分野別セミナー

Modular forms continue to attract attention for decades with many different application areas. To study statistical
properties of modular forms, including for instance Sato-Tate like problems, it is essential to be able to compute a
large number of Fourier coefficients. In this talk, firstly, we will show that this can be achieved in level 4 for a large
range of half-integral weights by making use of one of three explicit bases, the elements of which can be calculated
via fast power series operations.

After having "many" Fourier coefficients, it is time to ask the following question: Can the dis- tribution of normalised Fourier coefficients of half-integral weight level 4 Hecke eigenforms with bounded indices be approximated by a
distribution? We will suggest that they follow the generalised Gaussian distribution and give some numerical
evidence for that. Finally, we will see that the apparent symmetry around zero of the data lends strong evidence to
the Bruinier- Kohnen Conjecture on the equidistribution of signs and even suggests the strengthening that signs and
absolute values are distributed independently.

This is joint work with Gabor Wiese (Luxembourg), Zeynep Demirkol Ozkaya (Van) and Elif Tercan (Bilecik).