#### Vol. 11, No. 5, 2018

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Counting eta-quotients of prime level

### Allison Arnold-Roksandich, Kevin James and Rodney Keaton

Vol. 11 (2018), No. 5, 827–844
##### Abstract

It is known that a modular form on ${SL}_{2}\left(ℤ\right)$ 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.

##### Keywords
modular forms, eta-quotients, Dedekind eta-function, number theory
##### Mathematical Subject Classification 2010
Primary: 11F11, 11F20, 11F37