)&= \begin{cases}     \operatorname{diag}(\theta_0(z;\rho),\theta_0(z;\rho),\theta_0(z;\rho)), & \text{${\mathrm{Im}\,}z >0$,} \\     \operatorname{diag}(\theta_0(z;\rho),\theta_0(z;\rho),\theta_0(z;\rho)), & \text{${\mathrm{Im}\,}z <0$,} \\     \end{cases}\end{aligned}$$ with $$\label{eq: