Properties

Label 798.2.j.l
Level $798$
Weight $2$
Character orbit 798.j
Analytic conductor $6.372$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 798.j (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.37206208130\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.856615824.2
Defining polynomial: \(x^{8} + 11 x^{6} + 36 x^{4} + 32 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{2} + \beta_{1} q^{3} -\beta_{1} q^{4} + ( \beta_{5} - \beta_{6} ) q^{5} + q^{6} + ( \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{7} - q^{8} + ( -1 + \beta_{1} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{2} + \beta_{1} q^{3} -\beta_{1} q^{4} + ( \beta_{5} - \beta_{6} ) q^{5} + q^{6} + ( \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{7} - q^{8} + ( -1 + \beta_{1} ) q^{9} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} ) q^{10} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{11} + ( 1 - \beta_{1} ) q^{12} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{6} ) q^{13} + \beta_{7} q^{14} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{6} ) q^{15} + ( -1 + \beta_{1} ) q^{16} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{17} + \beta_{1} q^{18} + ( -1 + \beta_{1} ) q^{19} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} ) q^{20} + ( \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{21} + ( 1 + \beta_{5} + \beta_{7} ) q^{22} + ( -1 - 2 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} ) q^{23} -\beta_{1} q^{24} + ( -\beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{25} + ( 2 \beta_{5} - 2 \beta_{6} ) q^{26} - q^{27} + ( -\beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{28} + ( -3 + 3 \beta_{3} - 3 \beta_{5} ) q^{29} + ( \beta_{5} - \beta_{6} ) q^{30} + ( 3 \beta_{3} + \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{31} + \beta_{1} q^{32} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} ) q^{33} + ( -3 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{6} + 2 \beta_{7} ) q^{34} + ( -5 + 3 \beta_{1} + \beta_{2} + 3 \beta_{3} - 3 \beta_{5} + \beta_{6} ) q^{35} + q^{36} + ( 5 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{6} ) q^{37} + \beta_{1} q^{38} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{39} + ( -\beta_{5} + \beta_{6} ) q^{40} + ( 3 - \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{5} - \beta_{6} + \beta_{7} ) q^{41} + ( \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{42} + ( 7 + \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{43} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} ) q^{44} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} ) q^{45} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} + 3 \beta_{6} ) q^{46} + ( -2 + \beta_{1} - \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{47} - q^{48} + ( -1 + 5 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{49} + ( -2 + 2 \beta_{3} - \beta_{5} + \beta_{7} ) q^{50} + ( 3 - \beta_{1} + 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{51} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} ) q^{52} + ( 4 \beta_{1} + 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 3 \beta_{7} ) q^{53} + ( -1 + \beta_{1} ) q^{54} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{5} - \beta_{6} - \beta_{7} ) q^{55} + ( -\beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{56} - q^{57} + ( -3 + 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{5} + 3 \beta_{6} ) q^{58} + ( -2 \beta_{1} + 4 \beta_{2} + \beta_{3} + \beta_{5} + 5 \beta_{6} ) q^{59} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} ) q^{60} + ( -6 + 5 \beta_{1} - \beta_{4} - \beta_{5} ) q^{61} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{62} -\beta_{7} q^{63} + q^{64} + ( 14 - 10 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} ) q^{65} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{66} + ( -6 \beta_{1} - 6 \beta_{3} - 4 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} ) q^{67} + ( -3 + \beta_{1} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{68} + ( -1 - \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{5} - \beta_{6} ) q^{69} + ( -1 + 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + 3 \beta_{6} ) q^{70} + ( 3 + \beta_{1} - \beta_{2} - \beta_{3} + 4 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{71} + ( 1 - \beta_{1} ) q^{72} + ( -\beta_{1} - 