Properties

Label 688.4.a.i
Level 688
Weight 4
Character orbit 688.a
Self dual yes
Analytic conductor 40.593
Analytic rank 0
Dimension 6
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 688.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.5933140839\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{3} + ( 7 + \beta_{1} - \beta_{4} ) q^{5} + ( -2 - \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{7} + ( 12 - \beta_{1} + 2 \beta_{2} + \beta_{4} + 3 \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{3} + ( 7 + \beta_{1} - \beta_{4} ) q^{5} + ( -2 - \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{7} + ( 12 - \beta_{1} + 2 \beta_{2} + \beta_{4} + 3 \beta_{5} ) q^{9} + ( 3 + 2 \beta_{1} - 3 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{11} + ( 7 - 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} ) q^{13} + ( 22 + 8 \beta_{1} - \beta_{2} + 2 \beta_{4} + 3 \beta_{5} ) q^{15} + ( 4 + 3 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{17} + ( 11 - 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} ) q^{19} + ( 4 + 16 \beta_{1} - 10 \beta_{2} + 7 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{21} + ( -24 - 5 \beta_{1} - 2 \beta_{2} - \beta_{3} - 11 \beta_{4} - 6 \beta_{5} ) q^{23} + ( 18 + 13 \beta_{1} - 7 \beta_{2} - 3 \beta_{3} - 16 \beta_{4} - 7 \beta_{5} ) q^{25} + ( -30 + 8 \beta_{1} - 8 \beta_{2} + 6 \beta_{3} - \beta_{4} - 12 \beta_{5} ) q^{27} + ( 81 - 5 \beta_{1} + 13 \beta_{2} - \beta_{3} - 4 \beta_{4} - 6 \beta_{5} ) q^{29} + ( -42 - \beta_{1} + 11 \beta_{2} - 2 \beta_{3} - 8 \beta_{4} - 14 \beta_{5} ) q^{31} + ( 104 + 32 \beta_{1} - 16 \beta_{2} + 21 \beta_{3} + 3 \beta_{4} + 14 \beta_{5} ) q^{33} + ( -32 + 16 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} + 25 \beta_{4} + 33 \beta_{5} ) q^{35} + ( 47 - 3 \beta_{1} + 7 \beta_{2} - 16 \beta_{3} + 17 \beta_{4} - 14 \beta_{5} ) q^{37} + ( -26 + 46 \beta_{1} - 10 \beta_{2} + 23 \beta_{3} - \beta_{4} - 12 \beta_{5} ) q^{39} + ( 78 - 17 \beta_{1} + 3 \beta_{2} - 17 \beta_{3} + 4 \beta_{4} - 23 \beta_{5} ) q^{41} + 43 q^{43} + ( 56 + 8 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} + 28 \beta_{4} + 24 \beta_{5} ) q^{45} + ( -79 - 5 \beta_{1} + 29 \beta_{2} - 7 \beta_{3} - 4 \beta_{4} - 13 \beta_{5} ) q^{47} + ( 43 - 6 \beta_{1} - 2 \beta_{2} - 13 \beta_{3} - 35 \beta_{4} - 3 \beta_{5} ) q^{49} + ( 203 + 3 \beta_{1} + 25 \beta_{2} + 13 \beta_{3} + 7 \beta_{4} + 8 \beta_{5} ) q^{51} + ( 67 - 30 \beta_{1} - 9 \beta_{2} + 19 \beta_{3} - 35 \beta_{4} + 2 \beta_{5} ) q^{53} + ( 290 + 40 \beta_{1} - 25 \beta_{2} - 14 \beta_{3} - 17 \beta_{4} + 10 \beta_{5} ) q^{55} + ( -156 + 18 \beta_{1} - \beta_{2} - 5 \beta_{3} + \beta_{4} - 26 \beta_{5} ) q^{57} + ( -62 - 30 \beta_{1} - 10 \beta_{2} - 11 \beta_{3} - 3 \beta_{4} - 11 \beta_{5} ) q^{59} + ( -208 - 4 \beta_{1} + 3 \beta_{2} + 20 \beta_{3} + 29 \beta_{4} + 4 \beta_{5} ) q^{61} + ( 340 - 66 \beta_{1} + 9 \beta_{2} - 52 \beta_{3} - 29 \beta_{4} + 2 \beta_{5} ) q^{63} + ( -12 + 6 \beta_{1} + 13 \beta_{2} - 4 \beta_{3} + 45 \beta_{4} + 36 \beta_{5} ) q^{65} + ( 113 - 78 \beta_{1} - 7 \beta_{3} + 6 \beta_{4} - 63 \beta_{5} ) q^{67} + ( -186 - 48 \beta_{1} - 22 \beta_{2} + \beta_{3} + 8 \beta_{4} ) q^{69} + ( 62 + 28 \beta_{1} - 17 \beta_{2} + 45 \beta_{3} + 54 \beta_{4} + 16 \beta_{5} ) q^{71} + ( 148 + 92 \beta_{1} + 28 \beta_{2} + 45 \beta_{3} - 43 \beta_{4} + 8 \beta_{5} ) q^{73} + ( 402 - 8 \beta_{1} - 17 \beta_{2} + 22 \beta_{4} + 76 \beta_{5} ) q^{75} + ( 434 + 24 \beta_{1} - 11 \beta_{2} + 4 \beta_{3} + 35 \beta_{4} + 90 \beta_{5} ) q^{77} + ( 297 + 53 \beta_{1} - 60 \beta_{2} + 12 \beta_{3} + 23 \beta_{4} + 37 \beta_{5} ) q^{79} + ( -61 - 140 \beta_{1} + 13 \beta_{2} - 66 \beta_{3} - 22 \beta_{4} - 39 \beta_{5} ) q^{81} + ( 79 - 102 \beta_{1} + 42 \beta_{2} + 26 \beta_{3} + 11 \beta_{4} + 48 \beta_{5} ) q^{83} + ( 14 + 26 \beta_{1} + 29 \beta_{2} + 8 \beta_{3} + 14 \beta_{4} + 6 \beta_{5} ) q^{85} + ( -33 + 95 \beta_{1} + 44 \beta_{2} + 46 \beta_{3} + 31 \beta_{4} - 45 \beta_{5} ) q^{87} + ( 576 - 62 \beta_{1} + 9 \beta_{2} - 10 \beta_{3} + 53 \beta_{4} - 36 \beta_{5} ) q^{89} + ( 594 - 108 \beta_{1} + 15 \beta_{2} - 48 \beta_{3} - 51 \beta_{4} ) q^{91} + ( 313 - 71 \beta_{1} + 71 \beta_{2} + 47 \beta_{3} + 43 \beta_{4} - 16 \beta_{5} ) q^{93} + ( 13 - 37 \beta_{1} + 15 \beta_{2} + 17 \beta_{4} - 16 \beta_{5} ) q^{95} + ( 4 - 29 \beta_{1} - 3 \beta_{2} + 54 \beta_{3} - 79 \beta_{4} - 76 \beta_{5} ) q^{97} + ( 213 - 64 \beta_{1} + 18 \beta_{2} - 114 \beta_{3} + 37 \beta_{4} + 40 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 7q^{3} + 43q^{5} - 8q^{7} + 81q^{9} + O(q^{10}) \) \( 6q - 7q^{3} + 43q^{5} - 8q^{7} + 81q^{9} + 28q^{11} + 56q^{13} + 124q^{15} + 19q^{17} + 75q^{19} - 18q^{21} - 131q^{23} + 105q^{25} - 238q^{27} + 515q^{29} - 237q^{31} + 540q^{33} - 198q^{35} + 269q^{37} - 290q^{39} + 471q^{41} + 258q^{43} + 334q^{45} - 415q^{47} + 350q^{49} + 1241q^{51} + 450q^{53} + 1732q^{55} - 1000q^{57} - 356q^{59} - 1328q^{61} + 2290q^{63} - 62q^{65} + 632q^{67} - 1130q^{69} + 144q^{71} + 864q^{73} + 2494q^{75} + 2660q^{77} + 1613q^{79} - 102q^{81} + 682q^{83} + 84q^{85} - 449q^{87} + 3378q^{89} + 3900q^{91} + 1879q^{93} + 79q^{95} - 55q^{97} + 1612q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 32 x^{4} - 16 x^{3} + 251 x^{2} + 276 x + 60\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} + 2 \nu^{4} - 24 \nu^{3} - 36 \nu^{2} + 103 \nu + 30 \)\()/8\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} + 3 \nu^{3} - 17 \nu^{2} - 51 \nu - 16 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{4} - 3 \nu^{3} + 17 \nu^{2} + 59 \nu + 16 \)\()/2\)
