Let $N \in \mathbb{N}$, let $k \ge 2$, let $\varepsilon: (\mathbb{Z}/N\mathbb{Z})^\times \to \mathbb{C}^\times$ be a Dirichlet character, and let $f \in S_k(\Gamma_1(N), \varepsilon)$ be a normalized cuspidal Hecke eigenform. Let $K_f$ denote the number field generated over $\mathbb{Q}$ by the Hecke eigenvalues $a_n(f)$ and the values of $\varepsilon$.

For every prime $p \nmid N$ and every field embedding $\iota: K_f \hookrightarrow \mathbb{C}$,

\begin{align*} |\iota(a_p(f))| \le 2p^{(k-1)/2}. \end{align*}

More precisely, let $\overline{K_f}$ be an [algebraic closure](/page/Algebraic%20Closure) of $K_f$. If $\ell$ is a prime with $p \nmid N\ell$, and if $\alpha_p,\beta_p \in \overline{K_f}$ are the roots of

\begin{align*} X^2 - a_p(f)X + \varepsilon(p)p^{k-1} \in K_f[X], \end{align*}

then for every field embedding $\iota: \overline{K_f} \hookrightarrow \mathbb{C}$,

\begin{align*} |\iota(\alpha_p)| = |\iota(\beta_p)| = p^{(k-1)/2}. \end{align*}