(Sorry for my poor english..)

I have some questions about newforms of half-integral weight. In Mao's paper ("A generalized Shimura correspondence for newforms"), he said: "Ueda defined the set of newforms in the space of weight $k+1/2$ forms of level $4N$..." However, in Ueda's paper, he only defined newforms in $S_{k+\frac{1}{2}}(\Gamma_0(4N),\chi)$ with $\chi$ a "real" Dirichlet character modulo $4N$. In Mao's paper, there is no assumption on $\chi$. So, my questions are as follows:

Let $\chi$ be a Dirichlet character modulo $4N$. Then what is the definition of the set of newforms in $S_{k+\frac{1}{2}}(\Gamma_0(4N),\chi)$?

Furthermore, assume that $f\in S_{k+\frac{1}{2}}(\Gamma_0(4N),\chi)$ be a half-integral weight modular form. Let $t$ be a square-free integer and \begin{equation} Sh_t : S_{k+\frac{1}{2}}(\Gamma_0(4N),\chi) \to S_{2k}(\Gamma_0(4N),\chi^2) \end{equation} be a Shimura correspondence. I already know that if $f$ is a Hecke eigenform then $Sh_{t}(f)$ is also a Hecke eigenform. If $f$ is newform then is $Sh_{t}(f)$ also a newform in $S_{2k}(\Gamma_0(4N),\chi)$ ?

Thanks for reading.