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

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?

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.