Theta Series -- Sage

The following matrices are the ones we will need to use.

This worksheet is to calculate the generators as given in the Aoki-Ibukiyama paper. You probably want less terms below so they are easier to read if you want to do comparisons. This just shows how to do the easiest ones; one still needs $\theta_4$ and $\chi_{14}$. Note that $\theta_4$ and $\chi_{14}$ are cusp forms ($\chi_{14}$ from the Aoki-Ibukiyama paper and $\theta_{4}$ from an earlier paper on level 3), but we will still need them for the expansion.

A2=matrix([[2,1],[1,2]]);A2

 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} 2 & 1 \\ 1 & 2 \end{array}\right)$

E6=matrix([[2,-1,0,0,0,0],[-1,2,-1,0,0,0],[0,-1,2,-1,0,-1],[0,0,-1,2,-1,0],[0,0,0,-1,2,0],[0,0,-1,0,0,2]]);E6

 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrrr} 2 & -1 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & -1 \\ 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & -1 & 0 & 0 & 2 \end{array}\right)$

E6star=3*E6.inverse();E6star

 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrrr} 4 & 5 & 6 & 4 & 2 & 3 \\ 5 & 10 & 12 & 8 & 4 & 6 \\ 6 & 12 & 18 & 12 & 6 & 9 \\ 4 & 8 & 12 & 10 & 5 & 6 \\ 2 & 4 & 6 & 5 & 4 & 3 \\ 3 & 6 & 9 & 6 & 3 & 6 \end{array}\right)$

S4=matrix([[1,0,3/2,0],[0,1,0,3/2],[3/2,0,3,0],[0,3/2,0,3]]);S4

 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} 1 & 0 & \frac{3}{2} & 0 \\ 0 & 1 & 0 & \frac{3}{2} \\ \frac{3}{2} & 0 & 3 & 0 \\ 0 & \frac{3}{2} & 0 & 3 \end{array}\right)$

var('q1','q2','q3', 'z','tau','w')

 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(q_{1}, q_{2}, q_{3}, z, \tau, w\right)$

A2 = QuadraticForm(ZZ,2,[2,1,2]);A2

Quadratic form in 2 variables over Integer Ring with coefficients: 
[ 2 1 ]
[ * 2 ]

Quadratic form in 2 variables over Integer Ring with coefficients: 
[ 2 1 ]
[ * 2 ]

alpha1=A2.theta_series_degree_2(100);alpha1

{(0, 0, 0): 1,
 (0, 0, 1): 0,
 (0, 0, 2): 4,
 (0, 0, 3): 2,
 (0, 0, 4): 0,
 (0, 0, 5): 2,
 (0, 0, 6): 0,
 (0, 0, 7): 0,
 (0, 0, 8): 8,
 (0, 0, 9): 0,
 (0, 0, 10): 0,
 (0, 0, 11): 0,
 (0, 0, 12): 6,
 (0, 0, 13): 0,
 (0, 0, 14): 0,
 (0, 0, 15): 0,
 (0, 0, 16): 0,
 (0, 0, 17): 4,
 (0, 0, 18): 4,
 (0, 0, 19): 0,
 (0, 0, 20): 6,
 (0, 0, 21): 0,
 (0, 0, 22): 0,
 (0, 0, 23): 4,
 (0, 0, 24): 0,
 (0, 0, 25): 0,
 (2, 1, 2): 4,
 (2, 2, 8): 8,
 (3, 0, 5): 4}

{(0, 0, 0): 1,
 (0, 0, 1): 0,
 (0, 0, 2): 4,
 (0, 0, 3): 2,
 (0, 0, 4): 0,
 (0, 0, 5): 2,
 (0, 0, 6): 0,
 (0, 0, 7): 0,
 (0, 0, 8): 8,
 (0, 0, 9): 0,
 (0, 0, 10): 0,
 (0, 0, 11): 0,
 (0, 0, 12): 6,
 (0, 0, 13): 0,
 (0, 0, 14): 0,
 (0, 0, 15): 0,
 (0, 0, 16): 0,
 (0, 0, 17): 4,
 (0, 0, 18): 4,
 (0, 0, 19): 0,
 (0, 0, 20): 6,
 (0, 0, 21): 0,
 (0, 0, 22): 0,
 (0, 0, 23): 4,
 (0, 0, 24): 0,
 (0, 0, 25): 0,
 (2, 1, 2): 4,
 (2, 2, 8): 8,
 (3, 0, 5): 4}

Alpha1

 $\newcommand{\Bold}[1]{\mathbf{#1}}4 \, q_{3}^{23} + 6 \, q_{3}^{20} + 4 \, q_{3}^{18} + 4 \, q_{3}^{17} + 8 \, q_{1}^{2} q_{2}^{2} q_{3}^{8} + 6 \, q_{3}^{12} + 4 \, q_{1}^{3} q_{3}^{5} + 8 \, q_{3}^{8} + 4 \, q_{1}^{2} q_{2} q_{3}^{2} + 2 \, q_{3}^{5} + 2 \, q_{3}^{3} + 4 \, q_{3}^{2} + 1$

Alpha1=0; for j in [0..28]: Alpha1=Alpha1 + alpha1.items()[j][1]*q1^(alpha1.items()[j][0][0])*q2^(alpha1.items()[j][0][1])*q3^(alpha1.items()[j][0][2])

Since $\chi_{14}$ involves derivatives, it is probably best to encode these as functions with exponentials rather than formal $q$-series.

Alpha1exp=0; for j in [0..28]: Alpha1exp=Alpha1exp + alpha1.items()[j][1]*e^(2*pi*i*z*alpha1.items()[j][0][0])*e^(2*pi*i*tau*alpha1.items()[j][0][1])*e^(2*pi*i*w*alpha1.items()[j][0][2])

Alpha1exp;Alpha1exp.derivative(z,1);

 $\newcommand{\Bold}[1]{\mathbf{#1}}8 \, e^{\left(4 i \, \pi \tau + 16 i \, \pi w + 4 i \, \pi z\right)} + 4 \, e^{\left(2 i \, \pi \tau + 4 i \, \pi w + 4 i \, \pi z\right)} + 4 \, e^{\left(10 i \, \pi w + 6 i \, \pi z\right)} + 4 \, e^{\left(46 i \, \pi w\right)} + 6 \, e^{\left(40 i \, \pi w\right)} + 4 \, e^{\left(36 i \, \pi w\right)} + 4 \, e^{\left(34 i \, \pi w\right)} + 6 \, e^{\left(24 i \, \pi w\right)} + 8 \, e^{\left(16 i \, \pi w\right)} + 2 \, e^{\left(10 i \, \pi w\right)} + 2 \, e^{\left(6 i \, \pi w\right)} + 4 \, e^{\left(4 i \, \pi w\right)} + 1$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}32 i \, \pi e^{\left(4 i \, \pi \tau + 16 i \, \pi w + 4 i \, \pi z\right)} + 16 i \, \pi e^{\left(2 i \, \pi \tau + 4 i \, \pi w + 4 i \, \pi z\right)} + 24 i \, \pi e^{\left(10 i \, \pi w + 6 i \, \pi z\right)}$

E6=QuadraticForm(ZZ,6,[2,-1,0,0,0,0,2,-1,0,0,0,2,-1,0,-1,2,-1,0,2,0,2]);E6

Quadratic form in 6 variables over Integer Ring with coefficients: 
[ 2 -1 0 0 0 0 ]
[ * 2 -1 0 0 0 ]
[ * * 2 -1 0 -1 ]
[ * * * 2 -1 0 ]
[ * * * * 2 0 ]
[ * * * * * 2 ]

