Nonlinear twists of Rankin-Selberg coefficients in the eigenvalue aspect

Let $g$ be a fixed primitive Hecke--Maass cusp form, and let $f$ vary with spectral parameter $k_f=K$, at a fixed level and with trivial nebentypus. We study smooth sums of $λ_f(n)λ_g(n)e(αn^β)$ for fixed real $α\ne0$. The crucial input is the smooth spectral second moment of Lau, Liu, and Ye, for w