Expansions at Cusps and Petersson Products in Pari/GP
We begin by explaining how to compute Fourier expansions at all cusps of any modular form of integral or half-integral weight thanks to a theorem of Borisov–Gunnells and explicit expansions of Eisenstein series at all cusps. Using this, we then give a number of methods for computing arbitrary Petersson products. All this is available in the current release of the Pari/GP package.
- 1.D. Collins, Numerical computation of Petersson inner products and q-expansions. arXiv:1802.09740
- 3.The PARI Group, PARI/GP version 2.11.1, Univ. Bordeaux (2018), http://pari.math.u-bordeaux.fr/
- 4.L. Borisov, P. Gunnells, Toric modular forms and nonvanishing of L-functions. J. Reine Angew. Math. 539, 149–165 (2001)Google Scholar
- 6.H. Cohen, Haberland’s formula and numerical computation of Petersson scalar products, in Proceedings ANTS X, vol. 1 (Mathematical Sciences Publishers, 2013), pp. 249–270Google Scholar
- 7.H. Cohen, F. Strömberg, Modular Forms, A Classical Approach, Graduate studies in mathematics, vol. 179 (American Mathematical Society, Providence, 2017)Google Scholar