On the common zeros of quasi-modular forms for $\Gamma_0^+(N)$ of level $N=1,2,3$

Abstract

In this paper, we study common zeros of the iterated derivatives of the Eisenstein series for $\Gamma_0^+(N)$ of level $N=1,2$ and $3$, which are quasi-modular forms. More precisely, we investigate the common zeros of quasi-modular forms, and prove that all the zeros of the iterated derivatives of the Eisenstein series $\frac{d^m E_k^{(N)}(\tau)}{d\tau^m}$ of weight $k=2,4,6$ for $\Gamma_0^+(N)$ of level $N=2,3$ are simple by generalizaing the results of Meher \cite{MEH} and Gun and Oesterl\'{e} \cite{SJ20} for SL$_2(\mathbb{Z})$.