Explicit upper bound for the (analytic) rank of J0(q)

We refine the techniques of our previous paper [KM1] to prove that the average order of vanishing of L-functions of primitive automorphic forms of weight 2 and prime level q satisfies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum\limits_{f \in {S_2}(q)*} {{\text{or}}{{\text{d}}_s}} = {}_{1/2}L(f,s) \leqslant C\left| {{S_2}(q)*} \right|$$\end{document} with C = 6.5, for all q large enough. On the Birch and Swinnerton-Dyer conjecture, this implies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{rank}}{J_0}(q) \leqslant C\dim {J_0}(q)$$\end{document} for q prime large enough.