Properties

Label 392.3.g.j
Level 392
Weight 3
Character orbit 392.g
Analytic conductor 10.681
Analytic rank 0
Dimension 6
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 392.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.6812263629\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.15582448.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{3} + ( 1 + \beta_{1} - \beta_{4} ) q^{4} + ( -\beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{5} + ( -5 + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{6} + ( -2 \beta_{1} - 2 \beta_{5} ) q^{8} + ( 6 - 4 \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{2} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{3} + ( 1 + \beta_{1} - \beta_{4} ) q^{4} + ( -\beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{5} + ( -5 + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{6} + ( -2 \beta_{1} - 2 \beta_{5} ) q^{8} + ( 6 - 4 \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} + 2 \beta_{5} ) q^{10} + ( -6 - 3 \beta_{1} ) q^{11} + ( -5 + 3 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} + \beta_{4} ) q^{12} + ( -5 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{13} + ( -2 \beta_{2} + 2 \beta_{3} - 3 \beta_{5} ) q^{15} + ( -4 + 4 \beta_{2} - 8 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{16} + ( -5 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{17} + ( 5 + 5 \beta_{1} + 10 \beta_{2} - 8 \beta_{3} - \beta_{4} + 4 \beta_{5} ) q^{18} + ( -16 - \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{19} + ( 5 + 3 \beta_{1} + 3 \beta_{4} - 2 \beta_{5} ) q^{20} + ( 3 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} + 3 \beta_{5} ) q^{22} + ( -5 \beta_{2} + 5 \beta_{3} + 7 \beta_{5} ) q^{23} + ( 10 - 8 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} ) q^{24} + ( 16 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{25} + ( 19 - 5 \beta_{1} - 2 \beta_{2} + 9 \beta_{4} - 4 \beta_{5} ) q^{26} + ( 10 - 13 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{27} + ( -3 \beta_{2} + 3 \beta_{3} + 8 \beta_{4} - 2 \beta_{5} ) q^{29} + ( -5 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} ) q^{30} + ( 6 \beta_{2} - 6 \beta_{3} + 9 \beta_{5} ) q^{31} + ( -20 + 4 \beta_{1} - 8 \beta_{2} + 4 \beta_{4} - 8 \beta_{5} ) q^{32} + ( 15 - 6 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} ) q^{33} + ( 5 + 3 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{34} + ( -14 + 8 \beta_{1} - 4 \beta_{2} + 16 \beta_{3} - 2 \beta_{4} - 10 \beta_{5} ) q^{36} + ( -4 \beta_{2} + 4 \beta_{3} + 9 \beta_{4} + \beta_{5} ) q^{37} + ( -20 - 3 \beta_{1} - 15 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + \beta_{5} ) q^{38} + ( -17 \beta_{2} + 17 \beta_{3} - 14 \beta_{4} - 8 \beta_{5} ) q^{39} + ( -4 - 2 \beta_{1} + 4 \beta_{2} + 8 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{40} + ( -26 - 18 \beta_{1} - \beta_{2} - \beta_{3} ) q^{41} + ( -20 + 10 \beta_{1} - 10 \beta_{2} - 10 \beta_{3} ) q^{43} + ( -21 - 3 \beta_{1} - 6 \beta_{2} + 12 \beta_{3} + 9 \beta_{4} - 6 \beta_{5} ) q^{44} + ( -9 \beta_{2} + 9 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{45} + ( 29 + 2 \beta_{1} - 7 \beta_{2} + 14 \beta_{3} + 19 \beta_{4} - 7 \beta_{5} ) q^{46} + ( 2 \beta_{2} - 2 \beta_{3} - 16 \beta_{4} - 5 \beta_{5} ) q^{47} + ( 20 + 4 \beta_{1} + 20 \beta_{2} - 8 \beta_{3} + 4 \beta_{4} + 16 \beta_{5} ) q^{48} + ( 10 + 4 \beta_{1} + 18 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{50} + ( -3 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{51} + ( -10 + 8 \beta_{1} + 28 \beta_{2} - 6 \beta_{4} + 18 \beta_{5} ) q^{52} + ( 20 \beta_{2} - 20 \beta_{3} - 3 \beta_{4} + 7 \beta_{5} ) q^{53} + ( 10 + 15 \beta_{1} + 23 \beta_{2} - 26 \beta_{3} - 2 \beta_{4} + 13 \beta_{5} ) q^{54} + ( 3 \beta_{2} - 3 \beta_{3} - 12 \beta_{4} + 9 \beta_{5} ) q^{55} + ( 45 + 12 \beta_{1} + 20 \beta_{2} + 20 \beta_{3} ) q^{57} + ( 5 + 3 \beta_{1} + 2 \beta_{2} + 12 \beta_{3} - \beta_{4} + 10 \beta_{5} ) q^{58} + ( 12 - 9 \beta_{1} - 5 \beta_{2} - 5 \beta_{3} ) q^{59} + ( -15 + 7 \beta_{1} + 2 \beta_{2} - 12 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{60} + ( 10 \beta_{2} - 10 \beta_{3} - 15 \beta_{4} + 25 \beta_{5} ) q^{61} + ( 15 \beta_{1} - 9 \beta_{2} + 18 \beta_{3} + 12 \beta_{4} - 9 \beta_{5} ) q^{62} + ( -24 - 16 \beta_{1} - 16 \beta_{2} - 8 \beta_{4} + 8 \beta_{5} ) q^{64} + ( 4 + 18 \beta_{1} - 11 \beta_{2} - 11 \beta_{3} ) q^{65} + ( 30 + 12 \beta_{1} + 21 \beta_{2} - 12 \beta_{3} - 6 \beta_{4} + 6 \beta_{5} ) q^{66} + ( -70 + \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{67} + ( -15 - 5 \beta_{1} + 8 \beta_{3} + 7 \beta_{4} - 6 \beta_{5} ) q^{68} + ( 13 \beta_{2} - 13 \beta_{3} - 31 \beta_{4} - 3 \beta_{5} ) q^{69} + ( 22 \beta_{2} - 22 \beta_{3} - 6 \beta_{4} + 8 \beta_{5} ) q^{71} + ( 40 - 24 \beta_{1} - 12 \beta_{2} - 8 \beta_{3} - 16 \beta_{4} ) q^{72} + ( -15 - 18 \beta_{1} + 12 \beta_{2} + 12 \beta_{3} ) q^{73} + ( 14 + 6 \beta_{1} - \beta_{2} + 20 \beta_{3} + 6 \beta_{4} + 8 \beta_{5} ) q^{74} + ( -10 - 24 \beta_{1} - 18 \beta_{2} - 18 \beta_{3} ) q^{75} + ( -21 - 7 \beta_{1} - 18 \beta_{2} + 4 \beta_{3} + 17 \beta_{4} + 6 \beta_{5} ) q^{76} + ( 35 - 39 \beta_{1} + 8 \beta_{2} - 44 \beta_{3} + \beta_{4} - 6 \beta_{5} ) q^{78} + ( 5 \beta_{2} - 5 \beta_{3} + 16 \beta_{4} - 25 \beta_{5} ) q^{79} + ( 