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