Why the level of a half integral weight modular form must be a multiple of 4?

Question

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?

Answer 1

The problem isn't that $S_{k + 1/2}(\Gamma_0(N))$ is zero if $4 \nmid N$; it's that the space is not defined if $4 \nmid N$.

In order to make sense of what a "half-integer weight form of level $\Gamma$" means, for some subgroup $\Gamma \subseteq SL_2(\mathbf{Z})$, you need to have a consistent way of choosing square roots of $c\tau + d$, for all $\tau$ in the upper half-plane and $\begin{pmatrix}a&b\\c&d\end{pmatrix}\in \Gamma$. There is no sensible way of doing this for all $\Gamma$, but we can do it if $\Gamma \subseteq \Gamma_0(4)$; and that's why you need the $4\mid N$ condition.

This "choice of square roots" has a highbrow interpretation in terms of lifting $\Gamma$ to a subgroup of the metaplectic group, a double covering of $\operatorname{SL}_2(\mathbf{R})$. See my answer to the following Math.SE question: https://math.stackexchange.com/questions/2802562.