Properties

Label 70.3
Level 70
Weight 3
Dimension 84
Nonzero newspaces 6
Newform subspaces 9
Sturm bound 864
Trace bound 4

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 70\( 70 = 2 \cdot 5 \cdot 7 \) \)
Weight: \( k \) = \( 3 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 9 \)
Sturm bound: \(864\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(\Gamma_1(70))\).

Total New Old
Modular forms 336 84 252
Cusp forms 240 84 156
Eisenstein series 96 0 96

Trace form

\( 84q + 4q^{2} + 20q^{3} + 8q^{4} + 6q^{5} - 16q^{6} - 20q^{7} - 8q^{8} + O(q^{10}) \) \( 84q + 4q^{2} + 20q^{3} + 8q^{4} + 6q^{5} - 16q^{6} - 20q^{7} - 8q^{8} - 4q^{10} - 4q^{11} - 8q^{12} - 12q^{13} + 72q^{14} - 4q^{15} + 32q^{16} + 32q^{17} + 44q^{18} - 12q^{19} - 40q^{20} - 92q^{21} - 80q^{22} - 52q^{23} - 48q^{24} - 70q^{25} - 216q^{26} - 424q^{27} - 96q^{28} - 336q^{29} - 276q^{30} - 268q^{31} - 16q^{32} - 4q^{33} + 94q^{35} + 104q^{36} + 352q^{37} + 296q^{38} + 576q^{39} + 88q^{40} + 368q^{41} + 416q^{42} + 616q^{43} + 168q^{44} + 628q^{45} + 200q^{46} + 36q^{47} - 32q^{48} + 40q^{49} - 4q^{50} + 4q^{51} + 120q^{52} - 56q^{53} + 72q^{54} - 336q^{55} - 112q^{56} - 760q^{57} - 208q^{58} - 324q^{59} + 44q^{60} - 300q^{61} + 208q^{62} - 140q^{63} - 64q^{64} - 144q^{65} + 416q^{66} - 324q^{67} - 64q^{68} - 44q^{70} + 256q^{71} + 88q^{72} - 64q^{73} - 192q^{74} + 94q^{75} - 160q^{76} + 356q^{77} - 240q^{78} + 740q^{79} - 24q^{80} + 544q^{81} + 208q^{82} + 600q^{83} - 24q^{84} + 664q^{85} - 816q^{86} - 184q^{87} - 272q^{88} - 1044q^{89} - 1324q^{90} - 1320q^{91} - 320q^{92} - 1156q^{93} - 936q^{94} - 1298q^{95} - 32q^{96} - 756q^{97} - 444q^{98} - 1248q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(\Gamma_1(70))\)

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
70.3.b \(\chi_{70}(41, \cdot)\) 70.3.b.a 8 1
70.3.d \(\chi_{70}(69, \cdot)\) 70.3.d.a 8 1
70.3.f \(\chi_{70}(43, \cdot)\) 70.3.f.a 4 2
70.3.f.b 8
70.3.h \(\chi_{70}(19, \cdot)\) 70.3.h.a 16 2
70.3.j \(\chi_{70}(31, \cdot)\) 70.3.j.a 8 2
70.3.l \(\chi_{70}(23, \cdot)\) 70.3.l.a 8 4
70.3.l.b 8
70.3.l.c 16

Decomposition of \(S_{3}^{\mathrm{old}}(\Gamma_1(70))\) into lower level spaces

\( S_{3}^{\mathrm{old}}(\Gamma_1(70)) \cong \) \(S_{3}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(35))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - 2 T^{2} )^{4} \))(\( ( 1 + 2 T^{2} )^{4} \))(\( ( 1 + 2 T + 2 T^{2} )^{2} \))(\( ( 1 - 2 T + 2 T^{2} )^{4} \))(\( ( 1 - 2 T^{2} + 4 T^{4} )^{4} \))(\( ( 1 + 2 T^{2} + 4 T^{4} )^{2} \))(\( ( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )^{2} \))(\( ( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} )^{2} \))(\( ( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} )^{4} \))
$3$ (\( 1 - 18 T^{2} + 157 T^{4} - 702 T^{6} + 648 T^{8} - 56862 T^{10} + 1030077 T^{12} - 9565938 T^{14} + 43046721 T^{16} \))(\( ( 1 + 9 T^{2} + 98 T^{4} + 729 T^{6} + 6561 T^{8} )^{2} \))(\( 1 - 4 T + 8 T^{2} - 16 T^{3} + 7 T^{4} - 144 T^{5} + 648 T^{6} - 2916 T^{7} + 6561 T^{8} \))(\( 1 - 4 T + 8 T^{2} + 16 T^{3} - 119 T^{4} + 232 T^{5} + 152 T^{6} - 868 T^{7} + 6256 T^{8} - 7812 T^{9} + 12312 T^{10} + 169128 T^{11} - 780759 T^{12} + 944784 T^{13} + 4251528 T^{14} - 19131876 T^{15} + 43046721 T^{16} \))(\( 1 - 30 T^{2} + 458 T^{4} - 2796 T^{6} - 14835 T^{8} + 458004 T^{10} - 3071650 T^{12} - 6856410 T^{14} + 248875252 T^{16} - 555369210 T^{18} - 20153095650 T^{20} + 243402103764 T^{22} - 638598106035 