Properties

Label 35.3
Level 35
Weight 3
Dimension 70
Nonzero newspaces 6
Newform subspaces 9
Sturm bound 288
Trace bound 2

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Defining parameters

Level: \( N \) = \( 35\( 35 = 5 \cdot 7 \) \)
Weight: \( k \) = \( 3 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 9 \)
Sturm bound: \(288\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(\Gamma_1(35))\).

Total New Old
Modular forms 120 98 22
Cusp forms 72 70 2
Eisenstein series 48 28 20

Trace form

\( 70q - 6q^{2} - 12q^{3} - 22q^{4} - 12q^{5} - 24q^{6} + 2q^{7} - 18q^{8} - 30q^{9} + O(q^{10}) \) \( 70q - 6q^{2} - 12q^{3} - 22q^{4} - 12q^{5} - 24q^{6} + 2q^{7} - 18q^{8} - 30q^{9} - 12q^{10} - 12q^{11} - 12q^{12} - 12q^{13} - 54q^{14} - 24q^{15} - 2q^{16} - 12q^{17} + 42q^{18} - 12q^{19} - 12q^{20} - 24q^{21} - 60q^{22} - 48q^{23} + 180q^{24} + 46q^{25} + 168q^{26} + 240q^{27} + 346q^{28} + 204q^{29} + 276q^{30} + 48q^{31} + 42q^{32} + 48q^{33} - 48q^{35} - 330q^{36} - 44q^{37} - 204q^{38} - 312q^{39} - 444q^{40} - 192q^{41} - 588q^{42} - 356q^{43} - 432q^{44} - 408q^{45} - 204q^{46} - 168q^{47} - 444q^{48} - 110q^{49} + 126q^{50} - 24q^{51} + 180q^{52} + 420q^{54} + 324q^{55} + 558q^{56} + 552q^{57} + 720q^{58} + 468q^{59} + 1068q^{60} + 552q^{61} + 744q^{62} + 690q^{63} + 878q^{64} + 276q^{65} + 408q^{66} + 416q^{67} + 168q^{68} - 192q^{70} - 372q^{71} - 546q^{72} - 396q^{73} - 900q^{74} - 588q^{75} - 1032q^{76} - 672q^{77} - 1320q^{78} - 784q^{79} - 1008q^{80} - 1050q^{81} - 1572q^{82} - 684q^{83} - 1020q^{84} - 792q^{85} - 60q^{86} + 168q^{87} + 336q^{88} + 888q^{89} + 900q^{90} + 600q^{91} + 1524q^{92} + 912q^{93} + 948q^{94} + 852q^{95} + 1320q^{96} + 492q^{97} + 1146q^{98} + 564q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(\Gamma_1(35))\)

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
35.3.c \(\chi_{35}(34, \cdot)\) 35.3.c.a 1 1
35.3.c.b 1
35.3.c.c 4
35.3.d \(\chi_{35}(6, \cdot)\) 35.3.d.a 2 1
35.3.d.b 2
35.3.g \(\chi_{35}(8, \cdot)\) 35.3.g.a 12 2
35.3.h \(\chi_{35}(26, \cdot)\) 35.3.h.a 12 2
35.3.i \(\chi_{35}(19, \cdot)\) 35.3.i.a 12 2
35.3.l \(\chi_{35}(2, \cdot)\) 35.3.l.a 24 4

Decomposition of \(S_{3}^{\mathrm{old}}(\Gamma_1(35))\) into lower level spaces

\( S_{3}^{\mathrm{old}}(\Gamma_1(35)) \cong \) \(S_{3}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - 2 T )( 1 + 2 T ) \))(\( ( 1 - 2 T )( 1 + 2 T ) \))(\( ( 1 + T^{2} + 16 T^{4} )^{2} \))(\( ( 1 + T + 4 T^{2} )^{2} \))(\( ( 1 - 2 T + 4 T^{2} )^{2} \))(\( 1 + 4 T + 8 T^{2} + 8 T^{3} - 26 T^{4} - 104 T^{5} - 176 T^{6} - 208 T^{7} + 9 T^{8} + 468 T^{9} + 968 T^{10} + 2568 T^{11} + 5732 T^{12} + 10272 T^{13} + 15488 T^{14} + 29952 T^{15} + 2304 T^{16} - 212992 T^{17} - 720896 T^{18} - 1703936 T^{19} - 1703936 T^{20} + 2097152 T^{21} + 8388608 T^{22} + 16777216 T^{23} + 16777216 T^{24} \))(\( 1 + 2 T - 5 T^{2} - 6 T^{3} + 20 T^{4} + 10 T^{5} + 25 T^{6} + 58 T^{7} - 436 T^{8} - 470 T^{9} + 1107 T^{10} - 102 T^{11} - 5463 T^{12} - 408 T^{13} + 17712 T^{14} - 30080 T^{15} - 111616 T^{16} + 59392 T^{17} + 102400 T^{18} + 163840 T^{19} + 1310720 T^{20} - 1572864 T^{21} - 5242880 T^{22} + 8388608 T^{23} + 16777216 T^{24} \))(\( 1 + 9 T^{2} + 36 T^{4} + 11 T^{6} - 468 T^{8} - 1791 T^{10} - 6135 T^{12} - 28656 T^{14} - 119808 T^{16} + 45056 T^{18} + 2359296 T^{20} + 9437184 T^{22} + 16777216 T^{24} \))
$3$ (\( 1 + T + 9 T^{2} \))(\( 1 - T + 9 T^{2} \))(\( ( 1 + 8 T^{2} + 81 T^{4} )^{2} \))(\( ( 1 - 4 T + 9 T^{2} )( 1 + 4 T + 9 T^{2} ) \))(\( 1 - 13 T^{2} + 81 T^{4} \))(\( 1 + 4 T + 8 T^{2} - 32 T^{3} - 185 T^{4} - 104 T^{5} + 1576 T^{6} + 6004 T^{7} - 7229 T^{8} - 84904 T^{9} - 176944 T^{10} + 302584 T^{11} + 2631886 T^{12} + 2723256 T^{13} - 14332464 T^{14} - 61895016 T^{15} - 47429469 T^{16} + 354530196 T^{17} + 837551016 T^{18} - 497428776 T^{19} - 7963643385 T^{20} - 12397455648 T^{21} + 27894275208 T^{22} + 125524238436 T^{23} + 282429536481 T^{24} \))(\( 1 + 6 T + 38 T^{2} + 156 T^{3} + 660 T^{4} + 2070 T^{5} + 5744 T^{6} + 10314 T^{7} - 68 T^{8} - 114228 T^{9} - 708426 T^{10} - 2725974 T^{11} - 9235782 T^{12} - 24533766 T^{13} - 57382506 T^{14} - 83272212 T^{15} - 446148 T^{16} + 609031386 T^{17} + 3052597104 T^{18} + 9900745830 T^{19} + 28410835860 T^{20} + 60437596284 T^{21} + 132497807238 T^{22} + 188286357654 T^{23} + 282429536481 T^{24} \))(\( 1 - 18 T^{2} + 48 T^{4} + 700 T^{6} + 2808 T^{8} - 95466 T^{10} + 798418 T^{12} - 7732746 T^{14} + 18423288 T^{16} + 372008700 T^{18} + 2066242608 T^{20} - 62762119218 T^{22} + 282429536481 T^{24} \))
