Properties

Label 320.6.f.c
Level 320
Weight 6
Character orbit 320.f
Analytic conductor 51.323
Analytic rank 0
Dimension 16
CM no
Inner twists 8

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Newspace parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 320.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(51.3228223402\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{42}\cdot 5^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} -\beta_{10} q^{5} + ( -\beta_{8} + \beta_{12} ) q^{7} + ( 119 + \beta_{1} ) q^{9} +O(q^{10})\) \( q + \beta_{7} q^{3} -\beta_{10} q^{5} + ( -\beta_{8} + \beta_{12} ) q^{7} + ( 119 + \beta_{1} ) q^{9} + ( 16 \beta_{3} - \beta_{4} ) q^{11} + ( 4 \beta_{9} - 4 \beta_{10} + 3 \beta_{13} ) q^{13} + ( -\beta_{5} - 3 \beta_{8} - 5 \beta_{12} ) q^{15} + ( 5 \beta_{11} + \beta_{14} ) q^{17} + ( 17 \beta_{3} - 7 \beta_{4} ) q^{19} + ( \beta_{2} + \beta_{9} + \beta_{10} ) q^{21} + ( 23 \beta_{8} + 36 \beta_{12} ) q^{23} + ( 55 + 10 \beta_{1} - 8 \beta_{11} - \beta_{14} ) q^{25} + ( 53 \beta_{7} - \beta_{15} ) q^{27} + ( 3 \beta_{2} + 8 \beta_{9} + 8 \beta_{10} ) q^{29} + ( -6 \beta_{5} + 5 \beta_{6} ) q^{31} + ( 11 \beta_{11} - \beta_{14} ) q^{33} + ( -440 \beta_{3} - 15 \beta_{4} + 10 \beta_{7} + \beta_{15} ) q^{35} + ( 37 \beta_{9} - 37 \beta_{10} + 50 \beta_{13} ) q^{37} + ( -8 \beta_{5} + 15 \beta_{6} ) q^{39} + ( -6272 - 47 \beta_{1} ) q^{41} + ( -260 \beta_{7} + \beta_{15} ) q^{43} + ( 3 \beta_{2} + 220 \beta_{9} - 131 \beta_{10} + 43 \beta_{13} ) q^{45} + ( -219 \beta_{8} + 207 \beta_{12} ) q^{47} + ( 8017 - 5 \beta_{1} ) q^{49} + ( 1045 \beta_{3} + 226 \beta_{4} ) q^{51} + ( 150 \beta_{9} - 150 \beta_{10} - 101 \beta_{13} ) q^{53} + ( 3 \beta_{5} + 31 \beta_{6} + 8 \beta_{8} - 47 \beta_{12} ) q^{55} + ( -113 \beta_{11} - 7 \beta_{14} ) q^{57} + ( 393 \beta_{3} - 115 \beta_{4} ) q^{59} + ( 335 \beta_{9} + 335 \beta_{10} ) q^{61} + ( -321 \beta_{8} - 68 \beta_{12} ) q^{63} + ( 9360 + 70 \beta_{1} - 161 \beta_{11} + 8 \beta_{14} ) q^{65} + ( 162 \beta_{7} + 9 \beta_{15} ) q^{67} + ( -23 \beta_{2} + 567 \beta_{9} + 567 \beta_{10} ) q^{69} + ( 42 \beta_{5} + 150 \beta_{6} ) q^{71} + ( 561 \beta_{11} + 9 \beta_{14} ) q^{73} + ( -1600 \beta_{3} - 250 \beta_{4} + 1825 \beta_{7} - 10 \beta_{15} ) q^{75} + ( 31 \beta_{9} - 31 \beta_{10} - 105 \beta_{13} ) q^{77} + ( 54 \beta_{5} + 175 \beta_{6} ) q^{79} + ( -9661 - 5 \beta_{1} ) q^{81} + ( -2846 \beta_{7} - 11 \beta_{15} ) q^{83} + ( -24 \beta_{2} + 865 \beta_{9} + 187 \beta_{10} - 469 \beta_{13} ) q^{85} + ( -1662 \beta_{8} + 575 \beta_{12} ) q^{87} + ( -28706 + 144 \beta_{1} ) q^{89} + ( -835 \beta_{3} - 360 \beta_{4} ) q^{91} + ( 1902 \beta_{9} - 1902 \beta_{10} + 388 \beta_{13} ) q^{93} + ( 21 \beta_{5} + 122 \beta_{6} - 324 \beta_{8} - 139 \beta_{12} ) q^{95} + ( -439 \beta_{11} + 7 \beta_{14} ) q^{97} + ( -1973 \beta_{3} + 145 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 1904q^{9} + O(q^{10}) \) \( 16q + 1904q^{9} + 880q^{25} - 100352q^{41} + 128272q^{49} + 149760q^{65} - 154576q^{81} - 459296q^{89} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 39122 x^{12} + 391243971 x^{8} + 75462898750 x^{4} + 18538406640625\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{12} + 29029 \nu^{8} + 5616875 \nu^{4} - 2050595324964 \)\()/ 8101949622 \)
