(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 $T_{p^2}$?