[proofplan]
The proof uses Deligne's construction of the $\ell$-adic Galois representation attached to the normalized cuspidal eigenform $f$. At primes $p \nmid N\ell$, the characteristic polynomial of geometric Frobenius on this representation is exactly $X^2-a_p(f)X+\varepsilon(p)p^{k-1}$. Deligne's purity theorem, obtained from the [Weil conjectures](/theorems/2199), says that the Frobenius eigenvalues on the relevant cohomological piece have complex absolute value $p^{(k-1)/2}$. The stated Hecke bound then follows from the identity $a_p(f)=\alpha_p+\beta_p$ and the triangle inequality, and the choice of an auxiliary prime $\ell \ne p$ removes the restriction $p \ne \ell$.
[/proofplan]
[step:Attach Deligne's Galois representation to the eigenform]
Fix a prime $\ell$. Let $\overline{K_f}$ be an [algebraic closure](/page/Algebraic%20Closure) of $K_f$, and choose a prime $\lambda$ of $K_f$ above $\ell$. Write $K_{f,\lambda}$ for the completion of $K_f$ at $\lambda$, and let $\overline{K_{f,\lambda}}$ be an algebraic closure of $K_{f,\lambda}$.
We use the standard convention that $S_k(\Gamma_1(N),\varepsilon)$ denotes the space of holomorphic cusp forms of weight $k$ and nebentypus character $\varepsilon$ for $\Gamma_1(N)$, and that $f$ is normalized by $a_1(f)=1$ and is an eigenvector for all Hecke operators. Since $k \ge 2$, the algebraic local system used below is $\operatorname{Sym}^{k-2}$ of the relative first cohomology of the universal elliptic curve over the modular curve.
By Deligne's construction of the Galois representation attached to a normalized cuspidal Hecke eigenform of level $\Gamma_1(N)$ and nebentypus $\varepsilon$, obtained from the $f$-isotypic summand of parabolic étale cohomology of the modular curve with coefficients in this $\operatorname{Sym}^{k-2}$ local system, there is a continuous representation
\begin{align*}
\rho_{f,\lambda}: \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to GL_2(K_{f,\lambda})
\end{align*}
with the following property: for every prime $p \nmid N\ell$, the representation $\rho_{f,\lambda}$ is unramified at $p$, and for a geometric Frobenius element $\operatorname{Frob}_p$ at $p$,
\begin{align*}
\det\left(XI_2-\rho_{f,\lambda}(\operatorname{Frob}_p)\right)
=
X^2-a_p(f)X+\varepsilon(p)p^{k-1}.
\end{align*}
Here $I_2$ denotes the $2 \times 2$ identity matrix. The external input in this step is Deligne's construction of $\ell$-adic Galois representations attached to normalized cuspidal modular eigenforms, including the compatibility of the Hecke polynomial with geometric Frobenius at every prime $p \nmid N\ell$.
[/step]
[step:Apply Deligne purity to the Frobenius eigenvalues]
Let $p$ be a prime with $p \nmid N\ell$. Let $\alpha_p,\beta_p \in \overline{K_f}$ be the two roots of
\begin{align*}
X^2-a_p(f)X+\varepsilon(p)p^{k-1}.
\end{align*}
Choose an embedding of fields
\begin{align*}
jmath_\lambda:\overline{K_f} \hookrightarrow \overline{K_{f,\lambda}}
\end{align*}
extending the completion map $K_f \to K_{f,\lambda}$. Under this embedding, $jmath_\lambda(\alpha_p)$ and $jmath_\lambda(\beta_p)$ are the two roots in $\overline{K_{f,\lambda}}$ of the characteristic polynomial
\begin{align*}
\det\left(XI_2-\rho_{f,\lambda}(\operatorname{Frob}_p)\right).
\end{align*}
Thus they are precisely the eigenvalues of the $K_{f,\lambda}$-linear endomorphism $\rho_{f,\lambda}(\operatorname{Frob}_p)$ after extending scalars from $K_{f,\lambda}$ to $\overline{K_{f,\lambda}}$.
We now verify the hypotheses of the purity input. The prime condition $p \nmid N\ell$ gives good reduction of the modular curve and of the coefficient local system at $p$, and it also makes $\rho_{f,\lambda}$ unramified at $p$. The representation $\rho_{f,\lambda}$ occurs as the $f$-isotypic direct summand of parabolic étale cohomology of the smooth modular curve with coefficient system $\operatorname{Sym}^{k-2}$; equivalently, it is the two-dimensional summand cut out by the Hecke idempotent from the interior cohomology of the compactified modular curve. This construction removes the Eisenstein boundary contribution, so cuspidality of $f$ places the summand in pure cohomological degree $1$ rather than in boundary cohomology. Since the coefficient system $\operatorname{Sym}^{k-2}$ has weight $k-2$ and degree-$1$ cohomology contributes one additional weight, this Hecke summand is pure of weight $k-1$.
Deligne's purity theorem for this pure Hecke summand says that every Frobenius eigenvalue on it is a Weil $p$-number of weight $k-1$. Therefore, for every field embedding $\iota:\overline{K_f}\hookrightarrow\mathbb{C}$,
\begin{align*}
|\iota(\alpha_p)|=p^{(k-1)/2},
\qquad
|\iota(\beta_p)|=p^{(k-1)/2}.
\end{align*}
The external input in this step is Deligne's purity theorem for the parabolic étale cohomology summand of modular curves cut out by a normalized cuspidal Hecke eigenform.
[/step]
[step:Recover the Hecke eigenvalue bound from the roots]
Since $\alpha_p$ and $\beta_p$ are the roots of
\begin{align*}
X^2-a_p(f)X+\varepsilon(p)p^{k-1},
\end{align*}
Vieta's formula gives
\begin{align*}
a_p(f)=\alpha_p+\beta_p.
\end{align*}
Let $\iota:\overline{K_f}\hookrightarrow\mathbb{C}$ be any field embedding. Applying $\iota$ to the identity above and using the triangle inequality in $\mathbb{C}$ gives
\begin{align*}
|\iota(a_p(f))|
&=
|\iota(\alpha_p)+\iota(\beta_p)| \\
&\le
|\iota(\alpha_p)|+|\iota(\beta_p)| \\
&=
p^{(k-1)/2}+p^{(k-1)/2} \\
&=
2p^{(k-1)/2}.
\end{align*}
Thus the bound holds for every prime $p \nmid N\ell$.
[/step]
[step:Choose an auxiliary prime to handle every good prime]
Now let $p$ be any prime with $p \nmid N$. Choose a prime $\ell$ with $\ell \ne p$; then $p \nmid N\ell$. Applying the preceding steps with this auxiliary prime $\ell$ gives
\begin{align*}
|\iota(a_p(f))|\le 2p^{(k-1)/2}
\end{align*}
for every embedding $\iota:K_f\hookrightarrow\mathbb{C}$, after extending $\iota$ to an embedding $\overline{K_f}\hookrightarrow\mathbb{C}$. The extension exists because $\mathbb{C}$ is algebraically closed. The sharper root statement holds under the stated hypothesis $p \nmid N\ell$ by the purity step. This proves both assertions.
[/step]
