A local-parameter bound toward the generalized Ramanujan conjecture

Let $E$ be a number field, let $p$ be a rational prime that is unramified and does not split completely in $E$, and let $\nu\mid p$. Let $f_p$ denote the relevant residue-degree quantity and let $\alpha_\pi(j,\nu)$ be the local parameters of an automorphic representation $\pi$. The assumed bound. For every such $p$, every $\nu\mid p$, and every $j$, one has

$$|\alpha_\pi(j,\nu)|\leq p^{f_p\theta_p},$$

where

$$\theta_p=1/2-1/(2f_p)-\epsilon$$

for a small $\epsilon>0$. This is an explicit hypothesis used by the paper rather than an asserted resolution of the generalized Ramanujan conjecture; the source provides no independent resolution status for this bound.