\(\beta_{4}\)\(=\)\( \nu^{3} + \nu^{2} - 17 \nu - 19 \)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{5} + 26 \nu^{3} + 6 \nu^{2} - 153 \nu - 74 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{1} + 22\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{5} + 2 \beta_{4} + 15 \beta_{3} + 17 \beta_{2} - 4 \beta_{1} + 32\)\()/4\)
\(\nu^{4}\)\(=\)\(10 \beta_{5} + 7 \beta_{4} + 10 \beta_{3} + 2 \beta_{2} + 20 \beta_{1} + 179\)
\(\nu^{5}\)\(=\)\((\)\(-56 \beta_{5} + 64 \beta_{4} + 249 \beta_{3} + 289 \beta_{2} - 80 \beta_{1} + 800\)\()/4\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.847740
4.15653
−3.17112
−0.299707
−4.15251
4.31455
0 −9.49653 0 2.98245 0 26.0720 0 63.1842 0
1.2 0 −7.20925 0 1.36370 0 −13.0131 0 24.9733 0
1.3 0 −2.46717 0 −7.54340 0 −4.58222 0 −20.9131 0
1.4 0 −1.43046 0 20.4116 0 −29.9522 0 −24.9538 0
1.5 0 6.49933 0 17.2665 0 23.3206 0 15.2413 0
1.6 0 7.10409 0 8.51910 0 −9.84502 0 23.4681 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 688.4.a.i 6
4.b odd 2 1 43.4.a.b 6
12.b even 2 1 387.4.a.h 6
20.d odd 2 1 1075.4.a.b 6
28.d even 2 1 2107.4.a.c 6
172.d even 2 1 1849.4.a.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.4.a.b 6 4.b odd 2 1
387.4.a.h 6 12.b even 2 1
688.4.a.i 6 1.a even 1 1 trivial
1075.4.a.b 6 20.d odd 2 1
1849.4.a.c 6 172.d even 2 1
2107.4.a.c 6 28.d even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(43\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 7 T_{3}^{5} - 97 T_{3}^{4} - 588 T_{3}^{3} + 2140 T_{3}^{2} + 11756 T_{3} + 11156 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(688))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 7 T + 65 T^{2} + 357 T^{3} + 2599 T^{4} + 15158 T^{5} + 96098 T^{6} + 409266 T^{7} + 1894671 T^{8} + 7026831 T^{9} + 34543665 T^{10} + 100442349 T^{11} + 387420489 T^{12} \)
$5$ \( 1 - 43 T + 1247 T^{2} - 26367 T^{3} + 452519 T^{4} - 6421366 T^{5} + 77975134 T^{6} - 802670750 T^{7} + 7070609375 T^{8} - 51498046875 T^{9} + 304443359375 T^{10} - 1312255859375 T^{11} + 3814697265625 T^{12} \)
$7$ \( 1 + 8 T + 886 T^{2} + 4080 T^{3} + 442155 T^{4} + 3221432 T^{5} + 186242508 T^{6} + 1104951176 T^{7} + 52019093595 T^{8} + 164642716560 T^{9} + 12263380460086 T^{10} + 37980492079544 T^{11} + 1628413597910449 T^{12} \)
$11$ \( 1 - 28 T + 3144 T^{2} - 41080 T^{3} + 3977536 T^{4} - 3139972 T^{5} + 4111332998 T^{6} - 4179302732 T^{7} + 7046447653696 T^{8} - 96864491146280 T^{9} + 9867218816410824 T^{10} - 116962948743638228 T^{11} + 5559917313492231481 T^{12} \)
$13$ \( 1 - 56 T + 8776 T^{2} - 530884 T^{3} + 37473880 T^{4} - 2150765192 T^{5} + 100690118558 T^{6} - 4725231126824 T^{7} + 180879261248920 T^{8} - 5629759045135732 T^{9} + 204463995034893256 T^{10} - 2866410008789082392 T^{11} + \)\(11\!\cdots\!29\)\( T^{12} \)