Quadratic form in 6 variables over Integer Ring with coefficients: 
[ 2 -1 0 0 0 0 ]
[ * 2 -1 0 0 0 ]
[ * * 2 -1 0 -1 ]
[ * * * 2 -1 0 ]
[ * * * * 2 0 ]
[ * * * * * 2 ]

E6star=QuadraticForm(ZZ,6,[4,5,6,4,2,3,10,12,8,4,6,18,12,6,9,10,5,6,4,3,6]);E6star

Quadratic form in 6 variables over Integer Ring with coefficients: 
[ 4 5 6 4 2 3 ]
[ * 10 12 8 4 6 ]
[ * * 18 12 6 9 ]
[ * * * 10 5 6 ]
[ * * * * 4 3 ]
[ * * * * * 6 ]

Quadratic form in 6 variables over Integer Ring with coefficients: 
[ 4 5 6 4 2 3 ]
[ * 10 12 8 4 6 ]
[ * * 18 12 6 9 ]
[ * * * 10 5 6 ]
[ * * * * 4 3 ]
[ * * * * * 6 ]

thetaE6=E6.theta_series_degree_2(100);thetaE6

 $\newcommand{\Bold}[1]{\mathbf{#1}}\left\{\left(3, 0, 3\right) : 8, \left(0, 0, 8\right) : 126, \left(2, 0, 11\right) : 256, \left(3, 1, 8\right) : 146, \left(3, 0, 5\right) : 68, \left(0, 0, 22\right) : 1460, \left(2, 0, 5\right) : 160, \left(3, 1, 6\right) : 64, \left(3, 0, 7\right) : 152, \left(0, 0, 20\right) : 1000, \left(2, 0, 7\right) : 176, \left(5, 5, 5\right) : 216, \left(3, 1, 4\right) : 46, \left(0, 0, 18\right) : 842, \left(4, 1, 5\right) : 304, \left(2, 2, 4\right) : 20, \left(2, 0, 3\right) : 40, \left(0, 0, 4\right) : 50, \left(3, 2, 6\right) : 90, \left(3, 2, 4\right) : 50, \left(4, 3, 6\right) : 430, \left(3, 2, 8\right) : 112, \left(0, 0, 24\right) : 1510, \left(4, 3, 4\right) : 192, \left(0, 0, 11\right) : 218, \left(5, 1, 5\right) : 248, \left(0, 0, 7\right) : 120, \left(2, 1, 11\right) : 214, \left(3, 3, 5\right) : 32, \left(0, 0, 5\right) : 60, \left(2, 1, 9\right) : 184, \left(4, 0, 6\right) : 596, \left(3, 3, 7\right) : 120, \left(0, 0, 3\right) : 10, \left(2, 1, 7\right) : 142, \left(4, 0, 4\right) : 272, \left(3, 3, 9\right) : 88, \left(0, 0, 1\right) : 0, \left(2, 1, 5\right) : 86, \left(4, 2, 5\right) : 200, \left(0, 0, 15\right) : 472, \left(2, 1, 3\right) : 20, \left(5, 4, 5\right) : 256, \left(0, 0, 13\right) : 408, \left(2, 0, 12\right) : 328, \left(2, 2, 12\right) : 504, \left(2, 2, 7\right) : 72, \left(0, 0, 9\right) : 144, \left(2, 0, 8\right) : 248, \left(3, 0, 4\right) : 68, \left(0, 0, 23\right) : 974, \left(2, 0, 10\right) : 504, \left(3, 1, 7\right) : 88, \left(0, 0, 10\right) : 282, \left(3, 0, 6\right) : 28, \left(4, 4, 7\right) : 484, \left(0, 0, 21\right) : 832, \left(2, 0, 4\right) : 156, \left(2, 2, 5\right) : 32, \left(3, 1, 5\right) : 60, \left(3, 0, 8\right) : 96, \left(0, 0, 19\right) : 736, \left(2, 0, 6\right) : 92, \left(4, 1, 4\right) : 280, \left(3, 1, 3\right) : 16, \left(3, 2, 3\right) : 20, \left(0, 0, 17\right) : 514, \left(4, 1, 6\right) : 250, \left(5, 5, 6\right) : 260, \left(4, 4, 6\right) : 216, \left(2, 0, 2\right) : 80, \left(3, 2, 7\right) : 82, \left(3, 2, 5\right) : 68, \left(4, 4, 5\right) : 248, \left(0, 0, 25\right) : 1386, \left(5, 3, 5\right) : 272, \left(4, 3, 5\right) : 306, \left(2, 1, 12\right) : 558, \left(0, 0, 16\right) : 686, \left(0, 0, 12\right) : 432, \left(2, 1, 10\right) : 328, \left(0, 0, 6\right) : 86, \left(2, 1, 8\right) : 128, \left(3, 3, 4\right) : 80, \left(2, 2, 8\right) : 132, \left(2, 1, 6\right) : 156, \left(4, 0, 5\right) : 196, \left(3, 3, 6\right) : 120, \left(0, 0, 2\right) : 12, \left(2, 1, 4\right) : 102, \left(2, 2, 11\right) : 184, \left(3, 3, 8\right) : 24, \left(0, 0, 0\right) : 1, \left(2, 1, 2\right) : 20, \left(4, 2, 4\right) : 168, \left(0, 0, 14\right) : 418, \left(4, 2, 6\right) : 280, \left(2, 2, 9\right) : 104, \left(4, 4, 4\right) : 324, \left(2, 2, 10\right) : 304, \left(5, 2, 5\right) : 236, \left(2, 2, 6\right) : 52, \left(2, 0, 9\right) : 128\right\}$

thetaE6.items()[102]

 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\left(2, 0, 9\right), 128\right)$

ThetaE6=0; for j in [0..102]: ThetaE6=ThetaE6 + thetaE6.items()[j][1]*q1^(thetaE6.items()[j][0][0])*q2^(thetaE6.items()[j][0][1])*q3^(thetaE6.items()[j][0][2])

