Properties

Label 48.3
Level 48
Weight 3
Dimension 49
Nonzero newspaces 4
Newform subspaces 6
Sturm bound 384
Trace bound 1

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Defining parameters

Level: \( N \) = \( 48\( 48 = 2^{4} \cdot 3 \) \)
Weight: \( k \) = \( 3 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 6 \)
Sturm bound: \(384\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(\Gamma_1(48))\).

Total New Old
Modular forms 156 59 97
Cusp forms 100 49 51
Eisenstein series 56 10 46

Trace form

\( 49q - q^{3} + 8q^{4} + 12q^{5} + 4q^{6} + 10q^{7} - 12q^{8} - 11q^{9} + O(q^{10}) \) \( 49q - q^{3} + 8q^{4} + 12q^{5} + 4q^{6} + 10q^{7} - 12q^{8} - 11q^{9} - 80q^{10} + 32q^{11} - 56q^{12} - 34q^{13} - 44q^{14} - 36q^{15} + 16q^{16} - 12q^{17} - 48q^{18} - 98q^{19} + 80q^{20} - 26q^{21} + 72q^{22} - 128q^{23} - 20q^{24} + 33q^{25} - 100q^{26} + 23q^{27} + 92q^{29} + 156q^{30} + 82q^{31} + 160q^{32} + 100q^{33} + 232q^{34} + 96q^{35} + 208q^{36} - 34q^{37} + 168q^{38} - 86q^{39} + 136q^{40} - 108q^{41} + 220q^{42} - 50q^{43} + 88q^{44} - 24q^{45} - 64q^{46} - 32q^{48} + 7q^{49} - 236q^{50} + 128q^{51} - 368q^{52} - 196q^{53} - 164q^{54} - 192q^{55} - 224q^{56} - 50q^{57} - 424q^{58} - 128q^{59} - 648q^{60} - 306q^{61} - 276q^{62} + 90q^{63} - 640q^{64} - 200q^{65} - 500q^{66} + 318q^{67} - 448q^{68} + 12q^{69} - 72q^{70} + 512q^{71} - 252q^{72} + 282q^{73} + 348q^{74} + 459q^{75} + 848q^{76} + 512q^{77} + 632q^{78} + 370q^{79} + 552q^{80} - 15q^{81} + 696q^{82} - 160q^{83} + 952q^{84} + 280q^{85} + 528q^{86} - 96q^{87} + 832q^{88} + 228q^{89} + 984q^{90} - 28q^{91} + 496q^{92} - 46q^{93} - 24q^{94} + 8q^{96} - 126q^{97} - 440q^{98} - 260q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(\Gamma_1(48))\)

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
48.3.b \(\chi_{48}(7, \cdot)\) None 0 1
48.3.e \(\chi_{48}(17, \cdot)\) 48.3.e.a 1 1
48.3.e.b 2
48.3.g \(\chi_{48}(31, \cdot)\) 48.3.g.a 2 1
48.3.h \(\chi_{48}(41, \cdot)\) None 0 1
48.3.i \(\chi_{48}(5, \cdot)\) 48.3.i.a 8 2
48.3.i.b 20
48.3.l \(\chi_{48}(19, \cdot)\) 48.3.l.a 16 2

Decomposition of \(S_{3}^{\mathrm{old}}(\Gamma_1(48))\) into lower level spaces

\( S_{3}^{\mathrm{old}}(\Gamma_1(48)) \cong \) \(S_{3}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 4 T^{2} + 8 T^{4} + 64 T^{6} + 256 T^{8} \))(\( 1 - 2 T^{2} + 6 T^{4} - 24 T^{6} - 24 T^{8} + 1216 T^{10} - 384 T^{12} - 6144 T^{14} + 24576 T^{16} - 131072 T^{18} + 1048576 T^{20} \))(\( 1 - 6 T^{2} + 4 T^{3} + 10 T^{4} - 56 T^{5} + 88 T^{6} + 128 T^{7} - 496 T^{8} + 512 T^{9} + 1408 T^{10} - 3584 T^{11} + 2560 T^{12} + 4096 T^{13} - 24576 T^{14} + 65536 T^{16} \))
$3$ (\( 1 - 3 T \))(\( 1 + 2 T + 9 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 - 4 T + 8 T^{2} + 12 T^{3} - 126 T^{4} + 108 T^{5} + 648 T^{6} - 2916 T^{7} + 6561 T^{8} \))(\( 1 + 6 T + 18 T^{2} + 30 T^{3} + 65 T^{4} + 168 T^{5} + 288 T^{6} + 1560 T^{7} + 11278 T^{8} + 64332 T^{9} + 225180 T^{10} + 578988 T^{11} + 913518 T^{12} + 1137240 T^{13} + 1889568 T^{14} + 9920232 T^{15} + 34543665 T^{16} + 143489070 T^{17} + 774840978 T^{18} + 2324522934 T^{19} + 3486784401 T^{20} \))(\( ( 1 + 9 T^{4} )^{4} \))
$5$ (\( ( 1 - 5 T )( 1 + 5 T ) \))(\( 1 - 18 T^{2} + 625 T^{4} \))(\( ( 1 - 6 T + 25 T^{2} )^{2} \))(\( 1 + 372 T^{4} + 464614 T^{8} + 145312500 T^{12} + 152587890625 T^{16} \))(\( 1 - 946 T^{4} + 556509 T^{8} + 27762120 T^{12} - 359980854270 T^{16} + 279323158314196 T^{20} - 140617521199218750 T^{24} + 4236163330078125000 T^{28} + \)\(33\!\cdots\!25\)\( T^{32} - \)\(22\!\cdots\!50\)\( T^{36} + \)\(90\!\cdots\!25\)\( T^{40} \))(\( 1 + 32 T^{3} - 344 T^{4} + 5664 T^{5} + 512 T^{6} + 145600 T^{7} + 223452 T^{8} - 2255168 T^{9} + 20875776 T^{10} - 67282720 T^{11} + 753060504 T^{12} + 1881828576 T^{13} + 2740220928 T^{14} + 75471830656 T^{15} - 399298967994 T^{16} + 1886795766400 T^{17} + 1712638080000 T^{18} + 29403571500000 T^{19} + 294164259375000 T^{20} - 657057812500000 T^{21} + 5096625000000000 T^{22} - 13764453125000000 T^{23} + 34096069335937500 T^{24} + 555419921875000000 T^{25} + 48828125000000000 T^{26} + 13504028320312500000 T^{27} - 20503997802734375000 T^{28} + 47683715820312500000 T^{29} + \)\(23\!\cdots\!25\)\( T^{32} \))
