On an analogue of prime vectors among integer lattice points in ellipsoids for automorphic forms

In this paper, we firstly generalize a result of Friedlander and Iwaniec. By analogy, let [Formula: see text] denote the [Formula: see text]th coefficient of [Formula: see text]-function [Formula: see text] associated to a Hecke–Maass form [Formula: see text] for [Formula: see text] and [Formula: see text] denote the number of representations of [Formula: see text] by the quadratic form [Formula: see text], we investigate the sum with [Formula: see text] [Formula: see text] under a certain assumption.