Symmetric Squares of Hecke L-Functions and Fourier Coefficients of Cusp Forms

Let Sk(Г0(N)) be the space of cusp forms of even weight k for Г0(N), let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{F}_0 $$ \end{document} be the set of all newforms in Sk(Г0(N)), and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{H}_2 (s,f)$$ \end{document} be the symmetric square of the Hecke L-function of a form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$f \in \mathcal{F}_0 $$ \end{document}. It is proved that for N=p we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\sum\limits_{f \in \mathcal{F}_0 ,\mathcal{H}_2 (1/2,f) \ne 0} {1 \gg N^{1 - \varepsilon } ,} $$ \end{document} where the ≪-constant depends only on ɛ and k. Let f(z)∈Sk(Г0(N)): \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$f(x) = \sum\limits_{n = 1}^\infty {a_f (n)e^{2\pi inz} ,{\text{ }}a_f (n)n^{ - (k - 1)/2} = b_f (n).} $$ \end{document} The distribution of values of the sums \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\sum\limits_{n \leqslant X} {b_f (n){\text{ and }}\sum\limits_{n \leqslant X} {b_f (n)^2 } } $$ \end{document} for increasing X and N is studied. Bibliography: 13 titles.