T^{24} - 9749049185196 T^{26} + 129352727708298 T^{28} - 686303773648830 T^{30} + 1853020188851841 T^{32} \))(\( 1 - 12 T + 84 T^{2} - 432 T^{3} + 1697 T^{4} - 5184 T^{5} + 12756 T^{6} - 26724 T^{7} + 63760 T^{8} - 240516 T^{9} + 1033236 T^{10} - 3779136 T^{11} + 11134017 T^{12} - 25509168 T^{13} + 44641044 T^{14} - 57395628 T^{15} + 43046721 T^{16} \))(\( 1 + 4 T + 8 T^{2} + 64 T^{3} + 265 T^{4} + 824 T^{5} + 3224 T^{6} + 10004 T^{7} + 28048 T^{8} + 90036 T^{9} + 261144 T^{10} + 600696 T^{11} + 1738665 T^{12} + 3779136 T^{13} + 4251528 T^{14} + 19131876 T^{15} + 43046721 T^{16} \))(\( 1 - 2 T + 2 T^{2} + 12 T^{3} - 19 T^{4} - 60 T^{5} + 230 T^{6} - 1550 T^{7} - 1580 T^{8} - 13950 T^{9} + 18630 T^{10} - 43740 T^{11} - 124659 T^{12} + 708588 T^{13} + 1062882 T^{14} - 9565938 T^{15} + 43046721 T^{16} \))(\( 1 - 2 T + 2 T^{2} + 28 T^{3} - 2 T^{4} - 86 T^{5} + 568 T^{6} - 522 T^{7} - 4623 T^{8} - 5994 T^{9} + 11452 T^{10} - 106106 T^{11} + 135058 T^{12} - 1589180 T^{13} + 3237770 T^{14} + 5097334 T^{15} - 9280004 T^{16} + 45876006 T^{17} + 262259370 T^{18} - 1158512220 T^{19} + 886115538 T^{20} - 6265453194 T^{21} + 6086062332 T^{22} - 28669116186 T^{23} - 199004991183 T^{24} - 202233495258 T^{25} + 1980493539768 T^{26} - 2698771126374 T^{27} - 564859072962 T^{28} + 71172243193212 T^{29} + 45753584909922 T^{30} - 411782264189298 T^{31} + 1853020188851841 T^{32} \))
$5$ (\( ( 1 + 5 T^{2} )^{4} \))(\( 1 - 98 T^{4} + 390625 T^{8} \))(\( 1 - 12 T + 72 T^{2} - 300 T^{3} + 625 T^{4} \))(\( 1 + 4 T + 10 T^{2} + 140 T^{3} + 850 T^{4} + 3500 T^{5} + 6250 T^{6} + 62500 T^{7} + 390625 T^{8} \))(\( 1 + 6 T + 39 T^{2} + 162 T^{3} + 629 T^{4} - 300 T^{5} - 24150 T^{6} - 165000 T^{7} - 993750 T^{8} - 4125000 T^{9} - 15093750 T^{10} - 4687500 T^{11} + 245703125 T^{12} + 1582031250 T^{13} + 9521484375 T^{14} + 36621093750 T^{15} + 152587890625 T^{16} \))(\( ( 1 - 5 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - 6 T + 11 T^{2} - 150 T^{3} + 625 T^{4} )^{2} \))(\( 1 + 6 T + 21 T^{2} - 210 T^{3} - 1300 T^{4} - 5250 T^{5} + 13125 T^{6} + 93750 T^{7} + 390625 T^{8} \))(\( 1 + 2 T + 15 T^{2} + 102 T^{3} + 745 T^{4} + 1132 T^{5} - 12306 T^{6} + 14400 T^{7} - 187350 T^{8} + 360000 T^{9} - 7691250 T^{10} + 17687500 T^{11} + 291015625 T^{12} + 996093750 T^{13} + 3662109375 T^{14} + 12207031250 T^{15} + 152587890625 T^{16} \))
$7$ (\( 1 + 4 T + 6 T^{2} + 220 T^{3} + 4970 T^{4} + 10780 T^{5} + 14406 T^{6} + 470596 T^{7} + 5764801 T^{8} \))(\( 1 + 142 T^{2} + 9506 T^{4} + 340942 T^{6} + 5764801 T^{8} \))(\( 1 + 49 T^{4} \))(\( ( 1 + 49 T^{4} )^{2} \))(\( 1 - 148 T^{2} + 10397 T^{4} - 385080 T^{6} + 13653164 T^{8} - 924577080 T^{10} + 59936635997 T^{12} - 2048510505748 T^{14} + 33232930569601 T^{16} \))(\( 1 + 12 T + 60 T^{2} - 168 T^{3} - 2401 T^{4} - 8232 T^{5} + 144060 T^{6} + 1411788 T^{7} + 5764801 T^{8} \))(\( 1 + 4 T + 8 T^{2} - 336 T^{3} - 3577 T^{4} - 16464 T^{5} + 19208 T^{6} + 470596 T^{7} + 5764801 T^{8} \))(\( 1 + 12 T + 72 T^{2} - 252 T^{3} - 4018 T^{4} - 12348 T^{5} + 172872 T^{6} + 1411788 T^{7} + 5764801 T^{8} \))(\( 1 - 12 T + 72 T^{2} - 468 T^{3} - 59 T^{4} + 23880 T^{5} - 172800 T^{6} + 1375080 T^{7} - 10586940 T^{8} + 67378920 T^{9} - 414892800 T^{10} + 2809458120 T^{11} - 340123259 T^{12} - 132198416532 T^{13} + 996572678472 T^{14} - 8138676874188 T^{15} + 33232930569601 T^{16} \))
