Z9: FR19,8\mathbb{Z}_9:\ \text{FR}^{9,8}_{1}

Fusion Rules

123456789231689574312974856469817325587192643694725138758361492875243961946538217\begin{array}{|lllllllll|} \hline \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} & \mathbf{9} \\ \mathbf{2} & \mathbf{3} & \mathbf{1} & \mathbf{6} & \mathbf{8} & \mathbf{9} & \mathbf{5} & \mathbf{7} & \mathbf{4} \\ \mathbf{3} & \mathbf{1} & \mathbf{2} & \mathbf{9} & \mathbf{7} & \mathbf{4} & \mathbf{8} & \mathbf{5} & \mathbf{6} \\ \mathbf{4} & \mathbf{6} & \mathbf{9} & \mathbf{8} & \mathbf{1} & \mathbf{7} & \mathbf{3} & \mathbf{2} & \mathbf{5} \\ \mathbf{5} & \mathbf{8} & \mathbf{7} & \mathbf{1} & \mathbf{9} & \mathbf{2} & \mathbf{6} & \mathbf{4} & \mathbf{3} \\ \mathbf{6} & \mathbf{9} & \mathbf{4} & \mathbf{7} & \mathbf{2} & \mathbf{5} & \mathbf{1} & \mathbf{3} & \mathbf{8} \\ \mathbf{7} & \mathbf{5} & \mathbf{8} & \mathbf{3} & \mathbf{6} & \mathbf{1} & \mathbf{4} & \mathbf{9} & \mathbf{2} \\ \mathbf{8} & \mathbf{7} & \mathbf{5} & \mathbf{2} & \mathbf{4} & \mathbf{3} & \mathbf{9} & \mathbf{6} & \mathbf{1} \\ \mathbf{9} & \mathbf{4} & \mathbf{6} & \mathbf{5} & \mathbf{3} & \mathbf{8} & \mathbf{2} & \mathbf{1} & \mathbf{7} \\ \hline \end{array}

The fusion rules are invariant under the group generated by the following permutations:

{(2 3)(4 7 9 5 6 8),(2 3)(4 8 6 5 9 7)}\{(\mathbf{2} \ \mathbf{3}) (\mathbf{4} \ \mathbf{7} \ \mathbf{9} \ \mathbf{5} \ \mathbf{6} \ \mathbf{8}), (\mathbf{2} \ \mathbf{3}) (\mathbf{4} \ \mathbf{8} \ \mathbf{6} \ \mathbf{5} \ \mathbf{9} \ \mathbf{7})\}

The following particles form non-trivial sub fusion rings

Particles SubRing
{1,2,3}\{\mathbf{1},\mathbf{2},\mathbf{3}\} Z3: FR13,2\mathbb{Z}_3:\ \text{FR}^{3,2}_{1}

Quantum Dimensions

Particle Numeric Symbolic
1\mathbf{1} 1.1. 11
2\mathbf{2} 1.1. 11
3\mathbf{3} 1.1. 11
4\mathbf{4} 1.1. 11
5\mathbf{5} 1.1. 11
6\mathbf{6} 1.1. 11
7\mathbf{7} 1.1. 11
8\mathbf{8} 1.1. 11
9\mathbf{9} 1.1. 11
DFP2\mathcal{D}_{FP}^2 9.9. 99

