Moments of the zeros of Faber polynomials of the Miller basis

Abstract

We study the zeros of modular forms in the Miller basis, a natural basis for the space of modular forms. We show that the zeros of their Faber polynomials have linear moments. By analyzing the moments we can extend the known range of the forms in the Miller basis for which at least one of the zeros is not on the arc - the circular part of the boundary of the fundamental domain. Additionally, for forms in the Miller basis of an index asymptotically linear in the weight such that all zeros are on the arc, we compute the limit distribution of the zeros, which depends on the asymptotic ratio of the index to the weight.