Condition of Shimura correspondence

(Sorry for my poor english.)

Let $$N$$ be an integer and $$t$$ be a square-free integer. Let $$\chi$$ be a Dirichlet character modulo $$4N$$. In Shimura's paper, Shimura defined 'Shimura correspondence', $$$$S_{k+\frac{1}{2}}(\Gamma_0(4N),\chi)\to M_{2k}(\Gamma_0(2N),\chi^2).$$$$ In his main theorem, assume that "$$f$$ is a Hecke eigenform of $$T(p^2)$$ for prime factor $$p$$ of $$N$$ not dividing the conductor of $$\chi_t$$."

However, in Ono's book, "The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and $$q$$-series", this condition does not exist. (For all $$f\in S_{k+\frac{1}{2}}(\Gamma_0(4N),\chi)$$...without hecke eigenform)

Which is right? Do I need Hecke eigenform condition for Shimura correspondence?