Existence of $\eta$-Quotients For Squarefree Levels

In this paper, we investigate which modular form spaces can contain $\eta$-quotients, functions of the form $f(\tau) = \prod_{\delta \mid N} \eta(\delta\tau)^{r_\delta}$. For $k\geq 2$ even and $N$ coprime to 6, we give necessary conditions for the space $M_k(\Gamma_1(N))$ to contain $\eta$-quotients. We then show that these conditions are sufficient as well for a large family of squarefree levels $N$ coprime to 6.