Weight aspect exponential sums for Fourier coefficients of cusp forms

Let k≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 2$$\end{document} be an even integer. Let f be a primitive holomorphic cusp form of weight k, with λf(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _f(n)$$\end{document} being its n-th Fourier coefficient. We explicitly determine the dependence on the weight aspect by proving ∑n≤Xλf(n)en2α+nβ≪(Xk)εX2122k3122\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\sum _{n\le X}\lambda _f(n)e{\left(n^2\alpha +n\beta \right)}\ll (Xk)^\varepsilon X^{ \frac{21}{22} } k^{ \frac{31}{22} } \end{aligned}$$\end{document}uniformly for k, and any α,β∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha ,\beta \in {\mathbb {R}}$$\end{document}, where the implied constant depends only on ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}. In addition we obtain an analogue of the Prime Number Theorem associated to the Fourier coefficients of cusp forms ∑n≤XΛ(n)λf(n)en2α+nβ≪Xexp-clogXlogX+logk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\sum _{n\le X} \Lambda (n)\lambda _{f}(n)e{\left(n^2\alpha +n\beta \right)}\ll X\exp \left(-\frac{c\log X}{\sqrt{\log X}+\log k}\right) \end{aligned}$$\end{document}uniformly for k<X1/31-ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k< X^{1/31-\varepsilon }$$\end{document} and any α,β∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha ,\beta \in {\mathbb {R}}$$\end{document}, where the implied constant is absolute.