Strong orthogonality between the M\"obius function, additive characters, and Fourier coefficients of cusp forms

Let ${\nu }_f(n)$ be the $n$-th nomalized Fourier coefficient of a Hecke–Maass cusp form $f$ for $\mathrm{S}\mathrm{L}(2,\mathbb{Z})$ and let $\alpha$ be a real number. We prove strong oscillations of the argument of $\nu_{f}(n)\mu (n) \exp (2\pi i n \alpha)$as$n$ takes consecutive integral values.