$17$ \( 1 - 19 T + 23143 T^{2} - 543393 T^{3} + 241535186 T^{4} - 5650313095 T^{5} + 1493450611759 T^{6} - 27759988235735 T^{7} + 5830072218002834 T^{8} - 64439821973334321 T^{9} + 13483626436208358823 T^{10} - 54386037978686500067 T^{11} + \)\(14\!\cdots\!09\)\( T^{12} \)
$19$ \( 1 - 75 T + 38259 T^{2} - 2365781 T^{3} + 629552155 T^{4} - 31151517862 T^{5} + 5681041321490 T^{6} - 213668261015458 T^{7} + 29617835767423555 T^{8} - 763408424339300399 T^{9} + 84679215488552253699 T^{10} - \)\(11\!\cdots\!25\)\( T^{11} + \)\(10\!\cdots\!41\)\( T^{12} \)
$23$ \( 1 + 131 T + 52195 T^{2} + 3677795 T^{3} + 974730114 T^{4} + 33210519163 T^{5} + 11858751245947 T^{6} + 404072386656221 T^{7} + 144295038961061346 T^{8} + 6624270252565314085 T^{9} + \)\(11\!\cdots\!95\)\( T^{10} + \)\(34\!\cdots\!17\)\( T^{11} + \)\(32\!\cdots\!69\)\( T^{12} \)
$29$ \( 1 - 515 T + 204583 T^{2} - 58068807 T^{3} + 13900558631 T^{4} - 2733986934494 T^{5} + 464005745217070 T^{6} - 66679207345374166 T^{7} + 8268376448646633551 T^{8} - \)\(84\!\cdots\!83\)\( T^{9} + \)\(72\!\cdots\!03\)\( T^{10} - \)\(44\!\cdots\!35\)\( T^{11} + \)\(21\!\cdots\!61\)\( T^{12} \)
$31$ \( 1 + 237 T + 125373 T^{2} + 21016589 T^{3} + 7321362670 T^{4} + 1031063540341 T^{5} + 275109610824401 T^{6} + 30716413930298731 T^{7} + 6497736319560988270 T^{8} + \)\(55\!\cdots\!19\)\( T^{9} + \)\(98\!\cdots\!53\)\( T^{10} + \)\(55\!\cdots\!87\)\( T^{11} + \)\(69\!\cdots\!41\)\( T^{12} \)
$37$ \( 1 - 269 T + 126311 T^{2} - 30748693 T^{3} + 8177560635 T^{4} - 1302357216602 T^{5} + 425119347961066 T^{6} - 65968300092541106 T^{7} + 20981383282418309715 T^{8} - \)\(39\!\cdots\!61\)\( T^{9} + \)\(83\!\cdots\!91\)\( T^{10} - \)\(89\!\cdots\!17\)\( T^{11} + \)\(16\!\cdots\!29\)\( T^{12} \)
$41$ \( 1 - 471 T + 349763 T^{2} - 112600045 T^{3} + 50459129866 T^{4} - 12922113800443 T^{5} + 4372223136871043 T^{6} - 890605005240332003 T^{7} + \)\(23\!\cdots\!06\)\( T^{8} - \)\(36\!\cdots\!45\)\( T^{9} + \)\(78\!\cdots\!03\)\( T^{10} - \)\(73\!\cdots\!71\)\( T^{11} + \)\(10\!\cdots\!21\)\( T^{12} \)
$43$ \( ( 1 - 43 T )^{6} \)
$47$ \( 1 + 415 T + 421631 T^{2} + 116866317 T^{3} + 77411983523 T^{4} + 16052264514750 T^{5} + 9154052369892234 T^{6} + 1666594258714889250 T^{7} + \)\(83\!\cdots\!67\)\( T^{8} + \)\(13\!\cdots\!39\)\( T^{9} + \)\(48\!\cdots\!71\)\( T^{10} + \)\(50\!\cdots\!45\)\( T^{11} + \)\(12\!\cdots\!89\)\( T^{12} \)
$53$ \( 1 - 450 T + 321704 T^{2} - 149982378 T^{3} + 77929548632 T^{4} - 28654270442506 T^{5} + 12746558079363422 T^{6} - 4265961820668965762 T^{7} + \)\(17\!\cdots\!28\)\( T^{8} - \)\(49\!\cdots\!74\)\( T^{9} + \)\(15\!\cdots\!64\)\( T^{10} - \)\(32\!\cdots\!50\)\( T^{11} + \)\(10\!\cdots\!89\)\( T^{12} \)
