Zeros of certain weakly holomorphic modular forms for the Fricke group ${\mathrm{\Gamma }}_0^{+}(3)$

Let $M_k^{!}({\mathrm{\Gamma }}_0^{+}(3))$ be the space of weakly holomorphic modular forms of weight $k$ for the Fricke group of level $3$. We introduce a natural basis for $M_k^{!}({\mathrm{\Gamma }}_0^{+}(3))$ and prove that for almost all basis elements, all of their zeros in a fundamental domain lie on the circle centered at 0 with radius $\frac{1}{\sqrt{3}}$.