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

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