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

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