Half-integral weight modular forms (awaiting review)

Let $k$ be an odd integer, and let $N$ a positive integer divisible by $4$. Let $\chi$ be a character modulo $N$. A modular form of half-integral weight $k/2$ , level $N$ and character $\chi$ is an holomorphic function defined on the upper half plane $\mathcal{H}$, satisfying the transformation formula \[ f(\gamma z) = \chi(d) j(\gamma, z)^k f(z) \] for every $\gamma=\left(\begin{matrix} a & b \\ c & d \end{matrix}\right)\in\Gamma_0(N)$ and $z\in \mathcal{H}$, and being holomorphic at the cusps. Here the automorphy factor $j(\gamma, z)$ is given by \[ j(\gamma, z) = \theta(\gamma z)/\theta(z), \] where $\theta$ denotes the classical theta function. This space is denoted by $M_{k/2}(N)$, and the subspace of cusp forms is denoted by $S_{k/2}(N)$.

These modular forms are related with integral weight modular forms through the Shimura correspondence. In terms of this correspondence, the space of cusp forms can be decomposed according to the Shimura decomposition.