Properties

Label 600.2.b.e
Level 600
Weight 2
Character orbit 600.b
Analytic conductor 4.791
Analytic rank 0
Dimension 8
CM No
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 600.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1649659456.5
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \( + \beta_{4} q^{3} \) \( + ( \beta_{1} + \beta_{3} + \beta_{4} ) q^{4} \) \( + ( \beta_{3} - \beta_{7} ) q^{6} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} ) q^{7} \) \( + ( -1 + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{8} \) \( + ( -\beta_{1} - \beta_{5} - \beta_{7} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \( + \beta_{4} q^{3} \) \( + ( \beta_{1} + \beta_{3} + \beta_{4} ) q^{4} \) \( + ( \beta_{3} - \beta_{7} ) q^{6} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} ) q^{7} \) \( + ( -1 + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{8} \) \( + ( -\beta_{1} - \beta_{5} - \beta_{7} ) q^{9} \) \( + 2 \beta_{3} q^{11} \) \( + ( \beta_{1} + \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{12} \) \( + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{13} \) \( + ( -1 + \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{7} ) q^{14} \) \( + ( -1 + 2 \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{16} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{17} \) \( + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{6} + \beta_{7} ) q^{18} \) \( + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{19} \) \( + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{21} \) \( + ( -2 + 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{22} \) \( + ( \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{23} \) \( + ( -1 - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{24} \) \( + ( 2 - 2 \beta_{5} - 2 \beta_{6} ) q^{26} \) \( + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{27} \) \( + ( 2 \beta_{6} - 2 \beta_{7} ) q^{28} \) \( + ( -2 \beta_{1} - 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{29} \) \( + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{31} \) \( + ( -1 - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{32} \) \( + ( 4 \beta_{1} + 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{33} \) \( + ( -2 \beta_{3} + 2 \beta_{4} ) q^{34} \) \( + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{6} ) q^{36} \) \( + ( \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{37} \) \( + ( 2 + 2 \beta_{5} - 2 \beta_{6} ) q^{38} \) \( + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{39} \) \( + 2 \beta_{5} q^{41} \) \( + ( 1 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{42} \) \( + ( -\beta_{4} - \beta_{6} ) q^{43} \) \( + ( -4 + 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} ) q^{44} \) \( + ( -1 - \beta_{2} + \beta_{3} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{46} \) \( + ( -4 + 5 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{47} \) \( + ( 3 + 2 \beta_{1} + \beta_{3} - \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{48} \) \( + ( -3 + 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{49} \) \( + ( 2 + 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{51} \) \( + ( -2 - 2 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{52} \) \( + ( -2 - 2 \beta_{4} - 2 \beta_{6} ) q^{53} \) \( + ( 5 + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{54} \) \( + ( 4 - 2 \beta_{2} - 2 \beta_{5} + 2 \beta_{6} ) q^{56} \) \( + ( -\beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{57} \) \( + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{58} \) \( + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{5} ) q^{59} \) \( + ( 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{61} \) \( + ( 4 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{62} \) \( + ( 4 - \beta_{1} - \beta_{2} - 5 \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{63} \) \( + ( 