$5$ (\( 1 - 5 T \))(\( 1 + 5 T \))(\( 1 + 40 T^{2} + 625 T^{4} \))(\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 8 T - 328 T^{3} - 1231 T^{4} + 2600 T^{5} + 41200 T^{6} + 65000 T^{7} - 769375 T^{8} - 5125000 T^{9} + 78125000 T^{11} + 244140625 T^{12} \))(\( ( 1 - 5 T^{2} + 25 T^{4} )^{3} \))(\( 1 - 9 T^{2} + 306 T^{4} - 2160 T^{5} - 10025 T^{6} - 54000 T^{7} + 191250 T^{8} - 3515625 T^{10} + 244140625 T^{12} \))
$7$ (\( 1 + 7 T \))(\( 1 - 7 T \))(\( 1 - 62 T^{2} + 2401 T^{4} \))(\( ( 1 - 7 T )^{2} \))(\( 1 + 4 T + 49 T^{2} \))(\( ( 1 + 49 T^{4} )^{3} \))(\( 1 + 2 T + 14 T^{2} + 348 T^{3} + 1588 T^{4} + 14434 T^{5} + 136318 T^{6} + 707266 T^{7} + 3812788 T^{8} + 40941852 T^{9} + 80707214 T^{10} + 564950498 T^{11} + 13841287201 T^{12} \))(\( 1 + 162 T^{2} + 12348 T^{4} + 657874 T^{6} + 29647548 T^{8} + 933897762 T^{10} + 13841287201 T^{12} \))
$11$ (\( 1 + 13 T + 121 T^{2} \))(\( 1 + 13 T + 121 T^{2} \))(\( ( 1 - 14 T + 121 T^{2} )^{4} \))(\( ( 1 - 2 T + 121 T^{2} )^{2} \))(\( ( 1 + T + 121 T^{2} )^{2} \))(\( ( 1 + 6 T + 319 T^{2} + 1166 T^{3} + 55579 T^{4} + 201340 T^{5} + 7907818 T^{6} + 24362140 T^{7} + 813732139 T^{8} + 2065640126 T^{9} + 68380483039 T^{10} + 155624547606 T^{11} + 3138428376721 T^{12} )^{2} \))(\( 1 + 14 T - 179 T^{2} - 3378 T^{3} - 10510 T^{4} - 101510 T^{5} - 1940909 T^{6} + 49596994 T^{7} + 1461472640 T^{8} + 7377470974 T^{9} - 78163118391 T^{10} - 1070296414578 T^{11} - 9257941759776 T^{12} - 129505866163938 T^{13} - 1144386216362631 T^{14} + 13069639856170414 T^{15} + 313279639722515840 T^{16} + 1286418292311249394 T^{17} - 6091403882233179389 T^{18} - 38548405607034793910 T^{19} - \)\(48\!\cdots\!10\)\( T^{20} - \)\(18\!\cdots\!18\)\( T^{21} - \)\(12\!\cdots\!79\)\( T^{22} + \)\(11\!\cdots\!94\)\( T^{23} + \)\(98\!\cdots\!41\)\( T^{24} \))(\( ( 1 - 3 T - 294 T^{2} + 599 T^{3} + 52884 T^{4} - 52407 T^{5} - 7129020 T^{6} - 6341247 T^{7} + 774274644 T^{8} + 1061165039 T^{9} - 63021511014 T^{10} - 77812273803 T^{11} + 3138428376721 T^{12} )^{2} \))
$13$ (\( 1 - 19 T + 169 T^{2} \))(\( 1 + 19 T + 169 T^{2} \))(\( ( 1 + 328 T^{2} + 28561 T^{4} )^{2} \))(\( 1 - 158 T^{2} + 28561 T^{4} \))(\( 1 + 67 T^{2} + 28561 T^{4} \))(\( 1 + 4 T + 8 T^{2} - 2424 T^{3} + 42679 T^{4} + 821112 T^{5} + 5880904 T^{6} - 59122612 T^{7} - 455341821 T^{8} + 28627711344 T^{9} + 347588147888 T^{10} + 1969414999296 T^{11} - 34074574484818 T^{12} + 332831134881024 T^{13} + 9927465091829168 T^{14} + 138180494764621296 T^{15} - 371436311945782941 T^{16} - 8150554124493589588 T^{17} + \)\(13\!\cdots\!24\)\( T^{18} + \)\(32\!\cdots\!68\)\( T^{19} + \)\(28\!\cdots\!39\)\( T^{20} - \)\(27\!\cdots\!96\)\( T^{21} + \)\(15\!\cdots\!08\)\( T^{22} + \)\(12\!\cdots\!76\)\( T^{23} + \)\(54\!\cdots\!61\)\( T^{24} \))(\( 1 - 1482 T^{2} + 1042167 T^{4} - 464849386 T^{6} + 148108329555 T^{8} - 35857522447788 T^{10} + 6808838187253242 T^{12} - 1024126698631273068 T^{14} + \)\(12\!\cdots\!55\)\( T^{16} - \)\(10\!\cdots\!66\)\( T^{18} + \)\(69\!\cdots\!47\)\( T^{20} - \)\(28\!\cdots\!82\)\( T^{22} + \)\(54\!\cdots\!61\)\( T^{24} \))(\( ( 1 + 525 T^{2} + 176235 T^{4} + 35127322 T^{6} + 5033447835 T^{8} + 428258628525 T^{10} + 23298085122481 T^{12} )^{2} \))
$17$ (\( 1 + 29 T + 289 T^{2} \))(\( 1 - 29 T + 289 T^{2} \))(\( ( 1 + 538 T^{2} + 83521 T^{4} )^{2} \))(\( 1 + 142 T^{2} + 83521 T^{4} \))(\( 1 - 533 T^{2} + 83521 T^{4} \))(\( 1 + 12 T + 72 T^{2} + 2784 T^{3} + 76619 T^{4} + 611296 T^{5} + 5694312 T^{6} + 329614108 T^{7} - 1813984861 T^{8} - 125752615304 T^{9} - 710229888304 T^{10} - 21298315351968 T^{11} - 439455529642318 T^{12} - 6155213136718752 T^{13} - 59319110501038384 T^{14} - 3035362428830755976 T^{15} - 12653918391982100701 T^{16} + \)\(66\!\cdots\!92\)\( T^{17} + \)\(33\!\cdots\!32\)\( T^{18} + \)\(10\!\cdots\!84\)\( T^{19} + \)\(37\!\cdots\!39\)\( T^{20} + \)\(39\!\cdots\!56\)\( T^{21} + \)\(29\!\cdots\!72\)\( T^{22} + \)\(14\!\cdots\!68\)\( T^{23} + \)\(33\!\cdots\!21\)\( T^{24} \))(\( 1 - 48 T + 2382 T^{2} - 77472 T^{3} + 2498625 T^{4} - 67135632 T^{5} + 1727446034 T^{6} - 40136784912 T^{7} + 880931070942 T^{8} - 18174432672288 T^{9} + 351810098240190 T^{10} - 6513275972033712 T^{11} + 112870694757995013 T^{12} - 1882336755917742768 T^{13} + 29383531215118908990 T^{14} - \)\(43\!\cdots\!72\)\( T^{15} + \)\(61\!\cdots\!22\)\( T^{16} - \)\(80\!\cdots\!88\)\( T^{17} + \)\(10\!\cdots\!74\)\( T^{18} - \)\(11\!\cdots\!28\)\( T^{19} + \)\(12\!\cdots\!25\)\( T^{20} - \)\(10\!\cdots\!48\)\( T^{21} + \)\(96\!\cdots\!82\)\( T^{22} - \)\(56\!\cdots\!72\)\( T^{23} + \)\(33\!\cdots\!21\)\( T^{24} \))(\( 1 - 978 T^{2} + 403953 T^{4} - 153666190 T^{6} + 70041800718 T^{8} - 23391307509426 T^{10} + 6380863165523493 T^{12} - 1953665394494768946 T^{14} + \)\(48\!\cdots\!38\)\( T^{16} - \)\(89\!\cdots\!90\)\( T^{18} + \)\(19\!\cdots\!93\)\( T^{20} - \)\(39\!\cdots\!78\)\( T^{22} + \)\(33\!\cdots\!21\)\( T^{24} \))
