Sums of Hecke Eigenvalues over Values of Quadratic Polynomials

Let $f(z)=\sum\nolimits_{n}a(n)n^{(k-1)/2}e(nz)\in S_{k}(N,\chi)$ be a cusp form for Γ0(N), weight k ≥ 4, and character χ. Let $q(x) = x^{2} + sx + t \in \mathbb{Z}[x]$ be a quadratic polynomial. It is shown that$$\sum_{n\leq X}a(q(n)) = cX + O_{f,q,\varepsilon}(X^{{6/7}+\varepsilon})$$for some constant c = c(f, q). The constant vanishes in many (but not all) cases, for example, if k is even or if Δ = s² − 4t > 0. On the way, a Kuznetsov formula for half-integral weight and entries having different sign is derived.