(Sorry for my poor english...)

I apologize for the similarity or redundancy of the my previous questions.

My previous question 1, I was wondering about the definition of a newform in the space $S_{k+\frac{1}{2}}(\Gamma_0(4N),\chi)$ for general odd $N$ and a Dirichlet character $\chi$ modulo $4N$. Many reference defined only when $\chi$ is real character. My question is..

$\textbf{Q1.}$ What is the definition of newform in $S_{k+\frac{1}{2}}(\Gamma_0(4N),\chi)$?

What is the difficulty with non-real character $\chi$ if it is not defined in non -real case?

The next question is previous question 2, Mr. Cohen kindly referred me to the reference, however I do not understand it because I have poor math skill..

$\textbf{Q2.}$ Let $N$ be a square-free odd integer and $k$ be an integer. Let $\chi$ be a Dirichlet character modulo $4N$. Let $g\in S_{k+\frac{1}{2}}(\Gamma_0(4N),\chi)^{new}$ in newform subspace of half-integral weight modular forms $S_{k+\frac{1}{2}}(\Gamma_0(4N),\chi)$. Assume that for each prime $p\nmid 4N$, there is a complex number $d_p$ such that \begin{equation} T_{p^2}(g)=d_p g. \end{equation} Then for all prime $p$, (including $p$ divides level $4N$), is $g$ a Hecke eigenform of $T_{p^2}$?

Thanks for reading.

*Once again, I would like to thank Cohen for giving me a kindly answer.