On n-sectors of the Angles of an Arbitrary Triangle

Abstract

Morley’s theorem shows that the three points, each of which is the intersection of the two internal trisectors that are the closest to the same side of an arbitrary triangle $\Delta$, form an equilateral triangle. This beautiful theorem was proved mechanically by Wen-tsün Wu (J. Syst. Sci. Math. Sci. 4:207–235, 1984) in its most general form: the neighbouring trisectors of the three angles of $\Delta$ intersect to form 27 triangles in all, of which 18 are equilateral triangles, called Morley triangles. A natural question is: does there exist any equilateral triangle, other than Morley triangles, which is formed by three intersection points of the neighbouring angular n-sectors of $\Delta$ for $n>3$? In this paper, we approach this question using specialized techniques with interactive, semi-automatic algebraic computations and prove that for $n=4$ and 5 the three points, each of which is the intersection of the two internal (or two external) angular n-sectors closest to the same side of $\Delta$, form an equilateral triangle if and only if $\Delta$ is equilateral. The computational approach we present can also be applied to other cases for specific n. How to establish the non-existence of equilateral triangles formed by the intersection points of angular n-sectors for general n is a question that remains for further investigation.

Appendix A. List of Polynomials

$$\begin{aligned} f_1= & {} e_{{1}}^{2}k_{{n}}l_{{n}}^{3}y_{{n}}^{2} +e_{{2}}^{2}l_{{n}}^{3}s_{{n}}x_{{n}} -\sqrt{3}e_{{1}}e_{{2}}k_{{n}}l_{{n}}^{3}x_{{n}}y_{{n}}^{2} +\sqrt{3}e_{{1}}e_{{2}}l_{{n}}^{2}s_{{n}}t_{{n}}y_{{n}}^{3} +2\sqrt{3}e_{{1}}e_{{2}}s_{{n}}t_{{n}}^{3}x_{{n}}y_{{n}}^{2}\\&-2\sqrt{3}e_{{2}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}y_{{n}}^{2} -2\sqrt{3}e_{{1}}e_{{2}}l_{{n}}^{3}s_{{n}}x_{{n}}y_{{n}} +\sqrt{3}e_{{1}}e_{{2}}l_{{n}}s_{{n}}t_{{n}}^{2}y_{{n}}^{2} +2\sqrt{3}e_{{2}}^{2}l_{{n}}s_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}}\\&+\sqrt{3}e_{{1}}e_{{2}}l_{{n}}^{2}s_{{n}}t_{{n}}y_{{n}} +2\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}k_{{n}}l_{{n}}^{2}t_{{n}}x_{{n}}y_{{n}}^{3} +2\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}y_{{n}}^{2} -\sqrt{3}e_{{1}}^{2}e_{{2}}k_{{n}}l_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}}^{3}\\&-\sqrt{3}e_{{1}}e_{{2}}^{2}k_{{n}}l_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}}^{3} +\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}k_{{n}}l_{{n}}^{2}t_{{n}}x_{{n}}y_{{n}} -4\sqrt{3}e_{{1}}^{2}e_{{2}}l_{{n}}s_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}}^{2} -4\sqrt{3}e_{{1}}e_{{2}}^{2}l_{{n}}s_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}}^{2}\\&-\sqrt{3}e_{{1}}e_{{2}}k_{{n}}l_{{n}}^{2}t_{{n}}x_{{n}}y_{{n}}^{3} -4\sqrt{3}e_{{1}}^{2}e_{{2}}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}y_{{n}} -4\sqrt{3}e_{{1}}e_{{2}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}y_{{n}} -\sqrt{3}e_{{1}}e_{{2}}k_{{n}}l_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}}^{2}\\&-2\sqrt{3}e_{{1}}e_{{2}}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}y_{{n}}^{2} -\sqrt{3}e_{{1}}e_{{2}}k_{{n}}l_{{n}}^{2}t_{{n}}x_{{n}}y_{{n}} +2\sqrt{3}e_{{1}}e_{{2}}l_{{n}}s_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}} +\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}k_{{n}}l_{{n}}^{3}x_{{n}}y_{{n}}^{2}\\&-\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}k_{{n}}l_{{n}}^{2}t_{{n}}y_{{n}}^{2} +\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}l_{{n}}^{3}s_{{n}}x_{{n}}y_{{n}} +\sqrt{3}e_{{1}}^{2}e_{{2}}l_{{n}}s_{{n}}t_{{n}}^{2}y_{{n}}^{3} +\sqrt{3}e_{{1}}^{2}k_{{n}}l_{{n}}^{2}t_{{n}}x_{{n}}y_{{n}}^{3}\\&+\sqrt{3}e_{{1}}^{2}l_{{n}}s_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}}^{3} +\sqrt{3}e_{{1}}e_{{2}}^{2}l_{{n}}s_{{n}}t_{{n}}^{2}y_{{n}}^{3} -\sqrt{3}e_{{2}}^{2}l_{{n}}s_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}}^{3} +\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}\\&-\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}y_{{n}} +\sqrt{3}e_{{1}}^{2}e_{{2}}k_{{n}}l_{{n}}^{3}x_{{n}}y_{{n}} -\sqrt{3}e_{{1}}^{2}k_{{n}}l_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}}^{2} +2\sqrt{3}e_{{1}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}y_{{n}}^{2}\\&+\sqrt{3}e_{{1}}e_{{2}}^{2}k_{{n}}l_{{n}}^{3}x_{{n}}y_{{n}} -\sqrt{3}e_{{1}}^{2}e_{{2}}l_{{n}}^{3}s_{{n}}y_{{n}} -\sqrt{3}e_{{1}}^{2}k_{{n}}l_{{n}}^{2}t_{{n}}y_{{n}}^{2} +\sqrt{3}e_{{1}}^{2}l_{{n}}s_{{n}}t_{{n}}^{2}y_{{n}}^{2}\\&-\sqrt{3}e_{{1}}e_{{2}}^{2}l_{{n}}^{3}s_{{n}}y_{{n}} +\sqrt{3}e_{{1}}e_{{2}}l_{{n}}^{3}s_{{n}}y_{{n}}^{2} -\sqrt{3}e_{{2}}^{2}l_{{n}}^{3}s_{{n}}x_{{n}}y_{{n}} +\sqrt{3}e_{{2}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}\\&+\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}k_{{n}}l_{{n}}^{3}y_{{n}}^{3} +\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}l_{{n}}^{3}s_{{n}}y_{{n}}^{2} +\sqrt{3}e_{{1}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}y_{{n}}^{3} +\sqrt{3}e_{{1}}^{2}s_{{n}}t_{{n}}^{3}x_{{n}}y_{{n}}^{2}\\&+\sqrt{3}e_{{2}}^{2}s_{{n}}t_{{n}}^{3}x_{{n}}y_{{n}}^{2} -e_{{1}}e_{{2}}^{2}k_{{n}}l_{{n}}^{3}x_{{n}}y_{{n}}^{2} -e_{{1}}e_{{2}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}y_{{n}}^{3} +2e_{{1}}e_{{2}}s_{{n}}t_{{n}}^{3}x_{{n}}y_{{n}}^{3}\\&-2e_{{1}}^{2}e_{{2}}l_{{n}}^{3}s_{{n}}x_{{n}}y_{{n}} -e_{{1}}^{2}e_{{2}}l_{{n}}s_{{n}}t_{{n}}^{2}y_{{n}}^{2} +e_{{1}}^{2}k_{{n}}l_{{n}}^{2}t_{{n}}x_{{n}}y_{{n}}^{2} +e_{{1}}^{2}l_{{n}}s_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}}^{2}\\ \end{aligned}$$