3 + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{64} \) \( + ( -4 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{66} \) \( + ( 4 + 4 \beta_{1} + \beta_{4} + \beta_{6} + 4 \beta_{7} ) q^{67} \) \( + ( 2 - 2 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{68} \) \( + ( 3 - 3 \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{6} + \beta_{7} ) q^{69} \) \( + ( -4 + 6 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{71} \) \( + ( 1 + 2 \beta_{2} + 3 \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{72} \) \( + ( 2 - 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{73} \) \( + ( 4 - 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{74} \) \( + ( -2 - 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{76} \) \( + ( -4 \beta_{1} + 2 \beta_{4} + 2 \beta_{6} - 4 \beta_{7} ) q^{77} \) \( + ( -4 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} ) q^{78} \) \( + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{79} \) \( + ( 1 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} ) q^{81} \) \( + ( 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{82} \) \( + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{83} \) \( + ( 8 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} ) q^{84} \) \( + ( -1 + \beta_{2} - \beta_{3} + \beta_{7} ) q^{86} \) \( + ( -4 - 2 \beta_{2} - 2 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} ) q^{87} \) \( + ( -2 + 4 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{88} \) \( + ( -4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{89} \) \( + ( 4 - 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{91} \) \( + ( 4 + 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{92} \) \( + ( -\beta_{1} + 3 \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{5} + 3 \beta_{6} + 2 \beta_{7} ) q^{93} \) \( + ( -5 - \beta_{2} - 3 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{94} \) \( + ( -5 - 4 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{96} \) \( + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} - 4 \beta_{7} ) q^{97} \) \( + ( -6 + \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{98} \) \( + ( 2 + 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut +\mathstrut q^{6} \) \(\mathstrut -\mathstrut 7q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut -\mathstrut q^{2} \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut +\mathstrut q^{6} \) \(\mathstrut -\mathstrut 7q^{8} \) \(\mathstrut +\mathstrut q^{12} \) \(\mathstrut -\mathstrut 6q^{14} \) \(\mathstrut -\mathstrut 7q^{16} \) \(\mathstrut +\mathstrut 7q^{18} \) \(\mathstrut -\mathstrut 4q^{19} \) \(\mathstrut -\mathstrut 4q^{21} \) \(\mathstrut -\mathstrut 14q^{22} \) \(\mathstrut +\mathstrut 4q^{23} \) \(\mathstrut -\mathstrut 11q^{24} \) \(\mathstrut +\mathstrut 16q^{26} \) \(\mathstrut +\mathstrut 12q^{27} \) \(\mathstrut +\mathstrut 2q^{28} \) \(\mathstrut -\mathstrut 11q^{32} \) \(\mathstrut +\mathstrut 4q^{33} \) \(\mathstrut +\mathstrut 13q^{36} \) \(\mathstrut +\mathstrut 16q^{38} \) \(\mathstrut +\mathstrut 16q^{39} \) \(\mathstrut +\mathstrut 6q^{42} \) \(\mathstrut -\mathstrut 30q^{44} \) \(\mathstrut -\mathstrut 8q^{46} \) \(\mathstrut -\mathstrut 28q^{47} \) \(\mathstrut +\mathstrut 25q^{48} \) \(\mathstrut -\mathstrut 16q^{49} \) \(\mathstrut +\mathstrut 20q^{51} \) \(\mathstrut -\mathstrut 20q^{52} \) \(\mathstrut -\mathstrut 16q^{53} \) \(\mathstrut +\mathstrut 41q^{54} \) \(\mathstrut +\mathstrut 30q^{56} \) \(\mathstrut +\mathstrut 4q^{57} \) \(\mathstrut +\mathstrut 2q^{58} \) \(\mathstrut +\mathstrut 34q^{62} \) \(\mathstrut +\mathstrut 28q^{63} \) \(\mathstrut +\mathstrut 25q^{64} \) \(\mathstrut -\mathstrut 34q^{66} \) \(\mathstrut +\mathstrut 32q^{67} \) \(\mathstrut +\mathstrut 16q^{68} \) \(\mathstrut +\mathstrut 20q^{69} \) \(\mathstrut -\mathstrut 24q^{71} \) \(\mathstrut +\mathstrut 9q^{72} \) \(\mathstrut +\mathstrut 8q^{73} \) \(\mathstrut +\mathstrut 32q^{74} \) \(\mathstrut -\mathstrut 12q^{76} \) \(\mathstrut -\mathstrut 36q^{78} \) \(\mathstrut +\mathstrut 8q^{81} \) \(\mathstrut +\mathstrut 4q^{82} \) \(\mathstrut +\mathstrut 58q^{84} \) \(\mathstrut -\mathstrut 8q^{86} \) \(\mathstrut -\mathstrut 36q^{87} \) \(\mathstrut -\mathstrut 14q^{88} \) \(\mathstrut +\mathstrut 24q^{91} \) \(\mathstrut +\mathstrut 28q^{92} \) \(\mathstrut -\mathstrut 40q^{94} \) \(\mathstrut -\mathstrut 43q^{96} \) \(\mathstrut -\mathstrut 8q^{97} \) \(\mathstrut -\mathstrut 47q^{98} \) \(\mathstrut +\mathstrut 16q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(x^{7}\mathstrut -\mathstrut \) \(2\) \(x^{5}\mathstrut +\mathstrut \) \(4\) \(x^{4}\mathstrut -\mathstrut \) \(4\) \(x^{3}\mathstrut -\mathstrut \) \(8\) \(x\mathstrut +\mathstrut \) \(16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + \nu^{5} - 4 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + \nu^{6} + 4 \nu^{2} - 8 \nu - 8 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} - \nu^{6} + 4 \nu^{2} + 8 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} + \nu^{6} - 4 \nu^{4} + 4 \nu^{3} + 4 \nu^{2} - 16 \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - \nu^{6} + 2 \nu^{5} + 4 \nu^{3} - 4 \nu^{2} - 8 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} + \nu^{6} - 2 \nu^{5} + 4 \nu^{4} + 4 \nu^{2} - 8 \nu + 8 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)
\(\nu^{5}\)\(=\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(1\)
\(\nu^{6}\)\(=\)\(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{7}\)\(=\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(5\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
1.40014 + 0.199044i
1.40014 0.199044i
0.814732 + 1.15595i
0.814732 1.15595i
−0.578647 + 1.29041i
−0.578647 1.29041i
−1.13622 + 0.842022i
−1.13622 0.842022i
−1.40014 0.199044i 0.520627 1.65195i 1.92076 + 0.557378i 0 −1.05776 + 2.20933i 1.92736i −2.57839 1.16272i −2.45790 1.72010i 0
251.2 −1.40014 + 0.199044i 0.520627 + 1.65195i 1.92076 0.557378i 0 −1.05776 2.20933i 1.92736i −2.57839 + 1.16272i −2.45790 + 1.72010i 0
251.3 −0.814732 1.15595i −1.48716 + 0.887900i −0.672424 + 1.88357i 0 2.23800 + 0.995672i 0.797253i 2.72515 0.757320i 1.42327 2.64089i 0
251.4 −0.814732 + 1.15595i −1.48716 0.887900i −0.672424 1.88357i 0 2.23800 0.995672i 0.797253i 2.72515 + 0.757320i 1.42327 + 2.64089i 0
251.5 0.578647 1.29041i −0.751690 1.56044i −1.33034 1.49339i 0 −2.44857 + 0.0670494i 4.28591i −2.69688 + 0.852541i −1.86993 + 2.34593i 0
251.6 0.578647 + 1.29041i −0.751690 + 1.56044i −1.33034 + 1.49339i 0 −2.44857 0.0670494i 4.28591i −2.69688 0.852541i −1.86993 2.34593i 0
251.7 1.13622 0.842022i 1.71822 0.218455i 0.581998 1.91345i 0 1.76833 1.69499i 3.64426i −0.949886 2.66415i 2.90455 0.750707i 0
251.8 1.13622 + 0.842022i 1.71822 + 0.218455i 0.581998 + 1.91345i 0 1.76833 + 1.69499i 3.64426i −0.949886 + 2.66415i 2.90455 + 0.750707i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.8
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
24.f Even 1 no

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\):

\(T_{7}^{8} \) \(\mathstrut +\mathstrut 36 T_{7}^{6} \) \(\mathstrut +\mathstrut 384 T_{7}^{4} \) \(\mathstrut +\mathstrut 1136 T_{7}^{2} \) \(\mathstrut +\mathstrut 576 \)
\(T_{11}^{8} \) \(\mathstrut +\mathstrut 48 T_{11}^{6} \) \(\mathstrut +\mathstrut 672 T_{11}^{4} \) \(\mathstrut +\mathstrut 2560 T_{11}^{2} \) \(\mathstrut +\mathstrut 256 \)
\(T_{23}^{4} \) \(\mathstrut -\mathstrut 2 T_{23}^{3} \) \(\mathstrut -\mathstrut 44 T_{23}^{2} \) \(\mathstrut +\mathstrut 188 T_{23} \) \(\mathstrut -\mathstrut 192 \)
\(T_{43}^{4} \) \(\mathstrut -\mathstrut 12 T_{43}^{2} \) \(\mathstrut +\mathstrut 4 T_{43} \) \(\mathstrut +\mathstrut 16 \)