Properties

Label 20.5
Level 20
Weight 5
Dimension 22
Nonzero newspaces 3
Newform subspaces 5
Sturm bound 120
Trace bound 1

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Defining parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 5 \)
Nonzero newspaces: \( 3 \)
Newform subspaces: \( 5 \)
Sturm bound: \(120\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(\Gamma_1(20))\).

Total New Old
Modular forms 58 26 32
Cusp forms 38 22 16
Eisenstein series 20 4 16

Trace form

\( 22q + 6q^{2} - 10q^{3} - 36q^{4} + 4q^{5} + 32q^{6} + 110q^{7} + 216q^{8} - 170q^{9} + O(q^{10}) \) \( 22q + 6q^{2} - 10q^{3} - 36q^{4} + 4q^{5} + 32q^{6} + 110q^{7} + 216q^{8} - 170q^{9} - 210q^{10} - 300q^{11} - 200q^{12} - 8q^{13} - 184q^{14} + 542q^{15} - 592q^{16} + 912q^{17} + 286q^{18} + 100q^{20} - 2612q^{21} + 800q^{22} - 810q^{23} + 3216q^{24} + 2066q^{25} + 324q^{26} + 2120q^{27} + 40q^{28} + 1508q^{29} - 920q^{30} - 836q^{31} - 2304q^{32} - 2780q^{33} - 4308q^{34} - 2562q^{35} - 9220q^{36} - 6388q^{37} - 3360q^{38} + 4200q^{40} + 10280q^{41} + 12120q^{42} + 3270q^{43} + 9840q^{44} + 3344q^{45} + 14792q^{46} - 2250q^{47} + 8640q^{48} - 5130q^{49} - 11730q^{50} + 1948q^{51} - 12488q^{52} + 4572q^{53} - 26768q^{54} - 6780q^{55} - 25168q^{56} - 10000q^{57} - 7428q^{58} + 11320q^{60} + 22184q^{61} + 25680q^{62} + 12950q^{63} + 33984q^{64} - 3852q^{65} + 38000q^{66} - 4810q^{67} + 2712q^{68} - 18248q^{69} - 10800q^{70} - 5988q^{71} - 36264q^{72} - 28508q^{73} - 42108q^{74} - 11282q^{75} - 36000q^{76} + 1380q^{77} - 14480q^{78} + 19120q^{80} - 1062q^{81} + 27412q^{82} + 19950q^{83} + 80672q^{84} + 36536q^{85} + 46352q^{86} + 24800q^{87} + 18080q^{88} + 60068q^{89} - 29130q^{90} - 11124q^{91} - 52680q^{92} - 22700q^{93} - 67704q^{94} - 15576q^{95} - 78208q^{96} - 7548q^{97} - 21474q^{98} + O(q^{100}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(\Gamma_1(20))\)

We only show spaces with odd parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
20.5.b \(\chi_{20}(11, \cdot)\) 20.5.b.a 8 1
20.5.d \(\chi_{20}(19, \cdot)\) 20.5.d.a 1 1
20.5.d.b 1
20.5.d.c 8
20.5.f \(\chi_{20}(13, \cdot)\) 20.5.f.a 4 2

Decomposition of \(S_{5}^{\mathrm{old}}(\Gamma_1(20))\) into lower level spaces

\( S_{5}^{\mathrm{old}}(\Gamma_1(20)) \cong \) \(S_{5}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 6 T + 28 T^{2} - 168 T^{3} + 784 T^{4} - 2688 T^{5} + 7168 T^{6} - 24576 T^{7} + 65536 T^{8} \))(\( 1 + 4 T \))(\( 1 - 4 T \))(\( 1 + 24 T^{2} + 496 T^{4} + 6144 T^{6} + 65536 T^{8} \))(\( \))