Characters

The symbolic character table is the following

167238945111111111112(1+i3)12(1i3)1112(1i3)12(1+i3)12(1+i3)12(1i3)112(1i3)12(1+i3)1112(1+i3)12(1i3)12(1i3)12(1+i3)1Root[x6+x3+1,3]Root[x6+x3+1,4]12(1+i3)12(1i3)Root[x6+x3+1,5]Root[x6+x3+1,6]Root[x6+x3+1,2]Root[x6+x3+1,1]1Root[x6+x3+1,4]Root[x6+x3+1,3]12(1i3)12(1+i3)Root[x6+x3+1,6]Root[x6+x3+1,5]Root[x6+x3+1,1]Root[x6+x3+1,2]1Root[x6+x3+1,5]Root[x6+x3+1,6]12(1i3)12(1+i3)Root[x6+x3+1,2]Root[x6+x3+1,1]Root[x6+x3+1,4]Root[x6+x3+1,3]1Root[x6+x3+1,6]Root[x6+x3+1,5]12(1+i3)12(1i3)Root[x6+x3+1,1]Root[x6+x3+1,2]Root[x6+x3+1,3]Root[x6+x3+1,4]1Root[x6+x3+1,2]Root[x6+x3+1,1]12(1+i3)12(1i3)Root[x6+x3+1,4]Root[x6+x3+1,3]Root[x6+x3+1,6]Root[x6+x3+1,5]1Root[x6+x3+1,1]Root[x6+x3+1,2]12(1i3)12(1+i3)Root[x6+x3+1,3]Root[x6+x3+1,4]Root[x6+x3+1,5]Root[x6+x3+1,6]\begin{array}{|ccccccccc|} \hline \mathbf{1} & \mathbf{6} & \mathbf{7} & \mathbf{2} & \mathbf{3} & \mathbf{8} & \mathbf{9} & \mathbf{4} & \mathbf{5} \\ \hline 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 1 & 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 1 & 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) \\ 1 & \text{Root}\left[x^6+x^3+1,3\right] & \text{Root}\left[x^6+x^3+1,4\right] & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \text{Root}\left[x^6+x^3+1,5\right] & \text{Root}\left[x^6+x^3+1,6\right] & \text{Root}\left[x^6+x^3+1,2\right] & \text{Root}\left[x^6+x^3+1,1\right] \\ 1 & \text{Root}\left[x^6+x^3+1,4\right] & \text{Root}\left[x^6+x^3+1,3\right] & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \text{Root}\left[x^6+x^3+1,6\right] & \text{Root}\left[x^6+x^3+1,5\right] & \text{Root}\left[x^6+x^3+1,1\right] & \text{Root}\left[x^6+x^3+1,2\right] \\ 1 & \text{Root}\left[x^6+x^3+1,5\right] & \text{Root}\left[x^6+x^3+1,6\right] & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \text{Root}\left[x^6+x^3+1,2\right] & \text{Root}\left[x^6+x^3+1,1\right] & \text{Root}\left[x^6+x^3+1,4\right] & \text{Root}\left[x^6+x^3+1,3\right] \\ 1 & \text{Root}\left[x^6+x^3+1,6\right] & \text{Root}\left[x^6+x^3+1,5\right] & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \text{Root}\left[x^6+x^3+1,1\right] & \text{Root}\left[x^6+x^3+1,2\right] & \text{Root}\left[x^6+x^3+1,3\right] & \text{Root}\left[x^6+x^3+1,4\right] \\ 1 & \text{Root}\left[x^6+x^3+1,2\right] & \text{Root}\left[x^6+x^3+1,1\right] & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \text{Root}\left[x^6+x^3+1,4\right] & \text{Root}\left[x^6+x^3+1,3\right] & \text{Root}\left[x^6+x^3+1,6\right] & \text{Root}\left[x^6+x^3+1,5\right] \\ 1 & \text{Root}\left[x^6+x^3+1,1\right] & \text{Root}\left[x^6+x^3+1,2\right] & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \text{Root}\left[x^6+x^3+1,3\right] & \text{Root}\left[x^6+x^3+1,4\right] & \text{Root}\left[x^6+x^3+1,5\right] & \text{Root}\left[x^6+x^3+1,6\right] \\ \hline \end{array}

