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(Sorry for my poor english)

Let $f(z)\in S_{2k}(\Gamma_0(N))$ be a newform. Let $\chi$ be a Dirichlet character modulo $N$ and $\chi'$ be an unique even Dirichlet character modulo $4N$ associated to $\chi$ since $(\mathbb{Z}/4N\mathbb{Z})^{*}\cong (\mathbb{Z}/4\mathbb{Z})^{*}\times (\mathbb{Z}/N\mathbb{Z})^{*}$.

In Mao's paper ("A generalized Shimura correspondence for newforms") he defined the set $S_{k+\frac{1}{2}}(f,4M,\chi,e)$ of forms $g(z)\in S_{k+\frac{1}{2}}(\Gamma_0(4N),\chi')$ such that for prime $(p,4N)=1$, \begin{equation} T_{p^2}g(z)=\chi'(p)\frac{(-1)^k e}{p}\lambda_p g(z) \end{equation} with $\lambda_p$ is a Hecke eigenvalue of $F$ at $p$ and $e\in \{\pm 1\}$. For $g(z)\in S_{k+\frac{1}{2}}(f,4M,\chi,e)$, he said that $f(z)$ is the $\textbf{Shimura correspondence of $g(z)$ in a bit generalized sense}$ . My question is..

$\textbf{Q}$. Is there a relationship between the Shimura correpondence of $g_{\chi}\in S_{k+\frac{1}{2}}(f,4M,\chi,e)$ and the newform $f(z)$?

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