40 + 4 \beta_{1} - 4 \beta_{2} - 8 \beta_{3} - 4 \beta_{5} ) q^{80} + ( -9 - 26 \beta_{1} - 21 \beta_{2} - 21 \beta_{3} ) q^{81} + ( -5 + 17 \beta_{1} - 8 \beta_{2} - 36 \beta_{3} + \beta_{4} + 18 \beta_{5} ) q^{82} + ( -50 + 12 \beta_{1} - 14 \beta_{2} - 14 \beta_{3} ) q^{83} + ( 4 \beta_{2} - 4 \beta_{3} - 11 \beta_{4} + 11 \beta_{5} ) q^{85} + ( -50 - 20 \beta_{1} - 30 \beta_{2} + 20 \beta_{3} + 10 \beta_{4} - 10 \beta_{5} ) q^{86} + ( 23 \beta_{2} - 23 \beta_{3} - 8 \beta_{4} - 8 \beta_{5} ) q^{87} + ( 30 - 12 \beta_{2} - 6 \beta_{4} + 18 \beta_{5} ) q^{88} + ( -13 - 4 \beta_{1} + 36 \beta_{2} + 36 \beta_{3} ) q^{89} + ( 31 - 11 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 13 \beta_{4} - 6 \beta_{5} ) q^{90} + ( 35 + 3 \beta_{1} + 34 \beta_{2} + 28 \beta_{3} - 7 \beta_{4} + 24 \beta_{5} ) q^{92} + ( 54 \beta_{2} - 54 \beta_{3} - 15 \beta_{4} + 21 \beta_{5} ) q^{93} + ( -16 - 19 \beta_{1} + 5 \beta_{2} - 42 \beta_{3} - 12 \beta_{4} - 11 \beta_{5} ) q^{94} + ( 11 \beta_{2} - 11 \beta_{3} - 10 \beta_{4} + \beta_{5} ) q^{95} + ( 20 + 36 \beta_{1} + 48 \beta_{3} + 12 \beta_{4} - 16 \beta_{5} ) q^{96} + ( 8 \beta_{1} + 15 \beta_{2} + 15 \beta_{3} ) q^{97} + ( 24 - 12 \beta_{1} - 21 \beta_{2} - 21 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 2q^{2} + 6q^{3} + 4q^{4} - 28q^{6} + 4q^{8} + 40q^{9} + O(q^{10}) \) \( 6q - 2q^{2} + 6q^{3} + 4q^{4} - 28q^{6} + 4q^{8} + 40q^{9} + 6q^{10} - 30q^{11} - 32q^{12} - 16q^{16} - 30q^{17} + 16q^{18} - 78q^{19} + 24q^{20} + 12q^{22} + 76q^{24} + 92q^{25} + 128q^{26} + 78q^{27} + 16q^{30} - 112q^{32} + 78q^{33} + 38q^{34} - 124q^{36} - 80q^{38} - 44q^{40} - 116q^{41} - 100q^{43} - 132q^{44} + 156q^{46} + 88q^{48} + 24q^{50} - 10q^{51} - 132q^{52} + 36q^{54} + 166q^{57} - 4q^{58} + 110q^{59} - 84q^{60} - 48q^{62} - 80q^{64} + 32q^{65} + 138q^{66} - 434q^{67} - 96q^{68} + 328q^{72} - 102q^{73} + 34q^{74} + 60q^{75} - 84q^{76} + 360q^{78} + 256q^{80} + 82q^{81} + 24q^{82} - 268q^{83} - 240q^{86} + 204q^{88} - 214q^{89} + 220q^{90} + 80q^{92} + 16q^{94} - 48q^{96} - 76q^{97} + 252q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} + 13 x^{4} - 21 x^{3} + 20 x^{2} - 10 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{4} - 2 \nu^{3} + 10 \nu^{2} - 9 \nu + 3 \)
\(\beta_{2}\)\(=\)\( \nu^{5} - 2 \nu^{4} + 11 \nu^{3} - 10 \nu^{2} + 10 \nu - 2 \)
\(\beta_{3}\)\(=\)\( -\nu^{5} + 3 \nu^{4} - 13 \nu^{3} + 21 \nu^{2} - 20 \nu + 8 \)
\(\beta_{4}\)\(=\)\( -6 \nu^{5} + 15 \nu^{4} - 70 \nu^{3} + 90 \nu^{2} - 69 \nu + 20 \)