$7$ (\( 1 + 2 T + 49 T^{2} \))(\( ( 1 - 6 T + 49 T^{2} )^{2} \))(\( 1 - 50 T^{2} + 2401 T^{4} \))(\( ( 1 - 180 T^{2} + 12874 T^{4} - 432180 T^{6} + 5764801 T^{8} )^{2} \))(\( ( 1 - 154 T^{2} + 17009 T^{4} - 1179536 T^{6} + 71996590 T^{8} - 3527749420 T^{10} + 172863812590 T^{12} - 6799790312336 T^{14} + 235426454001809 T^{16} - 5117871307718554 T^{18} + 79792266297612001 T^{20} )^{2} \))(\( ( 1 + 168 T^{2} - 448 T^{3} + 15076 T^{4} - 56512 T^{5} + 1070392 T^{6} - 3649664 T^{7} + 60103046 T^{8} - 178833536 T^{9} + 2570011192 T^{10} - 6648580288 T^{11} + 86910139876 T^{12} - 126548911552 T^{13} + 2325336249768 T^{14} + 33232930569601 T^{16} )^{2} \))
$11$ (\( ( 1 - 11 T )( 1 + 11 T ) \))(\( 1 - 210 T^{2} + 14641 T^{4} \))(\( 1 + 190 T^{2} + 14641 T^{4} \))(\( 1 + 37380 T^{4} + 692870854 T^{8} + 8012734971780 T^{12} + 45949729863572161 T^{16} \))(\( 1 - 17026 T^{4} - 443639331 T^{8} + 11991832514824 T^{12} + 51739802427787650 T^{16} - \)\(36\!\cdots\!16\)\( T^{20} + \)\(11\!\cdots\!50\)\( T^{24} + \)\(55\!\cdots\!64\)\( T^{28} - \)\(43\!\cdots\!71\)\( T^{32} - \)\(35\!\cdots\!46\)\( T^{36} + \)\(45\!\cdots\!01\)\( T^{40} \))(\( 1 - 32 T + 512 T^{2} - 8480 T^{3} + 137032 T^{4} - 1636576 T^{5} + 18165248 T^{6} - 219655136 T^{7} + 2263228700 T^{8} - 21867108000 T^{9} + 253620152832 T^{10} - 2916953293728 T^{11} + 33797606438392 T^{12} - 431768458252384 T^{13} + 5293166227138048 T^{14} - 61910274521995104 T^{15} + 703855220885889990 T^{16} - 7491143217161407584 T^{17} + 77497246731528160768 T^{18} - \)\(76\!\cdots\!24\)\( T^{19} + \)\(72\!\cdots\!52\)\( T^{20} - \)\(75\!\cdots\!28\)\( T^{21} + \)\(79\!\cdots\!72\)\( T^{22} - \)\(83\!\cdots\!00\)\( T^{23} + \)\(10\!\cdots\!00\)\( T^{24} - \)\(12\!\cdots\!16\)\( T^{25} + \)\(12\!\cdots\!48\)\( T^{26} - \)\(13\!\cdots\!96\)\( T^{27} + \)\(13\!\cdots\!12\)\( T^{28} - \)\(10\!\cdots\!80\)\( T^{29} + \)\(73\!\cdots\!72\)\( T^{30} - \)\(55\!\cdots\!32\)\( T^{31} + \)\(21\!\cdots\!21\)\( T^{32} \))
$13$ (\( 1 + 22 T + 169 T^{2} \))(\( ( 1 - 10 T + 169 T^{2} )^{2} \))(\( ( 1 + 14 T + 169 T^{2} )^{2} \))(\( ( 1 + 48 T + 1152 T^{2} + 21264 T^{3} + 317422 T^{4} + 3593616 T^{5} + 32902272 T^{6} + 231686832 T^{7} + 815730721 T^{8} )^{2} \))(\( ( 1 - 46 T + 1058 T^{2} - 20510 T^{3} + 411997 T^{4} - 7447784 T^{5} + 117035288 T^{6} - 1803425192 T^{7} + 27622285106 T^{8} - 386182026516 T^{9} + 5044801970700 T^{10} - 65264762481204 T^{11} + 788920084912466 T^{12} - 8704788947572328 T^{13} + 95469279862682648 T^{14} - 1026740269857112616 T^{15} + 9598741176206804557 T^{16} - 80755589670692417390 T^{17} + \)\(70\!\cdots\!78\)\( T^{18} - \)\(51\!\cdots\!34\)\( T^{19} + \)\(19\!\cdots\!01\)\( T^{20} )^{2} \))(\( 1 + 3200 T^{3} + 7608 T^{4} + 95360 T^{5} + 5120000 T^{6} + 68335872 T^{7} + 2004669468 T^{8} + 7270355200 T^{9} + 184268595200 T^{10} + 4889456013184 T^{11} + 5354592144136 T^{12} + 669839496880000 T^{13} + 7008632866619392 T^{14} + 70586941744778752 T^{15} + 2398056097119178950 T^{16} + 11929193154867609088 T^{17} + \)\(20\!\cdots\!12\)\( T^{18} + \)\(32\!\cdots\!00\)\( T^{19} + \)\(43\!\cdots\!56\)\( T^{20} + \)\(67\!\cdots\!16\)\( T^{21} + \)\(42\!\cdots\!00\)\( T^{22} + \)\(28\!\cdots\!00\)\( T^{23} + \)\(13\!\cdots\!88\)\( T^{24} + \)\(76\!\cdots\!88\)\( T^{25} + \)\(97\!\cdots\!00\)\( T^{26} + \)\(30\!\cdots\!40\)\( T^{27} + \)\(41\!\cdots\!88\)\( T^{28} + \)\(29\!\cdots\!00\)\( T^{29} + \)\(44\!\cdots\!81\)\( T^{32} \))