$11$ (\( ( 1 - 10 T + 173 T^{2} + 570 T^{3} + 1208 T^{4} + 68970 T^{5} + 2532893 T^{6} - 17715610 T^{7} + 214358881 T^{8} )^{2} \))(\( ( 1 + 15 T + 214 T^{2} + 1815 T^{3} + 14641 T^{4} )^{4} \))(\( ( 1 - 2 T + 19 T^{2} - 242 T^{3} + 14641 T^{4} )^{2} \))(\( ( 1 - 10 T + 365 T^{2} - 2910 T^{3} + 60648 T^{4} - 352110 T^{5} + 5343965 T^{6} - 17715610 T^{7} + 214358881 T^{8} )^{2} \))(\( ( 1 + 6 T - 275 T^{2} - 1338 T^{3} + 33961 T^{4} + 54288 T^{5} - 6018974 T^{6} + 2522844 T^{7} + 1011276370 T^{8} + 305264124 T^{9} - 88123798334 T^{10} + 96174503568 T^{11} + 7279841957641 T^{12} - 34704274116138 T^{13} - 863067803598275 T^{14} + 2278499001499446 T^{15} + 45949729863572161 T^{16} )^{2} \))(\( 1 + 8 T - 228 T^{2} - 4688 T^{3} + 15770 T^{4} + 788088 T^{5} + 5396656 T^{6} - 54408328 T^{7} - 1014586317 T^{8} - 6583407688 T^{9} + 79012440496 T^{10} + 1396145965368 T^{11} + 3380439553370 T^{12} - 121594646529488 T^{13} - 715561669892388 T^{14} + 3037998668665928 T^{15} + 45949729863572161 T^{16} \))(\( ( 1 - 4 T - 200 T^{2} + 104 T^{3} + 30079 T^{4} + 12584 T^{5} - 2928200 T^{6} - 7086244 T^{7} + 214358881 T^{8} )^{2} \))(\( ( 1 + 8 T - 183 T^{2} + 40 T^{3} + 38624 T^{4} + 4840 T^{5} - 2679303 T^{6} + 14172488 T^{7} + 214358881 T^{8} )^{2} \))(\( ( 1 - 20 T + 53 T^{2} + 1868 T^{3} - 19793 T^{4} + 23808 T^{5} + 812056 T^{6} + 3236448 T^{7} - 190136502 T^{8} + 391610208 T^{9} + 11889311896 T^{10} + 42177324288 T^{11} - 4242805331633 T^{12} + 48451109154668 T^{13} + 166336703966213 T^{14} - 7594996671664820 T^{15} + 45949729863572161 T^{16} )^{2} \))
$13$ (\( 1 - 470 T^{2} + 96425 T^{4} - 7707750 T^{6} + 323093332 T^{8} - 220141047750 T^{10} + 78656834772425 T^{12} - 10950100007566070 T^{14} + 665416609183179841 T^{16} \))(\( ( 1 + 355 T^{2} + 84500 T^{4} + 10139155 T^{6} + 815730721 T^{8} )^{2} \))(\( 1 - 4 T + 8 T^{2} - 432 T^{3} + 19607 T^{4} - 73008 T^{5} + 228488 T^{6} - 19307236 T^{7} + 815730721 T^{8} \))(\( 1 + 16 T + 128 T^{2} + 4780 T^{3} + 86117 T^{4} + 621600 T^{5} + 10346824 T^{6} + 161691508 T^{7} + 1693276900 T^{8} + 27325864852 T^{9} + 295515640264 T^{10} + 3000344474400 T^{11} + 70248282500357 T^{12} + 658963591038220 T^{13} + 2982154895677568 T^{14} + 62998022171188624 T^{15} + 665416609183179841 T^{16} \))(\( ( 1 + 680 T^{2} + 225692 T^{4} + 53707032 T^{6} + 10194689414 T^{8} + 1533926540952 T^{10} + 184103897883932 T^{12} + 15842697883287080 T^{14} + 665416609183179841 T^{16} )^{2} \))(\( 1 - 296 T^{2} + 98076 T^{4} - 18130072 T^{6} + 3958421126 T^{8} - 517812986392 T^{10} + 80003606192796 T^{12} - 6896233196254376 T^{14} + 665416609183179841 T^{16} \))(\( ( 1 - 47518 T^{4} + 815730721 T^{8} )^{2} \))(\( ( 1 + 8 T + 32 T^{2} - 168 T^{3} - 35218 T^{4} - 28392 T^{5} + 913952 T^{6} + 38614472 T^{7} + 815730721 T^{8} )^{2} \))(\( ( 1 - 8 T + 32 T^{2} - 1112 T^{3} + 33680 T^{4} - 119592 T^{5} + 497248 T^{6} - 10550648 T^{7} + 68709598 T^{8} - 1783059512 T^{9} + 14201900128 T^{10} - 577247741928 T^{11} + 27473810683280 T^{12} - 153298642936088 T^{13} + 745538723919392 T^{14} - 31499011085594312 T^{15} + 665416609183179841 T^{16} )^{2} \))