\(\beta_{5}\)\(=\)\( -6 \nu^{5} + 15 \nu^{4} - 70 \nu^{3} + 90 \nu^{2} - 71 \nu + 21 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{5} + \beta_{4} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_{1} - 5\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(9 \beta_{5} - 8 \beta_{4} + 6 \beta_{2} - 3 \beta_{1} - 8\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(19 \beta_{5} - 17 \beta_{4} - 20 \beta_{3} - 8 \beta_{2} + 16 \beta_{1} + 37\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-61 \beta_{5} + 54 \beta_{4} - 20 \beta_{3} - 60 \beta_{2} + 45 \beta_{1} + 106\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(297\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
0.500000 2.94141i
0.500000 + 2.94141i
0.500000 0.759064i
0.500000 + 0.759064i
0.500000 0.148124i
0.500000 + 0.148124i
−1.88766 0.660851i 1.64878 3.12655 + 2.49493i 4.56111i −3.11234 1.08960i 0 −4.25310 6.77577i −6.28154 −3.01422 + 8.60985i
99.2 −1.88766 + 0.660851i 1.64878 3.12655 2.49493i 4.56111i −3.11234 + 1.08960i 0 −4.25310 + 6.77577i −6.28154 −3.01422 8.60985i
99.3 −0.789608 1.83753i 5.33225 −2.75304 + 2.90186i 2.15693i −4.21039 9.79818i 0 7.50608 + 2.76746i 19.4329 3.96343 1.70313i
99.4 −0.789608 + 1.83753i 5.33225 −2.75304 2.90186i 2.15693i −4.21039 + 9.79818i 0 7.50608 2.76746i 19.4329 3.96343 + 1.70313i
99.5 1.67727 1.08938i −3.98103 1.62649 3.65439i 1.88252i −6.67727 + 4.33687i 0 −1.25297 7.90127i 6.84860 2.05079 + 3.15750i
99.6 1.67727 + 1.08938i −3.98103 1.62649 + 3.65439i 1.88252i −6.67727 4.33687i 0 −1.25297 + 7.90127i 6.84860 2.05079 3.15750i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.3.g.j 6
4.b odd 2 1 1568.3.g.j 6
7.b odd 2 1 392.3.g.i 6
7.c even 3 2 56.3.k.d 12
7.d odd 6 2 392.3.k.l 12
8.b even 2 1 1568.3.g.j 6
8.d odd 2 1 inner 392.3.g.j 6
28.d even 2 1 1568.3.g.l 6
28.g odd 6 2 224.3.o.d 12
56.e even 2 1 392.3.g.i 6
56.h odd 2 1 1568.3.g.l 6
56.k odd 6 2 56.3.k.d 12
56.m even 6 2 392.3.k.l 12
56.p even 6 2 224.3.o.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.k.d 12 7.c even 3 2
56.3.k.d 12 56.k odd 6 2
224.3.o.d 12 28.g odd 6 2
224.3.o.d 12 56.p even 6 2
392.3.g.i 6 7.b odd 2 1
392.3.g.i 6 56.e even 2 1
392.3.g.j 6 1.a even 1 1 trivial
392.3.g.j 6 8.d odd 2 1 inner
392.3.k.l 12 7.d odd 6 2
392.3.k.l 12 56.m even 6 2
1568.3.g.j 6 4.b odd 2 1
1568.3.g.j 6 8.b even 2 1
1568.3.g.l 6 28.d even 2 1
1568.3.g.l 6 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 3 T_{3}^{2} - 19 T_{3} + 35 \) acting on \(S_{3}^{\mathrm{new}}(392, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T - 4 T^{3} + 32 T^{5} + 64 T^{6} \)
$3$ \( ( 1 - 3 T + 8 T^{2} - 19 T^{3} + 72 T^{4} - 243 T^{5} + 729 T^{6} )^{2} \)
$5$ \( 1 - 121 T^{2} + 6662 T^{4} - 212757 T^{6} + 4163750 T^{8} - 47265625 T^{10} + 244140625 T^{12} \)
$7$ 1