$17$ (\( ( 1 - 17 T )( 1 + 17 T ) \))(\( 1 - 66 T^{2} + 83521 T^{4} \))(\( ( 1 + 6 T + 289 T^{2} )^{2} \))(\( ( 1 - 236 T^{2} + 73334 T^{4} - 19710956 T^{6} + 6975757441 T^{8} )^{2} \))(\( ( 1 - 1938 T^{2} + 1855181 T^{4} - 1152381976 T^{6} + 513055082610 T^{8} - 170587207926956 T^{10} + 42850873554669810 T^{12} - 8038737143956283416 T^{14} + \)\(10\!\cdots\!41\)\( T^{16} - \)\(94\!\cdots\!78\)\( T^{18} + \)\(40\!\cdots\!01\)\( T^{20} )^{2} \))(\( ( 1 + 968 T^{2} - 2944 T^{3} + 516540 T^{4} - 3209600 T^{5} + 201700088 T^{6} - 1543904000 T^{7} + 63894476806 T^{8} - 446188256000 T^{9} + 16846193049848 T^{10} - 77471941462400 T^{11} + 3603257748574140 T^{12} - 5935086042921856 T^{13} + 563978325638408648 T^{14} + 48661191875666868481 T^{16} )^{2} \))
$19$ (\( 1 + 26 T + 361 T^{2} \))(\( ( 1 + 2 T + 361 T^{2} )^{2} \))(\( 1 - 674 T^{2} + 130321 T^{4} \))(\( ( 1 - 8 T + 32 T^{2} - 1160 T^{3} - 4606 T^{4} - 418760 T^{5} + 4170272 T^{6} - 376367048 T^{7} + 16983563041 T^{8} )^{2} \))(\( ( 1 + 26 T + 338 T^{2} + 13338 T^{3} + 349281 T^{4} + 2446864 T^{5} + 34512608 T^{6} + 926396720 T^{7} - 22197634642 T^{8} - 577965275108 T^{9} - 4229043710052 T^{10} - 208645464313988 T^{11} - 2892817944180082 T^{12} + 43583149847910320 T^{13} + 586147053677320928 T^{14} + 15001885307827986064 T^{15} + \)\(77\!\cdots\!41\)\( T^{16} + \)\(10\!\cdots\!98\)\( T^{17} + \)\(97\!\cdots\!78\)\( T^{18} + \)\(27\!\cdots\!66\)\( T^{19} + \)\(37\!\cdots\!01\)\( T^{20} )^{2} \))(\( 1 + 32 T + 512 T^{2} - 2656 T^{3} - 523448 T^{4} - 8424608 T^{5} + 1945088 T^{6} + 4454446304 T^{7} + 107916937244 T^{8} - 703649376 T^{9} - 30601835632128 T^{10} - 698985761087712 T^{11} + 998616856187896 T^{12} + 253693358084547040 T^{13} + 3161998119961945600 T^{14} - 30474951661580761248 T^{15} - \)\(19\!\cdots\!42\)\( T^{16} - \)\(11\!\cdots\!28\)\( T^{17} + \)\(41\!\cdots\!00\)\( T^{18} + \)\(11\!\cdots\!40\)\( T^{19} + \)\(16\!\cdots\!36\)\( T^{20} - \)\(42\!\cdots\!12\)\( T^{21} - \)\(67\!\cdots\!08\)\( T^{22} - \)\(56\!\cdots\!96\)\( T^{23} + \)\(31\!\cdots\!64\)\( T^{24} + \)\(46\!\cdots\!64\)\( T^{25} + \)\(73\!\cdots\!88\)\( T^{26} - \)\(11\!\cdots\!88\)\( T^{27} - \)\(25\!\cdots\!08\)\( T^{28} - \)\(46\!\cdots\!36\)\( T^{29} + \)\(32\!\cdots\!92\)\( T^{30} + \)\(73\!\cdots\!32\)\( T^{31} + \)\(83\!\cdots\!61\)\( T^{32} \))
$23$ (\( ( 1 - 23 T )( 1 + 23 T ) \))(\( 1 - 930 T^{2} + 279841 T^{4} \))(\( ( 1 - 23 T )^{2}( 1 + 23 T )^{2} \))(\( ( 1 + 996 T^{2} + 719878 T^{4} + 278721636 T^{6} + 78310985281 T^{8} )^{2} \))(\( ( 1 + 3054 T^{2} + 4762717 T^{4} + 4930394824 T^{6} + 3773391227074 T^{8} + 2245272418513300 T^{10} + 1055949574375615234 T^{12} + \)\(38\!\cdots\!44\)\( T^{14} + \)\(10\!\cdots\!57\)\( T^{16} + \)\(18\!\cdots\!94\)\( T^{18} + \)\(17\!\cdots\!01\)\( T^{20} )^{2} \))(\( ( 1 + 64 T + 3496 T^{2} + 127936 T^{3} + 4410332 T^{4} + 130001728 T^{5} + 3673719192 T^{6} + 94049622208 T^{7} + 2261818535238 T^{8} + 49752250148032 T^{9} + 1028057252408472 T^{10} + 19244921376016192 T^{11} + 345377444336323292 T^{12} + 5299942138629398464 T^{13} + 76613527014343042216 T^{14} + \)\(74\!\cdots\!76\)\( T^{15} + \)\(61\!\cdots\!61\)\( T^{16} )^{2} \))
$29$ (\( ( 1 - 29 T )( 1 + 29 T ) \))(\( 1 - 1394 T^{2} + 707281 T^{4} \))(\( ( 1 - 30 T + 841 T^{2} )^{2} \))(\( 1 + 147060 T^{4} + 623812106854 T^{8} + 73566237490044660 T^{12} + \)\(25\!\cdots\!21\)\( T^{16} \))(\( 1 + 865038 T^{4} - 341999726179 T^{8} - 621091035708977976 T^{12} - \)\(34\!\cdots\!02\)\( T^{16} + \)\(22\!\cdots\!76\)\( T^{20} - \)\(17\!\cdots\!22\)\( T^{24} - \)\(15\!\cdots\!96\)\( T^{28} - \)\(42\!\cdots\!99\)\( T^{32} + \)\(54\!\cdots\!58\)\( T^{36} + \)\(31\!\cdots\!01\)\( T^{40} \))(\( 1 - 32 T + 512 T^{2} + 18368 T^{3} - 1552984 T^{4} + 20596992 T^{5} + 304715776 T^{6} - 25469097376 T^{7} + 491466517980 T^{8} + 9791032230816 T^{9} - 347423504794624 T^{10} + 2649303176415616 T^{11} + 694517140133881240 T^{12} - 20658732330776531008 T^{13} + \)\(19\!