Let $\Gamma_0(N)$ be the Hecke congruence subgroup. Let $S_{k+1/2}(\Gamma_0(N))$ be the space of holomorphic forms of weight $k+1/2$ on $\Gamma_0(N)$, where $k\in\mathbb{N}$. How to prove that if $S_{k+1/2}(\Gamma_0(N))\neq0$ then $4\mid N$? It should be in Shimura's paper in 1973. But since I am not familiar with this paper, I don't know where exactly this statement and its proof are. Maybe somebody can provide a sketch if it's not too long. Thanks a lot.

What if forms with a character $\chi \ \mathrm{mod} \ N$? Do we still need $4\mid N$?

The last question is about half integral weight Maass forms. Do we also require $4\mid N$? Can someone provide a reference for this?