\(\beta_{2}\)\(=\)\((\)\( 14561 \nu^{12} - 630433567 \nu^{8} + 6876956439231 \nu^{4} + 648765163646250 \)\()/ 1687906171250 \)
\(\beta_{3}\)\(=\)\((\)\( -42872 \nu^{14} + 1660015884 \nu^{10} - 16273459572212 \nu^{6} - 6716350492827500 \nu^{2} \)\()/ 11180755397828125 \)
\(\beta_{4}\)\(=\)\((\)\( 21443033 \nu^{14} - 838625666026 \nu^{10} + 8292212087562418 \nu^{6} + 3422112718390069375 \nu^{2} \)\()/ 348839568412237500 \)
\(\beta_{5}\)\(=\)\((\)\( 142076123 \nu^{14} - 5626408460256 \nu^{10} + 58010716758878808 \nu^{6} - 1551252332852933125 \nu^{2} \)\()/ 1046518705236712500 \)
\(\beta_{6}\)\(=\)\((\)\( 409552846 \nu^{14} - 16108363381212 \nu^{10} + 163621302766477716 \nu^{6} - 4387553578919678750 \nu^{2} \)\()/ 1308148381545890625 \)
\(\beta_{7}\)\(=\)\((\)\(368758812 \nu^{15} - 2203274425 \nu^{13} - 14361898838689 \nu^{11} + 81848061919850 \nu^{9} + 142396967402520577 \nu^{7} - 772907074481616050 \nu^{5} + 39045763967463741875 \nu^{3} - 608626187272887734375 \nu\)\()/ \)\(28\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(91205484 \nu^{15} - 2203274425 \nu^{13} - 3614956005673 \nu^{11} + 81848061919850 \nu^{9} + 37042590132020089 \nu^{7} - 772907074481616050 \nu^{5} - 4435889123101493125 \nu^{3} - 29552503708573484375 \nu\)\()/ 57907368356431425000 \)
\(\beta_{9}\)\(=\)\((\)\(8401639129 \nu^{15} + 168863684675 \nu^{13} + 3010601501750 \nu^{12} - 330696613108488 \nu^{11} - 6798200017902225 \nu^{9} - 117090837330932250 \nu^{8} + 3382491912594489384 \nu^{7} + 70709994820713844425 \nu^{5} + 1170844192896108074250 \nu^{4} - 405098723407935974375 \nu^{3} + 2730345157724647156250 \nu + 110180743203375915312500\)\()/ \)\(43\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-8401639129 \nu^{15} - 168863684675 \nu^{13} + 3010601501750 \nu^{12} + 330696613108488 \nu^{11} + 6798200017902225 \nu^{9} - 117090837330932250 \nu^{8} - 3382491912594489384 \nu^{7} - 70709994820713844425 \nu^{5} + 1170844192896108074250 \nu^{4} + 405098723407935974375 \nu^{3} - 2730345157724647156250 \nu + 110180743203375915312500\)\()/ \)\(43\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(368758812 \nu^{15} + 2203274425 \nu^{13} - 14361898838689 \nu^{11} - 81848061919850 \nu^{9} + 142396967402520577 \nu^{7} + 772907074481616050 \nu^{5} + 39045763967463741875 \nu^{3} + 608626187272887734375 \nu\)\()/ 72384210445539281250 \)