\cdots\!28\)\( T^{14} + \)\(11\!\cdots\!40\)\( T^{15} - \)\(82\!\cdots\!10\)\( T^{16} + \)\(96\!\cdots\!40\)\( T^{17} + \)\(13\!\cdots\!68\)\( T^{18} - \)\(12\!\cdots\!68\)\( T^{19} + \)\(34\!\cdots\!40\)\( T^{20} + \)\(11\!\cdots\!16\)\( T^{21} - \)\(12\!\cdots\!84\)\( T^{22} + \)\(29\!\cdots\!96\)\( T^{23} + \)\(12\!\cdots\!80\)\( T^{24} - \)\(53\!\cdots\!36\)\( T^{25} + \)\(53\!\cdots\!76\)\( T^{26} + \)\(30\!\cdots\!72\)\( T^{27} - \)\(19\!\cdots\!04\)\( T^{28} + \)\(19\!\cdots\!28\)\( T^{29} + \)\(45\!\cdots\!32\)\( T^{30} - \)\(23\!\cdots\!32\)\( T^{31} + \)\(62\!\cdots\!41\)\( T^{32} \))
$31$ (\( 1 - 46 T + 961 T^{2} \))(\( ( 1 - 22 T + 961 T^{2} )^{2} \))(\( 1 - 1490 T^{2} + 923521 T^{4} \))(\( ( 1 - 18 T + 1996 T^{2} - 17298 T^{3} + 923521 T^{4} )^{4} \))(\( ( 1 + 20 T + 2055 T^{2} + 69364 T^{3} + 2243976 T^{4} + 102850448 T^{5} + 2156460936 T^{6} + 64059110644 T^{7} + 1823820064455 T^{8} + 17057820748820 T^{9} + 819628286980801 T^{10} )^{4} \))(\( 1 - 7312 T^{2} + 29025544 T^{4} - 80335806576 T^{6} + 171125889681052 T^{8} - 295006946315669072 T^{10} + \)\(42\!\cdots\!64\)\( T^{12} - \)\(51\!\cdots\!00\)\( T^{14} + \)\(53\!\cdots\!38\)\( T^{16} - \)\(47\!\cdots\!00\)\( T^{18} + \)\(36\!\cdots\!24\)\( T^{20} - \)\(23\!\cdots\!92\)\( T^{22} + \)\(12\!\cdots\!12\)\( T^{24} - \)\(53\!\cdots\!76\)\( T^{26} + \)\(18\!\cdots\!24\)\( T^{28} - \)\(41\!\cdots\!92\)\( T^{30} + \)\(52\!\cdots\!61\)\( T^{32} \))
$37$ (\( 1 - 26 T + 1369 T^{2} \))(\( ( 1 + 6 T + 1369 T^{2} )^{2} \))(\( ( 1 - 26 T + 1369 T^{2} )^{2} \))(\( ( 1 - 56 T + 1568 T^{2} - 79016 T^{3} + 3980078 T^{4} - 108172904 T^{5} + 2938684448 T^{6} - 143680678904 T^{7} + 3512479453921 T^{8} )^{2} \))(\( ( 1 + 58 T + 1682 T^{2} - 50742 T^{3} - 603459 T^{4} + 139637784 T^{5} + 10401384792 T^{6} + 114472119576 T^{7} + 338413283634 T^{8} + 111361866221948 T^{9} + 16107397431686060 T^{10} + 152454394857846812 T^{11} + 634240978068781074 T^{12} + \)\(29\!\cdots\!84\)\( T^{13} + \)\(36\!\cdots\!32\)\( T^{14} + \)\(67\!\cdots\!16\)\( T^{15} - \)\(39\!\cdots\!79\)\( T^{16} - \)\(45\!\cdots\!38\)\( T^{17} + \)\(20\!\cdots\!62\)\( T^{18} + \)\(97\!\cdots\!82\)\( T^{19} + \)\(23\!\cdots\!01\)\( T^{20} )^{2} \))(\( 1 + 96 T + 4608 T^{2} + 145952 T^{3} + 4040888 T^{4} + 217733344 T^{5} + 12932982272 T^{6} + 602883756192 T^{7} + 21839639792924 T^{8} + 655265530977504 T^{9} + 21703692469355008 T^{10} + 815191556064282016 T^{11} + 35433653736114978312 T^{12} + \)\(14\!\cdots\!40\)\( T^{13} + \)\(50\!\cdots\!52\)\( T^{14} + \)\(14\!\cdots\!84\)\( T^{15} + \)\(43\!\cdots\!90\)\( T^{16} + \)\(20\!\cdots\!96\)\( T^{17} + \)\(95\!\cdots\!72\)\( T^{18} + \)\(37\!\cdots\!60\)\( T^{19} + \)\(12\!\cdots\!52\)\( T^{20} + \)\(39\!\cdots\!84\)\( T^{21} + \)\(14\!\cdots\!48\)\( T^{22} + \)\(59\!\cdots\!56\)\( T^{23} + \)\(26\!\cdots\!84\)\( T^{24} + \)\(10\!\cdots\!68\)\( T^{25} + \)\(29\!\cdots\!72\)\( T^{26} + \)\(68\!\cdots\!36\)\( T^{27} + \)\(17\!\cdots\!68\)\( T^{28} + \)\(86\!\cdots\!68\)\( T^{29} + \)\(37\!\cdots\!68\)\( T^{30} + \)\(10\!\cdots\!04\)\( T^{31} + \)\(15\!\cdots\!81\)\( T^{32} \))
$41$ (\( ( 1 - 41 T )( 1 + 41 T ) \))(\( 1 - 2210 T^{2} + 2825761 T^{4} \))(\( ( 1 + 54 T + 1681 T^{2} )^{2} \))(\( ( 1 + 6060 T^{2} + 14724790 T^{4} + 17124111660 T^{6} + 7984925229121 T^{8} )^{2} \))(\( ( 1 + 8166 T^{2} + 35165469 T^{4} + 106382596584 T^{6} + 246760101929730 T^{8} + 458809512185363300 T^{10} + \)\(69\!\cdots\!30\)\( T^{12} + \)\(84\!\cdots\!64\)\( T^{14} + \)\(79\!\cdots\!89\)\( T^{16} + \)\(52\!\cdots\!06\)\( T^{18} + \)\(18\!\cdots\!01\)\( T^{20} )^{2} \))(\( 1 - 13840 T^{2} + 102706104 T^{4} - 524939980080 T^{6} + 2044068651261084 T^{8} - 6376104819902485008 T^{10} + \)\(16\!\cdots\!68\)\( T^{12} - \)\(35\!\cdots\!72\)\( T^{14} + \)\(64\!\cdots\!06\)\( T^{16} - \)\(99\!\cdots\!92\)\( T^{18} + \)\(13\!\cdots\!28\)\( T^{20} - \)\(14\!\cdots\!48\)\( T^{22} + \)\(13\!\cdots\!44\)\( T^{24} - \)\(94\!\cdots\!80\)\( T^{26} + \)\(52\!\cdots\!44\)\( T^{28} - \)\(19\!\cdots\!40\)\( T^{30} + \)\(40\!\cdots\!81\)\( T^{32} \))
