(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', \begin{equation} S_{k+\frac{1}{2}}(\Gamma_0(4N),\chi)\to M_{2k}(\Gamma_0(2N),\chi^2). \end{equation} 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?

Thanks for reading.

  • 1
    $\begingroup$ They are both finite-dimensional vector spaces, so you can write an cusp form in terms of a basis of eigenforms. $\endgroup$ – Peter Humphries Oct 15 '18 at 9:26

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