# Generalized Shimura correspondence

(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 $$(p,4N)=1$$, $$$$T_{p^2}g(z)=\chi'(p)\frac{(-1)^k e}{p}\lambda_p g(z)$$$$ with $$\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 $$f(z)$$?