$43$ (\( 1 - 22 T + 1849 T^{2} \))(\( ( 1 + 82 T + 1849 T^{2} )^{2} \))(\( 1 - 3266 T^{2} + 3418801 T^{4} \))(\( ( 1 + 120 T + 7200 T^{2} + 411000 T^{3} + 20977474 T^{4} + 759939000 T^{5} + 24615367200 T^{6} + 758563565880 T^{7} + 11688200277601 T^{8} )^{2} \))(\( ( 1 - 86 T + 3698 T^{2} - 270774 T^{3} + 16192641 T^{4} - 356953072 T^{5} + 7476857312 T^{6} - 199821444560 T^{7} - 33578430535506 T^{8} + 2320167893729020 T^{9} - 78617269367427492 T^{10} + 4289990435504957980 T^{11} - \)\(11\!\cdots\!06\)\( T^{12} - \)\(12\!\cdots\!40\)\( T^{13} + \)\(87\!\cdots\!12\)\( T^{14} - \)\(77\!\cdots\!28\)\( T^{15} + \)\(64\!\cdots\!41\)\( T^{16} - \)\(20\!\cdots\!26\)\( T^{17} + \)\(50\!\cdots\!98\)\( T^{18} - \)\(21\!\cdots\!14\)\( T^{19} + \)\(46\!\cdots\!01\)\( T^{20} )^{2} \))(\( 1 - 160 T + 12800 T^{2} - 978464 T^{3} + 71106632 T^{4} - 3813053664 T^{5} + 178619596288 T^{6} - 8719368905312 T^{7} + 336417491247900 T^{8} - 9339737479444512 T^{9} + 288453906337733120 T^{10} - 7137460469658328480 T^{11} - \)\(12\!\cdots\!76\)\( T^{12} + \)\(13\!\cdots\!84\)\( T^{13} - \)\(44\!\cdots\!20\)\( T^{14} + \)\(28\!\cdots\!76\)\( T^{15} - \)\(17\!\cdots\!30\)\( T^{16} + \)\(53\!\cdots\!24\)\( T^{17} - \)\(15\!\cdots\!20\)\( T^{18} + \)\(83\!\cdots\!16\)\( T^{19} - \)\(15\!\cdots\!76\)\( T^{20} - \)\(15\!\cdots\!20\)\( T^{21} + \)\(11\!\cdots\!20\)\( T^{22} - \)\(69\!\cdots\!88\)\( T^{23} + \)\(45\!\cdots\!00\)\( T^{24} - \)\(22\!\cdots\!88\)\( T^{25} + \)\(83\!\cdots\!88\)\( T^{26} - \)\(32\!\cdots\!36\)\( T^{27} + \)\(11\!\cdots\!32\)\( T^{28} - \)\(28\!\cdots\!36\)\( T^{29} + \)\(69\!\cdots\!00\)\( T^{30} - \)\(16\!\cdots\!40\)\( T^{31} + \)\(18\!\cdots\!01\)\( T^{32} \))
$47$ (\( ( 1 - 47 T )( 1 + 47 T ) \))(\( 1 + 190 T^{2} + 4879681 T^{4} \))(\( 1 - 2690 T^{2} + 4879681 T^{4} \))(\( ( 1 - 692 T^{2} + 7491686 T^{4} - 3376739252 T^{6} + 23811286661761 T^{8} )^{2} \))(\( ( 1 - 17146 T^{2} + 138875501 T^{4} - 703714777016 T^{6} + 2481001995058130 T^{8} - 6375856842165200540 T^{10} + \)\(12\!\cdots\!30\)\( T^{12} - \)\(16\!\cdots\!76\)\( T^{14} + \)\(16\!\cdots\!41\)\( T^{16} - \)\(97\!\cdots\!66\)\( T^{18} + \)\(27\!\cdots\!01\)\( T^{20} )^{2} \))(\( 1 - 24144 T^{2} + 280869112 T^{4} - 2097883923184 T^{6} + 11327375509374492 T^{8} - 47271044690493269328 T^{10} + \)\(15\!\cdots\!16\)\( T^{12} - \)\(44\!\cdots\!04\)\( T^{14} + \)\(10\!\cdots\!58\)\( T^{16} - \)\(21\!\cdots\!24\)\( T^{18} + \)\(37\!\cdots\!76\)\( T^{20} - \)\(54\!\cdots\!48\)\( T^{22} + \)\(64\!\cdots\!32\)\( T^{24} - \)\(58\!\cdots\!84\)\( T^{26} + \)\(37\!\cdots\!72\)\( T^{28} - \)\(15\!\cdots\!84\)\( T^{30} + \)\(32\!\cdots\!41\)\( T^{32} \))
$53$ (\( ( 1 - 53 T )( 1 + 53 T ) \))(\( 1 - 1746 T^{2} + 7890481 T^{4} \))(\( ( 1 + 18 T + 2809 T^{2} )^{2} \))(\( 1 + 1441524 T^{4} - 91064986479386 T^{8} + 89748837960546754164 T^{12} + \)\(38\!\cdots\!21\)\( T^{16} \))(\( 1 + 1437518 T^{4} + 119553426803037 T^{8} - \)\(89\!\cdots\!68\)\( T^{12} + \)\(36\!\cdots\!22\)\( T^{16} - \)\(10\!\cdots\!60\)\( T^{20} + \)\(22\!\cdots\!42\)\( T^{24} - \)\(34\!\cdots\!28\)\( T^{28} + \)\(28\!\cdots\!97\)\( T^{32} + \)\(21\!\cdots\!38\)\( T^{36} + \)\(93\!\cdots\!01\)\( T^{40} \))(\( 1 + 160 T + 12800 T^{2} + 602944 T^{3} + 3948712 T^{4} - 1481707264 T^{5} - 105845942272 T^{6} - 3791430241760 T^{7} + 34861972067036 T^{8} + 14471440004155872 T^{9} + 1098860393015073792 T^{10} + 54880211634179791488 T^{11} + \)\(13\!\cdots\!12\)\( T^{12} - \)\(17\!\cdots\!00\)\( T^{13} - \)\(18\!\cdots\!48\)\( T^{14} + \)\(16\!\cdots\!08\)\( T^{15} + \)\(51\!\cdots\!54\)\( T^{16} + \)\(46\!\cdots\!72\)\( T^{17} - \)\(14\!\cdots\!88\)\( T^{18} - \)\(38\!\cdots\!00\)\( T^{19} + \)\(84\!\cdots\!32\)\( T^{20} + \)\(95\!\cdots\!12\)\( T^{21} + \)\(53\!\cdots\!72\)\( T^{22} + \)\(19\!\cdots\!68\)\( T^{23} + \)\(13\!\cdots\!56\)\( T^{24} - \)\(41\!\cdots\!40\)\( T^{25} - \)\(32\!\cdots\!72\)\( T^{26} - \)\(12\!\cdots\!76\)\( T^{27} + \)\(95\!\cdots\!72\)\( T^{28} + \)\(40\!\cdots\!76\)\( T^{29} + \)\(24\!\cdots\!00\)\( T^{30} + \)\(85\!\cdots\!40\)\( T^{31} + \)\(15\!\cdots\!41\)\( T^{32} \))
