  From the Modularity Theorem proven by Wiles, Taylor, et al, 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 $N$
coprime to $6$ can be viewed as modular for $\Gamma_0(N)$. We then categorize
when even weight eta-quotients can exist in $M_k (\Gamma_1(p))$ and $M_k
(\Gamma_1(pq))$, for distinct primes $p,q$. 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.
