Section: New Results

Modular forms

Participants : Karim Belabas, Henri Cohen, Bill Allombert.

In [18], K. Belabas and H. Cohen give theoretical and practical information on the Pari/GP modular forms package, using the formalism of trace formulas. This huge package (about 70 exported public functions) handles standard operations on classical modular forms in $M_k({\Gamma }_0\left(N\right),\chi )$, also in weight 1 and non-integral weight (which are not cohomological, hence not directly handled by trace formulas). It is the first publicly available package which can compute Fourier expansions at any cusps, evaluate modular forms near the real axis, evaluate L-functions of non-eigenforms, and compute general Petersson scalar products.

In [39], H. Cohen explained how to compute Fourier expansions at all cusps of any modular form of integral or half-integral weight.

A complementary package using modular symbols is used in [17] by Karim Belabas, Dominique Bernardi and Bernadette Perrin-Riou to compute Manin's constant and the modular degree of elliptic curves defined over $\mathbb{Q}$.