$11$ \( ( 1 + 15 T + 354 T^{2} + 3117 T^{3} + 42834 T^{4} + 219615 T^{5} + 1771561 T^{6} )^{2} \)
$13$ \( 1 - 86 T^{2} + 60895 T^{4} - 4902788 T^{6} + 1739222095 T^{8} - 70152842006 T^{10} + 23298085122481 T^{12} \)
$17$ \( ( 1 + 15 T + 890 T^{2} + 8635 T^{3} + 257210 T^{4} + 1252815 T^{5} + 24137569 T^{6} )^{2} \)
$19$ \( ( 1 + 39 T + 1370 T^{2} + 28697 T^{3} + 494570 T^{4} + 5082519 T^{5} + 47045881 T^{6} )^{2} \)
$23$ \( 1 - 693 T^{2} + 495830 T^{4} - 329634845 T^{6} + 138753563030 T^{8} - 54269512799733 T^{10} + 21914624432020321 T^{12} \)
$29$ \( 1 - 3662 T^{2} + 6471151 T^{4} - 6858243380 T^{6} + 4576922150431 T^{8} - 1831902364263182 T^{10} + 353814783205469041 T^{12} \)
$31$ \( 1 - 3561 T^{2} + 6841842 T^{4} - 8041403549 T^{6} + 6318584765682 T^{8} - 3037144984327401 T^{10} + 787662783788549761 T^{12} \)
$37$ \( 1 - 5785 T^{2} + 15576398 T^{4} - 26031024189 T^{6} + 29192677652078 T^{8} - 20319693640932985 T^{10} + 6582952005840035281 T^{12} \)
$41$ \( ( 1 + 58 T + 3139 T^{2} + 88960 T^{3} + 5276659 T^{4} + 163894138 T^{5} + 4750104241 T^{6} )^{2} \)
$43$ \( ( 1 + 50 T + 4047 T^{2} + 107900 T^{3} + 7482903 T^{4} + 170940050 T^{5} + 6321363049 T^{6} )^{2} \)
$47$ \( 1 - 3905 T^{2} + 8711938 T^{4} - 15970210469 T^{6} + 42511478331778 T^{8} - 92983074414176705 T^{10} + \)\(11\!\cdots\!41\)\( T^{12} \)
$53$ \( 1 - 10561 T^{2} + 51800990 T^{4} - 168592871253 T^{6} + 408734727376190 T^{8} - 657524590434383521 T^{10} + \)\(49\!\cdots\!41\)\( T^{12} \)
$59$ \( ( 1 - 55 T + 10392 T^{2} - 376351 T^{3} + 36174552 T^{4} - 666454855 T^{5} + 42180533641 T^{6} )^{2} \)
$61$ \( 1 - 9201 T^{2} + 27716990 T^{4} - 50012187845 T^{6} + 383765036538590 T^{8} - 1763898986887982481 T^{10} + \)\(26\!\cdots\!21\)\( T^{12} \)
$67$ \( ( 1 + 217 T + 29036 T^{2} + 2316921 T^{3} + 130342604 T^{4} + 4372793257 T^{5} + 90458382169 T^{6} )^{2} \)
$71$ \( 1 - 23062 T^{2} + 245626031 T^{4} - 1559141837940 T^{6} + 6241770345068111 T^{8} - 14892367937589740182 T^{10} + \)\(16\!\cdots\!41\)\( T^{12} \)
$73$ \( ( 1 + 51 T + 11766 T^{2} + 589863 T^{3} + 62701014 T^{4} + 1448310291 T^{5} + 151334226289 T^{6} )^{2} \)
$79$ \( 1 - 16693 T^{2} + 129404966 T^{4} - 759834693213 T^{6} + 5040333907502246 T^{8} - 25325097363770222773 T^{10} + \)\(59\!\cdots\!41\)\( T^{12} \)
$83$ \( ( 1 + 134 T + 22583 T^{2} + 1661172 T^{3} + 155574287 T^{4} + 6359415014 T^{5} + 326940373369 T^{6} )^{2} \)
$89$ \( ( 1 + 107 T + 10054 T^{2} + 786431 T^{3} + 79637734 T^{4} + 6713419787 T^{5} + 496981290961 T^{6} )^{2} \)
$97$ \( ( 1 + 38 T + 25191 T^{2} + 597344 T^{3} + 237022119 T^{4} + 3364112678 T^{5} + 832972004929 T^{6} )^{2} \)
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