$19$ (\( ( 1 - 19 T )( 1 + 19 T ) \))(\( ( 1 - 19 T )( 1 + 19 T ) \))(\( ( 1 + 88 T^{2} + 130321 T^{4} )^{2} \))(\( 1 - 542 T^{2} + 130321 T^{4} \))(\( 1 - 542 T^{2} + 130321 T^{4} \))(\( 1 - 2208 T^{2} + 2306338 T^{4} - 1539873472 T^{6} + 753579200867 T^{8} - 300967197557024 T^{10} + 110083881158431556 T^{12} - 39222346152828924704 T^{14} + \)\(12\!\cdots\!47\)\( T^{16} - \)\(34\!\cdots\!92\)\( T^{18} + \)\(66\!\cdots\!78\)\( T^{20} - \)\(82\!\cdots\!08\)\( T^{22} + \)\(48\!\cdots\!21\)\( T^{24} \))(\( 1 + 30 T + 1401 T^{2} + 33030 T^{3} + 884826 T^{4} + 20001714 T^{5} + 344601839 T^{6} + 6453829386 T^{7} + 66537113172 T^{8} + 653188957326 T^{9} - 1006933743831 T^{10} - 384405046252650 T^{11} - 5242806750443784 T^{12} - 138770221697206650 T^{13} - 131224612429799751 T^{14} + 30729849956873074206 T^{15} + \)\(11\!\cdots\!52\)\( T^{16} + \)\(39\!\cdots\!86\)\( T^{17} + \)\(76\!\cdots\!79\)\( T^{18} + \)\(15\!\cdots\!94\)\( T^{19} + \)\(25\!\cdots\!06\)\( T^{20} + \)\(34\!\cdots\!30\)\( T^{21} + \)\(52\!\cdots\!01\)\( T^{22} + \)\(40\!\cdots\!30\)\( T^{23} + \)\(48\!\cdots\!21\)\( T^{24} \))(\( ( 1 - 9 T + 786 T^{2} - 6831 T^{3} + 329514 T^{4} - 5767407 T^{5} + 138225724 T^{6} - 2082033927 T^{7} + 42942593994 T^{8} - 321370413111 T^{9} + 13349080550226 T^{10} - 55179596320209 T^{11} + 2213314919066161 T^{12} )^{2} \))
$23$ (\( ( 1 - 23 T )( 1 + 23 T ) \))(\( ( 1 - 23 T )( 1 + 23 T ) \))(\( ( 1 - 914 T^{2} + 279841 T^{4} )^{2} \))(\( ( 1 - 26 T + 529 T^{2} )^{2} \))(\( ( 1 - 8 T + 529 T^{2} )^{2} \))(\( 1 + 16 T + 128 T^{2} + 12576 T^{3} + 414846 T^{4} - 3136832 T^{5} - 24211712 T^{6} - 1482627024 T^{7} + 34808010943 T^{8} + 851590367168 T^{9} + 5488492394624 T^{10} + 645053822277920 T^{11} + 70544571658994180 T^{12} + 341233471985019680 T^{13} + 1535905200203974784 T^{14} + \)\(12\!\cdots\!52\)\( T^{15} + \)\(27\!\cdots\!83\)\( T^{16} - \)\(61\!\cdots\!76\)\( T^{17} - \)\(53\!\cdots\!52\)\( T^{18} - \)\(36\!\cdots\!88\)\( T^{19} + \)\(25\!\cdots\!06\)\( T^{20} + \)\(40\!\cdots\!44\)\( T^{21} + \)\(21\!\cdots\!28\)\( T^{22} + \)\(14\!\cdots\!64\)\( T^{23} + \)\(48\!\cdots\!41\)\( T^{24} \))(\( 1 + 14 T - 1631 T^{2} - 3594 T^{3} + 1665797 T^{4} - 8196524 T^{5} - 890248094 T^{6} + 13524083944 T^{7} + 223091831501 T^{8} - 7643356730246 T^{9} + 95428737941649 T^{10} + 2006796517495686 T^{11} - 92105247923668542 T^{12} + 1061595357755217894 T^{13} + 26704873454328997809 T^{14} - \)\(11\!\cdots\!94\)\( T^{15} + \)\(17\!\cdots\!81\)\( T^{16} + \)\(56\!\cdots\!56\)\( T^{17} - \)\(19\!\cdots\!74\)\( T^{18} - \)\(95\!\cdots\!16\)\( T^{19} + \)\(10\!\cdots\!17\)\( T^{20} - \)\(11\!\cdots\!86\)\( T^{21} - \)\(27\!\cdots\!31\)\( T^{22} + \)\(12\!\cdots\!06\)\( T^{23} + \)\(48\!\cdots\!41\)\( T^{24} \))(\( 1 + 1791 T^{2} + 1735353 T^{4} + 949509974 T^{6} + 244638905769 T^{8} - 93367738199529 T^{10} - 102138108157049694 T^{12} - 26128121225494394889 T^{14} + \)\(19\!\cdots\!89\)\( T^{16} + \)\(20\!\cdots\!54\)\( T^{18} + \)\(10\!\cdots\!33\)\( T^{20} + \)\(30\!\cdots\!91\)\( T^{22} + \)\(48\!\cdots\!41\)\( T^{24} \))
$29$ (\( 1 - 23 T + 841 T^{2} \))(\( 1 - 23 T + 841 T^{2} \))(\( ( 1 - 14 T + 841 T^{2} )^{4} \))(\( ( 1 + 22 T + 841 T^{2} )^{2} \))(\( ( 1 - 41 T + 841 T^{2} )^{2} \))(\( 1 - 7018 T^{2} + 22990943 T^{4} - 47164435002 T^{6} + 68979050432827 T^{8} - 78004905252931004 T^{10} + 71933262446730943946 T^{12} - \)\(55\!\cdots\!24\)\( T^{14} + \)\(34\!\cdots\!47\)\( T^{16} - \)\(16\!\cdots\!82\)\( T^{18} + \)\(57\!\cdots\!03\)\( T^{20} - \)\(12\!\cdots\!18\)\( T^{22} + \)\(12\!\cdots\!81\)\( T^{24} \))(\( ( 1 - 32 T + 3552 T^{2} - 122256 T^{3} + 5916460 T^{4} - 196641728 T^{5} + 6117094942 T^{6} - 165375693248 T^{7} + 4184599745260 T^{8} - 72720719932176 T^{9} + 1776875258837472 T^{10} - 13462631465606432 T^{11} + 353814783205469041 T^{12} )^{2} \))(\( ( 1 + 1980 T^{2} - 4508 T^{3} + 1665180 T^{4} + 594823321 T^{6} )^{4} \))
