ZEROS OF A QUASI-MODULAR FORM OF WEIGHT $2$ FOR $\Gamma_0^+(N)$

Abstract

Basraoui and Sebbar showed that the Eisenstein series $E_2$ has infinitely many $\text{SL}_2(\Bbb{Z})$-inequivalent zeros in the upper half-plane $\Bbb{H}$, yet none in the standard fundamental domain $\mathfrak{F}$. They also found infinitely many such regions containing a zero of $E_2$ and infinitely many regions which do not have any zeros of $E_2$. In this paper we study the zeros of the quasi-modular form $E_2(z)+NE_2(Nz)$ of weight $2$ for $\Gamma_0^+(N)$.