$$\begin{aligned}&\qquad \, -2e_{{1}}e_{{2}}^{2}l_{{n}}^{3}s_{{n}}x_{{n}}y_{{n}} -e_{{1}}e_{{2}}^{2}l_{{n}}s_{{n}}t_{{n}}^{2}y_{{n}}^{2} +e_{{1}}e_{{2}}l_{{n}}s_{{n}}t_{{n}}^{2}y_{{n}}^{3} +3e_{{2}}^{2}l_{{n}}s_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}}^{2}\\&\qquad \, -e_{{1}}^{2}e_{{2}}l_{{n}}^{2}s_{{n}}t_{{n}}y_{{n}} -e_{{1}}e_{{2}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}y_{{n}} -e_{{1}}e_{{2}}k_{{n}}l_{{n}}^{3}x_{{n}}y_{{n}} +3e_{{2}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}y_{{n}}\\&\qquad \, +e_{{1}}^{2}e_{{2}}^{2}k_{{n}}l_{{n}}^{2}t_{{n}}y_{{n}}^{3} +e_{{1}}^{2}e_{{2}}^{2}k_{{n}}l_{{n}}^{3}x_{{n}}y_{{n}} +e_{{1}}^{2}e_{{2}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}y_{{n}}^{2} -e_{{1}}^{2}e_{{2}}k_{{n}}l_{{n}}^{3}x_{{n}}y_{{n}}^{2}\\&\qquad \, -e_{{1}}^{2}e_{{2}}l_{{n}}^{2}s_{{n}}t_{{n}}y_{{n}}^{3} +e_{{1}}^{2}k_{{n}}l_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}}^{3} +e_{{1}}^{2}e_{{2}}^{2}k_{{n}}l_{{n}}^{3}y_{{n}}^{2} +e_{{1}}^{2}s_{{n}}t_{{n}}^{3}x_{{n}}y_{{n}}^{3}\\&\qquad \, +e_{{2}}^{2}s_{{n}}t_{{n}}^{3}x_{{n}}y_{{n}}^{3} +e_{{1}}^{2}e_{{2}}^{2}l_{{n}}^{3}s_{{n}}x_{{n}} +e_{{1}}^{2}e_{{2}}^{2}l_{{n}}^{3}s_{{n}}y_{{n}} -e_{{1}}^{2}e_{{2}}l_{{n}}^{3}s_{{n}}y_{{n}}^{2} -e_{{1}}e_{{2}}^{2}k_{{n}}l_{{n}}^{2}t_{{n}}x_{{n}}y_{{n}}\\&\qquad \, +e_{{1}}^{2}k_{{n}}l_{{n}}^{2}t_{{n}}y_{{n}}^{3} +e_{{1}}^{2}l_{{n}}s_{{n}}t_{{n}}^{2}y_{{n}}^{3} -e_{{1}}e_{{2}}^{2}l_{{n}}^{3}s_{{n}}y_{{n}}^{2} +e_{{1}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}y_{{n}}^{2} +4e_{{1}}e_{{2}}l_{{n}}s_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}}^{2}\\&\qquad \, -e_{{1}}e_{{2}}l_{{n}}^{3}s_{{n}}y_{{n}} +\sqrt{3}e_{{1}}^{2}k_{{n}}l_{{n}}^{3}y_{{n}}^{3} +2\sqrt{3}e_{{1}}^{2}l_{{n}}^{3}s_{{n}}y_{{n}}^{2} +2e_{{1}}e_{{2}}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}y_{{n}}\\&\qquad \, +e_{{1}}^{2}e_{{2}}^{2}k_{{n}}l_{{n}}^{2}t_{{n}}x_{{n}}y_{{n}}^{2} -e_{{1}}^{2}e_{{2}}k_{{n}}l_{{n}}^{2}t_{{n}}x_{{n}}y_{{n}}^{3} -2e_{{1}}^{2}e_{{2}}l_{{n}}s_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}}^{3} -e_{{1}}e_{{2}}^{2}k_{{n}}l_{{n}}^{2}t_{{n}}x_{{n}}y_{{n}}^{3}\\&\qquad \, -2e_{{1}}e_{{2}}^{2}l_{{n}}s_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}}^{3} +e_{{1}}^{2}e_{{2}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}y_{{n}} -e_{{1}}^{2}e_{{2}}k_{{n}}l_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}}^{2} -4e_{{1}}^{2}e_{{2}}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}y_{{n}}^{2}\\&\qquad \, -e_{{1}}e_{{2}}^{2}k_{{n}}l_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}}^{2} -4e_{{1}}e_{{2}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}y_{{n}}^{2} +e_{{1}}e_{{2}}k_{{n}}l_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}}^{3} -e_{{1}}^{2}e_{{2}}k_{{n}}l_{{n}}^{2}t_{{n}}x_{{n}}y_{{n}}, \end{aligned}$$