The numeric character table is the following

1672389451.0001.0001.0001.0001.0001.0001.0001.0001.0001.0000.5000+0.8660i0.50000.8660i1.0001.0000.50000.8660i0.5000+0.8660i0.5000+0.8660i0.50000.8660i1.0000.50000.8660i0.5000+0.8660i1.0001.0000.5000+0.8660i0.50000.8660i0.50000.8660i0.5000+0.8660i1.0000.17360.9848i0.1736+0.9848i0.5000+0.8660i0.50000.8660i0.76600.6428i0.7660+0.6428i0.9397+0.3420i0.93970.3420i1.0000.1736+0.9848i0.17360.9848i0.50000.8660i0.5000+0.8660i0.7660+0.6428i0.76600.6428i0.93970.3420i0.9397+0.3420i1.0000.76600.6428i0.7660+0.6428i0.50000.8660i0.5000+0.8660i0.9397+0.3420i0.93970.3420i0.1736+0.9848i0.17360.9848i1.0000.7660+0.6428i0.76600.6428i0.5000+0.8660i0.50000.8660i0.93970.3420i0.9397+0.3420i0.17360.9848i0.1736+0.9848i1.0000.9397+0.3420i0.93970.3420i0.5000+0.8660i0.50000.8660i0.1736+0.9848i0.17360.9848i0.7660+0.6428i0.76600.6428i1.0000.93970.3420i0.9397+0.3420i0.50000.8660i0.5000+0.8660i0.17360.9848i0.1736+0.9848i0.76600.6428i0.7660+0.6428i\begin{array}{|rrrrrrrrr|} \hline \mathbf{1} & \mathbf{6} & \mathbf{7} & \mathbf{2} & \mathbf{3} & \mathbf{8} & \mathbf{9} & \mathbf{4} & \mathbf{5} \\ \hline 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 & 1.000 \\ 1.000 & -0.5000+0.8660 i & -0.5000-0.8660 i & 1.000 & 1.000 & -0.5000-0.8660 i & -0.5000+0.8660 i & -0.5000+0.8660 i & -0.5000-0.8660 i \\ 1.000 & -0.5000-0.8660 i & -0.5000+0.8660 i & 1.000 & 1.000 & -0.5000+0.8660 i & -0.5000-0.8660 i & -0.5000-0.8660 i & -0.5000+0.8660 i \\ 1.000 & 0.1736-0.9848 i & 0.1736+0.9848 i & -0.5000+0.8660 i & -0.5000-0.8660 i & 0.7660-0.6428 i & 0.7660+0.6428 i & -0.9397+0.3420 i & -0.9397-0.3420 i \\ 1.000 & 0.1736+0.9848 i & 0.1736-0.9848 i & -0.5000-0.8660 i & -0.5000+0.8660 i & 0.7660+0.6428 i & 0.7660-0.6428 i & -0.9397-0.3420 i & -0.9397+0.3420 i \\ 1.000 & 0.7660-0.6428 i & 0.7660+0.6428 i & -0.5000-0.8660 i & -0.5000+0.8660 i & -0.9397+0.3420 i & -0.9397-0.3420 i & 0.1736+0.9848 i & 0.1736-0.9848 i \\ 1.000 & 0.7660+0.6428 i & 0.7660-0.6428 i & -0.5000+0.8660 i & -0.5000-0.8660 i & -0.9397-0.3420 i & -0.9397+0.3420 i & 0.1736-0.9848 i & 0.1736+0.9848 i \\ 1.000 & -0.9397+0.3420 i & -0.9397-0.3420 i & -0.5000+0.8660 i & -0.5000-0.8660 i & 0.1736+0.9848 i & 0.1736-0.9848 i & 0.7660+0.6428 i & 0.7660-0.6428 i \\ 1.000 & -0.9397-0.3420 i & -0.9397+0.3420 i & -0.5000-0.8660 i & -0.5000+0.8660 i & 0.1736-0.9848 i & 0.1736+0.9848 i & 0.7660-0.6428 i & 0.7660+0.6428 i \\ \hline \end{array}

