# Question about newform of half-integral weight modular form

(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 $$$$T_{p^2}(g)=d_p g.$$$$ Then for all prime $$p$$, (including $$p$$ divides level $$4N$$), is $$g$$ a Hecke eigenform of $$T_{p^2}$$?