Extremal quasimodular forms have been introduced by Kaneko and Koike as quasimodular forms which have maximal possible order of vanishing at i∞. We show an asymptotic formula for the Fourier coefficients of such forms. This formula is then used to show that all but finitely many Fourier coefficients of such forms of depth ≤4 are positive, which partially solves a conjecture stated by Kaneko and Koike. Numerical experiments based on constructive estimates confirm the conjecture for weights ≤200 and depths between 1 and 4.
ASJC Scopus subject areas
- Algebra und Zahlentheorie