$31$ (\( ( 1 - 31 T )( 1 + 31 T ) \))(\( ( 1 - 31 T )( 1 + 31 T ) \))(\( ( 1 - 482 T^{2} + 923521 T^{4} )^{2} \))(\( 1 + 958 T^{2} + 923521 T^{4} \))(\( 1 - 302 T^{2} + 923521 T^{4} \))(\( ( 1 + 48 T + 3144 T^{2} + 79440 T^{3} + 3260215 T^{4} + 42545920 T^{5} + 2350062944 T^{6} + 40886629120 T^{7} + 3010877017015 T^{8} + 70503292418640 T^{9} + 2681489421714504 T^{10} + 39342157775078448 T^{11} + 787662783788549761 T^{12} )^{2} \))(\( 1 - 132 T + 12894 T^{2} - 935352 T^{3} + 58065489 T^{4} - 3050976288 T^{5} + 143999803298 T^{6} - 6064589395908 T^{7} + 236052211330302 T^{8} - 8471733336563436 T^{9} + 289433514790600350 T^{10} - 9384030507705444000 T^{11} + \)\(29\!\cdots\!29\)\( T^{12} - \)\(90\!\cdots\!00\)\( T^{13} + \)\(26\!\cdots\!50\)\( T^{14} - \)\(75\!\cdots\!16\)\( T^{15} + \)\(20\!\cdots\!82\)\( T^{16} - \)\(49\!\cdots\!08\)\( T^{17} + \)\(11\!\cdots\!78\)\( T^{18} - \)\(23\!\cdots\!48\)\( T^{19} + \)\(42\!\cdots\!09\)\( T^{20} - \)\(65\!\cdots\!32\)\( T^{21} + \)\(86\!\cdots\!94\)\( T^{22} - \)\(85\!\cdots\!52\)\( T^{23} + \)\(62\!\cdots\!21\)\( T^{24} \))(\( ( 1 + 54 T + 3969 T^{2} + 161838 T^{3} + 7973532 T^{4} + 261757926 T^{5} + 9562987675 T^{6} + 251549366886 T^{7} + 7363724246172 T^{8} + 143631820725678 T^{9} + 3385124527603329 T^{10} + 44259927496963254 T^{11} + 787662783788549761 T^{12} )^{2} \))
$37$ (\( ( 1 - 37 T )( 1 + 37 T ) \))(\( ( 1 - 37 T )( 1 + 37 T ) \))(\( ( 1 - 2414 T^{2} + 1874161 T^{4} )^{2} \))(\( ( 1 - 14 T + 1369 T^{2} )^{2} \))(\( ( 1 + 28 T + 1369 T^{2} )^{2} \))(\( 1 + 104 T + 5408 T^{2} + 174344 T^{3} + 2697254 T^{4} + 10557800 T^{5} + 1709176736 T^{6} + 138455876040 T^{7} + 4386451218735 T^{8} + 70184122125200 T^{9} + 3161821544956736 T^{10} + 334287451090079440 T^{11} + 17427941461580290900 T^{12} + \)\(45\!\cdots\!60\)\( T^{13} + \)\(59\!\cdots\!96\)\( T^{14} + \)\(18\!\cdots\!00\)\( T^{15} + \)\(15\!\cdots\!35\)\( T^{16} + \)\(66\!\cdots\!60\)\( T^{17} + \)\(11\!\cdots\!16\)\( T^{18} + \)\(95\!\cdots\!00\)\( T^{19} + \)\(33\!\cdots\!14\)\( T^{20} + \)\(29\!\cdots\!76\)\( T^{21} + \)\(12\!\cdots\!08\)\( T^{22} + \)\(32\!\cdots\!76\)\( T^{23} + \)\(43\!\cdots\!61\)\( T^{24} \))(\( 1 - 44 T - 4291 T^{2} + 241124 T^{3} + 9668742 T^{4} - 689019636 T^{5} - 12024976949 T^{6} + 1280625867436 T^{7} + 4009330536896 T^{8} - 1538422889903564 T^{9} + 15464652696139049 T^{10} + 849812472525660964 T^{11} - 34333638656751212332 T^{12} + \)\(11\!\cdots\!16\)\( T^{13} + \)\(28\!\cdots\!89\)\( T^{14} - \)\(39\!\cdots\!76\)\( T^{15} + \)\(14\!\cdots\!16\)\( T^{16} + \)\(61\!\cdots\!64\)\( T^{17} - \)\(79\!\cdots\!69\)\( T^{18} - \)\(62\!\cdots\!04\)\( T^{19} + \)\(11\!\cdots\!22\)\( T^{20} + \)\(40\!\cdots\!96\)\( T^{21} - \)\(99\!\cdots\!91\)\( T^{22} - \)\(13\!\cdots\!36\)\( T^{23} + \)\(43\!\cdots\!61\)\( T^{24} \))(\( 1 + 1311 T^{2} - 608082 T^{4} - 4472473351 T^{6} - 3878060107536 T^{8} + 1923001154428131 T^{10} + 11130752847775425036 T^{12} + \)\(36\!\cdots\!91\)\( T^{14} - \)\(13\!\cdots\!56\)\( T^{16} - \)\(29\!\cdots\!31\)\( T^{18} - \)\(75\!\cdots\!62\)\( T^{20} + \)\(30\!\cdots\!11\)\( T^{22} + \)\(43\!\cdots\!61\)\( T^{24} \))
$41$ (\( ( 1 - 41 T )( 1 + 41 T ) \))(\( ( 1 - 41 T )( 1 + 41 T ) \))(\( ( 1 - 3002 T^{2} + 2825761 T^{4} )^{2} \))(\( 1 - 2642 T^{2} + 2825761 T^{4} \))(\( 1 - 3182 T^{2} + 2825761 T^{4} \))(\( ( 1 + 104 T + 11808 T^{2} + 782920 T^{3} + 51161335 T^{4} + 2462767696 T^{5} + 114809897552 T^{6} + 4139912496976 T^{7} + 144569705150935 T^{8} + 3718951612363720 T^{9} + 94285997105460768 T^{10} + 1395956568255849704 T^{11} + 22563490300366186081 T^{12} )^{2} \))(\( 1 - 6978 T^{2} + 29395173 T^{4} - 89601849670 T^{6} + 218675773246986 T^{8} - 448976563577597898 T^{10} + \)\(80\!\cdots\!77\)\( T^{12} - \)\(12\!\cdots\!78\)\( T^{14} + \)\(17\!\cdots\!06\)\( T^{16} - \)\(20\!\cdots\!70\)\( T^{18} + \)\(18\!\cdots\!93\)\( T^{20} - \)\(12\!\cdots\!78\)\( T^{22} + \)\(50\!\cdots\!61\)\( T^{24} \))(\( ( 1 - 9009 T^{2} + 35490546 T^{4} - 77874429761 T^{6} + 100287800755506 T^{8} - 71936191389151089 T^{10} + 22563490300366186081 T^{12} )^{2} \))