\(\beta_{12}\)\(=\)\((\)\(23643689558 \nu^{15} - 502972951225 \nu^{13} - 932514926642451 \nu^{11} + 19735004679793200 \nu^{9} + 9543178085090485443 \nu^{7} - 199388020227548892600 \nu^{5} - 1142889131048483933125 \nu^{3} - 7677128093592305640625 \nu\)\()/ \)\(21\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-23643689558 \nu^{15} - 502972951225 \nu^{13} + 932514926642451 \nu^{11} + 19735004679793200 \nu^{9} - 9543178085090485443 \nu^{7} - 199388020227548892600 \nu^{5} + 1142889131048483933125 \nu^{3} - 7677128093592305640625 \nu\)\()/ \)\(10\!\cdots\!50\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-39315884039 \nu^{15} - 221984419225 \nu^{13} + 1535966810987508 \nu^{11} + 8626469159338575 \nu^{9} - 15282593232744638244 \nu^{7} - 82976143759118919975 \nu^{5} - 4190510036689937586875 \nu^{3} - 65323169452491358281250 \nu\)\()/ \)\(14\!\cdots\!00\)\( \)
\(\beta_{15}\)\(=\)\((\)\(157632294968 \nu^{15} - 890140951325 \nu^{13} - 6158229142788721 \nu^{11} + 34587724699274150 \nu^{9} + 61272769898381073553 \nu^{7} - 332677482110957295950 \nu^{5} + 16801085910727214089375 \nu^{3} - 261901303997238320859375 \nu\)\()/ \)\(28\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{13} + 5 \beta_{11} - 4 \beta_{10} + 4 \beta_{9} + 4 \beta_{8} - 20 \beta_{7}\)\()/80\)
\(\nu^{2}\)\(=\)\((\)\(-5 \beta_{6} + 8 \beta_{5} - 125 \beta_{3}\)\()/80\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{15} - 10 \beta_{14} + 131 \beta_{13} + 75 \beta_{12} - 485 \beta_{11} - 674 \beta_{10} + 674 \beta_{9} - 674 \beta_{8} - 1945 \beta_{7}\)\()/80\)
\(\nu^{4}\)\(=\)\((\)\(45 \beta_{10} + 45 \beta_{9} - 5 \beta_{2} + 156 \beta_{1} + 39122\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(905 \beta_{15} + 1810 \beta_{14} + 11186 \beta_{13} - 9825 \beta_{12} + 96910 \beta_{11} - 64394 \beta_{10} + 64394 \beta_{9} + 64394 \beta_{8} - 388545 \beta_{7}\)\()/80\)
\(\nu^{6}\)\(=\)\((\)\(-34590 \beta_{6} + 78744 \beta_{5} - 117000 \beta_{4} - 1911875 \beta_{3}\)\()/40\)
\(\nu^{7}\)\(=\)\((\)\(69180 \beta_{15} - 138360 \beta_{14} + 2677141 \beta_{13} + 2207700 \beta_{12} - 7418335 \beta_{11} - 15123964 \beta_{10} + 15123964 \beta_{9} - 15123964 \beta_{8} - 29742520 \beta_{7}\)\()/80\)
\(\nu^{8}\)\(=\)\((\)\(1507365 \beta_{10} + 1507365 \beta_{9} - 199985 \beta_{2} + 2954016 \beta_{1} + 748042942\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(19769705 \beta_{15} + 39539410 \beta_{14} + 160738321 \beta_{13} - 131931825 \beta_{12} + 2121427385 \beta_{11} - 906816934 \beta_{10} + 906816934 \beta_{9} + 906816934 \beta_{8} - 8505479245 \beta_{7}\)\()/80\)
\(\nu^{10}\)\(=\)\((\)\(-1304401855 \beta_{6} + 2943997768 \beta_{5} - 7628790000 \beta_{4} - 124565440625 \beta_{3}\)\()/80\)
\(\nu^{11}\)\(=\)\((\)\(886399855 \beta_{15} - 1772799710 \beta_{14} + 57541845626 \beta_{13} + 47259927825 \beta_{12} - 95129003810 \beta_{11} - 324687238154 \beta_{10} + 324687238154 \beta_{9} - 324687238154 \beta_{8} - 381402415095 \beta_{7}\)\()/80\)
\(\nu^{12}\)\(=\)\(11002514490 \beta_{10} + 11002514490 \beta_{9} - 1458362235 \beta_{2} + 13555141119 \beta_{1} + 3433075161803\)
