It is known that a modular form on
can be expressed as a rational function in
,
and
. 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
,
a prime. In this case, we will show an explicit count for the number of eta-quotients of
level
and show that they are linearly independent.
Keywords
modular forms, eta-quotients, Dedekind eta-function, number
theory