ThetaE6

 $\newcommand{\Bold}[1]{\mathbf{#1}}1386 \, q_{3}^{25} + 1510 \, q_{3}^{24} + 974 \, q_{3}^{23} + 1460 \, q_{3}^{22} + 832 \, q_{3}^{21} + 1000 \, q_{3}^{20} + 736 \, q_{3}^{19} + 842 \, q_{3}^{18} + 514 \, q_{3}^{17} + 260 \, q_{1}^{5} q_{2}^{5} q_{3}^{6} + 504 \, q_{1}^{2} q_{2}^{2} q_{3}^{12} + 686 \, q_{3}^{16} + 216 \, q_{1}^{5} q_{2}^{5} q_{3}^{5} + 484 \, q_{1}^{4} q_{2}^{4} q_{3}^{7} + 88 \, q_{1}^{3} q_{2}^{3} q_{3}^{9} + 184 \, q_{1}^{2} q_{2}^{2} q_{3}^{11} + 558 \, q_{1}^{2} q_{2} q_{3}^{12} + 472 \, q_{3}^{15} + 256 \, q_{1}^{5} q_{2}^{4} q_{3}^{5} + 216 \, q_{1}^{4} q_{2}^{4} q_{3}^{6} + 24 \, q_{1}^{3} q_{2}^{3} q_{3}^{8} + 304 \, q_{1}^{2} q_{2}^{2} q_{3}^{10} + 214 \, q_{1}^{2} q_{2} q_{3}^{11} + 328 \, q_{1}^{2} q_{3}^{12} + 418 \, q_{3}^{14} + 272 \, q_{1}^{5} q_{2}^{3} q_{3}^{5} + 248 \, q_{1}^{4} q_{2}^{4} q_{3}^{5} + 430 \, q_{1}^{4} q_{2}^{3} q_{3}^{6} + 120 \, q_{1}^{3} q_{2}^{3} q_{3}^{7} + 112 \, q_{1}^{3} q_{2}^{2} q_{3}^{8} + 104 \, q_{1}^{2} q_{2}^{2} q_{3}^{9} + 328 \, q_{1}^{2} q_{2} q_{3}^{10} + 256 \, q_{1}^{2} q_{3}^{11} + 408 \, q_{3}^{13} + 324 \, q_{1}^{4} q_{2}^{4} q_{3}^{4} + 236 \, q_{1}^{5} q_{2}^{2} q_{3}^{5} + 306 \, q_{1}^{4} q_{2}^{3} q_{3}^{5} + 280 \, q_{1}^{4} q_{2}^{2} q_{3}^{6} + 120 \, q_{1}^{3} q_{2}^{3} q_{3}^{6} + 82 \, q_{1}^{3} q_{2}^{2} q_{3}^{7} + 146 \, q_{1}^{3} q_{2} q_{3}^{8} + 132 \, q_{1}^{2} q_{2}^{2} q_{3}^{8} + 184 \, q_{1}^{2} q_{2} q_{3}^{9} + 504 \, q_{1}^{2} q_{3}^{10} + 432 \, q_{3}^{12} + 192 \, q_{1}^{4} q_{2}^{3} q_{3}^{4} + 248 \, q_{1}^{5} q_{2} q_{3}^{5} + 200 \, q_{1}^{4} q_{2}^{2} q_{3}^{5} + 32 \, q_{1}^{3} q_{2}^{3} q_{3}^{5} + 250 \, q_{1}^{4} q_{2} q_{3}^{6} + 90 \, q_{1}^{3} q_{2}^{2} q_{3}^{6} + 88 \, q_{1}^{3} q_{2} q_{3}^{7} + 72 \, q_{1}^{2} q_{2}^{2} q_{3}^{7} + 96 \, q_{1}^{3} q_{3}^{8} + 128 \, q_{1}^{2} q_{2} q_{3}^{8} + 128 \, q_{1}^{2} q_{3}^{9} + 218 \, q_{3}^{11} + 168 \, q_{1}^{4} q_{2}^{2} q_{3}^{4} + 80 \, q_{1}^{3} q_{2}^{3} q_{3}^{4} + 304 \, q_{1}^{4} q_{2} q_{3}^{5} + 68 \, q_{1}^{3} q_{2}^{2} q_{3}^{5} + 596 \, q_{1}^{4} q_{3}^{6} + 64 \, q_{1}^{3} q_{2} q_{3}^{6} + 52 \, q_{1}^{2} q_{2}^{2} q_{3}^{6} + 152 \, q_{1}^{3} q_{3}^{7} + 142 \, q_{1}^{2} q_{2} q_{3}^{7} + 248 \, q_{1}^{2} q_{3}^{8} + 282 \, q_{3}^{10} + 280 \, q_{1}^{4} q_{2} q_{3}^{4} + 50 \, q_{1}^{3} q_{2}^{2} q_{3}^{4} + 196 \, q_{1}^{4} q_{3}^{5} + 60 \, q_{1}^{3} q_{2} q_{3}^{5} + 32 \, q_{1}^{2} q_{2}^{2} q_{3}^{5} + 28 \, q_{1}^{3} q_{3}^{6} + 156 \, q_{1}^{2} q_{2} q_{3}^{6} + 176 \, q_{1}^{2} q_{3}^{7} + 144 \, q_{3}^{9} + 20 \, q_{1}^{3} q_{2}^{2} q_{3}^{3} + 272 \, q_{1}^{4} q_{3}^{4} + 46 \, q_{1}^{3} q_{2} q_{3}^{4} + 20 \, q_{1}^{2} q_{2}^{2} q_{3}^{4} + 68 \, q_{1}^{3} q_{3}^{5} + 86 \, q_{1}^{2} q_{2} q_{3}^{5} + 92 \, q_{1}^{2} q_{3}^{6} + 126 \, q_{3}^{8} + 16 \, q_{1}^{3} q_{2} q_{3}^{3} + 68 \, q_{1}^{3} q_{3}^{4} + 102 \, q_{1}^{2} q_{2} q_{3}^{4} + 160 \, q_{1}^{2} q_{3}^{5} + 120 \, q_{3}^{7} + 8 \, q_{1}^{3} q_{3}^{3} + 20 \, q_{1}^{2} q_{2} q_{3}^{3} + 156 \, q_{1}^{2} q_{3}^{4} + 86 \, q_{3}^{6} + 20 \, q_{1}^{2} q_{2} q_{3}^{2} + 40 \, q_{1}^{2} q_{3}^{3} + 60 \, q_{3}^{5} + 80 \, q_{1}^{2} q_{3}^{2} + 50 \, q_{3}^{4} + 10 \, q_{3}^{3} + 12 \, q_{3}^{2} + 1$

thetaE6star=E6star.theta_series_degree_2(120);thetaE6star

 $\newcommand{\Bold}[1]{\mathbf{#1}}\left\{\left(0, 0, 9\right) : 4, \left(0, 0, 8\right) : 0, \left(0, 0, 23\right) : 32, \left(0, 0, 22\right) : 56, \left(0, 0, 21\right) : 24, \left(0, 0, 7\right) : 4, \left(0, 0, 20\right) : 22, \left(0, 0, 6\right) : 4, \left(0, 0, 19\right) : 22, \left(0, 0, 5\right) : 0, \left(0, 0, 18\right) : 26, \left(0, 0, 4\right) : 4, \left(0, 0, 17\right) : 20, \left(0, 0, 3\right) : 0, \left(0, 0, 16\right) : 36, \left(0, 0, 2\right) : 0, \left(4, 1, 7\right) : 4, \left(0, 0, 1\right) : 0, \left(0, 0, 30\right) : 52, \left(0, 0, 0\right) : 1, \left(0, 0, 29\right) : 74, \left(4, 2, 4\right) : 4, \left(0, 0, 15\right) : 18, \left(0, 0, 28\right) : 72, \left(0, 0, 14\right) : 0, \left(0, 0, 27\right) : 28, \left(0, 0, 13\right) : 8, \left(0, 0, 26\right) : 50, \left(0, 0, 12\right) : 10, \left(4, 3, 6\right) : 4, \left(0, 0, 25\right) : 36, \left(0, 0, 11\right) : 4, \left(0, 0, 24\right) : 50, \left(0, 0, 10\right) : 16\right\}$

thetaE6star.items()[30]

 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\left(0, 0, 25\right), 36\right)$

ThetaE6star=0; for j in [0..30]: ThetaE6star=ThetaE6star + thetaE6star.items()[j][1]*q1^(thetaE6star.items()[j][0][0])*q2^(thetaE6star.items()[j][0][1])*q3^(thetaE6star.items()[j][0][2])

ThetaE6star;

