FABER POLYNOMIALS AND POINCARE SERIES

In this paper we consider weakly holomorphic modular forms (i.e., those meromorphic modular forms for which poles only possibly occur at the cusps) of weight 2-k is an element of 2Z for the full modular group SL2(Z). The space has a distinguished set of generators f(2-k), (m). Such weakly holomorphic modular forms have been classified in terms of finitely many Eisenstein series, the unique weight 12 newform Delta, and certain Faber polynomials in the modular invariant j(z), the Hauptmodul for SL2(Z). We employ the theory of harmonic weak Maass forms and (non-holomorphic) Maass-Poincare series in order to obtain the asymptotic growth of the coefficients of these Faber polynomials. Along the way, we obtain an asymptotic formula for the partial derivatives of the Maass-Poincare series with respect to y as well as extending an asymptotic for the growth of the lth repeated integral of the Gauss error function at x to include l is an element of R and a wider range of x.