*3.3. The Choice of Weights wc*

In the following, we present the basic condition of weights *wc*, according to (25), which satisfies that

$$\begin{split} 0 &= \sum\_{(\omega\_x, \omega\_y) \in \mathcal{C}\_R^\ell} w\_c(\omega\_x, \omega\_y) [\cos(\frac{2\pi}{Rm\_0 + 1} \mathbf{x} \omega\_x) \cos(\frac{2\pi}{Rm\_0 + 1} y \omega\_y) \\ &- \sin(\frac{2\pi}{Rm\_0 + 1} \mathbf{x} \omega\_x) \sin(\frac{2\pi}{Rm\_0 + 1} y \omega\_y)]; \\ 0 &= \sum\_{(\omega\_x, \omega\_y) \in \mathcal{C}\_R^\ell} w\_c(\omega\_x, \omega\_y) [\sin(\frac{2\pi}{Rm\_0 + 1} \mathbf{x} \omega\_x) \cos(\frac{2\pi}{Rm\_0 + 1} y \omega\_y) \\ &+ \cos(\frac{2\pi}{Rm\_0 + 1} \mathbf{x} \omega\_x) \sin(\frac{2\pi}{Rm\_0 + 1} y \omega\_y)]. \end{split} \tag{26}$$

According to the symmetry of the *CFT*, the weighting function *wc* is assumed to satisfy

$$\begin{aligned} w\_{\varepsilon}(\omega\_{\boldsymbol{X},\prime}\omega\_{\boldsymbol{y}}) &= w\_{\varepsilon}(\omega\_{\boldsymbol{y},\prime}\omega\_{\boldsymbol{x}}), (\omega\_{\boldsymbol{X},\prime}\omega\_{\boldsymbol{y}}) \in \mathcal{C}\_{\mathbb{R}}, \\ w\_{\varepsilon}(\omega\_{\boldsymbol{X},\prime}\omega\_{\boldsymbol{y}}) &= w\_{\varepsilon}(\omega\_{\boldsymbol{y},\prime} - \omega\_{\boldsymbol{x}}), (\omega\_{\boldsymbol{X},\prime}\omega\_{\boldsymbol{y}}) \in \mathcal{C}\_{\mathbb{R}}, \\ w\_{\varepsilon}(\omega\_{\boldsymbol{x},\prime}\omega\_{\boldsymbol{y}}) &= w\_{\varepsilon}(-\omega\_{\boldsymbol{y},\prime} - \omega\_{\boldsymbol{x}}), (\omega\_{\boldsymbol{x},\prime}\omega\_{\boldsymbol{y}}) \in \mathcal{C}\_{\mathbb{R}}. \end{aligned} \tag{27}$$

where four equations of (27) describe the (*<sup>ω</sup>y* = *<sup>ω</sup>x*)-symmetry, (*<sup>ω</sup>y* = <sup>−</sup>*ωx*)-symmetry and the *origin*-symmetry.

In addition,

$$\sum\_{\left(\omega\_{\mathbf{x}},\omega\_{\mathbf{y}}\right)\in\mathbb{C}\_{\mathbb{R}}} w\_{\mathfrak{c}}(\omega\_{\mathbf{x}},\omega\_{\mathbf{y}}) = 1. \tag{28}$$

To avoid high complexity, we choose the weight *<sup>w</sup>*(*<sup>ω</sup>x*, *<sup>ω</sup>y*) in the form:

$$w(\omega\_{\mathbf{x}\_{\prime}}\omega\_{\mathbf{y}}) = \frac{w\_{0}(\omega\_{\mathbf{x}\_{\prime}}\omega\_{\mathbf{y}})}{\sum\_{\omega\_{\mathbf{x}}\omega\_{\mathbf{y}}}w\_{0}(\omega\_{\mathbf{x}\_{\prime}}\omega\_{\mathbf{y}})} \tag{29}$$

where

$$w\_0(\omega\_{x\prime}\omega\_y) = \frac{1}{m\_0r^{\prime}}\{(\omega\_{x\prime}\omega\_y) : \omega\_x^2 + \omega\_y^2 = r^2\},\tag{30}$$

with *r* ∈ [1, *<sup>R</sup>*], and *w*(0, 0) = 1.