$17$ (\( 1 - 774 T^{2} + 378041 T^{4} - 152875638 T^{6} + 49129851156 T^{8} - 12768326161398 T^{10} + 2637122318753081 T^{12} - 450949611615835014 T^{14} + 48661191875666868481 T^{16} \))(\( ( 1 + 1027 T^{2} + 426596 T^{4} + 85776067 T^{6} + 6975757441 T^{8} )^{2} \))(\( 1 + 20 T + 200 T^{2} - 80 T^{3} - 85817 T^{4} - 23120 T^{5} + 16704200 T^{6} + 482751380 T^{7} + 6975757441 T^{8} \))(\( 1 - 8 T + 32 T^{2} + 12892 T^{3} + 48629 T^{4} - 2715024 T^{5} + 103265896 T^{6} + 888643468 T^{7} - 17470706396 T^{8} + 256817962252 T^{9} + 8624870899816 T^{10} - 65534079136656 T^{11} + 339224108598389 T^{12} + 25990193364588508 T^{13} + 18643911591352352 T^{14} - 1347022612475207432 T^{15} + 48661191875666868481 T^{16} \))(\( 1 - 710 T^{2} + 90371 T^{4} + 24973142 T^{6} + 3528484937 T^{8} - 1855718279020 T^{10} - 829313056399602 T^{12} - 134075919577703016 T^{14} + \)\(19\!\cdots\!98\)\( T^{16} - \)\(11\!\cdots\!36\)\( T^{18} - \)\(57\!\cdots\!82\)\( T^{20} - \)\(10\!\cdots\!20\)\( T^{22} + \)\(17\!\cdots\!97\)\( T^{24} + \)\(10\!\cdots\!42\)\( T^{26} + \)\(30\!\cdots\!91\)\( T^{28} - \)\(20\!\cdots\!10\)\( T^{30} + \)\(23\!\cdots\!61\)\( T^{32} \))(\( 1 + 48 T + 1844 T^{2} + 51648 T^{3} + 1261162 T^{4} + 27614256 T^{5} + 561236816 T^{6} + 10598780016 T^{7} + 187636972435 T^{8} + 3063047424624 T^{9} + 46875060109136 T^{10} + 666541009583664 T^{11} + 8797560205806442 T^{12} + 104122052970389952 T^{13} + 1074355405451679284 T^{14} + 8082135674851244592 T^{15} + 48661191875666868481 T^{16} \))(\( 1 + 16 T + 128 T^{2} - 544 T^{3} + 25790 T^{4} + 1504976 T^{5} + 20926464 T^{6} + 422304816 T^{7} + 4437400963 T^{8} + 122046091824 T^{9} + 1747799199744 T^{10} + 36326462043344 T^{11} + 179904784403390 T^{12} - 1096700681844256 T^{13} + 74575646365409408 T^{14} + 2694045224950414864 T^{15} + 48661191875666868481 T^{16} \))(\( 1 - 62 T + 1922 T^{2} - 40796 T^{3} + 583477 T^{4} - 2167148 T^{5} - 154922810 T^{6} + 5388805206 T^{7} - 109634151516 T^{8} + 1557364704534 T^{9} - 12939308014010 T^{10} - 52309684383212 T^{11} + 4070194024402357 T^{12} - 82244487162717404 T^{13} + 1119799939955600642 T^{14} - 10439425246682857598 T^{15} + 48661191875666868481 T^{16} \))(\( 1 - 46 T + 1058 T^{2} - 15228 T^{3} + 49795 T^{4} + 3577952 T^{5} - 101322910 T^{6} + 1912395662 T^{7} - 35538003267 T^{8} + 729089246500 T^{9} - 13614614462848 T^{10} + 149191614412260 T^{11} + 396919406750154 T^{12} - 52349464424861728 T^{13} + 1230645459219027588 T^{14} - 18772373078273663580 T^{15} + \)\(27\!\cdots\!30\)\( T^{16} - \)\(54\!\cdots\!20\)\( T^{17} + \)\(10\!\cdots\!48\)\( T^{18} - \)\(12\!\cdots\!32\)\( T^{19} + \)\(27\!\cdots\!14\)\( T^{20} + \)\(30\!\cdots\!40\)\( T^{21} - \)\(79\!\cdots\!28\)\( T^{22} + \)\(12\!\cdots\!00\)\( T^{23} - \)\(17\!\cdots\!27\)\( T^{24} + \)\(26\!\cdots\!58\)\( T^{25} - \)\(41\!\cdots\!10\)\( T^{26} + \)\(42\!\cdots\!28\)\( T^{27} + \)\(16\!\cdots\!95\)\( T^{28} - \)\(14\!\cdots\!32\)\( T^{29} + \)\(29\!\cdots\!78\)\( T^{30} - \)\(37\!\cdots\!54\)\( T^{31} + \)\(23\!\cdots\!61\)\( T^{32} \))
$19$ (\( 1 - 1712 T^{2} + 1528412 T^{4} - 900114768 T^{6} + 380400506758 T^{8} - 117303856680528 T^{10} + 25957881554620892 T^{12} - 3789195141441267632 T^{14} + \)\(28\!\cdots\!81\)\( T^{16} \))(\( ( 1 - 396 T^{2} + 35638 T^{4} - 51607116 T^{6} + 16983563041 T^{8} )^{2} \))(\( 1 - 768 T^{2} + 389954 T^{4} - 100086528 T^{6} + 16983563041 T^{8} \))(\( 1 - 1700 T^{2} + 1457128 T^{4} - 829976492 T^{6} + 345611101966 T^{8} - 108163366413932 T^{10} + 24747225246806248 T^{12} - 3762635362412473700 T^{14} + \)\(28\!\cdots\!81\)\( T^{16} \))(\( ( 1 + 6 T + 513 T^{2} + 3006 T^{3} + 50837 T^{4} + 1422120 T^{5} + 6939750 T^{6} + 1565615628 T^{7} + 10204655562 T^{8} + 565187241708 T^{9} + 904395159750 T^{10} + 66904888287720 T^{11} + 863393394315317 T^{12} + 18429985170949806 T^{13} + 1135430553480940593 T^{14} + 4794040114697304726 T^{15} + \)\(28\!