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

It is known that a modular form on SL2() can be expressed as a rational function in η(z), η(2z) and η(4z). 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.

modular forms, eta-quotients, Dedekind eta-function, number theory
Mathematical Subject Classification 2010
Primary: 11F11, 11F20, 11F37
Received: 18 December 2016
Revised: 30 July 2017
Accepted: 24 August 2017
Published: 2 April 2018

Communicated by Kenneth S. Berenhaut
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