#### 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
##### Milestones
Received: 18 December 2016
Revised: 30 July 2017
Accepted: 24 August 2017
Published: 2 April 2018

Communicated by Kenneth S. Berenhaut
##### Authors
 Allison Arnold-Roksandich Department of Mathematics Oregon State University Corvallis, OR United States Kevin James Department of Mathematical Sciences Clemson University Clemson, SC United States Rodney Keaton Department of Mathematics and Statistics East Tennessee State University Johnson City, TN United States