$43$ (\( ( 1 - 43 T )( 1 + 43 T ) \))(\( ( 1 - 43 T )( 1 + 43 T ) \))(\( ( 1 - 1934 T^{2} + 3418801 T^{4} )^{2} \))(\( ( 1 + 34 T + 1849 T^{2} )^{2} \))(\( ( 1 + 82 T + 1849 T^{2} )^{2} \))(\( 1 - 76 T + 2888 T^{2} - 67684 T^{3} + 11551602 T^{4} - 935583796 T^{5} + 40033903848 T^{6} - 1100447483004 T^{7} + 64203765803999 T^{8} - 5036169803944728 T^{9} + 247013912808013264 T^{10} - 7962007386466391112 T^{11} + \)\(24\!\cdots\!96\)\( T^{12} - \)\(14\!\cdots\!88\)\( T^{13} + \)\(84\!\cdots\!64\)\( T^{14} - \)\(31\!\cdots\!72\)\( T^{15} + \)\(75\!\cdots\!99\)\( T^{16} - \)\(23\!\cdots\!96\)\( T^{17} + \)\(15\!\cdots\!48\)\( T^{18} - \)\(69\!\cdots\!04\)\( T^{19} + \)\(15\!\cdots\!02\)\( T^{20} - \)\(17\!\cdots\!16\)\( T^{21} + \)\(13\!\cdots\!88\)\( T^{22} - \)\(65\!\cdots\!24\)\( T^{23} + \)\(15\!\cdots\!01\)\( T^{24} \))(\( ( 1 + 2 T + 7538 T^{2} + 1596 T^{3} + 27008008 T^{4} - 11306606 T^{5} + 60967237342 T^{6} - 20905914494 T^{7} + 92335004758408 T^{8} + 10088895426204 T^{9} + 88105653692556338 T^{10} + 43222964626568498 T^{11} + 39959630797262576401 T^{12} )^{2} \))(\( ( 1 - 4302 T^{2} + 12294324 T^{4} - 28688891246 T^{6} + 42031847185524 T^{8} - 50282637594239502 T^{10} + 39959630797262576401 T^{12} )^{2} \))
$47$ (\( 1 - 31 T + 2209 T^{2} \))(\( 1 + 31 T + 2209 T^{2} \))(\( ( 1 + 2458 T^{2} + 4879681 T^{4} )^{2} \))(\( 1 - 3698 T^{2} + 4879681 T^{4} \))(\( 1 - 4013 T^{2} + 4879681 T^{4} \))(\( 1 + 164 T + 13448 T^{2} + 868672 T^{3} + 63690059 T^{4} + 4678476896 T^{5} + 288061819304 T^{6} + 15844760579700 T^{7} + 894357153623779 T^{8} + 50754877518397512 T^{9} + 2657783539457568848 T^{10} + \)\(12\!\cdots\!80\)\( T^{11} + \)\(60\!\cdots\!22\)\( T^{12} + \)\(28\!\cdots\!20\)\( T^{13} + \)\(12\!\cdots\!88\)\( T^{14} + \)\(54\!\cdots\!48\)\( T^{15} + \)\(21\!\cdots\!19\)\( T^{16} + \)\(83\!\cdots\!00\)\( T^{17} + \)\(33\!\cdots\!64\)\( T^{18} + \)\(12\!\cdots\!24\)\( T^{19} + \)\(36\!\cdots\!39\)\( T^{20} + \)\(10\!\cdots\!08\)\( T^{21} + \)\(37\!\cdots\!48\)\( T^{22} + \)\(10\!\cdots\!76\)\( T^{23} + \)\(13\!\cdots\!81\)\( T^{24} \))(\( 1 - 204 T + 29397 T^{2} - 3167100 T^{3} + 290418402 T^{4} - 23370921444 T^{5} + 1705985151887 T^{6} - 114387377070276 T^{7} + 7121727778488432 T^{8} - 413844362647971516 T^{9} + 22564123030969504557 T^{10} - \)\(11\!\cdots\!96\)\( T^{11} + \)\(56\!\cdots\!04\)\( T^{12} - \)\(25\!\cdots\!64\)\( T^{13} + \)\(11\!\cdots\!17\)\( T^{14} - \)\(44\!\cdots\!64\)\( T^{15} + \)\(16\!\cdots\!52\)\( T^{16} - \)\(60\!\cdots\!24\)\( T^{17} + \)\(19\!\cdots\!67\)\( T^{18} - \)\(59\!\cdots\!36\)\( T^{19} + \)\(16\!\cdots\!42\)\( T^{20} - \)\(39\!\cdots\!00\)\( T^{21} + \)\(81\!\cdots\!97\)\( T^{22} - \)\(12\!\cdots\!36\)\( T^{23} + \)\(13\!\cdots\!81\)\( T^{24} \))(\( 1 - 4917 T^{2} + 4518390 T^{4} - 4665949483 T^{6} + 64359941563332 T^{8} - 77972699319432741 T^{10} - \)\(11\!\cdots\!28\)\( T^{12} - \)\(38\!\cdots\!21\)\( T^{14} + \)\(15\!\cdots\!52\)\( T^{16} - \)\(54\!\cdots\!03\)\( T^{18} + \)\(25\!\cdots\!90\)\( T^{20} - \)\(13\!\cdots\!17\)\( T^{22} + \)\(13\!\cdots\!81\)\( T^{24} \))
$53$ (\( ( 1 - 53 T )( 1 + 53 T ) \))(\( ( 1 - 53 T )( 1 + 53 T ) \))(\( ( 1 - 2702 T^{2} + 7890481 T^{4} )^{2} \))(\( ( 1 + 34 T + 2809 T^{2} )^{2} \))(\( ( 1 - 74 T + 2809 T^{2} )^{2} \))(\( 1 + 204 T + 20808 T^{2} + 1399092 T^{3} + 75768866 T^{4} + 3705029700 T^{5} + 157956707304 T^{6} + 4490093784828 T^{7} - 72880136612449 T^{8} - 22948283226553704 T^{9} - 1987483652559492912 T^{10} - \)\(11\!\cdots\!40\)\( T^{11} - \)\(61\!\cdots\!96\)\( T^{12} - \)\(33\!\cdots\!60\)\( T^{13} - \)\(15\!\cdots\!72\)\( T^{14} - \)\(50\!\cdots\!16\)\( T^{15} - \)\(45\!\cdots\!89\)\( T^{16} + \)\(78\!\cdots\!72\)\( T^{17} + \)\(77\!\cdots\!64\)\( T^{18} + \)\(51\!\cdots\!00\)\( T^{19} + \)\(29\!\cdots\!86\)\( T^{20} + \)\(15\!\cdots\!88\)\( T^{21} + \)\(63\!\cdots\!08\)\( T^{22} + \)\(17\!\cdots\!36\)\( T^{23} + \)\(24\!\cdots\!81\)\( T^{24} \))(\( 1 - 196 T + 13213 T^{2} - 531828 T^{3} + 46624490 T^{4} - 3460227932 T^{5} + 114727083151 T^{6} - 3187972234364 T^{7} + 109323786787880 T^{8} + 2196815143901836 T^{9} - 301463170054474635 T^{10} + 37355670780781789692 T^{11} - \)\(32\!\cdots\!76\)\( T^{12} + \)\(10\!\cdots\!28\)\( T^{13} - \)\(23\!\cdots\!35\)\( T^{14} + \)\(48\!\cdots\!44\)\( T^{15} + \)\(68\!\cdots\!80\)\( T^{16} - \)\(55\!\cdots\!36\)\( T^{17} + \)\(56\!\cdots\!91\)\( T^{18} - \)\(47\!\cdots\!08\)\( T^{19} + \)\(18\!\cdots\!90\)\( T^{20} - \)\(57\!\cdots\!92\)\( T^{21} + \)\(40\!\cdots\!13\)\( T^{22} - \)\(16\!\cdots\!64\)\( T^{23} + \)\(24\!\cdots\!81\)\( T^{24} \))(\( 1 + 9987 T^{2} + 43544766 T^{4} + 176791767125 T^{6} + 794854751690700 T^{8} + 2566788752469653379 T^{10} + \)\(68\!\cdots\!68\)\( T^{12} + \)\(20\!\cdots\!99\)\( T^{14} + \)\(49\!\cdots\!00\)\( T^{16} + \)\(86\!\cdots\!25\)\( T^{18} + \)\(16\!\cdots\!86\)\( T^{20} + \)\(30\!\cdots\!87\)\( T^{22} + \)\(24\!\cdots\!81\)\( T^{24} \))