$3$ (\( 1 - 160 T^{2} + 19676 T^{4} - 2014560 T^{6} + 180027846 T^{8} - 13217528160 T^{10} + 846987282396 T^{12} - 45188725836960 T^{14} + 1853020188851841 T^{16} \))(\( 1 - 2 T + 81 T^{2} \))(\( 1 + 2 T + 81 T^{2} \))(\( ( 1 + 84 T^{2} + 9126 T^{4} + 551124 T^{6} + 43046721 T^{8} )^{2} \))(\( 1 + 10 T + 50 T^{2} - 270 T^{3} - 10206 T^{4} - 21870 T^{5} + 328050 T^{6} + 5314410 T^{7} + 43046721 T^{8} \))
$5$ (\( ( 1 - 125 T^{2} )^{4} \))(\( 1 - 25 T \))(\( 1 - 25 T \))(\( ( 1 + 20 T + 710 T^{2} + 12500 T^{3} + 390625 T^{4} )^{2} \))(\( 1 + 6 T - 910 T^{2} + 3750 T^{3} + 390625 T^{4} \))
$7$ (\( 1 - 6720 T^{2} + 31323036 T^{4} - 102209887680 T^{6} + 281929138377926 T^{8} - 589219662707551680 T^{10} + \)\(10\!\cdots\!36\)\( T^{12} - \)\(12\!\cdots\!20\)\( T^{14} + \)\(11\!\cdots\!01\)\( T^{16} \))(\( 1 - 82 T + 2401 T^{2} \))(\( 1 + 82 T + 2401 T^{2} \))(\( ( 1 + 6804 T^{2} + 23087206 T^{4} + 39223706004 T^{6} + 33232930569601 T^{8} )^{2} \))(\( 1 - 110 T + 6050 T^{2} - 311190 T^{3} + 15823298 T^{4} - 747167190 T^{5} + 34877046050 T^{6} - 1522541592110 T^{7} + 33232930569601 T^{8} \))
$11$ (\( 1 - 77928 T^{2} + 2885754588 T^{4} - 68373804388056 T^{6} + 1162395919152747590 T^{8} - \)\(14\!\cdots\!36\)\( T^{10} + \)\(13\!\cdots\!68\)\( T^{12} - \)\(76\!\cdots\!48\)\( T^{14} + \)\(21\!\cdots\!21\)\( T^{16} \))(\( ( 1 - 121 T )( 1 + 121 T ) \))(\( ( 1 - 121 T )( 1 + 121 T ) \))(\( ( 1 - 10564 T^{2} + 433024326 T^{4} - 2264487218884 T^{6} + 45949729863572161 T^{8} )^{2} \))(\( ( 1 + 150 T + 28882 T^{2} + 2196150 T^{3} + 214358881 T^{4} )^{2} \))
$13$ (\( ( 1 - 176 T + 77628 T^{2} - 12420688 T^{3} + 2794789574 T^{4} - 354747269968 T^{5} + 63323544409788 T^{6} - 4100462981556656 T^{7} + 665416609183179841 T^{8} )^{2} \))(\( ( 1 - 169 T )( 1 + 169 T ) \))(\( ( 1 - 169 T )( 1 + 169 T ) \))(\( ( 1 - 93316 T^{2} + 3781796166 T^{4} - 76120727960836 T^{6} + 665416609183179841 T^{8} )^{2} \))(\( 1 + 360 T + 64800 T^{2} + 14552280 T^{3} + 3127330814 T^{4} + 415627669080 T^{5} + 52859350720800 T^{6} + 8387310644093160 T^{7} + 665416609183179841 T^{8} \))
$17$ (\( ( 1 + 24 T + 157788 T^{2} - 20618328 T^{3} + 13093999814 T^{4} - 1722063372888 T^{5} + 1100690815100508 T^{6} + 13982933693514264 T^{7} + 48661191875666868481 T^{8} )^{2} \))(\( ( 1 - 289 T )( 1 + 289 T ) \))(\( ( 1 - 289 T )( 1 + 289 T ) \))(\( ( 1 - 307716 T^{2} + 37466348806 T^{4} - 2146552176714756 T^{6} + 48661191875666868481 T^{8} )^{2} \))(\( 1 - 960 T + 460800 T^{2} - 168098880 T^{3} + 52934858494 T^{4} - 14039786556480 T^{5} + 3214429028812800 T^{6} - 559317347740570560 T^{7} + 48661191875666868481 T^{8} \))
