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(Sorry for my poor english skill..)

Let $N$ be an odd integer and $k$ be a positive integer. Let $\chi$ be a Dirichlet character modulo $4N$ and $f=\sum_{n=1}^{\infty} a(n)q^n \in S_{k+\frac{1}{2}}(\Gamma_0(4N),\chi)$ be a half-integral weight modular form.

I already know that for primes $p\nmid 4N$, \begin{equation} T_{p^2}(f)=\sum_{n=1}^{\infty} \left(a(p^2n)+\chi(p)\left(\frac{(-1)^kn}{p} \right)p^{k-1}+\chi(p^2)p^{2k-1}a(n/p^2) \right)q^n. \end{equation}

In Ono's book ("The web of Modularity: Arithmetic of the Coefficients of Modular Forms and $q$-series"), the Hecke operator $T(4,k,\chi_0)$ on $M^{+}_{k+\frac{1}{2}}(\Gamma_0(4),\chi_0)$ defined by \begin{equation} T_{4}(f)=\sum \left(a(4n)+\left(\frac{(-1)^kn)}{2}\right)2^{k-1}a(n)+2^{2k-1}a(n/4)\right) q^n \end{equation} where $\chi_0$ is trivial character and $M_{k+\frac{1}{2}}(\Gamma_0(4))$ is Kohnen plus space.

Is it possible to define a Hecke operator $T_4$ on $S_{k+\frac{1}{2}}(\Gamma_0(4N),\chi)$ as in the previous case?

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  • $\begingroup$ How is the Kohnen plus space defined? $\endgroup$ – ramanujan_dirac Feb 13 at 19:51

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