Generalized Ramanujan conjecture over general imaginary quadratic fields

The new bound toward the generalized Ramanujan conjecture for GL n (A Q )$\textup {GL}_n({\bf A}_{\bf Q})$ is given by Kim–Sarnak [4]. We generalize it to that for GL n (A F )$\textup {GL}_n({\bf A}_{F})$ with F$F$ being an imaginary quadratic field. Applying the obtained bound to the automorphic representation of GL 5 $\textup {GL}_5$ gives the following estimate for the first positive eigenvalue of the Laplacian on L 2 ( 3 )$L^2(\Gamma \backslash {\mathbb {H}}^3)$, where $\Gamma $ is any congruence subgroups of SL 2 ( F )$\textup {SL}_2({\mathcal {O}}_F)$ with F ${\mathcal {O}}_F$ the ring of integers of F$F$: 1 ()0.952$\lambda _1(\Gamma ) \ge 0.952\cdots $.