## Nagoya Mathematical Journal

### Modular forms of half-integral weights on $\operatorname{SL}(2,\mathbb{Z})$

Yifan Yang

#### Abstract

In this paper, we prove that, for an integer $r$ with $(r,6)=1$ and $0\lt r\lt 24$ and a nonnegative even integer $s$, the set $$\{\eta(24\tau)^{r}f(24\tau):f(\tau)\inM_{s}(1)\}$$ is isomorphic to $$S_{r+2s-1}^{\mathrm{new}}(6,-(\frac{8}{r}),-(\frac{12}{r}))\otimes (\frac{12}{\cdot})$$ as Hecke modules under the Shimura correspondence. Here $M_{s}(1)$ denotes the space of modular forms of weight $s$ on $\Gamma_{0}(1)=\operatorname{SL}(2,\mathbb{Z})$, $S_{2k}^{\mathrm{new}}(6,\epsilon_{2},\epsilon_{3})$ is the space of newforms of weight $2k$ on $\Gamma_{0}(6)$ that are eigenfunctions with eigenvalues $\epsilon_{2}$ and $\epsilon_{3}$ 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 $(r,6)=3$.

#### Article information

Source
Nagoya Math. J., Volume 215 (2014), 1-66.

Dates
First available in Project Euclid: 8 May 2014

https://projecteuclid.org/euclid.nmj/1399554604

Digital Object Identifier
doi:10.1215/00277630-2684452

Mathematical Reviews number (MathSciNet)
MR3263525

Zentralblatt MATH identifier
1303.11057

#### Citation

Yang, Yifan. Modular forms of half-integral weights on $\operatorname{SL}(2,\mathbb{Z})$. Nagoya Math. J. 215 (2014), 1--66. doi:10.1215/00277630-2684452. https://projecteuclid.org/euclid.nmj/1399554604

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