Zeros of weakly holomorphic modular forms of level 4

Let $M_k^\sharp(4)$ be the space of weakly holomorphic modular forms of weight $k$ and level $4$ that are holomorphic away from the cusp at $\infty$. We define a canonical basis for this space and show that for almost all of the basis elements, the majority of their zeros in a fundamental domain for $\Gamma_0(4)$ lie on the lower boundary of the fundamental domain. Additionally, we show that the Fourier coefficients of the basis elements satisfy an interesting duality property.