[proofplan]
We use Deligne's $l$-adic representation theorem for normalized cuspidal Hecke eigenforms, specialized to weight $2$ and nebentypus character $\varepsilon$. The hypotheses in the statement put $f$ exactly in the scope of that theorem, so it produces a continuous two-dimensional $K_{f,\lambda}$-linear Galois representation with the Euler factor prescribed by the Hecke polynomial at every prime $p \nmid Nl$. We then read the trace and determinant from that characteristic polynomial and replace the representation by its semisimplification, which preserves the Frobenius characteristic polynomials.
[/proofplan]
[step:Apply Deligne's representation theorem to the normalized newform]
Let $G_{\mathbb Q} := \operatorname{Gal}(\overline{\mathbb Q}/\mathbb Q)$ denote the absolute [Galois group](/page/Galois%20Group). Let $K_{f,\lambda}$ denote the completion of the coefficient field $K_f$ at the prime $\lambda \mid l$. Deligne's $l$-adic representation theorem for normalized cuspidal Hecke eigenforms states the following: if $g \in S_k(\Gamma_1(M),\psi)$ is a normalized Hecke eigenform with coefficient field $K_g$, and if $\mu \mid l$ is a prime of $K_g$, then there is a continuous representation
\begin{align*}
\rho_{g,\mu}: G_{\mathbb Q} \longrightarrow GL_2(K_{g,\mu})
\end{align*}
which is unramified at every prime $p \nmid Ml$ and satisfies
\begin{align*}
\det\left(I_2 - X\rho_{g,\mu}(\operatorname{Frob}_p)\right)
= 1 - a_p(g)X + \psi(p)p^{k-1}X^2
\end{align*}
for arithmetic Frobenius $\operatorname{Frob}_p$ and an indeterminate $X$.
The theorem statement assumes that $f \in S_2(\Gamma_1(N),\varepsilon)$ is a normalized newform. In particular, $f$ is a normalized cuspidal Hecke eigenform of weight $k = 2$, level $M = N$, nebentypus character $\psi = \varepsilon$, coefficient field $K_g = K_f$, and chosen prime $\mu = \lambda$. Therefore Deligne's theorem applies and gives a continuous representation
\begin{align*}
\rho^{\mathrm{Del}}_{f,\lambda}: G_{\mathbb Q} \longrightarrow GL_2(K_{f,\lambda})
\end{align*}
which is unramified at every prime $p \nmid Nl$ and satisfies
\begin{align*}
\det\left(I_2 - X\rho^{\mathrm{Del}}_{f,\lambda}(\operatorname{Frob}_p)\right)
= 1 - a_p(f)X + \varepsilon(p)pX^2.
\end{align*}
[/step]
[step:Read off the Frobenius trace and determinant]
Fix a prime $p \nmid Nl$. Since $\rho^{\mathrm{Del}}_{f,\lambda}$ is unramified at $p$, the conjugacy class of $\rho^{\mathrm{Del}}_{f,\lambda}(\operatorname{Frob}_p)$ is defined. For any matrix $A \in GL_2(K_{f,\lambda})$, the characteristic identity is
\begin{align*}
\det(I_2 - XA) = 1 - \operatorname{tr}(A)X + \det(A)X^2.
\end{align*}
Applying this identity to $A = \rho^{\mathrm{Del}}_{f,\lambda}(\operatorname{Frob}_p)$ and comparing coefficients with Deligne's formula gives
\begin{align*}
\operatorname{tr}\rho^{\mathrm{Del}}_{f,\lambda}(\operatorname{Frob}_p) &= a_p(f),\\
\det\rho^{\mathrm{Del}}_{f,\lambda}(\operatorname{Frob}_p) &= \varepsilon(p)p.
\end{align*}
[/step]
[step:Pass to the semisimplification without changing Frobenius polynomials]
Let $V := K_{f,\lambda}^2$ be the two-dimensional $K_{f,\lambda}$-[vector space](/page/Vector%20Space) on which $\rho^{\mathrm{Del}}_{f,\lambda}$ acts. Let $\rho_{f,\lambda}: G_{\mathbb Q} \to GL_2(K_{f,\lambda})$ denote the semisimplification of the $K_{f,\lambda}[G_{\mathbb Q}]$-module $V$. Semisimplification preserves the characteristic polynomial of every group element acting on a finite-dimensional representation, because the characteristic polynomial is the product of the characteristic polynomials on the successive composition factors. Hence, for every $\sigma \in G_{\mathbb Q}$,
\begin{align*}
\det(I_2 - X\rho_{f,\lambda}(\sigma))
= \det(I_2 - X\rho^{\mathrm{Del}}_{f,\lambda}(\sigma)).
\end{align*}
In particular, for every prime $p \nmid Nl$, the representation $\rho_{f,\lambda}$ is unramified at $p$ and satisfies
\begin{align*}
\operatorname{tr}\rho_{f,\lambda}(\operatorname{Frob}_p) &= a_p(f),\\
\det\rho_{f,\lambda}(\operatorname{Frob}_p) &= \varepsilon(p)p.
\end{align*}
By construction $\rho_{f,\lambda}$ is semisimple and continuous, so it is the representation required in the theorem statement.
[/step]
