Sum of the GL(3) Fourier coefficients over mixed powers

Let [Formula: see text] be the [Formula: see text]th Fourier coefficient of [Formula: see text] Hecke–Maass cusp form, denoted as [Formula: see text] or the triple divisor function, denoted as [Formula: see text]. Let [Formula: see text] be an integer. In this paper, we establish an asymptotic formula for the sum [Formula: see text] where [Formula: see text] is either von Mangoldt function or identity function, and [Formula: see text] is a binary quadratic polynomial. When [Formula: see text], then [Formula: see text] can be any bounded arithmetical function.