# Newform of Half-integral weight modular forms

(Sorry for my poor english..) Let $$k$$ and $$N$$ be integers. Let $$f\in S_{k+\frac{1}{2}}(\Gamma_1(4N))$$ be a half integral weight modular form.

I know that if $$g \in S_{k}(\Gamma_1(N))^{new}$$ in subspace of newforms and if for each prime $$p\nmid N$$, there is a complex number $$c_p$$ $$\begin{equation} T_{p}(g)=c_p g, \end{equation}$$ i.e. $$g$$ is a Hecke eigenform for $$T_{p}$$ ($$p\nmid N$$), then $$g$$ is a Hecke eigenform for all $$T_{p}$$. My question is..

$$\textbf{Q. Is there a similar theorem for half-integral weight modular form?}$$

More precisely, is subspace of newforms defined for general $$N$$?

If defined, if $$f\in S_{k+\frac{1}{2}}(\Gamma_1(4N))^{new}$$ such that for each prime $$p\nmid 4N$$, there is a complex number $$d_p$$ such that $$\begin{equation} T_{p^2}(f)=d_p f, \end{equation}$$ then for all prime $$p$$, $$f$$ is a Hecke eigenform for all $$T_{p^2}$$?

Yes, this is known when $$N$$ is odd and squarefree, by work of Kohnen, Cipra, and Purkait, see my book with Str\"omberg (AMS GSM 179) or S.~Purkait's 2012 Warwick PhD thesis. I believe nothing is known (in general) when $$N$$ is not squarefree.