$$\begin{aligned} f_2= & {} -3e_{{1}}^{2}l_{{n}}^{3}s_{{n}}y_{{n}}^{2} -3e_{{1}}^{2}e_{{2}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}} +3e_{{1}}^{2}e_{{2}}l_{{n}}^{3}s_{{n}}y_{{n}} +3e_{{1}}e_{{2}}^{2}l_{{n}}^{3}s_{{n}}y_{{n}} -3e_{{1}}e_{{2}}l_{{n}}^{3}s_{{n}}y_{{n}}^{2}\\&+\sqrt{3}e_{{1}}^{2}k_{{n}}l_{{n}}^{3}y_{{n}}^{2} +\sqrt{3}e_{{2}}^{2}l_{{n}}^{3}s_{{n}}x_{{n}} +3e_{{1}}^{2}k_{{n}}l_{{n}}^{2}t_{{n}}y_{{n}}^{2} +6e_{{1}}^{2}e_{{2}}^{2}k_{{n}}l_{{n}}^{2}t_{{n}}x_{{n}}^{2}y_{{n}}^{2}\\&+6e_{{1}}^{2}e_{{2}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}^{2}y_{{n}} -9e_{{1}}^{2}e_{{2}}k_{{n}}l_{{n}}t_{{n}}^{2}x_{{n}}^{2}y_{{n}}^{2} -9e_{{1}}e_{{2}}^{2}k_{{n}}l_{{n}}t_{{n}}^{2}x_{{n}}^{2}y_{{n}}^{2} -3e_{{1}}^{2}e_{{2}}k_{{n}}l_{{n}}^{2}t_{{n}}x_{{n}}^{2}y_{{n}}\\&-9e_{{1}}^{2}e_{{2}}k_{{n}}l_{{n}}^{2}t_{{n}}x_{{n}}y_{{n}}^{2} -3e_{{1}}e_{{2}}^{2}k_{{n}}l_{{n}}^{2}t_{{n}}x_{{n}}^{2}y_{{n}} -3e_{{1}}e_{{2}}k_{{n}}l_{{n}}^{2}t_{{n}}x_{{n}}^{2}y_{{n}}^{2} +3e_{{1}}e_{{2}}k_{{n}}l_{{n}}t_{{n}}^{2}x_{{n}}^{2}y_{{n}}\\&+\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}y_{{n}} -3\sqrt{3}e_{{1}}^{2}e_{{2}}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}y_{{n}}^{2} -\sqrt{3}e_{{1}}^{2}e_{{2}}k_{{n}}l_{{n}}^{2}t_{{n}}x_{{n}}y_{{n}} +\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}k_{{n}}l_{{n}}^{2}t_{{n}}x_{{n}}y_{{n}}^{2}\\&+\sqrt{3}e_{{1}}e_{{2}}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}y_{{n}} -3\sqrt{3}e_{{1}}e_{{2}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}y_{{n}}^{2} -\sqrt{3}e_{{1}}e_{{2}}^{2}k_{{n}}l_{{n}}^{2}t_{{n}}x_{{n}}y_{{n}} +3\sqrt{3}e_{{1}}e_{{2}}l_{{n}}s_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}}^{2}\\&+\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}l_{{n}}^{3}s_{{n}}x_{{n}} -\sqrt{3}e_{{1}}^{2}e_{{2}}l_{{n}}^{3}s_{{n}}y_{{n}}^{2} -\sqrt{3}e_{{1}}e_{{2}}l_{{n}}^{3}s_{{n}}y_{{n}} +\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}k_{{n}}l_{{n}}^{3}y_{{n}}^{2}\\&+\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}l_{{n}}^{3}s_{{n}}y_{{n}} +\sqrt{3}e_{{1}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}y_{{n}}^{2} -\sqrt{3}e_{{1}}e_{{2}}^{2}l_{{n}}^{3}s_{{n}}y_{{n}}^{2} +\sqrt{3}e_{{1}}^{2}s_{{n}}t_{{n}}^{3}x_{{n}}^{2}y_{{n}}^{2}\\&+\sqrt{3}e_{{2}}^{2}s_{{n}}t_{{n}}^{3}x_{{n}}^{2}y_{{n}}^{2} +\sqrt{3}e_{{2}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}^{2} -3e_{{1}}e_{{2}}l_{{n}}^{2}s_{{n}}t_{{n}}y_{{n}} +3e_{{1}}^{2}e_{{2}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}y_{{n}}\\&-3e_{{1}}^{2}e_{{2}}l_{{n}}^{2}s_{{n}}t_{{n}}y_{{n}}^{2} -3e_{{1}}e_{{2}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}y_{{n}}^{2} -3e_{{1}}e_{{2}}k_{{n}}l_{{n}}^{3}x_{{n}}y_{{n}}^{2} +3e_{{1}}^{2}e_{{2}}^{2}k_{{n}}l_{{n}}^{3}x_{{n}}y_{{n}}^{2}\\ \end{aligned}$$