3 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - 5 \beta_{6} - 2 \beta_{7} ) q^{73} + ( -4 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - 2 \beta_{6} ) q^{74} + ( 2 - \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{75} + q^{76} + ( -4 + 7 \beta_{1} - \beta_{2} + 3 \beta_{5} ) q^{77} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{6} ) q^{78} + ( 3 - 4 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{79} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} ) q^{80} -\beta_{1} q^{81} + ( 3 - \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - \beta_{4} + 3 \beta_{5} - 4 \beta_{6} ) q^{82} + ( 1 + 2 \beta_{3} - 4 \beta_{5} - 2 \beta_{7} ) q^{83} + \beta_{7} q^{84} + ( -1 + 3 \beta_{1} - 3 \beta_{2} + 7 \beta_{3} - 3 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{85} + ( 7 - 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{86} + ( -3 \beta_{2} - 3 \beta_{6} ) q^{87} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{88} + ( -1 + 5 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} + 5 \beta_{5} - 2 \beta_{6} ) q^{89} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{6} ) q^{90} + ( -8 + 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - 4 \beta_{6} ) q^{91} + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{5} + \beta_{6} ) q^{92} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{93} + ( 3 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{94} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} ) q^{95} + ( -1 + \beta_{1} ) q^{96} + ( 2 + \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{5} + \beta_{6} ) q^{97} + ( 4 + 4 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{98} + ( -1 - \beta_{5} - \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{2} + 4q^{3} - 4q^{4} + 8q^{6} - 2q^{7} - 8q^{8} - 4q^{9} + O(q^{10}) \) \( 8q + 4q^{2} + 4q^{3} - 4q^{4} + 8q^{6} - 2q^{7} - 8q^{8} - 4q^{9} + 2q^{11} + 4q^{12} - q^{14} - 4q^{16} - 10q^{17} + 4q^{18} - 4q^{19} - q^{21} + 4q^{22} + 5q^{23} - 4q^{24} - 4q^{25} - 8q^{27} + q^{28} - 6q^{29} - 9q^{31} + 4q^{32} - 2q^{33} - 20q^{34} - 9q^{35} + 8q^{36} + 14q^{37} + 4q^{38} + 8q^{41} - 2q^{42} + 42q^{43} + 2q^{44} - 5q^{46} - 7q^{47} - 8q^{48} - 4q^{49} - 8q^{50} + 10q^{51} + 7q^{53} - 4q^{54} + 2q^{56} - 8q^{57} - 3q^{58} - 7q^{59} - 23q^{61} - 18q^{62} + q^{63} + 8q^{64} + 48q^{65} + 2q^{66} - 6q^{67} - 10q^{68} + 10q^{69} + 15q^{70} + 4q^{71} + 4q^{72} + 5q^{73} - 14q^{74} + 4q^{75} + 8q^{76} - 17q^{77} + 11q^{79} - 4q^{81} + 4q^{82} + 28q^{83} - q^{84} + 12q^{85} + 21q^{86} - 3q^{87} - 2q^{88} - 10q^{89} - 48q^{91} - 10q^{92} + 9q^{93} + 7q^{94} - 4q^{96} - 2q^{97} + 25q^{98} - 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 11 x^{6} + 36 x^{4} + 32 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} + 7 \nu^{3} + 10 \nu + 2 \)\()/4\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + 9 \nu^{4} + 2 \nu^{3} + 22 \nu^{2} + 10 \nu + 12 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} - 11 \nu^{5} - 34 \nu^{3} - 2 \nu^{2} - 20 \nu - 4 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} - 10 \nu^{5} + 2 \nu^{4} - 29 \nu^{3} + 10 \nu^{2} - 26 \nu + 2 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} - 11 \nu^{5} - 34 \nu^{3} + 2 \nu^{2} - 20 \nu + 4 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - \nu^{6} + 10 \nu^{5} - 9 \nu^{4} + 29 \nu^{3} - 20 \nu^{2} + 20 \nu - 6 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} + 11 \nu^{5} - 4 \nu^{4} + 34 \nu^{3} - 22 \nu^{2} + 20 \nu - 12 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{7} - \beta_{6} - 2 \beta_{4} - \beta_{2} + \beta_{1} - 1\)\()/3\)
\(\nu^{2}\)\(=\)\(\beta_{5} - \beta_{3} - 2\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{7} + 8 \beta_{6} + 10 \beta_{4} + 3 \beta_{3} + 8 \beta_{2} - 2 \beta_{1} + 2\)\()/3\)
\(\nu^{4}\)\(=\)\(-\beta_{7} - 6 \beta_{5} + 5 \beta_{3} + 8\)
\(\nu^{5}\)\(=\)\((\)\(-25 \beta_{7} - 46 \beta_{6} - 50 \beta_{4} - 21 \beta_{3} - 46 \beta_{2} + 16 \beta_{1} - 10\)\()/3\)
\(\nu^{6}\)\(=\)\(9 \beta_{7} - 2 \beta_{6} + 32 \beta_{5} - 25 \beta_{3} + 2 \beta_{2} - 2 \beta_{1} - 38\)
\(\nu^{7}\)\(=\)\((\)\(125 \beta_{7} + 254 \beta_{6} - 6 \beta_{5} + 250 \beta_{4} + 123 \beta_{3} + 254 \beta_{2} - 128 \beta_{1} + 62\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/798\mathbb{Z}\right)^\times\).