$19$ (\( 1 - 730888 T^{2} + 257781714588 T^{4} - 57630026124579896 T^{6} + \)\(89\!\cdots\!90\)\( T^{8} - \)\(97\!\cdots\!36\)\( T^{10} + \)\(74\!\cdots\!28\)\( T^{12} - \)\(35\!\cdots\!48\)\( T^{14} + \)\(83\!\cdots\!61\)\( T^{16} \))(\( ( 1 - 361 T )( 1 + 361 T ) \))(\( ( 1 - 361 T )( 1 + 361 T ) \))(\( ( 1 - 73924 T^{2} - 4326450234 T^{4} - 1255492914242884 T^{6} + \)\(28\!\cdots\!81\)\( T^{8} )^{2} \))(\( 1 - 79796 T^{2} + 32223922086 T^{4} - 1355220396419636 T^{6} + \)\(28\!\cdots\!81\)\( T^{8} \))
$23$ (\( 1 - 885760 T^{2} + 416439331356 T^{4} - 136723474364764160 T^{6} + \)\(39\!\cdots\!66\)\( T^{8} - \)\(10\!\cdots\!60\)\( T^{10} + \)\(25\!\cdots\!16\)\( T^{12} - \)\(42\!\cdots\!60\)\( T^{14} + \)\(37\!\cdots\!21\)\( T^{16} \))(\( 1 + 878 T + 279841 T^{2} \))(\( 1 - 878 T + 279841 T^{2} \))(\( ( 1 + 769044 T^{2} + 304476266086 T^{4} + 60224593364441364 T^{6} + \)\(61\!\cdots\!61\)\( T^{8} )^{2} \))(\( 1 + 810 T + 328050 T^{2} + 264893490 T^{3} + 211669226338 T^{4} + 74128059135090 T^{5} + 25689918721432050 T^{6} + 17750845789936460010 T^{7} + \)\(61\!\cdots\!61\)\( T^{8} \))
$29$ (\( ( 1 - 600 T + 1936636 T^{2} - 649163880 T^{3} + 1640402898886 T^{4} - 459141278210280 T^{5} + 968795212211139196 T^{6} - \)\(21\!\cdots\!00\)\( T^{7} + \)\(25\!\cdots\!21\)\( T^{8} )^{2} \))(\( 1 + 1198 T + 707281 T^{2} \))(\( 1 + 1198 T + 707281 T^{2} \))(\( ( 1 - 676 T + 1518566 T^{2} - 478121956 T^{3} + 500246412961 T^{4} )^{4} \))(\( 1 - 1448516 T^{2} + 1055067026886 T^{4} - 724614933116615876 T^{6} + \)\(25\!\cdots\!21\)\( T^{8} \))
$31$ (\( 1 - 1907688 T^{2} + 1721712272988 T^{4} - 1980674566102865496 T^{6} + \)\(24\!\cdots\!90\)\( T^{8} - \)\(16\!\cdots\!36\)\( T^{10} + \)\(12\!\cdots\!28\)\( T^{12} - \)\(11\!\cdots\!48\)\( T^{14} + \)\(52\!\cdots\!61\)\( T^{16} \))(\( ( 1 - 961 T )( 1 + 961 T ) \))(\( ( 1 - 961 T )( 1 + 961 T ) \))(\( ( 1 - 2680324 T^{2} + 3246634805766 T^{4} - 2286024317038010884 T^{6} + \)\(72\!\cdots\!81\)\( T^{8} )^{2} \))(\( ( 1 + 418 T + 535098 T^{2} + 386031778 T^{3} + 852891037441 T^{4} )^{2} \))
$37$ (\( ( 1 + 2864 T + 5437308 T^{2} + 5448048592 T^{3} + 7051577121734 T^{4} + 10210520197231312 T^{5} + 19098432634640284668 T^{6} + \)\(18\!\cdots\!84\)\( T^{7} + \)\(12\!\cdots\!41\)\( T^{8} )^{2} \))(\( ( 1 - 1369 T )( 1 + 1369 T ) \))(\( ( 1 - 1369 T )( 1 + 1369 T ) \))(\( ( 1 - 4488196 T^{2} + 9924276565446 T^{4} - 15764696235170416516 T^{6} + \)\(12\!\cdots\!41\)\( T^{8} )^{2} \))(\( 1 + 660 T + 217800 T^{2} + 345245340 T^{3} - 1278103400242 T^{4} + 647045351659740 T^{5} + 765018025063993800 T^{6} + \)\(43\!\cdots\!60\)\( T^{7} + \)\(12\!\cdots\!41\)\( T^{8} \))
