[proofplan] We use Deligne's cohomological construction of the Galois representation attached to a normalized cuspidal eigenform. The construction realizes the $K_{f,\lambda}$-representation $\rho_{f,\lambda}$ as the $f$-isotypic direct summand of the first étale cohomology of the modular curve with coefficients in the symmetric-power local system. The Hodge-de Rham computation of the corresponding de Rham realization gives Hodge numbers in degrees $0$ and $k-1$, and the Hodge-Tate comparison theorem transfers exactly these graded degrees to the restriction to $G_{\mathbb Q_\ell}$. [/proofplan]

[step:Realize Deligne's representation in modular-curve cohomology] Let $E := K_f$ be the Hecke field of $f$, let $E_\lambda := K_{f,\lambda}$ be the completion at the finite place $\lambda \mid \ell$, and let \begin{align*} V_{f,\lambda}: G_{\mathbb Q} &\to \operatorname{Aut}_{E_\lambda}(E_\lambda^2) \end{align*} denote the two-dimensional $E_\lambda$-linear representation underlying $\rho_{f,\lambda}$. Since $f \in S_k(\Gamma_1(N),\varepsilon)$ is normalized and cuspidal with $k \geq 2$, Deligne's construction of Galois representations attached to cuspidal eigenforms applies to $f$ and gives $V_{f,\lambda}$ as the $f$-isotypic direct summand of the étale realization of the motive cut out from \begin{align*} H^1_{\mathrm{et}}\bigl(X_1(N)_{\overline{\mathbb Q}},\mathcal V_{k-2,E_\lambda}\bigr), \end{align*} where $X_1(N)$ is the compactified modular curve of level $\Gamma_1(N)$ and $\mathcal V_{k-2,E_\lambda}$ is the $E_\lambda$-local system obtained from the $(k-2)$-nd symmetric power of the relative first cohomology of the universal elliptic curve. This is precisely the content of the [Deligne construction for modular eigenforms](/theorems/???), applied with weight $k$, level $\Gamma_1(N)$, nebentypus $\varepsilon$, coefficient field $E$, and place $\lambda$. [/step]

[step:Compute the Hodge filtration on the de Rham realization] Let $M_f$ denote the rank-two $E$-motive over $\mathbb Q$ cut out by the Hecke eigensystem of $f$ in the same modular-curve cohomology. Its de Rham realization \begin{align*} M_{f,\mathrm{dR}}: \operatorname{Spec}\mathbb Q &\to \operatorname{Vect}_E \end{align*} is a two-dimensional filtered $E$-[vector space](/page/Vector%20Space). The Hodge-de Rham calculation for the modular motive attached to a weight-$k$ cuspidal eigenform states that the associated graded pieces satisfy \begin{align*} \operatorname{gr}^{0} M_{f,\mathrm{dR}} &\neq 0, & \operatorname{gr}^{k-1} M_{f,\mathrm{dR}} &\neq 0, & \operatorname{gr}^{i} M_{f,\mathrm{dR}} &= 0 \quad \text{for } i \notin \{0,k-1\}. \end{align*} Moreover each of the two nonzero graded pieces has $E$-dimension $1$. The hypotheses required by the [Hodge filtration computation for modular eigenform motives](/theorems/???) are exactly that $f$ is a normalized cuspidal Hecke eigenform of weight $k \geq 2$ and level $\Gamma_1(N)$ with nebentypus $\varepsilon$, so the theorem applies to the present $f$. [/step]

[step:Apply the Hodge-Tate comparison theorem at $\lambda \mid \ell$] Restrict $V_{f,\lambda}$ to the decomposition group $G_{\mathbb Q_\ell} \subset G_{\mathbb Q}$. Since $M_f$ is a pure motive over $\mathbb Q$ whose $\lambda$-adic étale realization is $V_{f,\lambda}$ and whose de Rham realization is $M_{f,\mathrm{dR}}$, the [Hodge-Tate comparison theorem for de Rham motives](/theorems/???) applies after base change from $E$ to $E_\lambda$. It gives a $G_{\mathbb Q_\ell}$-equivariant Hodge-Tate decomposition after extension of scalars to $\mathbb C_\ell$ and identifies the Hodge-Tate weights with the indices $i$ for which $\operatorname{gr}^{i} M_{f,\mathrm{dR}}$ is nonzero. By the preceding step, those indices are exactly $0$ and $k-1$, each with multiplicity $1$. [/step]

[step:Conclude the stated Hodge-Tate weights] The representation $V_{f,\lambda}$ is the representation underlying $\rho_{f,\lambda}$, and restriction to $G_{\mathbb Q_\ell}$ does not change the two-dimensional $E_\lambda$-vector space on which it acts. Therefore $\rho_{f,\lambda}|_{G_{\mathbb Q_\ell}}$ is Hodge-Tate, and its multiset of Hodge-Tate weights is \begin{align*} \{0,k-1\}. \end{align*} This is the asserted conclusion. [/step]