$$\begin{aligned}&\qquad \, +3e_{{1}}^{2}e_{{2}}^{2}k_{{n}}l_{{n}}^{2}t_{{n}}y_{{n}}^{2} -3e_{{1}}^{2}e_{{2}}k_{{n}}l_{{n}}^{3}x_{{n}}y_{{n}} +6e_{{1}}^{2}k_{{n}}l_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}}^{2} -3e_{{1}}e_{{2}}^{2}k_{{n}}l_{{n}}^{3}x_{{n}}y_{{n}}\nonumber \\&\qquad \,+\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}k_{{n}}l_{{n}}^{3}x_{{n}}y_{{n}} -\sqrt{3}e_{{1}}^{2}e_{{2}}k_{{n}}l_{{n}}^{3}x_{{n}}y_{{n}}^{2} -\sqrt{3}e_{{1}}^{2}e_{{2}}l_{{n}}^{2}s_{{n}}t_{{n}}y_{{n}} -\sqrt{3}e_{{1}}e_{{2}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}y_{{n}}\nonumber \\&\qquad \, -2\sqrt{3}e_{{1}}^{2}e_{{2}}l_{{n}}^{3}s_{{n}}x_{{n}}y_{{n}} +2\sqrt{3}e_{{1}}^{2}k_{{n}}l_{{n}}^{2}t_{{n}}x_{{n}}y_{{n}}^{2} +2\sqrt{3}e_{{1}}^{2}l_{{n}}s_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}}^{2} -\sqrt{3}e_{{1}}e_{{2}}^{2}k_{{n}}l_{{n}}^{3}x_{{n}}y_{{n}}^{2}\nonumber \\&\qquad \, -2\sqrt{3}e_{{1}}e_{{2}}^{2}l_{{n}}^{3}s_{{n}}x_{{n}}y_{{n}} -\sqrt{3}e_{{1}}e_{{2}}k_{{n}}l_{{n}}^{3}x_{{n}}y_{{n}} +\sqrt{3}e_{{2}}^{2}l_{{n}}s_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}}^{2} +2\sqrt{3}e_{{2}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}y_{{n}}\nonumber \\&\qquad \, +3e_{{2}}^{2}l_{{n}}^{3}s_{{n}}x_{{n}}^{2} -3e_{{2}}^{2}l_{{n}}s_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}} +3e_{{2}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}y_{{n}}^{2} -3e_{{1}}^{2}e_{{2}}l_{{n}}s_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}}^{2}\nonumber \\&\qquad \, -3e_{{1}}e_{{2}}^{2}l_{{n}}s_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}}^{2} +3e_{{1}}^{2}e_{{2}}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}y_{{n}} -3e_{{1}}e_{{2}}l_{{n}}s_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}} -3e_{{1}}^{2}e_{{2}}^{2}k_{{n}}l_{{n}}^{2}t_{{n}}x_{{n}}y_{{n}}\nonumber \\&\qquad \, +6e_{{1}}e_{{2}}k_{{n}}l_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}}^{2} +3e_{{1}}e_{{2}}k_{{n}}l_{{n}}^{2}t_{{n}}x_{{n}}y_{{n}} -9e_{{1}}e_{{2}}^{2}k_{{n}}l_{{n}}^{2}t_{{n}}x_{{n}}y_{{n}}^{2} +3e_{{2}}^{2}l_{{n}}^{3}s_{{n}}x_{{n}}y_{{n}}\nonumber \\&\qquad \, +3e_{{1}}^{2}e_{{2}}^{2}l_{{n}}^{3}s_{{n}}x_{{n}}^{2} +3e_{{1}}^{2}k_{{n}}t_{{n}}^{3}x_{{n}}^{2}y_{{n}}^{2} +3e_{{2}}^{2}k_{{n}}t_{{n}}^{3}x_{{n}}^{2}y_{{n}}^{2} +3e_{{2}}^{2}k_{{n}}l_{{n}}^{3}x_{{n}}^{2}y_{{n}} -3e_{{2}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}\nonumber \\&\qquad \, +3e_{{1}}^{2}e_{{2}}^{2}l_{{n}}^{3}s_{{n}}x_{{n}}y_{{n}} -6e_{{1}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}y_{{n}}^{2} +3e_{{2}}^{2}k_{{n}}l_{{n}}t_{{n}}^{2}x_{{n}}^{2}y_{{n}} +6e_{{2}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}^{2}y_{{n}}\nonumber \\&\qquad \, +3e_{{1}}^{2}e_{{2}}^{2}k_{{n}}l_{{n}}^{3}x_{{n}}^{2}y_{{n}} -3e_{{1}}^{2}l_{{n}}s_{{n}}t_{{n}}^{2}x_{{n}}^{2}y_{{n}}^{2} +6e_{{1}}e_{{2}}k_{{n}}t_{{n}}^{3}x_{{n}}^{2}y_{{n}}^{2} +3e_{{2}}^{2}k_{{n}}l_{{n}}^{2}t_{{n}}x_{{n}}^{2}y_{{n}}^{2}\nonumber \\&\qquad \, +3e_{{2}}^{2}l_{{n}}s_{{n}}t_{{n}}^{2}x_{{n}}^{2}y_{{n}}^{2} +3e_{{1}}e_{{2}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}y_{{n}} -3e_{{1}}e_{{2}}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}y_{{n}}^{2} +\sqrt{3}e_{{1}}e_{{2}}l_{{n}}^{2}s_{{n}}t_{{n}}y_{{n}}^{2}\nonumber \\&\qquad \, +\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}^{2} +\sqrt{3}e_{{1}}^{2}k_{{n}}l_{{n}}t_{{n}}^{2}x_{{n}}^{2}y_{{n}}^{2} +2\sqrt{3}e_{{1}}e_{{2}}s_{{n}}t_{{n}}^{3}x_{{n}}^{2}y_{{n}}^{2} +2\sqrt{3}e_{{2}}^{2}l_{{n}}s_{{n}}t_{{n}}^{2}x_{{n}}^{2}y_{{n}}\nonumber \\&\qquad \, -2\sqrt{3}e_{{1}}e_{{2}}^{2}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}^{2}y_{{n}} +\sqrt{3}e_{{1}}e_{{2}}k_{{n}}l_{{n}}t_{{n}}^{2}x_{{n}}^{2}y_{{n}}^{2} -\sqrt{3}e_{{1}}^{2}e_{{2}}l_{{n}}s_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}} -\sqrt{3}e_{{1}}e_{{2}}^{2}l_{{n}}s_{{n}}t_{{n}}^{2}x_{{n}}y_{{n}}\nonumber \\&\qquad \, -\sqrt{3}e_{{1}}e_{{2}}k_{{n}}l_{{n}}^{2}t_{{n}}x_{{n}}^{2}y_{{n}} +\sqrt{3}e_{{1}}e_{{2}}k_{{n}}l_{{n}}^{2}t_{{n}}x_{{n}}y_{{n}}^{2} +2\sqrt{3}e_{{1}}e_{{2}}l_{{n}}s_{{n}}t_{{n}}^{2}x_{{n}}^{2}y_{{n}} +\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}k_{{n}}l_{{n}}^{2}t_{{n}}x_{{n}}^{2}y_{{n}}\nonumber \\&\qquad \, -\sqrt{3}e_{{1}}^{2}e_{{2}}k_{{n}}l_{{n}}^{2}t_{{n}}x_{{n}}^{2}y_{{n}}^{2} -2\sqrt{3}e_{{1}}^{2}e_{{2}}l_{{n}}s_{{n}}t_{{n}}^{2}x_{{n}}^{2}y_{{n}}^{2} -\sqrt{3}e_{{1}}e_{{2}}^{2}k_{{n}}l_{{n}}^{2}t_{{n}}x_{{n}}^{2}y_{{n}}^{2} -2\sqrt{3}e_{{1}}e_{{2}}^{2}l_{{n}}s_{{n}}t_{{n}}^{2}x_{{n}}^{2}y_{{n}}^{2}\nonumber \\&\qquad \, -\sqrt{3}e_{{1}}^{2}e_{{2}}k_{{n}}l_{{n}}t_{{n}}^{2}x_{{n}}^{2}y_{{n}} -2\sqrt{3}e_{{1}}^{2}e_{{2}}l_{{n}}^{2}s_{{n}}t_{{n}}x_{{n}}^{2}y_{{n}} -\sqrt{3}e_{{1}}e_{{2}}^{2}k_{{n}}l_{{n}}t_{{n}}^{2}x_{{n}}^{2}y_{{n}}. \end{aligned}$$

