Sum of the Fourier coefficients of SL(3,Z) Hecke Maass forms over quadratics

Abstract

Let A(1,n) denote the (1,n)-th Fourier coefficients of a SL(3,Z) Hecke eigenform. Let Q(x,y) be a symmetric positive definite quadratic form. In this paper, we shall prove thatS:=∑m≤X∑n≤XA(1,Q(m,n))W1(mX)W2(nX)≪X2−168+ϵ, for any positive ϵ>0, where W1 and W2 are smooth bump functions supported on the interval [1,2].