 $\newcommand{\Bold}[1]{\mathbf{#1}}52 \, q_{3}^{30} + 74 \, q_{3}^{29} + 72 \, q_{3}^{28} + 28 \, q_{3}^{27} + 50 \, q_{3}^{26} + 36 \, q_{3}^{25} + 32 \, q_{3}^{23} + 56 \, q_{3}^{22} + 24 \, q_{3}^{21} + 22 \, q_{3}^{20} + 22 \, q_{3}^{19} + 26 \, q_{3}^{18} + 20 \, q_{3}^{17} + 36 \, q_{3}^{16} + 18 \, q_{3}^{15} + 4 \, q_{1}^{4} q_{2}^{3} q_{3}^{6} + 8 \, q_{3}^{13} + 4 \, q_{1}^{4} q_{2} q_{3}^{7} + 10 \, q_{3}^{12} + 4 \, q_{1}^{4} q_{2}^{2} q_{3}^{4} + 4 \, q_{3}^{9} + 4 \, q_{3}^{7} + 4 \, q_{3}^{6} + 4 \, q_{3}^{4} + 1$

Beta3 = ThetaE6 - 10*Alpha1^3 + 9*ThetaE6star; beta3.expand()

 $\newcommand{\Bold}[1]{\mathbf{#1}}-640 \, q_{3}^{69} - 2880 \, q_{3}^{66} - 1920 \, q_{3}^{64} - 6240 \, q_{3}^{63} - 5760 \, q_{3}^{61} - 7920 \, q_{3}^{60} - 1920 \, q_{3}^{59} - 3840 \, q_{1}^{2} q_{2}^{2} q_{3}^{54} - 11040 \, q_{3}^{58} - 6240 \, q_{3}^{57} - 2880 \, q_{3}^{56} - 11520 \, q_{1}^{2} q_{2}^{2} q_{3}^{51} - 14400 \, q_{3}^{55} - 1920 \, q_{1}^{3} q_{3}^{51} - 7360 \, q_{3}^{54} - 7680 \, q_{1}^{2} q_{2}^{2} q_{3}^{49} - 7680 \, q_{3}^{53} - 16320 \, q_{1}^{2} q_{2}^{2} q_{3}^{48} - 14160 \, q_{3}^{52} - 5760 \, q_{1}^{3} q_{3}^{48} - 1920 \, q_{1}^{2} q_{2} q_{3}^{48} - 13120 \, q_{3}^{51} - 11520 \, q_{1}^{2} q_{2}^{2} q_{3}^{46} - 8640 \, q_{3}^{50} - 11520 \, q_{1}^{2} q_{2}^{2} q_{3}^{45} - 3840 \, q_{1}^{3} q_{3}^{46} - 17280 \, q_{3}^{49} - 3840 \, q_{1}^{2} q_{2}^{2} q_{3}^{44} - 8160 \, q_{1}^{3} q_{3}^{45} - 5760 \, q_{1}^{2} q_{2} q_{3}^{45} - 24000 \, q_{3}^{48} - 7680 \, q_{1}^{4} q_{2}^{4} q_{3}^{39} - 19200 \, q_{1}^{2} q_{2}^{2} q_{3}^{43} - 10080 \, q_{3}^{47} - 3840 \, q_{1}^{2} q_{2}^{2} q_{3}^{42} - 5760 \, q_{1}^{3} q_{3}^{43} - 3840 \, q_{1}^{2} q_{2} q_{3}^{43} - 19680 \, q_{3}^{46} - 5760 \, q_{1}^{3} q_{3}^{42} - 8160 \, q_{1}^{2} q_{2} q_{3}^{42} - 21360 \, q_{3}^{45} - 11520 \, q_{1}^{4} q_{2}^{4} q_{3}^{36} - 17280 \, q_{1}^{2} q_{2}^{2} q_{3}^{40} - 1920 \, q_{1}^{3} q_{3}^{41} - 12240 \, q_{3}^{44} - 7680 \, q_{1}^{5} q_{2}^{2} q_{3}^{36} - 15360 \, q_{1}^{2} q_{2}^{2} q_{3}^{39} - 9600 \, q_{1}^{3} q_{3}^{40} - 5760 \, q_{1}^{2} q_{2} q_{3}^{40} - 31440 \, q_{3}^{43} - 7680 \, q_{1}^{4} q_{2}^{4} q_{3}^{34} - 11520 \, q_{1}^{2} q_{2}^{2} q_{3}^{38} - 1920 \, q_{1}^{3} q_{3}^{39} - 5760 \, q_{1}^{2} q_{2} q_{3}^{39} - 19200 \, q_{3}^{42} - 7680 \, q_{1}^{4} q_{2}^{4} q_{3}^{33} - 11520 \, q_{1}^{2} q_{2}^{2} q_{3}^{37} - 1920 \, q_{1}^{2} q_{2} q_{3}^{38} - 9120 \, q_{3}^{41} - 11520 \, q_{1}^{5} q_{2}^{2} q_{3}^{33} - 7680 \, q_{1}^{4} q_{2}^{3} q_{3}^{33} - 26880 \, q_{1}^{2} q_{2}^{2} q_{3}^{36} - 8640 \, q_{1}^{3} q_{3}^{37} - 9600 \, q_{1}^{2} q_{2} q_{3}^{37} - 32760 \, q_{3}^{40} - 1920 \, q_{1}^{6} q_{3}^{33} - 7680 \, q_{1}^{3} q_{3}^{36} - 1920 \, q_{1}^{2} q_{2} q_{3}^{36} - 15360 \, q_{3}^{39} - 7680 \, q_{1}^{5} q_{2}^{2} q_{3}^{31} - 19200 \, q_{1}^{2} q_{2}^{2} q_{3}^{34} - 5760 \, q_{1}^{3} q_{3}^{35} - 19680 \, q_{3}^{38} - 7680 \, q_{1}^{5} q_{2}^{2} q_{3}^{30} - 11520 \, q_{1}^{4} q_{2}^{3} q_{3}^{30} - 28800 \, q_{1}^{2} q_{2}^{2} q_{3}^{33} - 5760 \, q_{1}^{3} q_{3}^{34} - 8640 \, q_{1}^{2} q_{2} q_{3}^{34} - 27840 \, q_{3}^{37} - 5120 \, q_{1}^{6} q_{2}^{6} q_{3}^{24} - 11520 \, q_{1}^{4} q_{2}^{4} q_{3}^{28} - 2880 \, q_{1}^{6} q_{3}^{30} - 3840 \, q_{1}^{5} q_{2} q_{3}^{30} - 8640 \, q_{1}^{2} q_{2}^{2} q_{3}^{32} - 13440 \, q_{1}^{3} q_{3}^{33} - 7680 \, q_{1}^{2} q_{2} q_{3}^{33} - 19920 \, q_{3}^{36} - 7680 \, q_{1}^{4} q_{2}^{3} q_{3}^{28} - 11520 \, q_{1}^{2} q_{2}^{2} q_{3}^{31} - 5760 \, q_{1}^{2} q_{2} q_{3}^{32} - 9600 \, q_{3}^{35} - 7680 \, q_{1}^{4} q_{2}^{3} q_{3}^{27} - 1920 \, q_{1}^{6} q_{3}^{28} - 15360 \, q_{1}^{2} q_{2}^{2} q_{3}^{30} - 