$41$ (\( ( 1 - 2448 T + 9585788 T^{2} - 17538082416 T^{3} + 39790776226950 T^{4} - 49558429305918576 T^{5} + 76541800442205332348 T^{6} - \)\(55\!\cdots\!88\)\( T^{7} + \)\(63\!\cdots\!41\)\( T^{8} )^{2} \))(\( 1 - 482 T + 2825761 T^{2} \))(\( 1 - 482 T + 2825761 T^{2} \))(\( ( 1 - 364 T + 1690406 T^{2} - 1028577004 T^{3} + 7984925229121 T^{4} )^{4} \))(\( ( 1 - 1482 T + 6194578 T^{2} - 4187777802 T^{3} + 7984925229121 T^{4} )^{2} \))
$43$ (\( 1 - 18909280 T^{2} + 176096852184156 T^{4} - \)\(10\!\cdots\!20\)\( T^{6} + \)\(42\!\cdots\!86\)\( T^{8} - \)\(12\!\cdots\!20\)\( T^{10} + \)\(24\!\cdots\!56\)\( T^{12} - \)\(30\!\cdots\!80\)\( T^{14} + \)\(18\!\cdots\!01\)\( T^{16} \))(\( 1 + 2078 T + 3418801 T^{2} \))(\( 1 - 2078 T + 3418801 T^{2} \))(\( ( 1 + 3640404 T^{2} + 1840427380006 T^{4} + 42549771043379790804 T^{6} + \)\(13\!\cdots\!01\)\( T^{8} )^{2} \))(\( 1 - 3270 T + 5346450 T^{2} - 11205966270 T^{3} + 23487235317602 T^{4} - 38310968689842270 T^{5} + 62490378374179866450 T^{6} - \)\(13\!\cdots\!70\)\( T^{7} + \)\(13\!\cdots\!01\)\( T^{8} \))
$47$ (\( 1 - 19070080 T^{2} + 224159050635036 T^{4} - \)\(17\!\cdots\!40\)\( T^{6} + \)\(99\!\cdots\!26\)\( T^{8} - \)\(41\!\cdots\!40\)\( T^{10} + \)\(12\!\cdots\!56\)\( T^{12} - \)\(25\!\cdots\!80\)\( T^{14} + \)\(32\!\cdots\!41\)\( T^{16} \))(\( 1 - 4402 T + 4879681 T^{2} \))(\( 1 + 4402 T + 4879681 T^{2} \))(\( ( 1 + 18767124 T^{2} + 135582308428006 T^{4} + \)\(44\!\cdots\!64\)\( T^{6} + \)\(56\!\cdots\!21\)\( T^{8} )^{2} \))(\( 1 + 2250 T + 2531250 T^{2} + 7891319250 T^{3} + 22718088596834 T^{4} + 38507120609159250 T^{5} + 60272319362582531250 T^{6} + \)\(26\!\cdots\!50\)\( T^{7} + \)\(56\!\cdots\!21\)\( T^{8} \))
$53$ (\( ( 1 - 1296 T + 16848508 T^{2} - 32857590768 T^{3} + 141852843045574 T^{4} - 259262195660679408 T^{5} + \)\(10\!\cdots\!88\)\( T^{6} - \)\(63\!\cdots\!36\)\( T^{7} + \)\(38\!\cdots\!21\)\( T^{8} )^{2} \))(\( ( 1 - 2809 T )( 1 + 2809 T ) \))(\( ( 1 - 2809 T )( 1 + 2809 T ) \))(\( ( 1 - 7023876 T^{2} + 6572518202566 T^{4} - \)\(43\!\cdots\!36\)\( T^{6} + \)\(38\!\cdots\!21\)\( T^{8} )^{2} \))(\( 1 - 1980 T + 1960200 T^{2} - 16406396820 T^{3} + 137161065548878 T^{4} - 129454362386670420 T^{5} + \)\(12\!\cdots\!00\)\( T^{6} - \)\(97\!\cdots\!80\)\( T^{7} + \)\(38\!\cdots\!21\)\( T^{8} \))
