Oscillations of cusp form coefficients in exponential sums

Let f(z)=∑n=1∞λ(n)n(κ−1)/2e(nz) be a holomorphic cusp form of weight κ for the full modular group SL2(ℤ) and let μ(n) be the Möbius function. In this paper, we are concerned with the sum \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S(\alpha,X)=\sum _{n\leq X}\mu (n)\lambda(n)e(\alpha \sqrt{n}),\quad 0\neq \alpha \in \mathbb{R}.$$\end{document} It is proved that, unconditionally, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$S(\alpha,X)\ll X^{\frac{5}{6}}(\log X)^{20}$\end{document}, where the implied constant depends only on α and the cusp form f.