The transcendence of zeros of natural basis elements for the space of the weakly holomorphic modular forms for $\Gamma_{0}^{+}(3)$

Abstract

We consider a natural basis for the space of weakly holomorphic modular forms for $\Gamma_{0}^{+}(3)$. We prove that for some of the basis elements, if $z_{0}$ in the fundamental domain for $\Gamma_{0}^{+}(3)$ is one of zeroes of the elements, then either $z_{0}$ is transcendental or is in $\{\frac{i}{\sqrt{3}}, \frac{-1+\sqrt{2}i}{3}, \frac{-3+\sqrt{3}i}{6}, \frac{-1+\sqrt{11}i}{6}\}$.