$59$ \( 1 + 356 T + 1153950 T^{2} + 347607228 T^{3} + 570632518215 T^{4} + 139273869185096 T^{5} + 154351054402988548 T^{6} + 28603927979365831384 T^{7} + \)\(24\!\cdots\!15\)\( T^{8} + \)\(30\!\cdots\!92\)\( T^{9} + \)\(20\!\cdots\!50\)\( T^{10} + \)\(13\!\cdots\!44\)\( T^{11} + \)\(75\!\cdots\!21\)\( T^{12} \)
$61$ \( 1 + 1328 T + 1795994 T^{2} + 1454429624 T^{3} + 1136828745699 T^{4} + 649067768079368 T^{5} + 354455789917301172 T^{6} + \)\(14\!\cdots\!08\)\( T^{7} + \)\(58\!\cdots\!39\)\( T^{8} + \)\(17\!\cdots\!84\)\( T^{9} + \)\(47\!\cdots\!74\)\( T^{10} + \)\(80\!\cdots\!28\)\( T^{11} + \)\(13\!\cdots\!81\)\( T^{12} \)
$67$ \( 1 - 632 T + 927628 T^{2} - 253029932 T^{3} + 179674044568 T^{4} + 62778009874096 T^{5} - 5212390617577006 T^{6} + 18881302583762735248 T^{7} + \)\(16\!\cdots\!92\)\( T^{8} - \)\(68\!\cdots\!04\)\( T^{9} + \)\(75\!\cdots\!08\)\( T^{10} - \)\(15\!\cdots\!76\)\( T^{11} + \)\(74\!\cdots\!09\)\( T^{12} \)
$71$ \( 1 - 144 T + 863230 T^{2} - 72744496 T^{3} + 475313592223 T^{4} - 62333254020128 T^{5} + 209643801201434276 T^{6} - 22309757279598032608 T^{7} + \)\(60\!\cdots\!83\)\( T^{8} - \)\(33\!\cdots\!76\)\( T^{9} + \)\(14\!\cdots\!30\)\( T^{10} - \)\(84\!\cdots\!44\)\( T^{11} + \)\(21\!\cdots\!61\)\( T^{12} \)
$73$ \( 1 - 864 T + 1080658 T^{2} - 439706488 T^{3} + 458263245867 T^{4} - 209583356925592 T^{5} + 222637509631027524 T^{6} - 81531488761123023064 T^{7} + \)\(69\!\cdots\!63\)\( T^{8} - \)\(25\!\cdots\!44\)\( T^{9} + \)\(24\!\cdots\!18\)\( T^{10} - \)\(76\!\cdots\!48\)\( T^{11} + \)\(34\!\cdots\!69\)\( T^{12} \)
$79$ \( 1 - 1613 T + 2731081 T^{2} - 3130876751 T^{3} + 3268965374139 T^{4} - 2799662350208018 T^{5} + 2111366737833161318 T^{6} - \)\(13\!\cdots\!02\)\( T^{7} + \)\(79\!\cdots\!19\)\( T^{8} - \)\(37\!\cdots\!69\)\( T^{9} + \)\(16\!\cdots\!21\)\( T^{10} - \)\(46\!\cdots\!87\)\( T^{11} + \)\(14\!\cdots\!61\)\( T^{12} \)
$83$ \( 1 - 682 T + 1120004 T^{2} - 1783287782 T^{3} + 1291656570448 T^{4} - 1149121488958882 T^{5} + 1227368803599345682 T^{6} - \)\(65\!\cdots\!34\)\( T^{7} + \)\(42\!\cdots\!12\)\( T^{8} - \)\(33\!\cdots\!46\)\( T^{9} + \)\(11\!\cdots\!44\)\( T^{10} - \)\(41\!\cdots\!74\)\( T^{11} + \)\(34\!\cdots\!09\)\( T^{12} \)
$89$ \( 1 - 3378 T + 7851850 T^{2} - 13055260850 T^{3} + 17370332054203 T^{4} - 19017874978895668 T^{5} + 17369747195795800052 T^{6} - \)\(13\!\cdots\!92\)\( T^{7} + \)\(86\!\cdots\!83\)\( T^{8} - \)\(45\!\cdots\!50\)\( T^{9} + \)\(19\!\cdots\!50\)\( T^{10} - \)\(58\!\cdots\!22\)\( T^{11} + \)\(12\!\cdots\!81\)\( T^{12} \)
$97$ \( 1 + 55 T + 2496871 T^{2} - 940317011 T^{3} + 3317883854770 T^{4} - 1706175158728421 T^{5} + 3579842987764450575 T^{6} - \)\(15\!\cdots\!33\)\( T^{7} + \)\(27\!\cdots\!30\)\( T^{8} - \)\(71\!\cdots\!87\)\( T^{9} + \)\(17\!\cdots\!11\)\( T^{10} + \)\(34\!\cdots\!15\)\( T^{11} + \)\(57\!\cdots\!89\)\( T^{12} \)
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