## Abstract

In this paper, we prove that, for an integer $r$ with $(r,6)=1$ and $0<r<24$ and a nonnegative even integer $s$, the set $$\left\{\eta \right(24\tau {)}^{r}f\left(24\tau \right):f\left(\tau \right)\in {M}_{s}\left(1\right)\}$$ is isomorphic to $${S}_{r+2s-1}^{\mathrm{new}}(6,-(\frac{8}{r}),-(\frac{12}{r}\left)\right)\otimes \left(\frac{12}{\cdot}\right)$$ as Hecke modules under the Shimura correspondence. Here ${M}_{s}\left(1\right)$ denotes the space of modular forms of weight $s$ on ${\Gamma}_{0}\left(1\right)=SL(2,\mathbb{Z})$, ${S}_{2k}^{\mathrm{new}}(6,{\u03f5}_{2},{\u03f5}_{3})$ is the space of newforms of weight $2k$ on ${\Gamma}_{0}\left(6\right)$ that are eigenfunctions with eigenvalues ${\u03f5}_{2}$ and ${\u03f5}_{3}$ for Atkin–Lehner involutions ${W}_{2}$ and ${W}_{3}$, respectively, and the notation $\otimes (12/\cdot )$ means the twist by the quadratic character $(12/\cdot )$. There is also an analogous result for the cases $(r,6)=3$.

## Citation

Yifan Yang. "Modular forms of half-integral weights on $SL(2,\mathbb{Z})$." Nagoya Math. J. 215 1 - 66, September 2014. https://doi.org/10.1215/00277630-2684452

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