9600 \, q_{1}^{3} q_{3}^{31} - 5760 \, q_{1}^{2} q_{2} q_{3}^{31} - 23520 \, q_{3}^{34} - 1920 \, q_{1}^{6} q_{3}^{27} - 5760 \, q_{1}^{5} q_{2} q_{3}^{27} - 1920 \, q_{1}^{4} q_{2}^{2} q_{3}^{27} - 3840 \, q_{1}^{2} q_{2}^{2} q_{3}^{29} - 14400 \, q_{1}^{3} q_{3}^{30} - 13440 \, q_{1}^{2} q_{2} q_{3}^{30} - 24480 \, q_{3}^{33} - 7680 \, q_{1}^{7} q_{2}^{4} q_{3}^{21} - 15360 \, q_{1}^{4} q_{2}^{4} q_{3}^{24} - 11520 \, q_{1}^{5} q_{2}^{2} q_{3}^{25} - 37440 \, q_{1}^{2} q_{2}^{2} q_{3}^{28} - 4320 \, q_{1}^{3} q_{3}^{29} - 19440 \, q_{3}^{32} - 3840 \, q_{1}^{5} q_{2} q_{3}^{25} - 7680 \, q_{1}^{2} q_{2}^{2} q_{3}^{27} - 5760 \, q_{1}^{3} q_{3}^{28} - 9600 \, q_{1}^{2} q_{2} q_{3}^{28} - 18240 \, q_{3}^{31} - 3840 \, q_{1}^{5} q_{2} q_{3}^{24} - 2880 \, q_{1}^{4} q_{2}^{2} q_{3}^{24} - 1920 \, q_{1}^{2} q_{2}^{2} q_{3}^{26} - 7680 \, q_{1}^{3} q_{3}^{27} - 14400 \, q_{1}^{2} q_{2} q_{3}^{27} - 18972 \, q_{3}^{30} - 7680 \, q_{1}^{6} q_{2}^{5} q_{3}^{18} - 3840 \, q_{1}^{4} q_{2}^{4} q_{3}^{21} - 11520 \, q_{1}^{4} q_{2}^{3} q_{3}^{22} - 7680 \, q_{1}^{2} q_{2}^{2} q_{3}^{25} - 1920 \, q_{1}^{3} q_{3}^{26} - 4320 \, q_{1}^{2} q_{2} q_{3}^{26} - 7254 \, q_{3}^{29} - 3840 \, q_{1}^{8} q_{2}^{2} q_{3}^{18} - 15360 \, q_{1}^{5} q_{2}^{2} q_{3}^{21} - 2880 \, q_{1}^{6} q_{3}^{22} - 1920 \, q_{1}^{4} q_{2}^{2} q_{3}^{22} - 15360 \, q_{1}^{2} q_{2}^{2} q_{3}^{24} - 18720 \, q_{1}^{3} q_{3}^{25} - 5760 \, q_{1}^{2} q_{2} q_{3}^{25} - 29592 \, q_{3}^{28} - 3840 \, q_{1}^{4} q_{2}^{4} q_{3}^{19} - 1920 \, q_{1}^{4} q_{2}^{2} q_{3}^{21} - 5760 \, q_{1}^{2} q_{2}^{2} q_{3}^{23} - 3840 \, q_{1}^{3} q_{3}^{24} - 7680 \, q_{1}^{2} q_{2} q_{3}^{24} - 14868 \, q_{3}^{27} - 7680 \, q_{1}^{4} q_{2}^{4} q_{3}^{18} - 11520 \, q_{1}^{2} q_{2}^{2} q_{3}^{22} - 960 \, q_{1}^{3} q_{3}^{23} - 1920 \, q_{1}^{2} q_{2} q_{3}^{23} - 7950 \, q_{3}^{26} - 7680 \, q_{1}^{7} q_{2}^{3} q_{3}^{15} - 3840 \, q_{1}^{5} q_{2}^{2} q_{3}^{18} - 15360 \, q_{1}^{4} q_{2}^{3} q_{3}^{18} - 5760 \, q_{1}^{5} q_{2} q_{3}^{19} - 7680 \, q_{1}^{2} q_{2}^{2} q_{3}^{21} - 3840 \, q_{1}^{3} q_{3}^{22} - 18720 \, q_{1}^{2} q_{2} q_{3}^{22} - 13410 \, q_{3}^{25} - 640 \, q_{1}^{9} q_{3}^{15} - 1920 \, q_{1}^{4} q_{2}^{4} q_{3}^{16} - 3840 \, q_{1}^{6} q_{3}^{18} - 2880 \, q_{1}^{2} q_{2}^{2} q_{3}^{20} - 7680 \, q_{1}^{3} q_{3}^{21} - 3840 \, q_{1}^{2} q_{2} q_{3}^{21} - 9970 \, q_{3}^{24} - 3840 \, q_{1}^{5} q_{2}^{2} q_{3}^{16} - 7680 \, q_{1}^{2} q_{2}^{2} q_{3}^{19} - 2880 \, q_{1}^{3} q_{3}^{20} - 960 \, q_{1}^{2} q_{2} q_{3}^{20} - 8218 \, q_{3}^{23} - 3840 \, q_{1}^{6} q_{2}^{4} q_{3}^{12} - 7680 \, q_{1}^{5} q_{2}^{2} q_{3}^{15} - 3840 \, q_{1}^{4} q_{2}^{3} q_{3}^{15} - 2880 \, q_{1}^{4} q_{2}^{2} q_{3}^{16} - 16320 \, q_{1}^{2} q_{2}^{2} q_{3}^{18} - 5760 \, q_{1}^{3} q_{3}^{19} - 3840 \, q_{1}^{2} q_{2} q_{3}^{19} - 16036 \, q_{3}^{22} - 1920 \, q_{1}^{8} q_{2} q_{3}^{12} - 960 \, q_{1}^{6} q_{3}^{15} - 7680 \, q_{1}^{5} q_{2} q_{3}^{15} - 3840 \, q_{1}^{3} q_{3}^{18} - 7680 \, q_{1}^{2} q_{2} q_{3}^{18} - 5192 \, q_{3}^{21} - 1920 \, q_{1}^{5} q_{2}^{2} q_{3}^{13} - 3840 \, q_{1}^{4} q_{2}^{3} q_{3}^{13} - 5760 \, q_{1}^{2} q_{2}^{2} q_{3}^{16} - 1440 \, q_{1}^{3} q_{3}^{17} - 2880 \, q_{1}^{2} q_{2} q_{3}^{17} - 4742 \, q_{3}^{20} - 7680 \, q_{1}^{4} q_{2}^{3} q_{3}^{12} - 960 \, q_{1}^{6} q_{3}^{13} - 3840 \, q_{1}^{2} q_{2}^{2} q_{3}^{15} - 3840 \, q_{1}^{3} q_{3}^{16} - 5760 \, q_{1}^{2} q_{2} q_{3}^{16} - 6746 \, q_{3}^{19} - 1920 \, q_{1}^{7} q_{2}^{2} q_{3}^{9} - 1920 \, q_{1}^{6} q_{3}^{12} - 1920 \, q_{1}^{5} q_{2} q_{3}^{12} - 3840 \, q_{1}^{4} q_{2}^{2} q_{3}^{12} - 960 \, q_{1}^{2} q_{2}^{2} q_{3}^{14} - 8160 \, q_{1}^{3} q_{3}^{15} - 3840 \, q_{1}^{2} q_{2} q_{3}^{15} - 8404 \, q_{3}^{18} - 1920 \, q_{1}^{4} q_{2}^{3} q_{3}^{10} - 4800 \, q_{1}^{2} q_{2}^{2} q_{3}^{13} - 1440 \, q_{1}^{2} q_{2} q_{3}^{14} - 3026 \, q_{3}^{17} + 260 \, q_{1}^{5} q_{2}^{5} q_{3}^{6} - 480 \, q_{1}^{6} q_{3}^{10} - 1920 \, q_{1}^{5} q_{2} q_{3}^{10} - 3336 \, q_{1}^{2} q_{2}^{2} q_{3}^{12} - 2880 \, q_{1}^{3} q_{3}^{13} - 3840 \, q_{1}^{2} q_{2} q_{3}^{13} - 5710 \, q_{3}^{16} + 216 \, q_{1}^{5} q_{2}^{5} q_{3}^{5} - 640 \, q_{1}^{6} q_{2}^{3} q_{3}^{6} + 484 \, q_{1}^{4} q_{2}^{4} q_{3}^{7} - 3840 \, q_{1}^{5} q_{2} q_{3}^{9} - 960 \, q_{1}^{4} q_{2}^{2} q_{3}^{9} + 88 \, q_{1}^{3} q_{2}^{3} q_{3}^{9} - 776 \, q_{1}^{2} q_{2}^{2} q_{3}^{11} - 1920 \, q_{1}^{3} q_{3}^{12} - 7602 \, q_{1}^{2} q_{2} q_{3}^{12} - 4006 \, q_{3}^{15} + 256 \, q_{1}^{5} q_{2}^{4} q_{3}^{5} + 216 \, q_{1}^{4} q_{2}^{4} q_{3}^{6} + 24 \, q_{1}^{3} q_{2}^{3} q_{3}^{8} - 1616 \, q_{1}^{2} q_{2}^{2} q_{3}^{10} - 480 \, q_{1}^{3} q_{3}^{11} + 214 \, q_{1}^{2} q_{2} q_{3}^{11} + 328 \, q_{1}^{2} q_{3}^{12} - 1982 \, q_{3}^{14} + 272 \, q_{1}^{5} q_{2}^{3} q_{3}^{5} + 248 \, q_{1}^{4} q_{2}^{4} q_{3}^{5} + 466 \, q_{1}^{4} q_{2}^{3} q_{3}^{6} - 960 \, q_{1}^{5} q_{2} q_{3}^{7} - 960 \, q_{1}^{4} q_{2}^{2} q_{3}^{7} + 120 \, q_{1}^{3} q_{2}^{3} q_{3}^{7} + 112 \, q_{1}^{3} q_{2}^{2} q_{3}^{8} + 104 \, q_{1}^{2} q_{2}^{2} q_{3}^{9} - 2400 \, q_{1}^{3} q_{3}^{10} - 2552 \, q_{1}^{2} q_{2} q_{3}^{10} + 256 \, q_{1}^{2} q_{3}^{11} - 4560 \, q_{3}^{13} + 324 \, q_{1}^{4} q_{2}^{4} q_{3}^{4} + 236 \, q_{1}^{5} q_{2}^{2} q_{3}^{5} + 306 \, q_{1}^{4} q_{2}^{3} q_{3}^{5} - 1640 \, q_{1}^{4} q_{2}^{2} q_{3}^{6} + 120 \, q_{1}^{3} q_{2}^{3} q_{3}^{6} + 36 \, q_{1}^{4} q_{2} q_{3}^{7} + 82 \, q_{1}^{3} q_{2}^{2} q_{3}^{7} + 146 \, q_{1}^{3} q_{2} q_{3}^{8} - 108 \, q_{1}^{2} q_{2}^{2} q_{3}^{8} - 1920 \, q_{1}^{3} q_{3}^{9} - 1736 \, q_{1}^{2} q_{2} q_{3}^{9} + 504 \, q_{1}^{2} q_{3}^{10} - 3978 \, q_{3}^{12} + 192 \, q_{1}^{4} q_{2}^{3} q_{3}^{4} + 248 \, q_{1}^{5} q_{2} q_{3}^{5} + 200 \, q_{1}^{4} q_{2}^{2} q_{3}^{5} + 32 \, q_{1}^{3} q_{2}^{3} q_{3}^{5} + 250 \, q_{1}^{4} q_{2} q_{3}^{6} + 90 \, q_{1}^{3} q_{2}^{2} q_{3}^{6} + 88 \, q_{1}^{3} q_{2} q_{3}^{7} + 72 \, q_{1}^{2} q_{2}^{2} q_{3}^{7} - 384 \, q_{1}^{3} q_{3}^{8} - 352 \, q_{1}^{2} q_{2} q_{3}^{8} + 128 \, q_{1}^{2} q_{3}^{9} - 982 \, q_{3}^{11} - 276 \, q_{1}^{4} q_{2}^{2} q_{3}^{4} + 80 \, q_{1}^{3} q_{2}^{3} q_{3}^{4} + 304 \, q_{1}^{4} q_{2} q_{3}^{5} + 68 \, q_{1}^{3} q_{2}^{2} q_{3}^{5} + 596 \, q_{1}^{4} q_{3}^{6} + 64 \, q_{1}^{3} q_{2} q_{3}^{6} + 52 \, q_{1}^{2} q_{2}^{2} q_{3}^{6} - 808 \, q_{1}^{3} q_{3}^{7} - 2258 \, q_{1}^{2} q_{2} q_{3}^{7} + 248 \, q_{1}^{2} q_{3}^{8} - 2718 \, q_{3}^{10} + 280 \, q_{1}^{4} q_{2} q_{3}^{4} + 50 \, q_{1}^{3} q_{2}^{2} q_{3}^{4} + 196 \, q_{1}^{4} q_{3}^{5} + 60 \, q_{1}^{3} q_{2} q_{3}^{5} + 32 \, q_{1}^{2} q_{2}^{2} q_{3}^{5} + 28 \, q_{1}^{3} q_{3}^{6} - 1764 \, q_{1}^{2} q_{2} q_{3}^{6} + 176 \, q_{1}^{2} q_{3}^{7} - 860 \, q_{3}^{9} + 20 \, q_{1}^{3} q_{2}^{2} q_{3}^{3} + 272 \, q_{1}^{4} q_{3}^{4} + 46 \, q_{1}^{3} q_{2} q_{3}^{4} + 20 \, q_{1}^{2} q_{2}^{2} q_{3}^{4} - 52 \, q_{1}^{3} q_{3}^{5} - 394 \, q_{1}^{2} q_{2} q_{3}^{5} + 92 \, q_{1}^{2} q_{3}^{6} - 834 \, q_{3}^{8} + 16 \, q_{1}^{3} q_{2} q_{3}^{3} + 68 \, q_{1}^{3} q_{3}^{4} - 858 \, q_{1}^{2} q_{2} q_{3}^{4} + 160 \, q_{1}^{2} q_{3}^{5} - 1284 \, q_{3}^{7} + 8 \, q_{1}^{3} q_{3}^{3} + 20 \, q_{1}^{2} q_{2} q_{3}^{3} + 156 \, q_{1}^{2} q_{3}^{4} - 638 \, q_{3}^{6} - 100 \, q_{1}^{2} q_{2} q_{3}^{2} + 40 \, q_{1}^{2} q_{3}^{3} - 480 \, q_{3}^{5} + 80 \, q_{1}^{2} q_{3}^{2} - 394 \, q_{3}^{4} - 50 \, q_{3}^{3} - 108 \, q_{3}^{2}$