\(\nu^{13}\)\(=\)\((\)\(416939045880 \beta_{15} + 833878091760 \beta_{14} + 1770889401431 \beta_{13} - 1454449858200 \beta_{12} + 44744554623235 \beta_{11} - 9992457322124 \beta_{10} + 9992457322124 \beta_{9} + 9992457322124 \beta_{8} - 179395157538820 \beta_{7}\)\()/80\)
\(\nu^{14}\)\(=\)\((\)\(-23463988448155 \beta_{6} + 52959452609848 \beta_{5} - 206566594416000 \beta_{4} - 3373061336441875 \beta_{3}\)\()/80\)
\(\nu^{15}\)\(=\)\((\)\(7278833579155 \beta_{15} - 14557667158310 \beta_{14} + 1193404460483461 \beta_{13} + 980163927077325 \beta_{12} - 781139728334035 \beta_{11} - 6733945696088494 \beta_{10} + 6733945696088494 \beta_{9} - 6733945696088494 \beta_{8} - 3131837746915295 \beta_{7}\)\()/80\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
0.526271 11.8760i
0.526271 + 11.8760i
11.8760 + 0.526271i
11.8760 0.526271i
3.34569 + 1.86808i
3.34569 1.86808i
1.86808 3.34569i
1.86808 + 3.34569i
−1.86808 + 3.34569i
−1.86808 3.34569i
−3.34569 1.86808i
−3.34569 + 1.86808i
−11.8760 0.526271i
−11.8760 + 0.526271i
−0.526271 + 11.8760i
−0.526271 11.8760i
0 −24.8046 0 −53.4447 16.3911i 0 100.281i 0 372.267 0
289.2 0 −24.8046 0 −53.4447 + 16.3911i 0 100.281i 0 372.267 0
289.3 0 −24.8046 0 53.4447 16.3911i 0 100.281i 0 372.267 0
289.4 0 −24.8046 0 53.4447 + 16.3911i 0 100.281i 0 372.267 0
289.5 0 −10.4275 0 −17.9907 52.9276i 0 86.7391i 0 −134.267 0
289.6 0 −10.4275 0 −17.9907 + 52.9276i 0 86.7391i 0 −134.267 0
289.7 0 −10.4275 0 17.9907 52.9276i 0 86.7391i 0 −134.267 0
289.8 0 −10.4275 0 17.9907 + 52.9276i 0 86.7391i 0 −134.267 0
289.9 0 10.4275 0 −17.9907 52.9276i 0 86.7391i 0 −134.267 0
289.10 0 10.4275 0 −17.9907 + 52.9276i 0 86.7391i 0 −134.267 0
289.11 0 10.4275 0 17.9907 52.9276i 0 86.7391i 0 −134.267 0
289.12 0 10.4275 0 17.9907 + 52.9276i 0 86.7391i 0 −134.267 0
289.13 0 24.8046 0 −53.4447 16.3911i 0 100.281i 0 372.267 0
289.14 0 24.8046 0 −53.4447 + 16.3911i 0 100.281i 0 372.267 0
289.15 0 24.8046 0 53.4447 16.3911i 0 100.281i 0 372.267 0
289.16 0 24.8046 0 53.4447 + 16.3911i 0 100.281i 0 372.267 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
20.d odd 2 1 inner
40.e odd 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.6.f.c 16
4.b odd 2 1 inner 320.6.f.c 16
5.b even 2 1 inner 320.6.f.c 16
8.b even 2 1 inner 320.6.f.c 16
8.d odd 2 1 inner 320.6.f.c 16
20.d odd 2 1 inner 320.6.f.c 16
40.e odd 2 1 inner 320.6.f.c 16
40.f even 2 1 inner 320.6.f.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.6.f.c 16 1.a even 1 1 trivial
320.6.f.c 16 4.b odd 2 1 inner
320.6.f.c 16 5.b even 2 1 inner
320.6.f.c 16 8.b even 2 1 inner
320.6.f.c 16 8.d odd 2 1 inner
320.6.f.c 16 20.d odd 2 1 inner
320.6.f.c 16 40.e odd 2 1 inner
320.6.f.c 16 40.f even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 724 T_{3}^{2} + 66900 \) acting on \(S_{6}^{\mathrm{new}}(320, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + 248 T^{2} + 69330 T^{4} + 14644152 T^{6} + 3486784401 T^{8} )^{4} \)