(A.1)

$$\begin{aligned} {\bar{H}}_1^{(5)}= & {} -30+16\sqrt{5}{y}^{2}+94\sqrt{5}{x}^{2} +6\sqrt{3}x^{3}y^{4}+52\sqrt{3}{x}^{3}{y}^{2}-48\sqrt{3}{x}^{2}{y}^{3} -240\sqrt{3}{x}^{2}y\nonumber \\&+6\sqrt{3}x{y}^{4}+20\sqrt{3}x{y}^{2}-4\sqrt{3}{x}^{4}{y}^{3} +36\sqrt{3}{x}^{4}y+3\sqrt{150-30 \sqrt{5}}{y}^{4}\nonumber \\&-20\sqrt{15}{x}^{3}{y}^{2}-5\sqrt{150-30 \sqrt{5}}{x}^{2}+3{x}^{2}{y}^{4}\sqrt{30-6 \sqrt{5}}-12{x}^{2}{y}^{3}\sqrt{50-10\sqrt{5}}\nonumber \\&-6{x}^{2}{y}^{2}\sqrt{30-6\sqrt{5}} -28\sqrt{50-10\sqrt{5}}{x}^{2}y +64\sqrt{15}{x}^{2}y-\sqrt{50-10\sqrt{5}}x{y}^{4}\nonumber \\&+10\sqrt{15}x{y}^{4} +60\sqrt{30-6\sqrt{5}}xy -4\sqrt{30-6\sqrt{5}}x{y}^{3} +14\sqrt{50-10\sqrt{5}}x{y}^{2}\nonumber \\&+44\sqrt{15}x{y}^{2}-6\sqrt{150-30\sqrt{5}}{x}^{4} +4\sqrt{15}{x}^{4}{y}^{3} -2\sqrt{30-6\sqrt{5}}{x}^{4}{y}^{2}\nonumber \\&+8\sqrt{50-10\sqrt{5}}{x}^{4}y -20\sqrt{15}{x}^{4}y+3\sqrt{50-10\sqrt{5}}{x}^{3}{y}^{4} -6\sqrt{15}{x}^{3}{y}^{4}\nonumber \\&-4\sqrt{30-6\sqrt{5}}{x}^{3}y-4\sqrt{30-6\sqrt{5}}{x}^{3}{y}^{3} +6\sqrt{50-10\sqrt{5}}{x}^{3}{y}^{2}+44\sqrt{5}{x}^{4}\nonumber \\&+2\sqrt{150-30\sqrt{5}}{x}^{4}{y}^{2} -12\sqrt{150-30\sqrt{5}}{x}^{3}y +4\sqrt{150-30\sqrt{5}}{x}^{3}{y}^{3}\nonumber \\&-\sqrt{150-30\sqrt{5}}{x}^{2}{y}^{4}-6\sqrt{150-30\sqrt{5}}{x}^{2}{y}^{2} -12\sqrt{150-30\sqrt{5}}xy\nonumber \\&+4\sqrt{150-30\sqrt{5}}x{y}^{3} +20\sqrt{15}y+5\sqrt{150-30\sqrt{5}} +78\sqrt{10-2\sqrt{5}}x{y}^{2}\nonumber \\&+144\sqrt{5}xy-29\sqrt{50-10\sqrt{5}}{x}^{3} +66\sqrt{15}{x}^{3}+6\sqrt{5}{x}^{2}{y}^{4} -\sqrt{10-2\sqrt{5}}{x}^{3}{y}^{4}\nonumber \\&-32\sqrt{5}{x}^{3}{y}^{3}-42\sqrt{10-2\sqrt{5}}{x}^{3}{y}^{2} +48\sqrt{5}{x}^{3}y-12\sqrt{10-2\sqrt{5}}{x}^{2}{y}^{3}\nonumber \\&+36\sqrt{5}{x}^{2}{y}^{2}+132\sqrt{10-2\sqrt{5}}{x}^{2}y +55\sqrt{30-6\sqrt{5}}{x}^{2}-17\sqrt{5}\sqrt{10-2\sqrt{5}}x\nonumber \\&+3\sqrt{10-2\sqrt{5}}x{y}^{4}+8\sqrt{10-2\sqrt{5}}{x}^{4}{y}^{3} -12\sqrt{5}{x}^{4}{y}^{2}-24\sqrt{10-2\sqrt{5}}{x}^{4}y\nonumber \\&+14\sqrt{30-6\sqrt{5}}{x}^{4} -\sqrt{30-6\sqrt{5}}{y}^{4}+4\sqrt{50-10\sqrt{5}}{y}^{3} -4\sqrt{15}{y}^{3}\nonumber \\&+20\sqrt{30-6\sqrt{5}}{y}^{2}-20\sqrt{50-10\sqrt{5}}y +71\sqrt{10-2\sqrt{5}}{x}^{3}+5\sqrt{30-6\sqrt{5}}\nonumber \\&-10\sqrt{5}{y}^{4}-4\sqrt{10-2\sqrt{5}}{y}^{3} -20\sqrt{10-2\sqrt{5}}y-5\sqrt{10-2\sqrt{5}}x-130\sqrt{3}{x}^{3}\nonumber \\&+20\sqrt{3}{y}^{3}+30\sqrt{3}x-20\sqrt{3}y -30\sqrt{5}+22{y}^{4}-80{y}^{2}-104{x}^{4}\nonumber \\&-230{x}^{2} -4{x}^{4}{y}^{4}+28{x}^{4}{y}^{2}+32{x}^{3}{y}^{3}-14{x}^{2}{y}^{4} -80{x}^{3}y+50\sqrt{15}x\nonumber \\&+12{x}^{2}{y}^{2}+64x{y}^{3}-240xy,\nonumber \\ {\bar{H}}_2^{(5)}= & {} 30-94\sqrt{5}{y}^{2}-48\sqrt{5}{y}^{3}x -16\sqrt{5}{x}^{2}+4\sqrt{3}{x}^{3}{y}^{4}+48\sqrt{3}{x}^{3}{y}^{2} -52\sqrt{3}{x}^{2}{y}^{3}\nonumber \\&-20\sqrt{3}{x}^{2}y-36\sqrt{3}x{y}^{4}+240\sqrt{3}x{y}^{2} -6\sqrt{3}{x}^{4}{y}^{3}-6\sqrt{3}{x}^{4}y+6\sqrt{150-30\sqrt{5}}{y}^{4}\nonumber \\&+2\sqrt{30-6\sqrt{5}}{x}^{2}{y}^{4} -6\sqrt{50-10\sqrt{5}}{x}^{2}{y}^{3} +6\sqrt{30-6\sqrt{5}}{x}^{2}{y}^{2} -14\sqrt{50-10\sqrt{5}}{x}^{2}y\nonumber \\&-44\sqrt{15}{x}^{2}y-8\sqrt{50-10\sqrt{5}}x{y}^{4} +20\sqrt{15}x{y}^{4}-60\sqrt{30-6\sqrt{5}}xy +4\sqrt{30-6\sqrt{5}}x{y}^{3}\nonumber \\&+28\sqrt{50-10\sqrt{5}}x{y}^{2} -64\sqrt{15}x{y}^{2} -3\sqrt{150-30\sqrt{5}}{x}^{4} +6\sqrt{15}{x}^{4}{y}^{3}-4\sqrt{15}{x}^{3}{y}^{4}\nonumber \\&-3\sqrt{30-6\sqrt{5}}{x}^{4}{y}^{2} +\sqrt{50-10\sqrt{5}}{x}^{4}y -10\sqrt{15}{x}^{4}y +4\sqrt{30-6\sqrt{5}}{x}^{3}y\nonumber \\&+4\sqrt{30-6\sqrt{5}}{x}^{3}{y}^{3} +12\sqrt{50-10\sqrt{5}}{x}^{3}{y}^{2} +10\sqrt{5}{x}^{4}+20\sqrt{15}{y}^{3}{x}^{2}\nonumber \\&-3\sqrt{50-10\sqrt{5}}{y}^{3}{x}^{4} +5\sqrt{150-30\sqrt{5}}{y}^{2}+\sqrt{150-30\sqrt{5}}{x}^{4}{y}^{2}\nonumber \\&-4\sqrt{150-30\sqrt{5}}{x}^{3}{y}^{3} -2\sqrt{150-30\sqrt{5}}{x}^{2}{y}^{4} +6\sqrt{150-30\sqrt{5}}{x}^{2}{y}^{2}\nonumber \\&+12\sqrt{150-30\sqrt{5}}xy -4\sqrt{150-30\sqrt{5}}{x}^{3}y -20\sqrt{30-6\sqrt{5}}{x}^{2}-144\sqrt{5}xy\nonumber \\&+12\sqrt{150-30\sqrt{5}}x{y}^{3} -50\sqrt{15}y-5\sqrt{150-30\sqrt{5}} -132\sqrt{10-2\sqrt{5}}x{y}^{2}\nonumber \\&-4\sqrt{50-10\sqrt{5}}{x}^{3} +4\sqrt{15}{x}^{3}-8\sqrt{10-2\sqrt{5}}{x}^{3}{y}^{4} +32\sqrt{5}{x}^{3}{y}^{3} +12\sqrt{10-2\sqrt{5}}{x}^{3}{y}^{2}\nonumber \\&+12\sqrt{5}{x}^{2}{y}^{4} +42\sqrt{10-2\sqrt{5}}{x}^{2}{y}^{3} -36\sqrt{5}{x}^{2}{y}^{2} -78\sqrt{10-2\sqrt{5}}{x}^{2}y\nonumber \\&+20\sqrt{50-10\sqrt{5}}x -20\sqrt{15}x+24\sqrt{10-2\sqrt{5}}x{y}^{4} +\sqrt{10-2\sqrt{5}}{x}^{4}{y}^{3} -6\sqrt{5}{x}^{4}{y}^{2}\nonumber \\&-3\sqrt{10-2\sqrt{5}}{x}^{4}y +\sqrt{30-6\sqrt{5}}{x}^{4} -14\sqrt{30-6\sqrt{5}}{y}^{4} +29\sqrt{50-10\sqrt{5}}{y}^{3}\nonumber \\&-66\sqrt{15}{y}^{3}-55\sqrt{30-6\sqrt{5}}{y}^{2} +17\sqrt{50-10\sqrt{5}}y -44\sqrt{5}{y}^{4} -71\sqrt{10-2\sqrt{5}}{y}^{3}\nonumber \\&+5\sqrt{10-2\sqrt{5}}y+20\sqrt{10-2\sqrt{5}}x +4\sqrt{10-2\sqrt{5}}{x}^{3}-5\sqrt{30-6\sqrt{5}} -20\sqrt{3}{x}^{3}\nonumber \\&+130\sqrt{3}{y}^{3} +20\sqrt{3}x-30\sqrt{3}y +30\sqrt{5}+104{y}^{4}+230{y}^{2}-22{x}^{4} +80{x}^{2}+4{x}^{4}{y}^{4}\nonumber \\&+14{x}^{4}{y}^{2}-32{x}^{3}{y}^{3} -28{x}^{2}{y}^{4}-64{x}^{3}y-12{x}^{2}{y}^{2}+80x{y}^{3}+240xy. \end{aligned}$$