$59$ (\( 1 - 57863048 T^{2} + 1797979000179868 T^{4} - \)\(36\!\cdots\!16\)\( T^{6} + \)\(52\!\cdots\!90\)\( T^{8} - \)\(53\!\cdots\!36\)\( T^{10} + \)\(38\!\cdots\!88\)\( T^{12} - \)\(18\!\cdots\!28\)\( T^{14} + \)\(46\!\cdots\!81\)\( T^{16} \))(\( ( 1 - 3481 T )( 1 + 3481 T ) \))(\( ( 1 - 3481 T )( 1 + 3481 T ) \))(\( ( 1 - 18192964 T^{2} + 372799944667206 T^{4} - \)\(26\!\cdots\!44\)\( T^{6} + \)\(21\!\cdots\!41\)\( T^{8} )^{2} \))(\( 1 + 162508 T^{2} + 44050763806758 T^{4} + 23861120754202997068 T^{6} + \)\(21\!\cdots\!41\)\( T^{8} \))
$61$ (\( ( 1 - 3968 T + 48790428 T^{2} - 144590839936 T^{3} + 984937331366150 T^{4} - 2001981779810306176 T^{5} + \)\(93\!\cdots\!68\)\( T^{6} - \)\(10\!\cdots\!28\)\( T^{7} + \)\(36\!\cdots\!61\)\( T^{8} )^{2} \))(\( 1 + 4078 T + 13845841 T^{2} \))(\( 1 + 4078 T + 13845841 T^{2} \))(\( ( 1 - 1644 T + 17384326 T^{2} - 22762562604 T^{3} + 191707312997281 T^{4} )^{4} \))(\( ( 1 - 7914 T + 36788306 T^{2} - 109575985674 T^{3} + 191707312997281 T^{4} )^{2} \))
$67$ (\( 1 - 73661280 T^{2} + 1417160078043996 T^{4} + \)\(25\!\cdots\!60\)\( T^{6} - \)\(13\!\cdots\!74\)\( T^{8} + \)\(10\!\cdots\!60\)\( T^{10} + \)\(23\!\cdots\!76\)\( T^{12} - \)\(49\!\cdots\!80\)\( T^{14} + \)\(27\!\cdots\!61\)\( T^{16} \))(\( 1 + 4478 T + 20151121 T^{2} \))(\( 1 - 4478 T + 20151121 T^{2} \))(\( ( 1 + 69567444 T^{2} + 1999220647242406 T^{4} + \)\(28\!\cdots\!04\)\( T^{6} + \)\(16\!\cdots\!81\)\( T^{8} )^{2} \))(\( 1 + 4810 T + 11568050 T^{2} + 108537019890 T^{3} + 1012520446566818 T^{4} + 2187142620782796690 T^{5} + \)\(46\!\cdots\!50\)\( T^{6} + \)\(39\!\cdots\!10\)\( T^{7} + \)\(16\!\cdots\!81\)\( T^{8} \))
$71$ (\( 1 - 93361128 T^{2} + 5034263138166108 T^{4} - \)\(18\!\cdots\!76\)\( T^{6} + \)\(54\!\cdots\!90\)\( T^{8} - \)\(12\!\cdots\!36\)\( T^{10} + \)\(20\!\cdots\!68\)\( T^{12} - \)\(25\!\cdots\!68\)\( T^{14} + \)\(17\!\cdots\!41\)\( T^{16} \))(\( ( 1 - 5041 T )( 1 + 5041 T ) \))(\( ( 1 - 5041 T )( 1 + 5041 T ) \))(\( ( 1 - 36881284 T^{2} + 596367162990726 T^{4} - \)\(23\!\cdots\!24\)\( T^{6} + \)\(41\!\cdots\!21\)\( T^{8} )^{2} \))(\( ( 1 + 2994 T + 49877146 T^{2} + 76082572914 T^{3} + 645753531245761 T^{4} )^{2} \))
$73$ (\( ( 1 + 7224 T + 116019228 T^{2} + 579521463432 T^{3} + 4936809828072134 T^{4} + 16457390183214623112 T^{5} + \)\(93\!\cdots\!68\)\( T^{6} + \)\(16\!\cdots\!04\)\( T^{7} + \)\(65\!\cdots\!61\)\( T^{8} )^{2} \))(\( ( 1 - 5329 T )( 1 + 5329 T ) \))(\( ( 1 - 5329 T )( 1 + 5329 T ) \))(\( ( 1 - 45412996 T^{2} + 2127886061569926 T^{4} - \)\(36\!\cdots\!76\)\( T^{6} + \)\(65\!\cdots\!61\)\( T^{8} )^{2} \))(\( 1 + 14060 T + 98841800 T^{2} + 628627141380 T^{3} + 3731941946353934 T^{4} + 17851905060050312580 T^{5} + \)\(79\!\cdots\!00\)\( T^{6} + \)\(32\!\cdots\!60\)\( T^{7} + \)\(65\!\cdots\!61\)\( T^{8} \))