$59$ (\( ( 1 - 59 T )( 1 + 59 T ) \))(\( 1 - 1554 T^{2} + 12117361 T^{4} \))(\( 1 - 6530 T^{2} + 12117361 T^{4} \))(\( 1 + 15787044 T^{4} + 346992256453126 T^{8} + \)\(23\!\cdots\!24\)\( T^{12} + \)\(21\!\cdots\!41\)\( T^{16} \))(\( 1 - 19699682 T^{4} + 5882421086685 T^{8} + \)\(56\!\cdots\!08\)\( T^{12} + \)\(29\!\cdots\!74\)\( T^{16} - \)\(57\!\cdots\!12\)\( T^{20} + \)\(43\!\cdots\!54\)\( T^{24} + \)\(12\!\cdots\!28\)\( T^{28} + \)\(18\!\cdots\!85\)\( T^{32} - \)\(91\!\cdots\!42\)\( T^{36} + \)\(68\!\cdots\!01\)\( T^{40} \))(\( 1 + 128 T + 8192 T^{2} + 1121408 T^{3} + 136226184 T^{4} + 9279937408 T^{5} + 700645040128 T^{6} + 71627082366848 T^{7} + 5234572115355804 T^{8} + 316007889653226112 T^{9} + 25502997282495045632 T^{10} + \)\(19\!\cdots\!80\)\( T^{11} + \)\(10\!\cdots\!40\)\( T^{12} + \)\(69\!\cdots\!16\)\( T^{13} + \)\(51\!\cdots\!56\)\( T^{14} + \)\(29\!\cdots\!24\)\( T^{15} + \)\(15\!\cdots\!38\)\( T^{16} + \)\(10\!\cdots\!44\)\( T^{17} + \)\(62\!\cdots\!16\)\( T^{18} + \)\(29\!\cdots\!56\)\( T^{19} + \)\(16\!\cdots\!40\)\( T^{20} + \)\(98\!\cdots\!80\)\( T^{21} + \)\(45\!\cdots\!92\)\( T^{22} + \)\(19\!\cdots\!32\)\( T^{23} + \)\(11\!\cdots\!64\)\( T^{24} + \)\(53\!\cdots\!08\)\( T^{25} + \)\(18\!\cdots\!28\)\( T^{26} + \)\(84\!\cdots\!48\)\( T^{27} + \)\(43\!\cdots\!24\)\( T^{28} + \)\(12\!\cdots\!28\)\( T^{29} + \)\(31\!\cdots\!32\)\( T^{30} + \)\(17\!\cdots\!28\)\( T^{31} + \)\(46\!\cdots\!81\)\( T^{32} \))
$61$ (\( 1 - 74 T + 3721 T^{2} \))(\( ( 1 + 86 T + 3721 T^{2} )^{2} \))(\( ( 1 + 70 T + 3721 T^{2} )^{2} \))(\( ( 1 - 104 T + 5408 T^{2} - 491192 T^{3} + 43609454 T^{4} - 1827725432 T^{5} + 74878308128 T^{6} - 5358118933544 T^{7} + 191707312997281 T^{8} )^{2} \))(\( ( 1 + 122 T + 7442 T^{2} + 699306 T^{3} + 60909597 T^{4} + 2415315288 T^{5} + 85893685080 T^{6} + 3780099265368 T^{7} - 289655325423054 T^{8} - 34035914575403972 T^{9} - 1668196322377933396 T^{10} - \)\(12\!\cdots\!12\)\( T^{11} - \)\(40\!\cdots\!14\)\( T^{12} + \)\(19\!\cdots\!48\)\( T^{13} + \)\(16\!\cdots\!80\)\( T^{14} + \)\(17\!\cdots\!88\)\( T^{15} + \)\(16\!\cdots\!37\)\( T^{16} + \)\(69\!\cdots\!46\)\( T^{17} + \)\(27\!\cdots\!62\)\( T^{18} + \)\(16\!\cdots\!82\)\( T^{19} + \)\(50\!\cdots\!01\)\( T^{20} )^{2} \))(\( 1 + 32 T + 512 T^{2} - 38048 T^{3} - 60439624 T^{4} - 1520787552 T^{5} - 16996289024 T^{6} + 2981900088544 T^{7} + 2018049968078364 T^{8} + 40394929489472928 T^{9} + 275088896591278592 T^{10} - \)\(10\!\cdots\!56\)\( T^{11} - \)\(45\!\cdots\!20\)\( T^{12} - \)\(71\!\cdots\!16\)\( T^{13} - \)\(18\!\cdots\!28\)\( T^{14} + \)\(23\!\cdots\!64\)\( T^{15} + \)\(72\!\cdots\!42\)\( T^{16} + \)\(88\!\cdots\!44\)\( T^{17} - \)\(26\!\cdots\!48\)\( T^{18} - \)\(36\!\cdots\!76\)\( T^{19} - \)\(86\!\cdots\!20\)\( T^{20} - \)\(75\!\cdots\!56\)\( T^{21} + \)\(73\!\cdots\!32\)\( T^{22} + \)\(39\!\cdots\!48\)\( T^{23} + \)\(74\!\cdots\!04\)\( T^{24} + \)\(40\!\cdots\!64\)\( T^{25} - \)\(86\!\cdots\!24\)\( T^{26} - \)\(28\!\cdots\!92\)\( T^{27} - \)\(42\!\cdots\!84\)\( T^{28} - \)\(99\!\cdots\!28\)\( T^{29} + \)\(49\!\cdots\!72\)\( T^{30} + \)\(11\!\cdots\!32\)\( T^{31} + \)\(13\!\cdots\!21\)\( T^{32} \))