\cdots\!81\)\( T^{16} )^{2} \))(\( 1 + 484 T^{2} + 193050 T^{4} + 2952000 T^{5} + 85792976 T^{6} + 988944000 T^{7} + 23497895219 T^{8} + 357008784000 T^{9} + 11180626425296 T^{10} + 138879440712000 T^{11} + 3278676845065050 T^{12} + 1071244420828021924 T^{14} + \)\(28\!\cdots\!81\)\( T^{16} \))(\( 1 + 896 T^{2} + 345790 T^{4} + 175960064 T^{6} + 87265417699 T^{8} + 22931291500544 T^{10} + 5872746263947390 T^{12} + 1983130167483280256 T^{14} + \)\(28\!\cdots\!81\)\( T^{16} \))(\( 1 + 1338 T^{2} + 1083625 T^{4} + 596717226 T^{6} + 249020459796 T^{8} + 77764785609546 T^{10} + 18403813500303625 T^{12} + 2961415361710523418 T^{14} + \)\(28\!\cdots\!81\)\( T^{16} \))(\( 1 + 962 T^{2} + 285903 T^{4} - 9599234 T^{6} - 15336277051 T^{8} + 2324255772132 T^{10} + 984840646865914 T^{12} - 821244130299946144 T^{14} - \)\(53\!\cdots\!82\)\( T^{16} - \)\(10\!\cdots\!24\)\( T^{18} + \)\(16\!\cdots\!74\)\( T^{20} + \)\(51\!\cdots\!52\)\( T^{22} - \)\(44\!\cdots\!31\)\( T^{24} - \)\(36\!\cdots\!34\)\( T^{26} + \)\(14\!\cdots\!63\)\( T^{28} + \)\(61\!\cdots\!42\)\( T^{30} + \)\(83\!\cdots\!61\)\( T^{32} \))
$23$ (\( ( 1 + 36 T + 878 T^{2} + 19356 T^{3} + 572138 T^{4} + 10239324 T^{5} + 245700398 T^{6} + 5329292004 T^{7} + 78310985281 T^{8} )^{2} \))(\( ( 1 - 1610 T^{2} + 1150754 T^{4} - 450544010 T^{6} + 78310985281 T^{8} )^{2} \))(\( 1 + 40 T + 800 T^{2} + 1160 T^{3} - 248318 T^{4} + 613640 T^{5} + 223872800 T^{6} + 5921435560 T^{7} + 78310985281 T^{8} \))(\( 1 - 24 T + 288 T^{2} - 4856 T^{3} - 1596 T^{4} - 2433192 T^{5} + 70646624 T^{6} - 2515925128 T^{7} + 95356456966 T^{8} - 1330924392712 T^{9} + 19769821906784 T^{10} - 360199740827688 T^{11} - 124984332508476 T^{12} - 201167138453479544 T^{13} + 6311411836421852448 T^{14} - \)\(27\!\cdots\!16\)\( T^{15} + \)\(61\!\cdots\!61\)\( T^{16} \))(\( 1 + 1906 T^{2} + 1525754 T^{4} + 889421012 T^{6} + 533272315133 T^{8} + 198611962449908 T^{10} - 12646802794504338 T^{12} - 38464547268982252842 T^{14} - \)\(19\!\cdots\!92\)\( T^{16} - \)\(10\!\cdots\!22\)\( T^{18} - \)\(99\!\cdots\!78\)\( T^{20} + \)\(43\!\cdots\!68\)\( T^{22} + \)\(32\!\cdots\!13\)\( T^{24} + \)\(15\!\cdots\!12\)\( T^{26} + \)\(73\!\cdots\!14\)\( T^{28} + \)\(25\!\cdots\!86\)\( T^{30} + \)\(37\!\cdots\!21\)\( T^{32} \))(\( 1 + 36 T - 396 T^{2} - 35424 T^{3} - 158639 T^{4} + 14284368 T^{5} + 171452916 T^{6} - 1838171412 T^{7} - 65570951088 T^{8} - 972392676948 T^{9} + 47979555466356 T^{10} + 2114599115683152 T^{11} - 12423176393992559 T^{12} - 1467492733232302176 T^{13} - 8678191275080047116 T^{14} + \)\(41\!\cdots\!24\)\( T^{15} + \)\(61\!\cdots\!61\)\( T^{16} \))(\( 1 + 4 T + 8 T^{2} - 1216 T^{3} + 88025 T^{4} + 1058744 T^{5} + 4270104 T^{6} + 539371764 T^{7} - 65615257712 T^{8} + 285327663156 T^{9} + 1194950173464 T^{10} + 156732109263416 T^{11} + 6893324479360025 T^{12} - 50374637635797184 T^{13} + 175316995456162568 T^{14} + 46371345298154999236 T^{15} + \)\(61\!\cdots\!61\)\( T^{16} \))(\( 1 - 22 T + 242 T^{2} + 8756 T^{3} - 257435 T^{4} + 1661396 T^{5} + 64082326 T^{6} - 4078447362 T^{7} + 48340392708 T^{8} - 2157498654498 T^{9} + 17932862190166 T^{10} + 245946233841044 T^{11} - 20159988495814235 T^{12} + 362730532186710644 T^{13} + 5303339112548917682 T^{14} - \)\(25\!\cdots\!98\)\( T^{15} + \)\(61\!\cdots\!61\)\( T^{16} \))(\( 1 - 54 T + 1458 T^{2} - 41404 T^{3} + 1531542 T^{4} - 51187722 T^{5} + 1388294360 T^{6} - 34788210422 T^{7} + 978568184441 T^{8} - 28666106605246 T^{9} + 745448729152316 T^{10} - 17609421034026766 T^{11} + 432143271417763898 T^{12} - 11387865059627737708 T^{13} + \)\(28\!