Let $M$ be an odd positive integer, $\chi$ an even quadratic character defined modulo $32 M$, and $\psi$ a quadratic primitive character of conductor divisible by 8. Then, we can define twisted Hecke operators $R_{\psi} \tilde{T}(n^{2})$ on the space of cusp forms of weight $k+1/2$, level $32M$, and character $\chi$, under certain conditions on the conductors of $\chi$ and $\psi$. This is a specific feature of the case of half-integral weight. We give explicit trace formulas of the twisted Hecke operators and their trace identities.

We shall present partial solutions to the conjecture such that $(q^{p}-1)/(q-1)$ does not divide $(p^{q}-1)/(p-1)$ for distinct primes $p < q$.