$59$ (\( ( 1 - 59 T )( 1 + 59 T ) \))(\( ( 1 - 59 T )( 1 + 59 T ) \))(\( ( 1 - 6872 T^{2} + 12117361 T^{4} )^{2} \))(\( 1 - 5342 T^{2} + 12117361 T^{4} \))(\( 1 + 1858 T^{2} + 12117361 T^{4} \))(\( 1 - 27144 T^{2} + 365962322 T^{4} - 3231050981288 T^{6} + 20796912147061923 T^{8} - \)\(10\!\cdots\!08\)\( T^{10} + \)\(40\!\cdots\!08\)\( T^{12} - \)\(12\!\cdots\!88\)\( T^{14} + \)\(30\!\cdots\!83\)\( T^{16} - \)\(57\!\cdots\!28\)\( T^{18} + \)\(78\!\cdots\!02\)\( T^{20} - \)\(70\!\cdots\!44\)\( T^{22} + \)\(31\!\cdots\!61\)\( T^{24} \))(\( 1 - 72 T + 17682 T^{2} - 1148688 T^{3} + 164787345 T^{4} - 10566780840 T^{5} + 1093761678926 T^{6} - 70159293997992 T^{7} + 5748831154849182 T^{8} - 368559907967058624 T^{9} + 25324575120331389906 T^{10} - \)\(15\!\cdots\!28\)\( T^{11} + \)\(95\!\cdots\!93\)\( T^{12} - \)\(54\!\cdots\!68\)\( T^{13} + \)\(30\!\cdots\!66\)\( T^{14} - \)\(15\!\cdots\!84\)\( T^{15} + \)\(84\!\cdots\!22\)\( T^{16} - \)\(35\!\cdots\!92\)\( T^{17} + \)\(19\!\cdots\!06\)\( T^{18} - \)\(65\!\cdots\!40\)\( T^{19} + \)\(35\!\cdots\!45\)\( T^{20} - \)\(86\!\cdots\!48\)\( T^{21} + \)\(46\!\cdots\!82\)\( T^{22} - \)\(65\!\cdots\!32\)\( T^{23} + \)\(31\!\cdots\!61\)\( T^{24} \))(\( ( 1 - 198 T + 20817 T^{2} - 1534302 T^{3} + 81357012 T^{4} - 3269775942 T^{5} + 144715968451 T^{6} - 11382090054102 T^{7} + 985832284285332 T^{8} - 64717677126453582 T^{9} + 3056569219609150257 T^{10} - \)\(10\!\cdots\!98\)\( T^{11} + \)\(17\!\cdots\!81\)\( T^{12} )^{2} \))
$61$ (\( ( 1 - 61 T )( 1 + 61 T ) \))(\( ( 1 - 61 T )( 1 + 61 T ) \))(\( ( 1 - 3032 T^{2} + 13845841 T^{4} )^{2} \))(\( 1 + 1378 T^{2} + 13845841 T^{4} \))(\( 1 - 962 T^{2} + 13845841 T^{4} \))(\( ( 1 - 140 T + 18986 T^{2} - 1612508 T^{3} + 146964673 T^{4} - 10078896872 T^{5} + 706860431120 T^{6} - 37503575260712 T^{7} + 2034849494974993 T^{8} - 83077015820107388 T^{9} + 3639755044566377066 T^{10} - 99868007632803564140 T^{11} + \)\(26\!\cdots\!21\)\( T^{12} )^{2} \))(\( 1 - 72 T + 19176 T^{2} - 1256256 T^{3} + 196053300 T^{4} - 13507864560 T^{5} + 1475851108916 T^{6} - 102662314431264 T^{7} + 8714109428620692 T^{8} - 590990079276027360 T^{9} + 42479838876985802160 T^{10} - \)\(27\!\cdots\!84\)\( T^{11} + \)\(17\!\cdots\!82\)\( T^{12} - \)\(10\!\cdots\!64\)\( T^{13} + \)\(58\!\cdots\!60\)\( T^{14} - \)\(30\!\cdots\!60\)\( T^{15} + \)\(16\!\cdots\!52\)\( T^{16} - \)\(73\!\cdots\!64\)\( T^{17} + \)\(39\!\cdots\!36\)\( T^{18} - \)\(13\!\cdots\!60\)\( T^{19} + \)\(72\!\cdots\!00\)\( T^{20} - \)\(17\!\cdots\!36\)\( T^{21} + \)\(97\!\cdots\!76\)\( T^{22} - \)\(13\!\cdots\!12\)\( T^{23} + \)\(70\!\cdots\!41\)\( T^{24} \))(\( ( 1 + 54 T + 9270 T^{2} + 448092 T^{3} + 43173882 T^{4} + 1646729082 T^{5} + 167471347078 T^{6} + 6127478914122 T^{7} + 597778705524762 T^{8} + 23085867588169212 T^{9} + 1777126791484794870 T^{10} + 38520517229795660454 T^{11} + \)\(26\!\cdots\!21\)\( T^{12} )^{2} \))
$67$ (\( ( 1 - 67 T )( 1 + 67 T ) \))(\( ( 1 - 67 T )( 1 + 67 T ) \))(\( ( 1 + 1426 T^{2} + 20151121 T^{4} )^{2} \))(\( ( 1 - 14 T + 4489 T^{2} )^{2} \))(\( ( 1 - 2 T + 4489 T^{2} )^{2} \))(\( 1 - 324 T + 52488 T^{2} - 5922460 T^{3} + 578352418 T^{4} - 55559962188 T^{5} + 5182632258728 T^{6} - 447217160038868 T^{7} + 35870583162738015 T^{8} - 2754817067496129224 T^{9} + \)\(20\!\cdots\!04\)\( T^{10} - \)\(14\!\cdots\!28\)\( T^{11} + \)\(99\!\cdots\!72\)\( T^{12} - \)\(65\!\cdots\!92\)\( T^{13} + \)\(41\!\cdots\!84\)\( T^{14} - \)\(24\!\cdots\!56\)\( T^{15} + \)\(14\!\cdots\!15\)\( T^{16} - \)\(81\!\cdots\!32\)\( T^{17} + \)\(42\!\cdots\!08\)\( T^{18} - \)\(20\!\cdots\!52\)\( T^{19} + \)\(95\!\cdots\!58\)\( T^{20} - \)\(43\!\cdots\!40\)\( T^{21} + \)\(17\!\cdots\!88\)\( T^{22} - \)\(48\!\cdots\!36\)\( T^{23} + \)\(66\!\cdots\!21\)\( T^{24} \))(\( 1 + 138 T - 9270 T^{2} - 1437548 T^{3} + 115886436 T^{4} + 10557292578 T^{5} - 1139954744456 T^{6} - 58817128757610 T^{7} + 8471104062197196 T^{8} + 229786064617296868 T^{9} - 50182738990417249110 T^{10} - \)\(37\!\cdots\!50\)\( T^{11} + \)\(25\!\cdots\!50\)\( T^{12} - \)\(16\!\cdots\!50\)\( T^{13} - \)\(10\!\cdots\!10\)\( T^{14} + \)\(20\!\cdots\!92\)\( T^{15} + \)\(34\!\cdots\!36\)\( T^{16} - \)\(10\!\cdots\!90\)\( T^{17} - \)\(93\!\cdots\!16\)\( T^{18} + \)\(38\!\cdots\!62\)\( T^{19} + \)\(19\!\cdots\!16\)\( T^{20} - \)\(10\!\cdots\!32\)\( T^{21} - \)\(30\!\cdots\!70\)\( T^{22} + \)\(20\!\cdots\!82\)\( T^{23} + \)\(66\!\cdots\!21\)\( T^{24} \))(\( 1 + 18594 T^{2} + 182612664 T^{4} + 1241779502180 T^{6} + 6668167043256336 T^{8} + 31414801721204265066 T^{10} + \)\(14\!\cdots\!18\)\( T^{12} + \)\(63\!\cdots\!86\)\( T^{14} + \)\(27\!\cdots\!76\)\( T^{16} + \)\(10\!\cdots\!80\)\( T^{18} + \)\(30\!\cdots\!84\)\( T^{20} + \)\(61\!\cdots\!94\)\( T^{22} + \)\(66\!\cdots\!21\)\( T^{24} \))