$67$ (\( 1 + 122 T + 4489 T^{2} \))(\( ( 1 + 2 T + 4489 T^{2} )^{2} \))(\( 1 + 4894 T^{2} + 20151121 T^{4} \))(\( ( 1 + 116 T + 6728 T^{2} + 350436 T^{3} + 16097858 T^{4} + 1573107204 T^{5} + 135576742088 T^{6} + 10493172331604 T^{7} + 406067677556641 T^{8} )^{2} \))(\( ( 1 - 178 T + 15842 T^{2} - 1501482 T^{3} + 163953249 T^{4} - 13424405736 T^{5} + 919420948512 T^{6} - 73920183453528 T^{7} + 6211456955376366 T^{8} - 413849456030337652 T^{9} + 25618409953668575740 T^{10} - \)\(18\!\cdots\!28\)\( T^{11} + \)\(12\!\cdots\!86\)\( T^{12} - \)\(66\!\cdots\!32\)\( T^{13} + \)\(37\!\cdots\!92\)\( T^{14} - \)\(24\!\cdots\!64\)\( T^{15} + \)\(13\!\cdots\!89\)\( T^{16} - \)\(55\!\cdots\!78\)\( T^{17} + \)\(26\!\cdots\!02\)\( T^{18} - \)\(13\!\cdots\!02\)\( T^{19} + \)\(33\!\cdots\!01\)\( T^{20} )^{2} \))(\( 1 - 320 T + 51200 T^{2} - 6047552 T^{3} + 641735304 T^{4} - 64228593856 T^{5} + 5982745065472 T^{6} - 525110406070976 T^{7} + 43992629224199580 T^{8} - 3502096836597496384 T^{9} + \)\(26\!\cdots\!12\)\( T^{10} - \)\(19\!\cdots\!80\)\( T^{11} + \)\(14\!\cdots\!20\)\( T^{12} - \)\(10\!\cdots\!88\)\( T^{13} + \)\(71\!\cdots\!04\)\( T^{14} - \)\(48\!\cdots\!52\)\( T^{15} + \)\(32\!\cdots\!74\)\( T^{16} - \)\(21\!\cdots\!28\)\( T^{17} + \)\(14\!\cdots\!84\)\( T^{18} - \)\(93\!\cdots\!72\)\( T^{19} + \)\(58\!\cdots\!20\)\( T^{20} - \)\(36\!\cdots\!20\)\( T^{21} + \)\(21\!\cdots\!32\)\( T^{22} - \)\(12\!\cdots\!36\)\( T^{23} + \)\(72\!\cdots\!80\)\( T^{24} - \)\(38\!\cdots\!84\)\( T^{25} + \)\(19\!\cdots\!72\)\( T^{26} - \)\(95\!\cdots\!84\)\( T^{27} + \)\(42\!\cdots\!84\)\( T^{28} - \)\(18\!\cdots\!88\)\( T^{29} + \)\(69\!\cdots\!00\)\( T^{30} - \)\(19\!\cdots\!80\)\( T^{31} + \)\(27\!\cdots\!61\)\( T^{32} \))
$71$ (\( ( 1 - 71 T )( 1 + 71 T ) \))(\( 1 + 5406 T^{2} + 25411681 T^{4} \))(\( 1 - 3170 T^{2} + 25411681 T^{4} \))(\( ( 1 + 3604 T^{2} + 10180454 T^{4} + 91583698324 T^{6} + 645753531245761 T^{8} )^{2} \))(\( ( 1 + 37534 T^{2} + 679811933 T^{4} + 7799021001352 T^{6} + 62618275556164866 T^{8} + \)\(36\!\cdots\!28\)\( T^{10} + \)\(15\!\cdots\!46\)\( T^{12} + \)\(50\!\cdots\!72\)\( T^{14} + \)\(11\!\cdots\!53\)\( T^{16} + \)\(15\!\cdots\!14\)\( T^{18} + \)\(10\!\cdots\!01\)\( T^{20} )^{2} \))(\( ( 1 - 256 T + 68104 T^{2} - 10692864 T^{3} + 1610923548 T^{4} - 179723087616 T^{5} + 18972832358712 T^{6} - 1588998739085056 T^{7} + 125568612540426694 T^{8} - 8010142643727767296 T^{9} + \)\(48\!\cdots\!72\)\( T^{10} - \)\(23\!\cdots\!36\)\( T^{11} + \)\(10\!\cdots\!28\)\( T^{12} - \)\(34\!\cdots\!64\)\( T^{13} + \)\(11\!\cdots\!64\)\( T^{14} - \)\(21\!\cdots\!36\)\( T^{15} + \)\(41\!\cdots\!21\)\( T^{16} )^{2} \))
$73$ (\( 1 + 46 T + 5329 T^{2} \))(\( ( 1 - 82 T + 5329 T^{2} )^{2} \))(\( ( 1 - 82 T + 5329 T^{2} )^{2} \))(\( ( 1 - 17604 T^{2} + 131615494 T^{4} - 499922634564 T^{6} + 806460091894081 T^{8} )^{2} \))(\( ( 1 - 37130 T^{2} + 665990797 T^{4} - 7634966781176 T^{6} + 62349742704798482 T^{8} - \)\(38\!\cdots\!28\)\( T^{10} + \)\(17\!\cdots\!62\)\( T^{12} - \)\(61\!\cdots\!56\)\( T^{14} + \)\(15\!\cdots\!37\)\( T^{16} - \)\(24\!\cdots\!30\)\( T^{18} + \)\(18\!\cdots\!01\)\( T^{20} )^{2} \))(\( 1 - 42768 T^{2} + 946714744 T^{4} - 14391245893936 T^{6} + 167549428359087132 T^{8} - \)\(15\!\cdots\!24\)\( T^{10} + \)\(12\!\cdots\!76\)\( T^{12} - \)\(83\!\cdots\!96\)\( T^{14} + \)\(47\!\cdots\!22\)\( T^{16} - \)\(23\!\cdots\!36\)\( T^{18} + \)\(10\!\cdots\!56\)\( T^{20} - \)\(36\!\cdots\!04\)\( T^{22} + \)\(10\!\cdots\!52\)\( T^{24} - \)\(26\!\cdots\!36\)\( T^{26} + \)\(49\!\cdots\!04\)\( T^{28} - \)\(63\!\cdots\!08\)\( T^{30} + \)\(42\!\cdots\!21\)\( T^{32} \))
