Counting Eta-Quotients of Prime Level

TL;DR

This paper investigates the enumeration of eta-quotients at prime levels, leveraging known theorems and dimension formulas to understand their linear independence and span within modular forms.

Contribution

It provides a method to count eta-quotients at prime levels and analyzes their linear independence and span in the space of modular forms.

Findings

Counted eta-quotients for prime levels.

Determined linear independence of eta-quotients.

Identified the subspace spanned by eta-quotients.

Abstract

It is known that all modular forms on SL_2(Z) can be expressed as a rational function in eta(z), eta(2z) and eta(4z). By utilizing 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 N=p a prime.