$71$ (\( 1 - 2 T + 5041 T^{2} \))(\( 1 - 2 T + 5041 T^{2} \))(\( ( 1 + 16 T + 5041 T^{2} )^{4} \))(\( ( 1 - 62 T + 5041 T^{2} )^{2} \))(\( ( 1 - 14 T + 5041 T^{2} )^{2} \))(\( ( 1 - 72 T + 18698 T^{2} - 925624 T^{3} + 153299139 T^{4} - 5976553984 T^{5} + 867379084884 T^{6} - 30127808633344 T^{7} + 3895588817842659 T^{8} - 118572697204091704 T^{9} + 12074299527233239178 T^{10} - \)\(23\!\cdots\!72\)\( T^{11} + \)\(16\!\cdots\!41\)\( T^{12} )^{2} \))(\( ( 1 + 4 T + 14826 T^{2} - 442692 T^{3} + 122154931 T^{4} - 3956292368 T^{5} + 781022121076 T^{6} - 19943669827088 T^{7} + 3104162139149011 T^{8} - 56708970889555332 T^{9} + 9573941854249652586 T^{10} + 13020974204039524804 T^{11} + \)\(16\!\cdots\!41\)\( T^{12} )^{2} \))(\( ( 1 - 48 T + 10821 T^{2} - 468428 T^{3} + 54548661 T^{4} - 1219760688 T^{5} + 128100283921 T^{6} )^{4} \))
$73$ (\( 1 - 34 T + 5329 T^{2} \))(\( 1 + 34 T + 5329 T^{2} \))(\( ( 1 + 6658 T^{2} + 28398241 T^{4} )^{2} \))(\( 1 - 7778 T^{2} + 28398241 T^{4} \))(\( 1 - 6158 T^{2} + 28398241 T^{4} \))(\( 1 + 248 T + 30752 T^{2} + 3080952 T^{3} + 254433510 T^{4} + 15775678040 T^{5} + 834161467552 T^{6} + 36580397057880 T^{7} + 633277440059215 T^{8} - 33802054838880848 T^{9} - 2957631245611274944 T^{10} - \)\(15\!\cdots\!12\)\( T^{11} - \)\(73\!\cdots\!92\)\( T^{12} - \)\(81\!\cdots\!48\)\( T^{13} - \)\(83\!\cdots\!04\)\( T^{14} - \)\(51\!\cdots\!72\)\( T^{15} + \)\(51\!\cdots\!15\)\( T^{16} + \)\(15\!\cdots\!20\)\( T^{17} + \)\(19\!\cdots\!92\)\( T^{18} + \)\(19\!\cdots\!60\)\( T^{19} + \)\(16\!\cdots\!10\)\( T^{20} + \)\(10\!\cdots\!88\)\( T^{21} + \)\(56\!\cdots\!52\)\( T^{22} + \)\(24\!\cdots\!92\)\( T^{23} + \)\(52\!\cdots\!41\)\( T^{24} \))(\( 1 + 528 T + 147318 T^{2} + 28717920 T^{3} + 4362863697 T^{4} + 544924287120 T^{5} + 57462036213962 T^{6} + 5186233581298704 T^{7} + 403863565075955790 T^{8} + 27374688378441994752 T^{9} + \)\(16\!\cdots\!86\)\( T^{10} + \)\(95\!\cdots\!24\)\( T^{11} + \)\(62\!\cdots\!73\)\( T^{12} + \)\(51\!\cdots\!96\)\( T^{13} + \)\(46\!\cdots\!26\)\( T^{14} + \)\(41\!\cdots\!28\)\( T^{15} + \)\(32\!\cdots\!90\)\( T^{16} + \)\(22\!\cdots\!96\)\( T^{17} + \)\(13\!\cdots\!02\)\( T^{18} + \)\(66\!\cdots\!80\)\( T^{19} + \)\(28\!\cdots\!17\)\( T^{20} + \)\(99\!\cdots\!80\)\( T^{21} + \)\(27\!\cdots\!18\)\( T^{22} + \)\(51\!\cdots\!12\)\( T^{23} + \)\(52\!\cdots\!41\)\( T^{24} \))(\( 1 - 28914 T^{2} + 473704881 T^{4} - 5432049531118 T^{6} + 48062871940123566 T^{8} - \)\(34\!\cdots\!54\)\( T^{10} + \)\(20\!\cdots\!57\)\( T^{12} - \)\(97\!\cdots\!14\)\( T^{14} + \)\(38\!\cdots\!46\)\( T^{16} - \)\(12\!\cdots\!78\)\( T^{18} + \)\(30\!\cdots\!41\)\( T^{20} - \)\(53\!\cdots\!14\)\( T^{22} + \)\(52\!\cdots\!41\)\( T^{24} \))
$79$ (\( 1 + 157 T + 6241 T^{2} \))(\( 1 + 157 T + 6241 T^{2} \))(\( ( 1 + 76 T + 6241 T^{2} )^{4} \))(\( ( 1 - 38 T + 6241 T^{2} )^{2} \))(\( ( 1 + 19 T + 6241 T^{2} )^{2} \))(\( 1 - 30162 T^{2} + 585976951 T^{4} - 7906141610906 T^{6} + 83787919669756051 T^{8} - \)\(70\!\cdots\!12\)\( T^{10} + \)\(48\!\cdots\!94\)\( T^{12} - \)\(27\!\cdots\!72\)\( T^{14} + \)\(12\!\cdots\!11\)\( T^{16} - \)\(46\!\cdots\!46\)\( T^{18} + \)\(13\!\cdots\!71\)\( T^{20} - \)\(27\!\cdots\!62\)\( T^{22} + \)\(34\!\cdots\!81\)\( T^{24} \))(\( 1 + 12 T - 26574 T^{2} - 366848 T^{3} + 390887217 T^{4} + 5621569728 T^{5} - 3680496208466 T^{6} - 54811745357412 T^{7} + 24735046819850526 T^{8} + 328584508359816652 T^{9} - \)\(13\!\cdots\!54\)\( T^{10} - \)\(88\!\cdots\!28\)\( T^{11} + \)\(71\!\cdots\!73\)\( T^{12} - \)\(55\!\cdots\!48\)\( T^{13} - \)\(50\!\cdots\!74\)\( T^{14} + \)\(79\!\cdots\!92\)\( T^{15} + \)\(37\!\cdots\!86\)\( T^{16} - \)\(51\!\cdots\!12\)\( T^{17} - \)\(21\!\cdots\!06\)\( T^{18} + \)\(20\!\cdots\!68\)\( T^{19} + \)\(89\!\cdots\!57\)\( T^{20} - \)\(52\!\cdots\!28\)\( T^{21} - \)\(23\!\cdots\!74\)\( T^{22} + \)\(67\!\cdots\!92\)\( T^{23} + \)\(34\!\cdots\!81\)\( T^{24} \))(\( ( 1 + 96 T - 3885 T^{2} - 1175224 T^{3} - 26907300 T^{4} + 3977488764 T^{5} + 535804365705 T^{6} + 24823507376124 T^{7} - 1048041514491300 T^{8} - 285682211827211704 T^{9} - 5893967726486989485 T^{10} + \)\(90\!\cdots\!96\)\( T^{11} + \)\(59\!\cdots\!41\)\( T^{12} )^{2} \))