\cdots\!34\)\( T^{14} - \)\(63\!\cdots\!38\)\( T^{15} + \)\(13\!\cdots\!36\)\( T^{16} - \)\(33\!\cdots\!02\)\( T^{17} + \)\(80\!\cdots\!94\)\( T^{18} - \)\(16\!\cdots\!12\)\( T^{19} + \)\(33\!\cdots\!38\)\( T^{20} - \)\(72\!\cdots\!34\)\( T^{21} + \)\(16\!\cdots\!36\)\( T^{22} - \)\(33\!\cdots\!14\)\( T^{23} + \)\(60\!\cdots\!01\)\( T^{24} - \)\(11\!\cdots\!18\)\( T^{25} + \)\(23\!\cdots\!60\)\( T^{26} - \)\(46\!\cdots\!38\)\( T^{27} + \)\(73\!\cdots\!22\)\( T^{28} - \)\(10\!\cdots\!56\)\( T^{29} + \)\(19\!\cdots\!98\)\( T^{30} - \)\(38\!\cdots\!46\)\( T^{31} + \)\(37\!\cdots\!21\)\( T^{32} \))
$29$ (\( ( 1 - 6 T + 2321 T^{2} - 22062 T^{3} + 2551316 T^{4} - 18554142 T^{5} + 1641599201 T^{6} - 3568939926 T^{7} + 500246412961 T^{8} )^{2} \))(\( ( 1 + 23 T + 1056 T^{2} + 19343 T^{3} + 707281 T^{4} )^{4} \))(\( 1 - 466 T^{2} + 371251 T^{4} - 329592946 T^{6} + 500246412961 T^{8} \))(\( 1 - 2646 T^{2} + 2985729 T^{4} - 2543114574 T^{6} + 2203254062180 T^{8} - 1798696619013294 T^{10} + 1493600222323633569 T^{12} - \)\(93\!\cdots\!86\)\( T^{14} + \)\(25\!\cdots\!21\)\( T^{16} \))(\( ( 1 + 34 T + 1333 T^{2} - 9890 T^{3} - 84272 T^{4} - 8317490 T^{5} + 942805573 T^{6} + 20223992914 T^{7} + 500246412961 T^{8} )^{4} \))(\( ( 1 + 60 T + 4458 T^{2} + 157200 T^{3} + 6089363 T^{4} + 132205200 T^{5} + 3153058698 T^{6} + 35689399260 T^{7} + 500246412961 T^{8} )^{2} \))(\( ( 1 - 626 T^{2} + 264051 T^{4} - 442757906 T^{6} + 500246412961 T^{8} )^{2} \))(\( ( 1 - 1756 T^{2} + 1903846 T^{4} - 1241985436 T^{6} + 500246412961 T^{8} )^{2} \))(\( ( 1 - 4830 T^{2} + 11400705 T^{4} - 16870918974 T^{6} + 17005870320740 T^{8} - 11932480442849694 T^{10} + 5703161781476537505 T^{12} - \)\(17\!\cdots\!30\)\( T^{14} + \)\(25\!\cdots\!21\)\( T^{16} )^{2} \))
$31$ (\( 1 - 2336 T^{2} + 3598076 T^{4} - 4780408032 T^{6} + 5102420377606 T^{8} - 4414807206120672 T^{10} + 3068766772431563516 T^{12} - \)\(18\!\cdots\!96\)\( T^{14} + \)\(72\!\cdots\!81\)\( T^{16} \))(\( ( 1 - 2748 T^{2} + 3729526 T^{4} - 2537835708 T^{6} + 852891037441 T^{8} )^{2} \))(\( ( 1 - 24 T + 1940 T^{2} - 23064 T^{3} + 923521 T^{4} )^{2} \))(\( ( 1 + 40 T + 3878 T^{2} + 108728 T^{3} + 5633866 T^{4} + 104487608 T^{5} + 3581414438 T^{6} + 35500147240 T^{7} + 852891037441 T^{8} )^{2} \))(\( ( 1 - 42 T + 2721 T^{2} - 89586 T^{3} + 2991941 T^{4} - 12678696 T^{5} - 400552650 T^{6} + 101519581308 T^{7} - 3139475731878 T^{8} + 97560317636988 T^{9} - 369918783880650 T^{10} - 11252389370279976 T^{11} + 2551799663452262981 T^{12} - 73427219717462038386 T^{13} + \)\(21\!\cdots\!81\)\( T^{14} - \)\(31\!\cdots\!82\)\( T^{15} + \)\(72\!\cdots\!81\)\( T^{16} )^{2} \))(\( 1 + 72 T + 3940 T^{2} + 159264 T^{3} + 5902026 T^{4} + 166723272 T^{5} + 3604087568 T^{6} + 80807933736 T^{7} + 2044410282323 T^{8} + 77656424320296 T^{9} + 3328450554886928 T^{10} + 147967517608364232 T^{11} + 5033785078143755466 T^{12} + \)\(13\!\cdots\!64\)\( T^{13} + \)\(31\!\cdots\!40\)\( T^{14} + \)\(54\!\cdots\!12\)\( T^{15} + \)\(72\!\cdots\!81\)\( T^{16} \))(\( ( 1 + 16 T - 705 T^{2} + 15376 T^{3} + 923521 T^{4} )^{4} \))(\( ( 1 - 12 T - 923 T^{2} + 10260 T^{3} + 76584 T^{4} + 9859860 T^{5} - 852409883 T^{6} - 10650044172 T^{7} + 852891037441 T^{8} )^{2} \))(\( ( 1 + 104 T + 3335 T^{2} + 80216 T^{3} + 6958945 T^{4} + 299604368 T^{5} + 5519548642 T^{6} + 232310443200 T^{7} + 11482603425294 T^{8} + 223250335915200 T^{9} + 5097419081408482 T^{10} + 265899979443678608 T^{11} + 5935221820544859745 T^{12} + 65747302668451933016 T^{13} + \)\(26\!\cdots\!35\)\( T^{14} + \)\(78\!\cdots\!84\)\( T^{15} + \)\(72\!\cdots\!81\)\( T^{16} )^{2} \))
