the constant term’s coefficient of $\Psi _{00}$. Nextly we introduce rotation $R$ in noncommutative space $R^{2}$ $$R(\theta )=e^{-i\theta \frac{\hat{y_{1}}^{2}+\hat{y_{2}}^{2}}{2}+i\frac{%
\theta }{2}}$$with $$R^{-1}\hat{y}_{1}R =\cos \theta \hat{y}_{1}+\sin \theta
\hat{y}_{2},~~~~ R^{-1}\hat{y}_{2}R =\cos \theta \hat{y}_{2}-\sin
\theta \hat{y}_{1}.$$When $\tau =\tau _{1}+\tau _{2}=e^{2\pi i/N}$, setting $\theta
=2\pi /N(N\in
Z)$. The noncommutative torus $T^{2}$ keep invariant under rotation $%
R_{N}\equiv R(2\pi /N)$ [@16; @11; @9]. Namely $R_{N}^{-1}\hat{O}%
_{T^{2}}R_{N} $ is still the operators on the noncommutative Torus $T^{2}$. Now $U_{i}^{\prime }\equiv R_{N}^{-1}U_{i}R_{N}$ can be expressed by monomial of $\{U_{i}\}$ and their inverses [@11].In this case, we call the operators invariant under rotation $R_{N}$ on the noncommutative torus as operators on noncommutative orbifold $T^{2}/Z_{N}$. We can also realize these operators in Fock space. Introduce $$a=\frac{\hat{y}_{2}-i\hat{y}_{1}}{\sqrt{2}},~~~~~~~~~a^{+}=\frac{\hat{y}%
_{2}+i\hat{y}_{1}}{\sqrt{2}},$$then $$\begin{aligned}
\lbrack a,a^{+}] &=&1, \\
R_{N} &=&e^{-i\theta a^{+}a}.\end{aligned}$$In this paper, we study the projector $P$ on the orbifold $T^{2}/Z_{4}$: $$\begin{aligned}
\tau &=&i, \\
P^{2} &=&P, \\
U_{j}^{-1}PU_{j} &=&P,~~~~~~j=1,2 \\
R_{4}^{-1}PR_{4} &=&P.\end{aligned}$$

The $|k,q>$ representation, standard form and eigen value function
==================================================================

From the above discussion, we know that the operators $U_{1}$ and $U_{2}$ commute with each other on the integral torus $T^{2}$ when $A$ is an integer. So we can introduce a complete set of their common eigenstates, namely $|k,q>$ representation [@21; 