Delta3=ThetaE6 - 9*ThetaE6star; Delta3.expand()

 $\newcommand{\Bold}[1]{\mathbf{#1}}-468 \, q_{3}^{30} - 666 \, q_{3}^{29} - 648 \, q_{3}^{28} - 252 \, q_{3}^{27} - 450 \, q_{3}^{26} + 1062 \, q_{3}^{25} + 1510 \, q_{3}^{24} + 686 \, q_{3}^{23} + 956 \, q_{3}^{22} + 616 \, q_{3}^{21} + 802 \, q_{3}^{20} + 538 \, q_{3}^{19} + 608 \, q_{3}^{18} + 334 \, q_{3}^{17} + 260 \, q_{1}^{5} q_{2}^{5} q_{3}^{6} + 504 \, q_{1}^{2} q_{2}^{2} q_{3}^{12} + 362 \, q_{3}^{16} + 216 \, q_{1}^{5} q_{2}^{5} q_{3}^{5} + 484 \, q_{1}^{4} q_{2}^{4} q_{3}^{7} + 88 \, q_{1}^{3} q_{2}^{3} q_{3}^{9} + 184 \, q_{1}^{2} q_{2}^{2} q_{3}^{11} + 558 \, q_{1}^{2} q_{2} q_{3}^{12} + 310 \, q_{3}^{15} + 256 \, q_{1}^{5} q_{2}^{4} q_{3}^{5} + 216 \, q_{1}^{4} q_{2}^{4} q_{3}^{6} + 24 \, q_{1}^{3} q_{2}^{3} q_{3}^{8} + 304 \, q_{1}^{2} q_{2}^{2} q_{3}^{10} + 214 \, q_{1}^{2} q_{2} q_{3}^{11} + 328 \, q_{1}^{2} q_{3}^{12} + 418 \, q_{3}^{14} + 272 \, q_{1}^{5} q_{2}^{3} q_{3}^{5} + 248 \, q_{1}^{4} q_{2}^{4} q_{3}^{5} + 394 \, q_{1}^{4} q_{2}^{3} q_{3}^{6} + 120 \, q_{1}^{3} q_{2}^{3} q_{3}^{7} + 112 \, q_{1}^{3} q_{2}^{2} q_{3}^{8} + 104 \, q_{1}^{2} q_{2}^{2} q_{3}^{9} + 328 \, q_{1}^{2} q_{2} q_{3}^{10} + 256 \, q_{1}^{2} q_{3}^{11} + 336 \, q_{3}^{13} + 324 \, q_{1}^{4} q_{2}^{4} q_{3}^{4} + 236 \, q_{1}^{5} q_{2}^{2} q_{3}^{5} + 306 \, q_{1}^{4} q_{2}^{3} q_{3}^{5} + 280 \, q_{1}^{4} q_{2}^{2} q_{3}^{6} + 120 \, q_{1}^{3} q_{2}^{3} q_{3}^{6} - 36 \, q_{1}^{4} q_{2} q_{3}^{7} + 82 \, q_{1}^{3} q_{2}^{2} q_{3}^{7} + 146 \, q_{1}^{3} q_{2} q_{3}^{8} + 132 \, q_{1}^{2} q_{2}^{2} q_{3}^{8} + 184 \, q_{1}^{2} q_{2} q_{3}^{9} + 504 \, q_{1}^{2} q_{3}^{10} + 342 \, q_{3}^{12} + 192 \, q_{1}^{4} q_{2}^{3} q_{3}^{4} + 248 \, q_{1}^{5} q_{2} q_{3}^{5} + 200 \, q_{1}^{4} q_{2}^{2} q_{3}^{5} + 32 \, q_{1}^{3} q_{2}^{3} q_{3}^{5} + 250 \, q_{1}^{4} q_{2} q_{3}^{6} + 90 \, q_{1}^{3} q_{2}^{2} q_{3}^{6} + 88 \, q_{1}^{3} q_{2} q_{3}^{7} + 72 \, q_{1}^{2} q_{2}^{2} q_{3}^{7} + 96 \, q_{1}^{3} q_{3}^{8} + 128 \, q_{1}^{2} q_{2} q_{3}^{8} + 128 \, q_{1}^{2} q_{3}^{9} + 218 \, q_{3}^{11} + 132 \, q_{1}^{4} q_{2}^{2} q_{3}^{4} + 80 \, q_{1}^{3} q_{2}^{3} q_{3}^{4} + 304 \, q_{1}^{4} q_{2} q_{3}^{5} + 68 \, q_{1}^{3} q_{2}^{2} q_{3}^{5} + 596 \, q_{1}^{4} q_{3}^{6} + 64 \, q_{1}^{3} q_{2} q_{3}^{6} + 52 \, q_{1}^{2} q_{2}^{2} q_{3}^{6} + 152 \, q_{1}^{3} q_{3}^{7} + 142 \, q_{1}^{2} q_{2} q_{3}^{7} + 248 \, q_{1}^{2} q_{3}^{8} + 282 \, q_{3}^{10} + 280 \, q_{1}^{4} q_{2} q_{3}^{4} + 50 \, q_{1}^{3} q_{2}^{2} q_{3}^{4} + 196 \, q_{1}^{4} q_{3}^{5} + 60 \, q_{1}^{3} q_{2} q_{3}^{5} + 32 \, q_{1}^{2} q_{2}^{2} q_{3}^{5} + 28 \, q_{1}^{3} q_{3}^{6} + 156 \, q_{1}^{2} q_{2} q_{3}^{6} + 176 \, q_{1}^{2} q_{3}^{7} + 108 \, q_{3}^{9} + 20 \, q_{1}^{3} q_{2}^{2} q_{3}^{3} + 272 \, q_{1}^{4} q_{3}^{4} + 46 \, q_{1}^{3} q_{2} q_{3}^{4} + 20 \, q_{1}^{2} q_{2}^{2} q_{3}^{4} + 68 \, q_{1}^{3} q_{3}^{5} + 86 \, q_{1}^{2} q_{2} q_{3}^{5} + 92 \, q_{1}^{2} q_{3}^{6} + 126 \, q_{3}^{8} + 16 \, q_{1}^{3} q_{2} q_{3}^{3} + 68 \, q_{1}^{3} q_{3}^{4} + 102 \, q_{1}^{2} q_{2} q_{3}^{4} + 160 \, q_{1}^{2} q_{3}^{5} + 84 \, q_{3}^{7} + 8 \, q_{1}^{3} q_{3}^{3} + 20 \, q_{1}^{2} q_{2} q_{3}^{3} + 156 \, q_{1}^{2} q_{3}^{4} + 50 \, q_{3}^{6} + 20 \, q_{1}^{2} q_{2} q_{3}^{2} + 40 \, q_{1}^{2} q_{3}^{3} + 60 \, q_{3}^{5} + 80 \, q_{1}^{2} q_{3}^{2} + 14 \, q_{3}^{4} + 10 \, q_{3}^{3} + 12 \, q_{3}^{2} - 8$

var('x1','x2','x3','x4','y1','y2','y3','y4');

 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(x_{1}, x_{2}, x_{3}, x_{4}, y_{1}, y_{2}, y_{3}, y_{4}\right)$

X=matrix([[x1],[x2],[x3],[x4]]);Y=matrix([[y1],[y2],[y3],[y4]]);Y

 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{r} y_{1} \\ y_{2} \\ y_{3} \\ y_{4} \end{array}\right)$