$79$ (\( 1 - 254373768 T^{2} + 29821166949528348 T^{4} - \)\(21\!\cdots\!76\)\( T^{6} + \)\(99\!\cdots\!90\)\( T^{8} - \)\(32\!\cdots\!36\)\( T^{10} + \)\(68\!\cdots\!08\)\( T^{12} - \)\(88\!\cdots\!08\)\( T^{14} + \)\(52\!\cdots\!41\)\( T^{16} \))(\( ( 1 - 6241 T )( 1 + 6241 T ) \))(\( ( 1 - 6241 T )( 1 + 6241 T ) \))(\( ( 1 - 128520964 T^{2} + 7130553234151686 T^{4} - \)\(19\!\cdots\!04\)\( T^{6} + \)\(23\!\cdots\!21\)\( T^{8} )^{2} \))(\( 1 - 127811332 T^{2} + 7111883058262278 T^{4} - \)\(19\!\cdots\!52\)\( T^{6} + \)\(23\!\cdots\!21\)\( T^{8} \))
$83$ (\( 1 - 315969760 T^{2} + 46108155295297116 T^{4} - \)\(40\!\cdots\!60\)\( T^{6} + \)\(23\!\cdots\!86\)\( T^{8} - \)\(91\!\cdots\!60\)\( T^{10} + \)\(23\!\cdots\!96\)\( T^{12} - \)\(36\!\cdots\!60\)\( T^{14} + \)\(25\!\cdots\!61\)\( T^{16} \))(\( 1 - 8002 T + 47458321 T^{2} \))(\( 1 + 8002 T + 47458321 T^{2} \))(\( ( 1 + 71614164 T^{2} + 3304864070081446 T^{4} + \)\(16\!\cdots\!24\)\( T^{6} + \)\(50\!\cdots\!81\)\( T^{8} )^{2} \))(\( 1 - 19950 T + 199001250 T^{2} - 1392338210550 T^{3} + 9242910030806018 T^{4} - 66078033736847486550 T^{5} + \)\(44\!\cdots\!50\)\( T^{6} - \)\(21\!\cdots\!50\)\( T^{7} + \)\(50\!\cdots\!81\)\( T^{8} \))
$89$ (\( ( 1 - 11880 T + 292095196 T^{2} - 2292250543320 T^{3} + 28806792351684166 T^{4} - \)\(14\!\cdots\!20\)\( T^{5} + \)\(11\!\cdots\!76\)\( T^{6} - \)\(29\!\cdots\!80\)\( T^{7} + \)\(15\!\cdots\!61\)\( T^{8} )^{2} \))(\( 1 - 4322 T + 62742241 T^{2} \))(\( 1 - 4322 T + 62742241 T^{2} \))(\( ( 1 - 6916 T + 63458246 T^{2} - 433925338756 T^{3} + 3936588805702081 T^{4} )^{4} \))(\( 1 - 213169796 T^{2} + 19233126298796166 T^{4} - \)\(83\!\cdots\!76\)\( T^{6} + \)\(15\!\cdots\!61\)\( T^{8} \))
$97$ (\( ( 1 + 2184 T + 140081788 T^{2} - 468341237448 T^{3} + 7891029582726534 T^{4} - 41461913013921714888 T^{5} + \)\(10\!\cdots\!68\)\( T^{6} + \)\(15\!\cdots\!44\)\( T^{7} + \)\(61\!\cdots\!21\)\( T^{8} )^{2} \))(\( ( 1 - 9409 T )( 1 + 9409 T ) \))(\( ( 1 - 9409 T )( 1 + 9409 T ) \))(\( ( 1 - 267347716 T^{2} + 33011083228247046 T^{4} - \)\(20\!\cdots\!76\)\( T^{6} + \)\(61\!\cdots\!21\)\( T^{8} )^{2} \))(\( 1 + 3180 T + 5056200 T^{2} - 122563753980 T^{3} - 13176145025425522 T^{4} - 10850481016510288380 T^{5} + \)\(39\!\cdots\!00\)\( T^{6} + \)\(22\!\cdots\!80\)\( T^{7} + \)\(61\!\cdots\!21\)\( T^{8} \))
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