$83$ (\( 1 + 86 T + 6889 T^{2} \))(\( 1 - 86 T + 6889 T^{2} \))(\( ( 1 + 8488 T^{2} + 47458321 T^{4} )^{2} \))(\( 1 - 12158 T^{2} + 47458321 T^{4} \))(\( 1 - 4958 T^{2} + 47458321 T^{4} \))(\( 1 + 224 T + 25088 T^{2} + 4058120 T^{3} + 584315970 T^{4} + 50039102632 T^{5} + 4783608901408 T^{6} + 554051313045392 T^{7} + 38707506803414307 T^{8} + 2439696738179139952 T^{9} + \)\(28\!\cdots\!24\)\( T^{10} + \)\(19\!\cdots\!36\)\( T^{11} + \)\(97\!\cdots\!44\)\( T^{12} + \)\(13\!\cdots\!04\)\( T^{13} + \)\(13\!\cdots\!04\)\( T^{14} + \)\(79\!\cdots\!88\)\( T^{15} + \)\(87\!\cdots\!87\)\( T^{16} + \)\(85\!\cdots\!08\)\( T^{17} + \)\(51\!\cdots\!88\)\( T^{18} + \)\(36\!\cdots\!28\)\( T^{19} + \)\(29\!\cdots\!70\)\( T^{20} + \)\(14\!\cdots\!80\)\( T^{21} + \)\(60\!\cdots\!88\)\( T^{22} + \)\(37\!\cdots\!36\)\( T^{23} + \)\(11\!\cdots\!21\)\( T^{24} \))(\( 1 - 43200 T^{2} + 987057204 T^{4} - 15370870045228 T^{6} + 180607989502531968 T^{8} - \)\(16\!\cdots\!36\)\( T^{10} + \)\(12\!\cdots\!66\)\( T^{12} - \)\(79\!\cdots\!56\)\( T^{14} + \)\(40\!\cdots\!88\)\( T^{16} - \)\(16\!\cdots\!08\)\( T^{18} + \)\(50\!\cdots\!24\)\( T^{20} - \)\(10\!\cdots\!00\)\( T^{22} + \)\(11\!\cdots\!21\)\( T^{24} \))(\( ( 1 + 17814 T^{2} + 211711020 T^{4} + 1761588062494 T^{6} + 10047449546397420 T^{8} + 40122333823324876374 T^{10} + \)\(10\!\cdots\!61\)\( T^{12} )^{2} \))
$89$ (\( ( 1 - 89 T )( 1 + 89 T ) \))(\( ( 1 - 89 T )( 1 + 89 T ) \))(\( ( 1 - 12602 T^{2} + 62742241 T^{4} )^{2} \))(\( 1 - 15122 T^{2} + 62742241 T^{4} \))(\( ( 1 - 142 T + 7921 T^{2} )( 1 + 142 T + 7921 T^{2} ) \))(\( 1 - 55228 T^{2} + 1551237378 T^{4} - 29097366482252 T^{6} + 406334885845730607 T^{8} - \)\(44\!\cdots\!64\)\( T^{10} + \)\(39\!\cdots\!36\)\( T^{12} - \)\(27\!\cdots\!24\)\( T^{14} + \)\(15\!\cdots\!67\)\( T^{16} - \)\(71\!\cdots\!92\)\( T^{18} + \)\(24\!\cdots\!58\)\( T^{20} - \)\(53\!\cdots\!28\)\( T^{22} + \)\(61\!\cdots\!41\)\( T^{24} \))(\( 1 - 204 T + 49104 T^{2} - 7187328 T^{3} + 1045283496 T^{4} - 127643879244 T^{5} + 15255527568428 T^{6} - 1684635845896692 T^{7} + 183174646224057864 T^{8} - 18563328978636389184 T^{9} + \)\(18\!\cdots\!76\)\( T^{10} - \)\(17\!\cdots\!52\)\( T^{11} + \)\(16\!\cdots\!74\)\( T^{12} - \)\(13\!\cdots\!92\)\( T^{13} + \)\(11\!\cdots\!16\)\( T^{14} - \)\(92\!\cdots\!24\)\( T^{15} + \)\(72\!\cdots\!84\)\( T^{16} - \)\(52\!\cdots\!92\)\( T^{17} + \)\(37\!\cdots\!88\)\( T^{18} - \)\(24\!\cdots\!04\)\( T^{19} + \)\(16\!\cdots\!56\)\( T^{20} - \)\(88\!\cdots\!68\)\( T^{21} + \)\(47\!\cdots\!04\)\( T^{22} - \)\(15\!\cdots\!84\)\( T^{23} + \)\(61\!\cdots\!41\)\( T^{24} \))(\( ( 1 - 342 T + 66990 T^{2} - 9576684 T^{3} + 1074495570 T^{4} - 104351555322 T^{5} + 9499427045782 T^{6} - 826568669705562 T^{7} + 67416260006372370 T^{8} - 4759432777445553324 T^{9} + \)\(26\!\cdots\!90\)\( T^{10} - \)\(10\!\cdots\!42\)\( T^{11} + \)\(24\!\cdots\!21\)\( T^{12} )^{2} \))
$97$ (\( 1 + 149 T + 9409 T^{2} \))(\( 1 - 149 T + 9409 T^{2} \))(\( ( 1 + 13978 T^{2} + 88529281 T^{4} )^{2} \))(\( 1 - 18098 T^{2} + 88529281 T^{4} \))(\( 1 - 15173 T^{2} + 88529281 T^{4} \))(\( 1 - 564 T + 159048 T^{2} - 28992320 T^{3} + 3611144475 T^{4} - 287383360192 T^{5} + 8016218179688 T^{6} + 1344437255154268 T^{7} - 202827254806258013 T^{8} + 6263140923061522168 T^{9} + \)\(20\!\cdots\!44\)\( T^{10} - \)\(45\!\cdots\!96\)\( T^{11} + \)\(54\!\cdots\!74\)\( T^{12} - \)\(43\!\cdots\!64\)\( T^{13} + \)\(18\!\cdots\!64\)\( T^{14} + \)\(52\!\cdots\!72\)\( T^{15} - \)\(15\!\cdots\!93\)\( T^{16} + \)\(99\!\cdots\!32\)\( T^{17} + \)\(55\!\cdots\!08\)\( T^{18} - \)\(18\!\cdots\!48\)\( T^{19} + \)\(22\!\cdots\!75\)\( T^{20} - \)\(16\!\cdots\!80\)\( T^{21} + \)\(86\!\cdots\!48\)\( T^{22} - \)\(28\!\cdots\!76\)\( T^{23} + \)\(48\!\cdots\!81\)\( T^{24} \))(\( 1 - 64284 T^{2} + 2051576418 T^{4} - 44019752488108 T^{6} + 711686325387159663 T^{8} - \)\(91\!\cdots\!92\)\( T^{10} + \)\(94\!\cdots\!68\)\( T^{12} - \)\(80\!\cdots\!52\)\( T^{14} + \)\(55\!\cdots\!43\)\( T^{16} - \)\(30\!\cdots\!28\)\( T^{18} + \)\(12\!\cdots\!78\)\( T^{20} - \)\(34\!\cdots\!84\)\( T^{22} + \)\(48\!\cdots\!81\)\( T^{24} \))(\( ( 1 + 38226 T^{2} + 661356303 T^{4} + 7342899199036 T^{6} + 58549397989408143 T^{8} + \)\(29\!\cdots\!86\)\( T^{10} + \)\(69\!\cdots\!41\)\( T^{12} )^{2} \))
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