From the modularity theorem proven by Wiles, Taylor, Conrad, Diamond, and
Breuil, we know that all elliptic curves are modular. It has been shown by
Martin and Ono exactly which are represented by eta-quotients, and some
examples of elliptic curves represented by modular forms that are linear
combinations of eta-quotients have been given by Pathakjee, RosnBrick, and
Yoong.
In this paper, we first show that eta-quotients which are modular for any congruence subgroup
of level
coprime
to
can be viewed
as modular for
.
We then categorize when even-weight eta-quotients can exist in
and
for distinct
primes
.
We conclude by providing some new examples of elliptic curves whose corresponding
modular forms can be written as a linear combination of eta-quotients, and describe
an algorithmic method for finding additional examples.