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It is known that all modular forms on $SL_2(Z)$ can be expressed as arational function in $\eta(z)$, $\eta(2z)$ and $\eta(4z)$. By using a theoremby Gordon, Hughes, and Newman, and calculating the order of vanishing, we cancompute the $\eta$-quotients for a given level. Using this count, knowing howmany $\eta$-quotients are linearly independent and using the dimension formula,we can figure out how the $\eta$-quotients span higher levels. In this paper,we primarily focus on the case where $N=p$ a prime, and some discussion fornon-prime indicies.

Proceedings Paper for REU Project Involving Counting Eta-Quotients