$5$ \( ( 1 - 220 T^{2} - 6114250 T^{4} - 2148437500 T^{6} + 95367431640625 T^{8} )^{2} \)
$7$ \( ( 1 - 49648 T^{2} + 1179577874 T^{4} - 14024331162352 T^{6} + 79792266297612001 T^{8} )^{4} \)
$11$ \( ( 1 - 577284 T^{2} + 134071408310 T^{4} - 14973260223363684 T^{6} + \)\(67\!\cdots\!01\)\( T^{8} )^{4} \)
$13$ \( ( 1 + 1011892 T^{2} + 531287817014 T^{4} + 139497905034068308 T^{6} + \)\(19\!\cdots\!01\)\( T^{8} )^{4} \)
$17$ \( ( 1 + 554828 T^{2} + 3105096833510 T^{4} + 1118529863798317772 T^{6} + \)\(40\!\cdots\!01\)\( T^{8} )^{4} \)
$19$ \( ( 1 - 8279076 T^{2} + 29313371734870 T^{4} - 50759563509370071876 T^{6} + \)\(37\!\cdots\!01\)\( T^{8} )^{4} \)
$23$ \( ( 1 - 265392 T^{2} + 72532989580114 T^{4} - 10994264664012735408 T^{6} + \)\(17\!\cdots\!01\)\( T^{8} )^{4} \)
$29$ \( ( 1 - 18262996 T^{2} + 467877918484406 T^{4} - \)\(76\!\cdots\!96\)\( T^{6} + \)\(17\!\cdots\!01\)\( T^{8} )^{4} \)
$31$ \( ( 1 + 39407004 T^{2} + 689994929185606 T^{4} + \)\(32\!\cdots\!04\)\( T^{6} + \)\(67\!\cdots\!01\)\( T^{8} )^{4} \)
$37$ \( ( 1 + 171738148 T^{2} + 16366808202619574 T^{4} + \)\(82\!\cdots\!52\)\( T^{6} + \)\(23\!\cdots\!01\)\( T^{8} )^{4} \)
$41$ \( ( 1 + 12544 T + 129356290 T^{2} + 1453300185344 T^{3} + 13422659310152401 T^{4} )^{8} \)
$43$ \( ( 1 + 515310072 T^{2} + 108506205300479794 T^{4} + \)\(11\!\cdots\!28\)\( T^{6} + \)\(46\!\cdots\!01\)\( T^{8} )^{4} \)
$47$ \( ( 1 - 112031408 T^{2} + 100831510268064114 T^{4} - \)\(58\!\cdots\!92\)\( T^{6} + \)\(27\!\cdots\!01\)\( T^{8} )^{4} \)
$53$ \( ( 1 + 823212052 T^{2} + 458582260389137174 T^{4} + \)\(14\!\cdots\!48\)\( T^{6} + \)\(30\!\cdots\!01\)\( T^{8} )^{4} \)
$59$ \( ( 1 - 2409567364 T^{2} + 2462718757251169526 T^{4} - \)\(12\!\cdots\!64\)\( T^{6} + \)\(26\!\cdots\!01\)\( T^{8} )^{4} \)
$61$ \( ( 1 - 2000262204 T^{2} + 2103804668157499606 T^{4} - \)\(14\!\cdots\!04\)\( T^{6} + \)\(50\!\cdots\!01\)\( T^{8} )^{4} \)
$67$ \( ( 1 + 3461519512 T^{2} + 6000732802449327570 T^{4} + \)\(63\!\cdots\!88\)\( T^{6} + \)\(33\!\cdots\!01\)\( T^{8} )^{4} \)
$71$ \( ( 1 + 823411004 T^{2} + 3069578968927394406 T^{4} + \)\(26\!\cdots\!04\)\( T^{6} + \)\(10\!\cdots\!01\)\( T^{8} )^{4} \)
$73$ \( ( 1 - 4148991508 T^{2} + 8650887997979942790 T^{4} - \)\(17\!\cdots\!92\)\( T^{6} + \)\(18\!\cdots\!01\)\( T^{8} )^{4} \)
$79$ \( ( 1 + 2847963996 T^{2} + 15373907451301926406 T^{4} + \)\(26\!\cdots\!96\)\( T^{6} + \)\(89\!\cdots\!01\)\( T^{8} )^{4} \)
$83$ \( ( 1 + 7032005368 T^{2} + 39846447958779448850 T^{4} + \)\(10\!\cdots\!32\)\( T^{6} + \)\(24\!\cdots\!01\)\( T^{8} )^{4} \)
$89$ \( ( 1 + 57412 T + 10662063350 T^{2} + 320592021085988 T^{3} + 31181719929966183601 T^{4} )^{8} \)
$97$ \( ( 1 - 31837059828 T^{2} + \)\(40\!\cdots\!94\)\( T^{4} - \)\(23\!\cdots\!72\)\( T^{6} + \)\(54\!\cdots\!01\)\( T^{8} )^{4} \)
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