Modular Data

The matching SS-matrices and twist factors are the following

SS-matrix Twist factors
13(11111111111112(1i3)12(1+i3)12(1i3)12(1+i3)12(1+i3)12(1i3)11112(1+i3)12(1i3)12(1+i3)12(1i3)12(1i3)12(1+i3)112(1i3)12(1+i3)3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,4]112(1+i3)12(1i3)3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,3]112(1i3)12(1+i3)3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,5]112(1+i3)12(1i3)3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,6]112(1+i3)12(1i3)3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,2]112(1i3)12(1+i3)3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,1])\frac{1}{3}\left(\begin{array}{ccccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) \\ 1 & 1 & 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] \\\end{array}\right) (0,0,0,49,49,19,19,29,29)\begin{array}{l}\left(0,0,0,-\frac{4}{9},-\frac{4}{9},-\frac{1}{9},-\frac{1}{9},\frac{2}{9},\frac{2}{9}\right)\end{array}
13(11111111111112(1i3)12(1+i3)12(1i3)12(1+i3)12(1+i3)12(1i3)11112(1+i3)12(1i3)12(1+i3)12(1i3)12(1i3)12(1+i3)112(1i3)12(1+i3)3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,1]112(1+i3)12(1i3)3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,2]112(1i3)12(1+i3)3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,4]112(1+i3)12(1i3)3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,3]112(1+i3)12(1i3)3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,6]112(1i3)12(1+i3)3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,5])\frac{1}{3}\left(\begin{array}{ccccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) \\ 1 & 1 & 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] \\\end{array}\right) (0,0,0,19,19,29,29,49,49)\begin{array}{l}\left(0,0,0,-\frac{1}{9},-\frac{1}{9},\frac{2}{9},\frac{2}{9},-\frac{4}{9},-\frac{4}{9}\right)\end{array}
13(11111111111112(1i3)12(1+i3)12(1i3)12(1+i3)12(1+i3)12(1i3)11112(1+i3)12(1i3)12(1+i3)12(1i3)12(1i3)12(1+i3)112(1i3)12(1+i3)3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,5]112(1+i3)12(1i3)3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,6]112(1i3)12(1+i3)3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,1]112(1+i3)12(1i3)3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,2]112(1+i3)12(1i3)3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,3]112(1i3)12(1+i3)3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,4])\frac{1}{3}\left(\begin{array}{ccccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) \\ 1 & 1 & 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] \\\end{array}\right) (0,0,0,29,29,49,49,19,19)\begin{array}{l}\left(0,0,0,\frac{2}{9},\frac{2}{9},-\frac{4}{9},-\frac{4}{9},-\frac{1}{9},-\frac{1}{9}\right)\end{array}
13(11111111111112(1+i3)12(1i3)12(1+i3)12(1i3)12(1i3)12(1+i3)11112(1i3)12(1+i3)12(1i3)12(1+i3)12(1+i3)12(1i3)112(1+i3)12(1i3)3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,3]112(1i3)12(1+i3)3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,4]112(1+i3)12(1i3)3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,6]112(1i3)12(1+i3)3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,5]112(1i3)12(1+i3)3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,1]112(1+i3)12(1i3)3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,2])\frac{1}{3}\left(\begin{array}{ccccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) \\ 1 & 1 & 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] \\\end{array}\right) (0,0,0,49,49,19,19,29,29)\begin{array}{l}\left(0,0,0,\frac{4}{9},\frac{4}{9},\frac{1}{9},\frac{1}{9},-\frac{2}{9},-\frac{2}{9}\right)\end{array}
13(11111111111112(1+i3)12(1i3)12(1+i3)12(1i3)12(1i3)12(1+i3)11112(1i3)12(1+i3)12(1i3)12(1+i3)12(1+i3)12(1i3)112(1+i3)12(1i3)3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,2]112(1i3)12(1+i3)3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,1]112(1+i3)12(1i3)3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,3]112(1i3)12(1+i3)3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,4]112(1i3)12(1+i3)3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,5]112(1+i3)12(1i3)3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,6])\frac{1}{3}\left(\begin{array}{ccccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) \\ 1 & 1 & 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] \\\end{array}\right) (0,0,0,19,19,29,29,49,49)\begin{array}{l}\left(0,0,0,\frac{1}{9},\frac{1}{9},-\frac{2}{9},-\frac{2}{9},\frac{4}{9},\frac{4}{9}\right)\end{array}
13(11111111111112(1+i3)12(1i3)12(1+i3)12(1i3)12(1i3)12(1+i3)11112(1i3)12(1+i3)12(1i3)12(1+i3)12(1+i3)12(1i3)112(1+i3)12(1i3)3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,6]112(1i3)12(1+i3)3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,5]112(1+i3)12(1i3)3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,2]112(1i3)12(1+i3)3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,1]112(1i3)12(1+i3)3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,3]3Root[729x6+27x3+1,4]112(1+i3)12(1i3)3Root[729x6+27x3+1,6]3Root[729x6+27x3+1,5]3Root[729x6+27x3+1,2]3Root[729x6+27x3+1,1]3Root[729x6+27x3+1,4]3Root[729x6+27x3+1,3])\frac{1}{3}\left(\begin{array}{ccccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) \\ 1 & 1 & 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] \\ 1 & \frac{1}{2} \left(-1-i \sqrt{3}\right) & \frac{1}{2} \left(-1+i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] \\ 1 & \frac{1}{2} \left(-1+i \sqrt{3}\right) & \frac{1}{2} \left(-1-i \sqrt{3}\right) & 3 \text{Root}\left[729 x^6+27 x^3+1,6\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,5\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,2\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,1\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,4\right] & 3 \text{Root}\left[729 x^6+27 x^3+1,3\right] \\\end{array}\right) (0,0,0,29,29,49,49,19,19)\begin{array}{l}\left(0,0,0,-\frac{2}{9},-\frac{2}{9},\frac{4}{9},\frac{4}{9},\frac{1}{9},\frac{1}{9}\right)\end{array}

