Non-split Sums of Coefficients of GL(2)-Automorphic Forms

Given a cuspidal automorphic form $\pi$ on ${\text{GL}}_2$, we study smoothed sums of the form ${\sum }_{n\in \mathbb{N}}{a}_{\pi }(n^2+d)W(\frac{n}{Y})$. The error term we get is sharp in that it is uniform in both $d$ and $Y$ and depends directly on bounds towards Ramanujan for forms of half-integral weight and Selberg eigenvalue conjecture. Moreover, we identify (at least in the case where the level is square-free) the main term as a simple factor times the residue as $s=1$ of the symmetric square L-function $L(s,{\text{Msym}}^2\pi )$. In particular there is no main term unless $d>0$ and $\pi$ is a dihedral form.