(A.2)

$$\begin{aligned} G_1= & {} \frac{1}{16777216}\left( x_5+\sqrt{3}\right) \left( x_5^2+1\right) ^3\Big [\sqrt{10-2\sqrt{5}} \left( \sqrt{5}-3\right) -2\sqrt{15}+6\sqrt{3}-4x_5\Big ]\nonumber \\&\Big [\sqrt{10-2\sqrt{5}}\left( \sqrt{5}+2\right) -\sqrt{15}-3\sqrt{3}+2x_5\Big ] \Big [\sqrt{10-2\sqrt{5}}\left( \sqrt{5}-1\right) +4x_5\Big ]^3\nonumber \\&\Big [\sqrt{10-2\sqrt{5}}\left( \sqrt{5}+1\right) +2\sqrt{15}+2\sqrt{3}-4x_5\Big ] \Big [\sqrt{10-2\sqrt{5}}\left( \sqrt{5}+3\right) -4x_5\Big ]^3\nonumber \\&\Big [\sqrt{10-2\sqrt{5}}\left( \sqrt{5}+3\right) +4x_5\Big ]^3 \Big [\sqrt{10-2\sqrt{5}}\left( \sqrt{5}+1\right) -2\sqrt{15}-2\sqrt{3}-4x_5\Big ]. \end{aligned}$$

(A.3)