$37$ (\( ( 1 - 68 T + 6144 T^{2} - 261644 T^{3} + 12851966 T^{4} - 358190636 T^{5} + 11514845184 T^{6} - 174469395812 T^{7} + 3512479453921 T^{8} )^{2} \))(\( ( 1 - 3932 T^{2} + 7349270 T^{4} - 7369201052 T^{6} + 3512479453921 T^{8} )^{2} \))(\( 1 - 104 T + 5408 T^{2} - 256776 T^{3} + 10981922 T^{4} - 351526344 T^{5} + 10135462688 T^{6} - 266835546536 T^{7} + 3512479453921 T^{8} \))(\( 1 - 44 T + 968 T^{2} - 596 T^{3} + 1621968 T^{4} - 158586284 T^{5} + 5407909080 T^{6} - 137987642580 T^{7} + 2130290543326 T^{8} - 188905082692020 T^{9} + 10135292289281880 T^{10} - 406889016963974156 T^{11} + 5697129274917336528 T^{12} - 2865916285961038004 T^{13} + \)\(63\!\cdots\!08\)\( T^{14} - \)\(39\!\cdots\!16\)\( T^{15} + \)\(12\!\cdots\!41\)\( T^{16} \))(\( 1 + 9154 T^{2} + 45306851 T^{4} + 157704458750 T^{6} + 428483590459289 T^{8} + 959659332166684052 T^{10} + \)\(18\!\cdots\!70\)\( T^{12} + \)\(30\!\cdots\!64\)\( T^{14} + \)\(44\!\cdots\!82\)\( T^{16} + \)\(56\!\cdots\!04\)\( T^{18} + \)\(64\!\cdots\!70\)\( T^{20} + \)\(63\!\cdots\!12\)\( T^{22} + \)\(52\!\cdots\!49\)\( T^{24} + \)\(36\!\cdots\!50\)\( T^{26} + \)\(19\!\cdots\!11\)\( T^{28} + \)\(74\!\cdots\!34\)\( T^{30} + \)\(15\!\cdots\!81\)\( T^{32} \))(\( 1 - 96 T + 3324 T^{2} - 73536 T^{3} + 1295146 T^{4} + 95028192 T^{5} - 7452950544 T^{6} + 251675740512 T^{7} - 8124255291213 T^{8} + 344544088760928 T^{9} - 13968029244493584 T^{10} + 243816341813922528 T^{11} + 4549173714827967466 T^{12} - \)\(35\!\cdots\!64\)\( T^{13} + \)\(21\!\cdots\!44\)\( T^{14} - \)\(86\!\cdots\!44\)\( T^{15} + \)\(12\!\cdots\!41\)\( T^{16} \))(\( 1 - 144 T + 10368 T^{2} - 421344 T^{3} + 8176990 T^{4} + 93049776 T^{5} - 9412816896 T^{6} + 169435939536 T^{7} - 564618587997 T^{8} + 231957801224784 T^{9} - 17641134326624256 T^{10} + 238740267634734384 T^{11} + 28721509369917477790 T^{12} - \)\(20\!\cdots\!56\)\( T^{13} + \)\(68\!\cdots\!08\)\( T^{14} - \)\(12\!\cdots\!16\)\( T^{15} + \)\(12\!\cdots\!41\)\( T^{16} \))(\( 1 + 134 T + 8978 T^{2} + 354028 T^{3} + 6883829 T^{4} - 119634932 T^{5} - 15166185258 T^{6} - 689629336926 T^{7} - 25693504224764 T^{8} - 944102562251694 T^{9} - 28423872929318538 T^{10} - 306950504471319188 T^{11} + 24179307926805543509 T^{12} + \)\(17\!\cdots\!72\)\( T^{13} + \)\(59\!\cdots\!18\)\( T^{14} + \)\(12\!\cdots\!26\)\( T^{15} + \)\(12\!\cdots\!41\)\( T^{16} \))(\( 1 + 38 T + 722 T^{2} + 36204 T^{3} + 921843 T^{4} + 54419040 T^{5} + 2057717682 T^{6} + 7359428170 T^{7} + 1748254042141 T^{8} - 98996461623124 T^{9} - 5173840906504960 T^{10} - 59928313854983156 T^{11} - 5578617333398806678 T^{12} + 6554354740117891808 T^{13} - \)\(20\!\cdots\!96\)\( T^{14} - \)\(42\!\cdots\!16\)\( T^{15} - \)\(83\!\cdots\!54\)\( T^{16} - \)\(57\!\cdots\!04\)\( T^{17} - \)\(39\!\cdots\!56\)\( T^{18} + \)\(16\!\cdots\!72\)\( T^{19} - \)\(19\!\cdots\!38\)\( T^{20} - \)\(28\!\cdots\!44\)\( T^{21} - \)\(34\!\cdots\!60\)\( T^{22} - \)\(89\!\cdots\!36\)\( T^{23} + \)\(21\!\cdots\!81\)\( T^{24} + \)\(12\!\cdots\!30\)\( T^{25} + \)\(47\!\cdots\!82\)\( T^{26} + \)\(17\!\cdots\!60\)\( T^{27} + \)\(39\!\cdots\!23\)\( T^{28} + \)\(21\!\cdots\!36\)\( T^{29} + \)\(58\!\cdots\!62\)\( T^{30} + \)\(42\!\cdots\!62\)\( T^{31} + \)\(15\!\cdots\!81\)\( T^{32} \))
