# Hecke operator on half integral weight modular form

(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 $$p\nmid 4N$$, $$$$T_{p^2}(f)=\sum_{n=1}^{\infty} \left(a(p^2n)+\chi(p)\left(\frac{(-1)^kn}{p} \right)p^{k-1}+\chi(p^2)p^{2k-1}a(n/p^2) \right)q^n.$$$$

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 $$$$T_{4}(f)=\sum \left(a(4n)+\left(\frac{(-1)^kn)}{2}\right)2^{k-1}a(n)+2^{2k-1}a(n/4)\right) q^n$$$$ where $$\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?

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