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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 {\eta(24\tau)^rf(24\tau):f(\tau)\inM_s(1)} is isomorphic toS_{r+2s-1}^{\text{new}}(6,-(\frac8r),-(\frac{12}r))\otimes(\frac{12}\cdot) asHecke modules under the Shimura correspondence. Here $M_s(1)$ denotes the spaceof modular forms of weight $s$ on $\Gamma_0(1)=\mathrm{SL}(2,\Z)$,$S_{2k}^{\text{new}}(6,\epsilon_2,\epsilon_3)$ is the space of newforms ofweight $2k$ 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 $\otimes(\frac{12}\cdot)$ means the twist by thequadratic character $\frac{12}\cdot)$. There is also an analogous result forthe cases $(r,6)=3$.

Modular forms of half-integral weights on SL(2,Z)