The structure of irreducible cuspidal eta quotients

Abstract

We show that an ir{\-}reducible cuspidal eta quotient of level~$N$ is uniquely identified by its orders of vanishing at $O(({\log }_{2}N{)}^{r-1})$ cusps, where $r$ denotes the number of distinct prime divisors of $N$. In particular, all such eta quotients which are Fricke eigenforms of a prime power level, are uniquely identified only by their orders of vanishing at~$\infty$. We also prove that for all $k\in \mathbb{N}$, there exists a primitive and irreducible cuspidal eta quotient of weight $k/2$. It follows from the exhaustiveness of Zagier's list of the primitive holomorphic eta quotients of weight~$1/2$ that the 1st Mersmann level, $\cl{M}_1=12$. Here we show that for all $k\in \mathbb{N}$, the $k$-th Mersmann level is divisible by $2^{2k}$.