$$\begin{aligned} g_1= & {} \sqrt{3}e_{{1}}^{2}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{3}{\bar{y}}_{{n}}^{3} +\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{y}}_{{n}}^{2} +\sqrt{3}e_{{1}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{y}}_{{n}}^{3} +\sqrt{3}e_{{1}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{3}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} +\sqrt{3}e_{{2}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{3}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2}\\&+\sqrt{3}e_{{1}}^{2}e_{{2}}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{y}}_{{n}} -\sqrt{3}e_{{1}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{y}}_{{n}}^{2} +\sqrt{3}e_{{1}}^{2}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{y}}_{{n}}^{2} +\sqrt{3}e_{{1}}e_{{2}}^{2}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{y}}_{{n}} +\sqrt{3}e_{{1}}e_{{2}}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{y}}_{{n}}^{2}\\&-\sqrt{3}e_{{2}}^{2}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}} +\sqrt{3}e_{{2}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}} -e_{{1}}e_{{2}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{y}}_{{n}} +e_{{1}}e_{{2}}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{3}{\bar{x}}_{{n}}{\bar{y}}_{{n}} -3e_{{2}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}}\\&-e_{{1}}^{2}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{y}}_{{n}}^{3} -e_{{1}}^{2}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{3}{\bar{x}}_{{n}}{\bar{y}}_{{n}} -e_{{1}}^{2}e_{{2}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{y}}_{{n}}^{2} -e_{{1}}^{2}e_{{2}}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{3}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} -e_{{1}}^{2}e_{{2}}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{y}}_{{n}}^{3}\\&-e_{{1}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{3} -e_{{1}}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{3}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} -e_{{1}}e_{{2}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}y_{{n}}^{3} -2e_{{1}}e_{{2}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{3}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{3} -2e_{{1}}^{2}e_{{2}}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}}\\&-e_{{1}}^{2}e_{{2}}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{y}}_{{n}}^{2} -e_{{1}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} -e_{{1}}^{2}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} -2e_{{1}}e_{{2}}^{2}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}} -e_{{1}}e_{{2}}^{2}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{y}}_{{n}}^{2}\\&-e_{{1}}e_{{2}}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{y}}_{{n}}^{3} -3e_{{2}}^{2}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} -e_{{1}}^{2}e_{{2}}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{y}}_{{n}} -e_{{1}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{3}{\bar{y}}_{{n}}^{2} -e_{{2}}^{2}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{x}}_{{n}}\\&-\sqrt{3}e_{{2}}^{2}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{3} +\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}} -\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{y}}_{{n}} -\sqrt{3}e_{{1}}^{2}e_{{2}}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{3}{\bar{x}}_{{n}}{\bar{y}}_{{n}}\\&-\sqrt{3}e_{{1}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} +2\sqrt{3}e_{{1}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} -\sqrt{3}e_{{1}}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{3}{\bar{x}}_{{n}}{\bar{y}}_{{n}} -\sqrt{3}e_{{1}}e_{{2}}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{3}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2}\\&+\sqrt{3}e_{{1}}e_{{2}}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{y}}_{{n}}^{3} +2\sqrt{3}e_{{1}}e_{{2}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{3}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} -2\sqrt{3}e_{{2}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} -2\sqrt{3}e_{{1}}e_{{2}}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}}\\&+\sqrt{3}e_{{1}}e_{{2}}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{y}}_{{n}}^{2} +2\sqrt{3}e_{{2}}^{2}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}} +\sqrt{3}e_{{1}}e_{{2}}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{y}}_{{n}} +\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{3}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2}\\&-\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{y}}_{{n}}^{2} +\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}} -\sqrt{3}e_{{1}}^{2}e_{{2}}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{y}}_{{n}}^{3} +\sqrt{3}e_{{1}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{3}\\ \end{aligned}$$

$$\begin{aligned}&\qquad \, +\sqrt{3}e_{{1}}^{2}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{3} -\sqrt{3}e_{{1}}e_{{2}}^{2}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{y}}_{{n}}^{3} -e_{{1}}^{2}e_{{2}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}x_{{n}}y_{{n}} -e_{{1}}^{2}e_{{2}}{\bar{k}}_{{n}}{\bar{l}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2}\\&\qquad \, -4e_{{1}}^{2}e_{{2}}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} -e_{{1}}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} -4e_{{1}}e_{{2}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} -e_{{1}}e_{{2}}{\bar{k}}_{{n}}{\bar{l}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{3} \\&\qquad \, -e_{{1}}^{2}e_{{2}}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}} -e_{{1}}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}} -4e_{{1}}e_{{2}}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} -2e_{{1}}e_{{2}}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}}\\&\qquad \, -e_{{1}}^{2}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} -e_{{1}}^{2}e_{{2}}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{3} -2e_{{1}}^{2}e_{{2}}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{3} -e_{{1}}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{3}\\&\qquad \, -2e_{{1}}e_{{2}}^{2}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{3} +4\sqrt{3}e_{{1}}^{2}e_{{2}}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}} +4\sqrt{3}e_{{1}}e_{{2}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}} -\sqrt{3}e_{{1}}e_{{2}}{\bar{k}}_{{n}}{\bar{l}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2}\\&\qquad \, -2\sqrt{3}e_{{1}}e_{{2}}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} -\sqrt{3}e_{{1}}e_{{2}}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}} +2\sqrt{3}e_{{1}}e_{{2}}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}} +2\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{3}\\&\qquad \, +2\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} +\sqrt{3}e_{{1}}^{2}e_{{2}}{\bar{k}}_{{n}}{\bar{l}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{3} +\sqrt{3}e_{{1}}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{3} +\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}}\\&\qquad \, +4\sqrt{3}e_{{1}}^{2}e_{{2}}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} +4\sqrt{3}e_{{1}}e_{{2}}^{2}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} -\sqrt{3}e_{{1}}e_{{2}}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{3} -e_{{1}}^{2}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{3}{\bar{y}}_{{n}}^{2}\\&\qquad \, -e_{{1}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{3}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{3} -e_{{2}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{3}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{3} -e_{{1}}^{2}e_{{2}}^{2}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{x}}_{{n}} -e_{{1}}^{2}e_{{2}}^{2}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{y}}_{{n}} -e_{{1}}^{2}e_{{2}}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{y}}_{{n}}^{2}\\&\qquad \, -e_{{1}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{y}}_{{n}}^{3} -e_{{1}}^{2}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{y}}_{{n}}^{3} -e_{{1}}e_{{2}}^{2}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{y}}_{{n}}^{2} -e_{{1}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{y}}_{{n}}^{2} +e_{{1}}e_{{2}}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{y}}_{{n}} +\sqrt{3}e_{{1}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{3}{\bar{y}}_{{n}}^{3}\\&\qquad \, +2\sqrt{3}e_{{1}}^{2}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{y}}_{{n}}^{2}, \end{aligned}$$

