Modular forms of half-integral weights on \(\mathrm{SL}(2,\mathbb Z)\) (Q2926245)

Let \(M_s(1)\) be the space of modular forms of weight \(s\) on \(\mathrm{SL}_2(\mathbb Z)\) and \(S^{\mathrm{new}}_{2k}(6,\varepsilon_2,\varepsilon_3)\) the space of new forms of weight \(2k\) on \(\Gamma_0(6)\) that are eigenfunctions with eigenvalues \(\varepsilon_2\) and \(\varepsilon_3\) for Atkin-Lehner involutions \(W_2\) and \(W_3\) respectively. Consider a space of modular forms of half-integral weight \(S_{r,s}=\{\eta(24\tau)^rf(24\tau)~|~f\in M_s(1)\}\subset S_{r/2+s}(576,\left(\frac{12}\cdot\right))\) for integers \(r\) and \(s\) such that \((r,6)=1,0<r<24\) and \(s\) are non negative and even. In [\textit{L. Guo} and \textit{K. Ono}, Int. Math. Res. Not. 1999, No. 21, 1179--1197 (1999; Zbl 0970.11037)], for the few \((r,s)\), they showed that the Shimura correspondence gives an isomorphism as Hecke modules of \(S_{r,s}\) to \(S^{\mathrm{new}}_{r+2s-1}\left(6,-\left(\frac 8r\right),-\left(\frac{12}r\right)\right)\otimes\left(\frac{12}\cdot\right)\) in proving the congruence properties of the partition function. In this article the author proves that this isomorphism holds for general \(r\) and \(s\). To prove his result, the author considers a space of modular forms of \(\eta\)-type \(S_{r,s}(1)=\{\eta(\tau)^rf(\tau)~|f\in M_s(1)\}\) instead of \(S_{r,s}\) and shows the equality of trace of Hecke operator on both spaces, thus, \(\mathrm{tr}(T_{n^2}|S_{r,s}(1))=\left(\frac{12}n\right)\mathrm{tr}(T_n|S^{\mathrm{new}}_{r+2s-1}\left(6,-\left(\frac 8r\right),-\left(\frac{12}r\right)\right))\) for positive integers \(n\) prime to \(6\). In the case that \((r,6)=3\), \(0<r<8\), a similar results is obtained.