Sums of Hecke Eigenvalues over Values of Quadratic Polynomials

Let be a cusp form for Γ0( N ), weight k ≥ 4, and character χ. Let be a quadratic polynomial. It is shown that 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 = s 2 − 4 t > 0. On the way, a Kuznetsov formula for half-integral weight and entries having different sign is derived.