$$\begin{aligned} g_2= & {} \sqrt{3}e_{{1}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{3}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}}^{2} +\sqrt{3}e_{{2}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{3}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}}^{2} +\sqrt{3}e_{{2}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}}^{2} +\sqrt{3}e_{{1}}^{2}e_{{2}}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{y}}_{{n}}^{2} +\sqrt{3}e_{{1}}e_{{2}}^{2}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{y}}_{{n}}^{2}\\&+\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{x}}_{{n}} +\sqrt{3}e_{{1}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{y}}_{{n}}^{2} -\sqrt{3}e_{{1}}e_{{2}}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{y}}_{{n}} +\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{3}{\bar{y}}_{{n}}^{2} +\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{y}}_{{n}}\\&+3e_{{1}}^{2}e_{{2}}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{y}}_{{n}} -3e_{{1}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{y}}_{{n}}^{2} +3e_{{1}}e_{{2}}^{2}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{y}}_{{n}} +3e_{{1}}e_{{2}}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{y}}_{{n}}^{2} -3e_{{2}}^{2}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}} +3e_{{2}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}}\\&+\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}}^{2} +\sqrt{3}e_{{1}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}}^{2} +2\sqrt{3}e_{{1}}e_{{2}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{3}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}}^{2} +2\sqrt{3}e_{{2}}^{2}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}}\\&+\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{3}{\bar{x}}_{{n}}{\bar{y}}_{{n}} +\sqrt{3}e_{{1}}^{2}e_{{2}}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{3}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} +2\sqrt{3}e_{{1}}^{2}e_{{2}}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}} +2\sqrt{3}e_{{1}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2}\\&-\sqrt{3}e_{{1}}e_{{2}}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{3}{\bar{x}}_{{n}}{\bar{y}}_{{n}} +\sqrt{3}e_{{1}}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{3}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} +2\sqrt{3}e_{{1}}e_{{2}}^{2}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}} +2\sqrt{3}e_{{1}}^{2}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2}\\&+2\sqrt{3}e_{{2}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}} +\sqrt{3}e_{{2}}^{2}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} +\sqrt{3}e_{{1}}^{2}e_{{2}}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{y}}_{{n}} +\sqrt{3}e_{{1}}e_{{2}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{y}}_{{n}}\\&+\sqrt{3}e_{{1}}e_{{2}}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{y}}_{{n}}^{2} +\sqrt{3}e_{{1}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{3}{\bar{y}}_{{n}}^{2} +\sqrt{3}e_{{2}}^{2}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{x}}_{{n}} +\sqrt{3}e_{{1}}^{2}e_{{2}}{\bar{k}}_{{n}}{\bar{l}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}} -3e_{{2}}^{2}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{x}}_{{n}}^{2}\\&-3e_{{1}}^{2}e_{{2}}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{y}}_{{n}}^{2} -3e_{{1}}e_{{2}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{y}}_{{n}}^{2} +3e_{{1}}^{2}e_{{2}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}} -3e_{{1}}^{2}e_{{2}}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{3}{\bar{x}}_{{n}}{\bar{y}}_{{n}} -6e_{{1}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2}\\&+6e_{{1}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} -3e_{{1}}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{3}{\bar{x}}_{{n}}{\bar{y}}_{{n}} +3e_{{1}}e_{{2}}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{3}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} -3e_{{2}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} +3e_{{2}}^{2}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}}\\&+3e_{{1}}e_{{2}}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{y}}_{{n}} -3e_{{1}}^{2}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{3}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} -3e_{{1}}^{2}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{y}}_{{n}}^{2} -3e_{{1}}^{2}e_{{2}}^{2}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}} -3e_{{1}}^{2}e_{{2}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{y}}_{{n}}\\&+2\sqrt{3}e_{{1}}^{2}e_{{2}}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}} +\sqrt{3}e_{{1}}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}} +2\sqrt{3}e_{{1}}e_{{2}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}} +\sqrt{3}e_{{1}}e_{{2}}{\bar{k}}_{{n}}{\bar{l}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}}^{2}\\&+\sqrt{3}e_{{1}}^{2}e_{{2}}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}} +\sqrt{3}e_{{1}}e_{{2}}^{2}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}} -\sqrt{3}e_{{1}}e_{{2}}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}} +\sqrt{3}e_{{1}}e_{{2}}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2}\\ \end{aligned}$$

$$\begin{aligned}&+2\sqrt{3}e_{{1}}e_{{2}}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}} +\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}} +\sqrt{3}e_{{1}}^{2}e_{{2}}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}}^{2} +2\sqrt{3}e_{{1}}^{2}e_{{2}}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}}^{2}\nonumber \\&+\sqrt{3}e_{{1}}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}}^{2} +2\sqrt{3}e_{{1}}e_{{2}}^{2}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}}^{2} +\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}} +3\sqrt{3}e_{{1}}^{2}e_{{2}}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2}\nonumber \\&+\sqrt{3}e_{{1}}^{2}e_{{2}}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}} +\sqrt{3}e_{{1}}^{2}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} +3\sqrt{3}e_{{1}}e_{{2}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} +\sqrt{3}e_{{1}}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}}\nonumber \\&+3\sqrt{3}e_{{1}}e_{{2}}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} +\sqrt{3}e_{{1}}e_{{2}}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}} +3e_{{1}}^{2}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}} -3e_{{1}}^{2}e_{{2}}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2}\nonumber \\&-3e_{{1}}e_{{2}}^{2}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} +3e_{{1}}^{2}e_{{2}}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}} +3e_{{1}}e_{{2}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}} -6e_{{1}}e_{{2}}{\bar{k}}_{{n}}{\bar{l}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2}\nonumber \\&+3e_{{1}}e_{{2}}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} -3e_{{1}}e_{{2}}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}} +3e_{{1}}e_{{2}}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}{\bar{y}}_{{n}} -6e_{{1}}^{2}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}}^{2}\nonumber \\&-6e_{{1}}^{2}e_{{2}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}} -9e_{{1}}^{2}e_{{2}}{\bar{k}}_{{n}}{\bar{l}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}}^{2} -9e_{{1}}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}}^{2} -3e_{{1}}^{2}e_{{2}}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}}\nonumber \\&-9e_{{1}}^{2}e_{{2}}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} -3e_{{1}}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}} +3e_{{1}}e_{{2}}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}}^{2} -3e_{{1}}e_{{2}}{\bar{k}}_{{n}}{\bar{l}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}}\nonumber \\&-3e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}} -6e_{{2}}^{2}{\bar{l}}_{{n}}^{2}{\bar{s}}_{{n}}{\bar{t}}_{{n}}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}} -3e_{{1}}^{2}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{3}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}} +3e_{{1}}^{2}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}}^{2} -6e_{{1}}e_{{2}}{\bar{k}}_{{n}}{\bar{t}}_{{n}}^{3}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}}^{2}\nonumber \\&-3e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}}^{2} -3e_{{2}}^{2}{\bar{l}}_{{n}}{\bar{s}}_{{n}}{\bar{t}}_{{n}}^{2}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}}^{2} -3e_{{1}}^{2}e_{{2}}^{2}{\bar{l}}_{{n}}^{3}{\bar{s}}_{{n}}{\bar{x}}_{{n}}^{2} -3e_{{1}}^{2}{\bar{k}}_{{n}}{\bar{t}}_{{n}}^{3}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}}^{2} -3e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{t}}_{{n}}^{3}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}}^{2}\nonumber \\&-3e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{3}{\bar{x}}_{{n}}^{2}{\bar{y}}_{{n}} -9e_{{1}}e_{{2}}^{2}{\bar{k}}_{{n}}{\bar{l}}_{{n}}^{2}{\bar{t}}_{{n}}{\bar{x}}_{{n}}{\bar{y}}_{{n}}^{2} +3e_{{1}}^{2}{\bar{l}}_{{n}}^{3}{\bar{s}}_{ {n}}{\bar{y}}_{{n}}^{2}. \end{aligned}$$

(A.4)