On the common zeros of quasi-modular forms for ${\mathrm{\Gamma }}_0^{+}(N)$ of level $N=1,2,3$

In this paper, we study common zeros of the iterated derivatives of the Eisenstein series for ${\mathrm{\Gamma }}_0^{+}(N)$ 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 ${\mathrm{\Gamma }}_0^{+}(N)$ of level $N=2,3$ are simple by generalizaing the results of Meher \cite{MEH} and Gun and Oesterlé \cite{SJ20} for SL${}_2(\mathbb{Z})$.