\(n\) \(115\) \(211\) \(533\)
\(\chi(n)\) \(-1 + \beta_{1}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
457.1
1.07834i
2.33086i
2.06288i
0.385731i
1.07834i
2.33086i
2.06288i
0.385731i
0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −1.24624 + 2.15855i 1.00000 2.47720 + 0.929227i −1.00000 −0.500000 + 0.866025i 1.24624 + 2.15855i
457.2 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −1.05903 + 1.83430i 1.00000 −1.11699 2.39840i −1.00000 −0.500000 + 0.866025i 1.05903 + 1.83430i
457.3 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 0.574618 0.995268i 1.00000 0.00953166 + 2.64573i −1.00000 −0.500000 + 0.866025i −0.574618 0.995268i
457.4 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 1.73065 2.99758i 1.00000 −2.36975 1.17656i −1.00000 −0.500000 + 0.866025i −1.73065 2.99758i
571.1 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −1.24624 2.15855i 1.00000 2.47720 0.929227i −1.00000 −0.500000 0.866025i 1.24624 2.15855i
571.2 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −1.05903 1.83430i 1.00000 −1.11699 + 2.39840i −1.00000 −0.500000 0.866025i 1.05903 1.83430i
571.3 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 0.574618 + 0.995268i 1.00000 0.00953166 2.64573i −1.00000 −0.500000 0.866025i −0.574618 + 0.995268i
571.4 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 1.73065 + 2.99758i 1.00000 −2.36975 + 1.17656i −1.00000 −0.500000 0.866025i −1.73065 + 2.99758i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 571.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 798.2.j.l 8
7.c even 3 1 inner 798.2.j.l 8
7.c even 3 1 5586.2.a.bw 4
7.d odd 6 1 5586.2.a.bz 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.j.l 8 1.a even 1 1 trivial
798.2.j.l 8 7.c even 3 1 inner
5586.2.a.bw 4 7.c even 3 1
5586.2.a.bz 4 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(798, [\chi])\):

\( T_{5}^{8} + 12 T_{5}^{6} + 12 T_{5}^{5} + 123 T_{5}^{4} + 72 T_{5}^{3} + 288 T_{5}^{2} - 126 T_{5} + 441 \)
\(T_{11}^{8} - \cdots\)
\( T_{13}^{4} - 48 T_{13}^{2} - 48 T_{13} + 336 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{4} \)
$3$ \( ( 1 - T + T^{2} )^{4} \)
$5$ \( 441 - 126 T + 288 T^{2} + 72 T^{3} + 123 T^{4} + 12 T^{5} + 12 T^{6} + T^{8} \)
$7$ \( 2401 + 686 T + 196 T^{2} - 70 T^{3} - 41 T^{4} - 10 T^{5} + 4 T^{6} + 2 T^{7} + T^{8} \)
$11$ \( 49 + 154 T + 568 T^{2} - 236 T^{3} + 181 T^{4} - 20 T^{5} + 16 T^{6} - 2 T^{7} + T^{8} \)
$13$ \( ( 336 - 48 T - 48 T^{2} + T^{4} )^{2} \)
$17$ \( 204304 + 114808 T + 61804 T^{2} + 10564 T^{3} + 3028 T^{4} + 448 T^{5} + 106 T^{6} + 10 T^{7} + T^{8} \)
$19$ \( ( 1 + T + T^{2} )^{4} \)
$23$ \( 850084 + 175180 T + 91420 T^{2} - 2180 T^{3} + 3628 T^{4} - 80 T^{5} + 85 T^{6} - 5 T^{7} + T^{8} \)
$29$ \( ( 1296 - 135 T - 81 T^{2} + 3 T^{3} + T^{4} )^{2} \)
$31$ \( 1140624 + 509436 T + 192285 T^{2} + 34965 T^{3} + 6450 T^{4} + 657 T^{5} + 114 T^{6} + 9 T^{7} + T^{8} \)
$37$ \( 64 - 176 T + 820 T^{2} + 700 T^{3} + 2080 T^{4} - 632 T^{5} + 154 T^{6} - 14 T^{7} + T^{8} \)
$41$ \( ( -344 + 506 T - 96 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$43$ \( ( -654 + 24 T + 114 T^{2} - 21 T^{3} + T^{4} )^{2} \)
$47$ \( 65536 - 32768 T + 22528 T^{2} - 512 T^{3} + 1216 T^{4} + 88 T^{5} + 73 T^{6} + 7 T^{7} + T^{8} \)
$53$ \( 952576 + 408944 T + 266329 T^{2} - 25303 T^{3} + 10606 T^{4} - 187 T^{5} + 142 T^{6} - 7 T^{7} + T^{8} \)
$59$ \( 84676804 - 6008906 T + 2110375 T^{2} - 9329 T^{3} + 28858 T^{4} + 25 T^{5} + 232 T^{6} + 7 T^{7} + T^{8} \)
$61$ \( 506944 + 441440 T + 251968 T^{2} + 82568 T^{3} + 19624 T^{4} + 3038 T^{5} + 343 T^{6} + 23 T^{7} + T^{8} \)
$67$ \( 63489024 - 8031744 T + 2928384 T^{2} + 146304 T^{3} + 55680 T^{4} + 576 T^{5} + 276 T^{6} + 6 T^{7} + T^{8} \)
$71$ \( ( 28 + 46 T - 126 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$73$ \( 2808976 - 2078240 T + 1225864 T^{2} - 247400 T^{3} + 42472 T^{4} - 1550 T^{5} + 211 T^{6} - 5 T^{7} + T^{8} \)
$79$ \( 29584 + 25972 T + 22285 T^{2} + 4237 T^{3} + 1498 T^{4} - 335 T^{5} + 118 T^{6} - 11 T^{7} + T^{8} \)
$83$ \( ( 43 + 286 T - 60 T^{2} - 14 T^{3} + T^{4} )^{2} \)
$89$ \( 23464336 + 9484552 T + 3078100 T^{2} + 402328 T^{3} + 48760 T^{4} + 2356 T^{5} + 256 T^{6} + 10 T^{7} + T^{8} \)
$97$ \( ( 1048 - 59 T - 69 T^{2} + T^{3} + T^{4} )^{2} \)
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