Let \(f\) be a normalized newform of weight 2 for \(\Gamma_0(N)\) and let \(L(f,s)\) be its \(L\)-function. Yu. I. Manin [Izv. Akad. Nauk SSSR, Ser. Mat. 36, 19-66 (1972; Zbl 0243.14008)] gave an explicit formula for \(L(f,1)\) in terms of period integrals. D. Goldfeld [Can. Math. Soc. Conf. Proc. 15, 159-173 (1995; Zbl 0845.11021)] has recently given an explicit formula for \(L'(f,1)\) (when \(L(f,1)=0\)) as a linear combination of 1-cocycles defined in a similar, cohomological way.
The author generalizes the above results to include all derivatives of \(L(f,s)\). He obtains closed formulas for \(L^{(i)} (f,s)\), analogous to those of Manin and Goldfeld, in terms of \(i\)-cycles (Theorem 1).