Q=matrix([[(X.transpose()*S4*X), X.transpose()*S4*Y],[Y.transpose()*S4*X, Y.transpose()*S4*Y]]);Q

 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} \left(\begin{array}{r} \frac{1}{2} \, {\left(2 \, x_{1} + 3 \, x_{3}\right)} x_{1} + \frac{1}{2} \, {\left(2 \, x_{2} + 3 \, x_{4}\right)} x_{2} + \frac{3}{2} \, {\left(x_{1} + 2 \, x_{3}\right)} x_{3} + \frac{3}{2} \, {\left(x_{2} + 2 \, x_{4}\right)} x_{4} \end{array}\right) & \left(\begin{array}{r} \frac{1}{2} \, {\left(2 \, x_{1} + 3 \, x_{3}\right)} y_{1} + \frac{1}{2} \, {\left(2 \, x_{2} + 3 \, x_{4}\right)} y_{2} + \frac{3}{2} \, {\left(x_{1} + 2 \, x_{3}\right)} y_{3} + \frac{3}{2} \, {\left(x_{2} + 2 \, x_{4}\right)} y_{4} \end{array}\right) \\ \left(\begin{array}{r} \frac{1}{2} \, x_{1} {\left(2 \, y_{1} + 3 \, y_{3}\right)} + \frac{3}{2} \, x_{3} {\left(y_{1} + 2 \, y_{3}\right)} + \frac{1}{2} \, x_{2} {\left(2 \, y_{2} + 3 \, y_{4}\right)} + \frac{3}{2} \, x_{4} {\left(y_{2} + 2 \, y_{4}\right)} \end{array}\right) & \left(\begin{array}{r} \frac{1}{2} \, {\left(2 \, y_{1} + 3 \, y_{3}\right)} y_{1} + \frac{1}{2} \, {\left(2 \, y_{2} + 3 \, y_{4}\right)} y_{2} + \frac{3}{2} \, {\left(y_{1} + 2 \, y_{3}\right)} y_{3} + \frac{3}{2} \, {\left(y_{2} + 2 \, y_{4}\right)} y_{4} \end{array}\right) \end{array}\right)$

c=x1*y3-x3*y1+x2*y4-y2*x4;c;d=x1*y4-y1*x4+x3*y2-y3*x2+x1*y2-y1*x2;d

 $\newcommand{\Bold}[1]{\mathbf{#1}}-x_{3} y_{1} - x_{4} y_{2} + x_{1} y_{3} + x_{2} y_{4}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-x_{2} y_{1} - x_{4} y_{1} + x_{1} y_{2} + x_{3} y_{2} - x_{2} y_{3} + x_{1} y_{4}$

Z=matrix([[tau,z],[z,w]]);Z

 $\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} \tau & z \\ z & w \end{array}\right)$

j=1 Theta4=0; for x1 in [-j..j]: for x2 in [-j..j]: for x2 in [-j..j]: for x3 in [-j..j]: for x4 in [-j..j]: for y1 in [-j..j]: for y2 in [-j..j]: for y3 in [-j..j]: for y4 in [-j..j]: Theta4 = Theta4 + ((x1*y3-x3*y1+x2*y4-y2*x4)^2 - (x1*y4-y1*x4+x3*y2-y3*x2+x1*y2-y1*x2)^2); Q1=(matrix([[x1],[x2],[x3],[x4]]).transpose()*S4*matrix([[x1],[x2],[x3],[x4]]))

*e^(2*pi*i*(matrix([[(matrix([[x1],[x2],[x3],[x4]]).transpose()*S4*matrix([[x1],[x2],[x3],[x4]])), matrix([[x1],[x2],[x3],[x4]]).transpose()*S4*matrix([[y1],[y2],[y3],[y4]])],[matrix([[y1],[y2],[y3],[y4]]).transpose()*S4*matrix([[x1],[x2],[x3],[x4]]), matrix([[y1],[y2],[y3],[y4]]).transpose()*S4*matrix([[y1],[y2],[y3],[y4]])]])*Z).trace()))

 $\newcommand{\Bold}[1]{\mathbf{#1}}0$

for x1 in [-2..2]: for x2 in [-2..2]: for x3 in [-2..2]: for x4 in [-2..2]: for y1 in [-2..2]: for y2 in [-2..2]: for y3 in [-2..2]: for y4 in [-2..2]: (c^2-d^2)

WARNING: Output truncated!  
full_output.txt



 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 

...

 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
 $\newcommand{\Bold}[1]{\mathbf{#1}}-{\left(x_{2} y_{1} + x_{4} y_{1} - x_{1} y_{2} - x_{3} y_{2} + x_{2} y_{3} - x_{1} y_{4}\right)}^{2} + {\left(x_{3} y_{1} + x_{4} y_{2} - x_{1} y_{3} - x_{2} y_{4}\right)}^{2}$ 
^C
Traceback (click to the left of this block for traceback)
...
__SAGE__

WARNING: Output truncated!  
full_output.txt































































...











































^C
Traceback (most recent call last):                            for y3 in [-2..2]:
  File "", line 1, in <module>
    
  File "/tmp/tmpjK1XTI/___code___.py", line 3, in <module>
    exec compile(u'for x1 in (ellipsis_range(-_sage_const_2 ,Ellipsis,_sage_const_2 )):\n    for x2 in (ellipsis_range(-_sage_const_2 ,Ellipsis,_sage_const_2 )):\n        for x3 in (ellipsis_range(-_sage_const_2 ,Ellipsis,_sage_const_2 )):\n            for x4 in (ellipsis_range(-_sage_const_2 ,Ellipsis,_sage_const_2 )):\n                for y1 in (ellipsis_range(-_sage_const_2 ,Ellipsis,_sage_const_2 )):\n                    for y2 in (ellipsis_range(-_sage_const_2 ,Ellipsis,_sage_const_2 )):\n                        for y3 in (ellipsis_range(-_sage_const_2 ,Ellipsis,_sage_const_2 )):\n                            for y4 in (ellipsis_range(-_sage_const_2 ,Ellipsis,_sage_const_2 )):\n                                (c**_sage_const_2 -d**_sage_const_2 )
  File "", line 9, in <module>
    
  File "/usr/local/sage-6.10/local/lib/python2.7/site-packages/sage/repl/rich_output/display_manager.py", line 746, in displayhook
    return self._backend.displayhook(plain_text, rich_output)
  File "/usr/local/sage-6.10/local/lib/python2.7/site-packages/sage/repl/rich_output/backend_base.py", line 538, in displayhook
    return self.display_immediately(plain_text, rich_output)
  File "/usr/local/sage-6.10/local/lib/python2.7/site-packages/sage/repl/rich_output/backend_sagenb.py", line 368, in display_immediately
    print(rich_output.mathjax())
  File "sage/ext/interrupt/interrupt.pyx", line 203, in sage.ext.interrupt.interrupt.sage_python_check_interrupt (/usr/local/sage-6.10/src/build/cythonized/sage/ext/interrupt/interrupt.c:1891)
  File "sage/ext/interrupt/interrupt.pyx", line 88, in sage.ext.interrupt.interrupt.sig_raise_exception (/usr/local/sage-6.10/src/build/cythonized/sage/ext/interrupt/interrupt.c:925)
KeyboardInterrupt
__SAGE__

full_output.txt

Theta Series

1039 days ago by jimlb