Counting integer points on quadrics with arithmetic weights

Abstract: Let $F \in \mathbf{Z}[\boldsymbol{x}]$ be a diagonal, non-singular quadratic form in $4$ variables. Let $\lambda(n)$ be the normalised Fourier coefficients of a holomorphic Hecke form of full level. We give an upper bound for the problem of counting integer zeros of $F$ with $|\boldsymbol{x}| \leq X$, weighted by $\lambda(x_1)$.