Adjoint Subring

The adjoint subring is the trivial ring.

The upper central series is the following: Z91Trivial\mathbb{Z}_9 \underset{ \mathbf{1} }{\supset} \text{Trivial}

Universal grading

Each particle can be graded as follows: deg(1)=1,deg(2)=2,deg(3)=3,deg(4)=4,deg(5)=5,deg(6)=6,deg(7)=7,deg(8)=8,deg(9)=9\text{deg}(\mathbf{1}) = \mathbf{1}', \text{deg}(\mathbf{2}) = \mathbf{2}', \text{deg}(\mathbf{3}) = \mathbf{3}', \text{deg}(\mathbf{4}) = \mathbf{4}', \text{deg}(\mathbf{5}) = \mathbf{5}', \text{deg}(\mathbf{6}) = \mathbf{6}', \text{deg}(\mathbf{7}) = \mathbf{7}', \text{deg}(\mathbf{8}) = \mathbf{8}', \text{deg}(\mathbf{9}) = \mathbf{9}', where the degrees form the group Z9\mathbb{Z}_9 with multiplication table:

123456789231689574312974856469817325587192643694725138758361492875243961946538217\begin{array}{|lllllllll|} \hline \mathbf{1}' & \mathbf{2}' & \mathbf{3}' & \mathbf{4}' & \mathbf{5}' & \mathbf{6}' & \mathbf{7}' & \mathbf{8}' & \mathbf{9}' \\ \mathbf{2}' & \mathbf{3}' & \mathbf{1}' & \mathbf{6}' & \mathbf{8}' & \mathbf{9}' & \mathbf{5}' & \mathbf{7}' & \mathbf{4}' \\ \mathbf{3}' & \mathbf{1}' & \mathbf{2}' & \mathbf{9}' & \mathbf{7}' & \mathbf{4}' & \mathbf{8}' & \mathbf{5}' & \mathbf{6}' \\ \mathbf{4}' & \mathbf{6}' & \mathbf{9}' & \mathbf{8}' & \mathbf{1}' & \mathbf{7}' & \mathbf{3}' & \mathbf{2}' & \mathbf{5}' \\ \mathbf{5}' & \mathbf{8}' & \mathbf{7}' & \mathbf{1}' & \mathbf{9}' & \mathbf{2}' & \mathbf{6}' & \mathbf{4}' & \mathbf{3}' \\ \mathbf{6}' & \mathbf{9}' & \mathbf{4}' & \mathbf{7}' & \mathbf{2}' & \mathbf{5}' & \mathbf{1}' & \mathbf{3}' & \mathbf{8}' \\ \mathbf{7}' & \mathbf{5}' & \mathbf{8}' & \mathbf{3}' & \mathbf{6}' & \mathbf{1}' & \mathbf{4}' & \mathbf{9}' & \mathbf{2}' \\ \mathbf{8}' & \mathbf{7}' & \mathbf{5}' & \mathbf{2}' & \mathbf{4}' & \mathbf{3}' & \mathbf{9}' & \mathbf{6}' & \mathbf{1}' \\ \mathbf{9}' & \mathbf{4}' & \mathbf{6}' & \mathbf{5}' & \mathbf{3}' & \mathbf{8}' & \mathbf{2}' & \mathbf{1}' & \mathbf{7}' \\ \hline \end{array}

Categorifications

Data

Download links for numeric data: