&=&\frac{A}{4\pi }e^{2\pi i\phi
}\sum_{a,d=0,1}(-1)^{ad}\theta
(-u+\frac{1}{2}(Ad+t)+\frac{i}{2}(s+\mu -\nu ),\frac{Ai}{2})  \nonumber \\
&&\times \theta (-Av+\frac{1}{2}(Aa+s-\mu -\nu
)+\frac{i}{2}t,\frac{Ai}{2}).\end{aligned}$$They can be uniformly written as

$$\begin{aligned}
G_{st}(u,v) &=&\frac{A}{4\pi }e^{2\pi i\phi
}\sum_{a,d=0,1}(-1)^{ad}\theta
(-u+\frac{1}{2}(Ad+t)+\frac{i}{2}(s+\mu -\nu ),\frac{Ai}{2})  \nonumber \\
&&\times \theta (-Av+\frac{1}{2}(Aa+s-\mu -\nu
)+\frac{i}{2}t,\frac{Ai}{2}),\end{aligned}$$

which is $Z_{4}$ covariant. So we have from (\[eq:30\])(\[eq:31\]) and (\[eq:32\]), $$\psi _{st}(k,q)=\frac{1}{A}\{\frac{\int d^{2}\mu d^{2}\nu
G_{st}(u,v)f_{1}(\mu )f_{2}(\nu )}{\int d^{2}\mu d^{2}\nu
G_{00}(u,v)f_{1}(\mu )f_{2}(\nu )}\} \label{eq:33}$$where functions $f_{i}$ should satisfy$$f_{i}(\omega _{N}\xi )=e^{i\alpha _{i}}f_{i}(\xi ).$$ Let $\hat{u}=\frac{l}{2\pi }\hat{y}_{2}$ and $A\hat{v}=\frac{l}{2\pi }\hat{y}%
_{1}$, we may replace $u,v$ by $\hat{u}$ and $\hat{v}$ in (\[eq:33\]) and get $$\Psi _{st}(u_{1}^{A},u_{2}^{A})=\frac{1}{A}\frac{\int d^{2}\mu
d^{2}\nu G_{st}(\frac{l}{2\pi }\hat{y}_{2},\frac{l}{2A\pi
}\hat{y}_{1})f_{1}(\mu
)f_{2}(\nu )}{\int d^{2}\mu d^{2}\nu G_{00}(\frac{l}{2\pi }\hat{y}_{2},\frac{%
l}{2A\pi }\hat{y}_{1})f_{1}(\mu )f_{2}(\nu )}.$$The operators $u_{1}$ and $u_{2}$ commute with the operators $u_{1}^{A}$ and $u_{2}^{A}$,and from (\[eq:40\]) $$\begin{aligned}
u_{1}^{s} &=&e^{-2\pi i\frac{s}{A}\hat{u}}, \\
u_{2}^{t} &=&e^{-2\pi it\hat{v}}.\end{aligned}$$Further takeing $u_{1}^{s}$ and $u_{2}^{t}$ into account, we can insert them to the corresponding operator form of Eq.(\[eq:13\]). This leads to the function of $\hat{u}$ and $\hat{v}$ $$\begin{aligned}
h_{ad} &\equiv &u_{1}^{s}u_{2}^{t}G_{st}(\hat{u},\hat{v})\nonumber\\
&=&\frac{A}{4\pi}\sum_{a,d=0}^{1}(-1)^{ad}e^{2\pi i\phi}e^{-2\pi i\frac{s}{A}\hat{u}}\theta (-%
\hat{u}+\frac{1}{2}(Ad+t)+\frac{i}{2}(s+\mu -\nu ),\frac{Ai}{2})
\nonumber
\\
&&\times e^{-2\pi i\hat{v}t}\theta (-A\hat{v}+\frac{1}{2}(Aa+s-\mu -\nu )+\frac{i}{%
2}t,\frac{Ai}{2})  \nonumber \\
&=&\frac{A}{4\pi}\sum_{a,d=0}^{1}(-1)^{ad}e^{2\pi
i(-\frac{3st}{2A}+\frac{i}{4A}(|\mu |^{2}+|\nu |^{2}-2\mu \nu
))}e^{-2\pi i(\frac{sd}{2}+\frac{ta}{2})}  \nonumber \\
&&\times \theta \left[
\begin{array}{l}
\frac{s}{A} \\
\frac{t}{2}%
\end{array}%
\right] (-\hat{u}+\frac{1}{2}Ad+\frac{i}{2}(\mu -\nu
),\frac{Ai}{2})
\nonumber \\
&&\times \theta \left[
\begin{array}{l}
\frac{t}{A} 