Modular forms in Pari/GP
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We give theoretical and practical information on the Pari/GP modular forms package available since the spring of 2018. Thanks to the use of products of two Eisenstein series, this package is the first which can compute Fourier expansions at any cusps, evaluate modular forms near the real axis, evaluate L-functions of noneigenforms, and compute general Petersson scalar products.
We would like to thank B. Allombert, J. Bober, A. Booker, M. Lee, and B. Perrin-Riou for very helpful discussions and help in algorithms and programming, F. Brunault and M. Neururer for Theorem 7, as well as K. Khuri-Makdisi, N. Mascot, N. Billerey and E. Royer. Last but not least, we thank Don Zagier for his continuous input on this package and on Pari/GP in general. In addition, note that a much smaller program written 30 years ago by Don, N. Skoruppa, and the second author can be considered as an ancestor to the present package.
- 3.Cohen, H., Oesterlé, J.: Dimensions des espaces de formes modulaires. In: Serre, J.-P., Zagier, D.B. (eds.) Modular Functions of One Variable VI. Lecture Notes in Mathematics, vol. 627, pp. 69–78. Springer, Berlin (1977)Google Scholar
- 5.Collins, D.: Numerical computation of Petersson inner products and \(q\)-expansions. arXiv:802.09740 [math]
- 8.The PARI Group, PARI/GP version 2.11.0, University of Bordeaux (2018). http://pari.math.u-bordeaux.fr/
- 9.Weisinger, J.: Some results on classical Eisenstein series and modular forms over function fields. Ph.D. thesis, Harvard University (1977)Google Scholar