Counting eta-quotients of prime level

Abstract

It is known that a modular form on ${SL}_2\left(\mathbb{Z}\right)$ can be expressed as a rational function in $\eta \left(z\right)$, $\eta \left(2z\right)$ and $\eta \left(4z\right)$. By using known theorems and calculating the order of vanishing, we can compute the eta-quotients for a given level. Using this count, knowing how many eta-quotients are linearly independent, and using the dimension formula, we can figure out a subspace spanned by the eta-quotients. In this paper, we primarily focus on the case where the level is $N=p$, a prime. In this case, we will show an explicit count for the number of eta-quotients of level $p$ and show that they are linearly independent.