Bounds for orders of zeros of a class of Eisenstein series and their applications on dual pairs of eta quotients

Abstract

Let k k be an even positive integer, p p be a prime and m m be a nonnegative integer. We find an upper bound for orders of zeros (at cusps) of a linear combination of classical Eisenstein series of weight k k and level p m p^m . As an immediate consequence we find the set of all eta quotients that are linear combinations of these Eisenstein series and hence the set of all eta quotients of level p m p^m whose derivatives are also eta quotients.