$79$ (\( 1 - 142 T + 6241 T^{2} \))(\( ( 1 + 10 T + 6241 T^{2} )^{2} \))(\( 1 - 6674 T^{2} + 38950081 T^{4} \))(\( ( 1 + 34 T + 11588 T^{2} + 212194 T^{3} + 38950081 T^{4} )^{4} \))(\( ( 1 - 96 T + 27671 T^{2} - 2133500 T^{3} + 327107352 T^{4} - 19299334696 T^{5} + 2041476983832 T^{6} - 83099997813500 T^{7} + 6726472981721591 T^{8} - 145642445751029856 T^{9} + 9468276082626847201 T^{10} )^{4} \))(\( 1 - 62928 T^{2} + 1905826568 T^{4} - 37296559235888 T^{6} + 534425714020543644 T^{8} - \)\(60\!\cdots\!08\)\( T^{10} + \)\(55\!\cdots\!96\)\( T^{12} - \)\(43\!\cdots\!36\)\( T^{14} + \)\(29\!\cdots\!62\)\( T^{16} - \)\(16\!\cdots\!16\)\( T^{18} + \)\(84\!\cdots\!56\)\( T^{20} - \)\(35\!\cdots\!28\)\( T^{22} + \)\(12\!\cdots\!24\)\( T^{24} - \)\(33\!\cdots\!88\)\( T^{26} + \)\(66\!\cdots\!08\)\( T^{28} - \)\(85\!\cdots\!08\)\( T^{30} + \)\(52\!\cdots\!41\)\( T^{32} \))
$83$ (\( ( 1 - 83 T )( 1 + 83 T ) \))(\( 1 - 8370 T^{2} + 47458321 T^{4} \))(\( 1 - 13346 T^{2} + 47458321 T^{4} \))(\( 1 - 64624380 T^{4} + 2364945173919814 T^{8} - \)\(14\!\cdots\!80\)\( T^{12} + \)\(50\!\cdots\!81\)\( T^{16} \))(\( 1 + 11186750 T^{4} - 558936874100067 T^{8} - \)\(49\!\cdots\!36\)\( T^{12} + \)\(20\!\cdots\!66\)\( T^{16} + \)\(44\!\cdots\!52\)\( T^{20} + \)\(46\!\cdots\!06\)\( T^{24} - \)\(24\!\cdots\!16\)\( T^{28} - \)\(63\!\cdots\!07\)\( T^{32} + \)\(28\!\cdots\!50\)\( T^{36} + \)\(57\!\cdots\!01\)\( T^{40} \))(\( 1 + 160 T + 12800 T^{2} + 895904 T^{3} + 107479624 T^{4} + 16432771168 T^{5} + 1654826188288 T^{6} + 174484645067104 T^{7} + 18280323695716892 T^{8} + 1483531366054758688 T^{9} + \)\(11\!\cdots\!96\)\( T^{10} + \)\(11\!\cdots\!36\)\( T^{11} + \)\(13\!\cdots\!00\)\( T^{12} + \)\(12\!\cdots\!44\)\( T^{13} + \)\(89\!\cdots\!76\)\( T^{14} + \)\(74\!\cdots\!76\)\( T^{15} + \)\(61\!\cdots\!66\)\( T^{16} + \)\(51\!\cdots\!64\)\( T^{17} + \)\(42\!\cdots\!96\)\( T^{18} + \)\(39\!\cdots\!36\)\( T^{19} + \)\(30\!\cdots\!00\)\( T^{20} + \)\(17\!\cdots\!64\)\( T^{21} + \)\(12\!\cdots\!56\)\( T^{22} + \)\(10\!\cdots\!52\)\( T^{23} + \)\(92\!\cdots\!52\)\( T^{24} + \)\(60\!\cdots\!36\)\( T^{25} + \)\(39\!\cdots\!88\)\( T^{26} + \)\(27\!\cdots\!52\)\( T^{27} + \)\(12\!\cdots\!04\)\( T^{28} + \)\(70\!\cdots\!76\)\( T^{29} + \)\(69\!\cdots\!00\)\( T^{30} + \)\(59\!\cdots\!40\)\( T^{31} + \)\(25\!\cdots\!61\)\( T^{32} \))
$89$ (\( ( 1 - 89 T )( 1 + 89 T ) \))(\( 1 - 14690 T^{2} + 62742241 T^{4} \))(\( ( 1 - 114 T + 7921 T^{2} )^{2} \))(\( ( 1 + 25444 T^{2} + 277784198 T^{4} + 1596413580004 T^{6} + 3936588805702081 T^{8} )^{2} \))(\( ( 1 + 29470 T^{2} + 451584989 T^{4} + 4107082352008 T^{6} + 26252370181550850 T^{8} + \)\(16\!\cdots\!64\)\( T^{10} + \)\(16\!\cdots\!50\)\( T^{12} + \)\(16\!\cdots\!48\)\( T^{14} + \)\(11\!\cdots\!69\)\( T^{16} + \)\(45\!\cdots\!70\)\( T^{18} + \)\(97\!\cdots\!01\)\( T^{20} )^{2} \))(\( 1 - 81008 T^{2} + 3201135736 T^{4} - 82544801381712 T^{6} + 1567286911309649436 T^{8} - \)\(23\!\cdots\!04\)\( T^{10} + \)\(28\!\cdots\!72\)\( T^{12} - \)\(29\!\cdots\!36\)\( T^{14} + \)\(25\!\cdots\!10\)\( T^{16} - \)\(18\!\cdots\!76\)\( T^{18} + \)\(11\!\cdots\!32\)\( T^{20} - \)\(57\!\cdots\!84\)\( T^{22} + \)\(24\!\cdots\!96\)\( T^{24} - \)\(80\!\cdots\!12\)\( T^{26} + \)\(19\!\cdots\!76\)\( T^{28} - \)\(31\!\cdots\!48\)\( T^{30} + \)\(24\!\cdots\!21\)\( T^{32} \))
$97$ (\( 1 - 2 T + 9409 T^{2} \))(\( ( 1 + 94 T + 9409 T^{2} )^{2} \))(\( ( 1 - 34 T + 9409 T^{2} )^{2} \))(\( ( 1 + 120 T + 21970 T^{2} + 1129080 T^{3} + 88529281 T^{4} )^{4} \))(\( ( 1 - 118 T + 35265 T^{2} - 3292640 T^{3} + 583062270 T^{4} - 43725541204 T^{5} + 5486032898430 T^{6} - 291495051791840 T^{7} + 29374757753821185 T^{8} - 924817164136481398 T^{9} + 73742412689492826049 T^{10} )^{4} \))(\( ( 1 + 38216 T^{2} + 116224 T^{3} + 770481564 T^{4} + 3485408768 T^{5} + 10857255215864 T^{6} + 49274039499776 T^{7} + 116292098553803590 T^{8} + 463619437653392384 T^{9} + \)\(96\!\cdots\!84\)\( T^{10} + \)\(29\!\cdots\!72\)\( T^{11} + \)\(60\!\cdots\!04\)\( T^{12} + \)\(85\!\cdots\!76\)\( T^{13} + \)\(26\!\cdots\!56\)\( T^{14} + \)\(61\!\cdots\!21\)\( T^{16} )^{2} \))
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