Existence of $\hat{I}\cdot$-Quotients For Squarefree Levels

Abstract

In this paper, we investigate which modular form spaces can contain $\hat{I}\cdot$-quotients, functions of the form $f(\ddot{I}„)={\prod }_{\hat{I}´\mid N}\hat{I}\cdot (\hat{I}´\ddot{I}„{)}^{r_{\hat{I}}´}$. For $k\ge 2$ even and $N$ coprime to 6, we give necessary conditions for the space $M_k(\hat{I}{“}_1(N))$ to contain $\hat{I}\cdot$-quotients. We then show that these conditions are sufficient as well for a large family of squarefree levels $N$ coprime to 6.

10 pages