The purpose of this note is to study the homotopy groups of $U(n)$ and $S_{p}(n)$ of the first dimension over the meta-stable range. Our method is to use the classical $\mathrm{EHP}$-sequence.

In the previous paper [U], we calculated the trace of the Hecke operator $\tilde{T}_{k+1/2,N,\chi }(n^{2})$ on the space of cusp forms $S(k + 1/2, N, \chi )$ of half-integral weight under the assumption $\chi ^{2} = 1$. The purpose of this calculation is to find a relation between these traces and those of the Hecke operators of integral weight $2k$. When the 2-order of the level $N$ ($= \mathrm{ord}_{2}(N))$ is small, we found certain relations between the traces, in [U].

In this paper, we report relations for the remaining cases.