[proofplan] We use Deligne's construction of the Galois representation attached to a normalized cuspidal Hecke eigenform of weight at least $2$. The construction realizes the Hecke eigensystem of $f$ inside the $\ell$-adic etale cohomology of the modular curve of level $\Gamma_1(N)$ with the usual weight-$k$ coefficient sheaf. The Eichler-Shimura relation identifies the Frobenius characteristic polynomials away from $N\ell$, and taking the semisimplification gives the continuous semisimple representation required by the theorem. [/proofplan] [step:Extract the Hecke eigensystem determined by the normalized eigenform] Let $a_n(f) \in \overline{\mathbb Q}$ denote the $n$th Fourier coefficient of the normalized eigenform $f$, so $a_1(f)=1$. Since $f$ is a Hecke eigenform, for each prime $p \nmid N$ the Hecke operator $T_p$ acts on the line spanned by $f$ through the scalar $a_p(f)$. Let $\varepsilon_f:(\mathbb Z/N\mathbb Z)^\times \to \overline{\mathbb Q}^{\times}$ denote the nebentypus character of $f$, equivalently the system of eigenvalues for the diamond operators on the line generated by $f$. Choose an embedding $\iota_\ell:\overline{\mathbb Q}\hookrightarrow \overline{\mathbb Q}_\ell$. This gives an $\ell$-adic Hecke eigensystem by sending $a_p(f)$ and $\varepsilon_f(p)$ to $\iota_\ell(a_p(f))$ and $\iota_\ell(\varepsilon_f(p))$ for every prime $p\nmid N$. [/step] [step:Apply Deligne's construction on the cohomology of the modular curve] Let $G_{\mathbb Q}:=\operatorname{Gal}(\overline{\mathbb Q}/\mathbb Q)$ be the absolute [Galois group](/page/Galois%20Group). By Deligne's construction of Galois representations attached to normalized cuspidal Hecke eigenforms of weight $k\ge 2$, applied to the modular curve of level $\Gamma_1(N)$ and to the coefficient sheaf of weight $k-2$, the $\iota_\ell$-localized $f$-isotypic quotient of etale cohomology yields a continuous representation \begin{align*} \rho_{f,\ell}:G_{\mathbb Q}&\longrightarrow GL_2(\overline{\mathbb Q}_\ell) \end{align*} which is unramified at every prime $p\nmid N\ell$ and satisfies \begin{align*} \det\left(1-X\rho_{f,\ell}(\operatorname{Frob}_p)\right) &=1-\iota_\ell(a_p(f))X+\iota_\ell(\varepsilon_f(p))p^{k-1}X^2 \end{align*} for every prime $p\nmid N\ell$, where $\operatorname{Frob}_p\in G_{\mathbb Q}$ denotes an arithmetic Frobenius element at $p$. The hypotheses of Deligne's construction are exactly the hypotheses in the theorem: $f$ is cuspidal because $f\in S_k(\Gamma_1(N))$, $f$ is normalized by assumption, $f$ is a simultaneous Hecke eigenform by assumption, and the condition $k\ge 2$ is the weight range in which the construction uses the algebraic coefficient sheaf of weight $k-2$ on the modular curve. [guided] We now use the deep input in the theorem: Deligne's construction. The input data required by that construction are a normalized cuspidal Hecke eigenform of level $\Gamma_1(N)$ and weight $k\ge 2$, together with an embedding of its coefficient field into $\overline{\mathbb Q}_\ell$. These are available here: $f\in S_k(\Gamma_1(N))$ gives cuspidality and level, the theorem assumes $f$ is normalized and a Hecke eigenform, the theorem assumes $k\ge 2$, and we fixed an embedding $\iota_\ell:\overline{\mathbb Q}\hookrightarrow \overline{\mathbb Q}_\ell$ in the previous step. Deligne's construction realizes the Hecke eigensystem of $f$ in the etale cohomology of the modular curve of level $\Gamma_1(N)$ with the coefficient sheaf corresponding to the algebraic representation of weight $k-2$. The output is a continuous representation \begin{align*} \rho_{f,\ell}:G_{\mathbb Q}&\longrightarrow GL_2(\overline{\mathbb Q}_\ell). \end{align*} The Eichler-Shimura relation in that cohomology identifies the action of an arithmetic Frobenius element at every prime $p\nmid N\ell$ with the Hecke operator $T_p$ and the diamond operator eigenvalue. Consequently the characteristic polynomial is \begin{align*} \det\left(1-X\rho_{f,\ell}(\operatorname{Frob}_p)\right) &=1-\iota_\ell(a_p(f))X+\iota_\ell(\varepsilon_f(p))p^{k-1}X^2. \end{align*} This formula is the precise sense in which the representation is attached to the eigenform $f$: the Frobenius traces recover the Hecke eigenvalues, and the Frobenius determinants recover the nebentypus times $p^{k-1}$. [/guided] [/step] [step:Pass to the semisimplification and keep the same Frobenius data] Let $\rho_{f,\ell}^{\operatorname{ss}}:G_{\mathbb Q}\to GL_2(\overline{\mathbb Q}_\ell)$ denote the semisimplification of the representation obtained from Deligne's cohomological construction. Semisimplification preserves characteristic polynomials of every group element, so for every prime $p\nmid N\ell$ we still have \begin{align*} \det\left(1-X\rho_{f,\ell}^{\operatorname{ss}}(\operatorname{Frob}_p)\right) &=1-\iota_\ell(a_p(f))X+\iota_\ell(\varepsilon_f(p))p^{k-1}X^2. \end{align*} The semisimplification is semisimple by definition. We replace $\rho_{f,\ell}$ by $\rho_{f,\ell}^{\operatorname{ss}}$ and keep the notation $\rho_{f,\ell}$ for this semisimple representation. [/step] [step:Conclude that the representation is attached to $f$] The preceding steps produce, for the fixed prime $\ell$, a continuous semisimple representation \begin{align*} \rho_{f,\ell}:G_{\mathbb Q}&\longrightarrow GL_2(\overline{\mathbb Q}_\ell) \end{align*} whose Frobenius characteristic polynomial at each prime $p\nmid N\ell$ is determined by the Hecke eigenvalue $a_p(f)$ and the nebentypus value $\varepsilon_f(p)$ as above. This is exactly the attachment condition described in the theorem. Since $\ell$ was arbitrary, the construction gives such a representation for every prime $\ell$. [/step]