lambda}$. Consequently, the good Euler factor of $f$ at $p$ is

\begin{align*} L_p(f,s)=\det\left(I_2-p^{-s}\rho_{f,\lambda}(\operatorname{Frob}_p)\right)^{-1}. \end{align*}