#### Vol. 274, No. 1, 2015

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Theory of newforms of half-integral weight

### Murugesan Manickam, Jaban Meher and Balakrishnan Ramakrishnan

Vol. 274 (2015), No. 1, 125–139
##### Abstract

We set up the theory of newforms of half-integral weight on ${\Gamma }_{0}\left(8N\right)$ and ${\Gamma }_{0}\left(16N\right)$, where $N$ is odd and squarefree. Further, we extend the definition of the Kohnen plus space in general for trivial character and also study the theory of newforms in the plus spaces on ${\Gamma }_{0}\left(8N\right)$, ${\Gamma }_{0}\left(16N\right)$, where $N$ is odd and squarefree. Finally, we show that the Atkin–Lehner $W$-operator ${W}_{4}$ acts as the identity operator on ${S}_{2k}^{new}\left(4N\right)$, where $N$ is odd and squarefree. This proves that ${S}_{2k}^{-}\left(4\right)={S}_{2k}\left(4\right)$.

##### Keywords
modular forms of half-integral weight, theory of newforms, Kohnen plus space
##### Mathematical Subject Classification 2010
Primary: 11F37
Secondary: 11F11, 11F32