$41$ (\( 1 - 7016 T^{2} + 25625276 T^{4} - 64434523992 T^{6} + 122819632739206 T^{8} - 182076564950157912 T^{10} + \)\(20\!\cdots\!96\)\( T^{12} - \)\(15\!\cdots\!96\)\( T^{14} + \)\(63\!\cdots\!41\)\( T^{16} \))(\( ( 1 + 3912 T^{2} + 9411406 T^{4} + 11054377032 T^{6} + 7984925229121 T^{8} )^{2} \))(\( ( 1 - 8 T + 2244 T^{2} - 13448 T^{3} + 2825761 T^{4} )^{2} \))(\( ( 1 - 120 T + 9958 T^{2} - 532072 T^{3} + 24823946 T^{4} - 894413032 T^{5} + 28138928038 T^{6} - 570012508920 T^{7} + 7984925229121 T^{8} )^{2} \))(\( ( 1 - 8238 T^{2} + 35834897 T^{4} - 101403728814 T^{6} + 201449086750692 T^{8} - 286542702137177454 T^{10} + \)\(28\!\cdots\!37\)\( T^{12} - \)\(18\!\cdots\!78\)\( T^{14} + \)\(63\!\cdots\!41\)\( T^{16} )^{2} \))(\( 1 - 3716 T^{2} + 10727274 T^{4} - 24786880336 T^{6} + 46146959629235 T^{8} - 70041799765135696 T^{10} + 85656480802293746154 T^{12} - \)\(83\!\cdots\!96\)\( T^{14} + \)\(63\!\cdots\!41\)\( T^{16} \))(\( ( 1 + 34 T + 3531 T^{2} + 57154 T^{3} + 2825761 T^{4} )^{4} \))(\( ( 1 - 80 T + 3378 T^{2} - 134480 T^{3} + 2825761 T^{4} )^{4} \))(\( ( 1 + 18 T + 4237 T^{2} + 27182 T^{3} + 8225864 T^{4} + 45692942 T^{5} + 11972749357 T^{6} + 85501876338 T^{7} + 7984925229121 T^{8} )^{4} \))
$43$ (\( ( 1 - 64 T + 7682 T^{2} - 352392 T^{3} + 21563770 T^{4} - 651572808 T^{5} + 26263229282 T^{6} - 404567235136 T^{7} + 11688200277601 T^{8} )^{2} \))(\( ( 1 - 5834 T^{2} + 14933666 T^{4} - 19945285034 T^{6} + 11688200277601 T^{8} )^{2} \))(\( 1 - 188 T + 17672 T^{2} - 1157140 T^{3} + 57226414 T^{4} - 2139551860 T^{5} + 60417051272 T^{6} - 1188416253212 T^{7} + 11688200277601 T^{8} \))(\( 1 + 84 T + 3528 T^{2} + 141372 T^{3} + 3824016 T^{4} + 98755524 T^{5} + 4797356760 T^{6} + 198286867884 T^{7} + 8208413810270 T^{8} + 366632418717516 T^{9} + 16401208088444760 T^{10} + 624269520298232676 T^{11} + 44695864872750665616 T^{12} + \)\(30\!\cdots\!28\)\( T^{13} + \)\(14\!\cdots\!28\)\( T^{14} + \)\(62\!\cdots\!16\)\( T^{15} + \)\(13\!\cdots\!01\)\( T^{16} \))(\( ( 1 - 6556 T^{2} + 23291117 T^{4} - 58436738184 T^{6} + 118955808601868 T^{8} - 199783578940197384 T^{10} + \)\(27\!\cdots\!17\)\( T^{12} - \)\(26\!\cdots\!56\)\( T^{14} + \)\(13\!\cdots\!01\)\( T^{16} )^{2} \))(\( ( 1 - 52 T + 3564 T^{2} - 177176 T^{3} + 7084511 T^{4} - 327598424 T^{5} + 12184606764 T^{6} - 328710878548 T^{7} + 11688200277601 T^{8} )^{2} \))(\( ( 1 - 60 T + 1800 T^{2} - 65040 T^{3} + 1764887 T^{4} - 120258960 T^{5} + 6153841800 T^{6} - 379281782940 T^{7} + 11688200277601 T^{8} )^{2} \))(\( ( 1 - 8 T + 32 T^{2} + 14888 T^{3} - 6837458 T^{4} + 27527912 T^{5} + 109401632 T^{6} - 50570904392 T^{7} + 11688200277601 T^{8} )^{2} \))(\( ( 1 - 72 T + 2592 T^{2} - 12288 T^{3} - 872859 T^{4} - 17006160 T^{5} + 3562391520 T^{6} + 141343265880 T^{7} - 11372920448860 T^{8} + 261343698612120 T^{9} + 12179107690967520 T^{10} - 107502111429381840 T^{11} - 10202150806106531259 T^{12} - \)\(26\!\cdots\!12\)\( T^{13} + \)\(10\!\cdots\!92\)\( T^{14} - \)\(53\!\cdots\!28\)\( T^{15} + \)\(13\!\cdots\!01\)\( T^{16} )^{2} \))
$47$ (\( 1 - 5618 T^{2} + 16643837 T^{4} - 38947889982 T^{6} + 83393318217448 T^{8} - 190053278735255742 T^{10} + \)\(39\!\cdots\!57\)\( T^{12} - \)\(65\!\cdots\!38\)\( T^{14} + \)\(56\!\cdots\!21\)\( T^{16} \))(\( ( 1 + 3609 T^{2} + 6828178 T^{4} + 17610768729 T^{6} + 23811286661761 T^{8} )^{2} \))(\( 1 + 84 T + 3528 T^{2} - 51408 T^{3} - 7208953 T^{4} - 113560272 T^{5} + 17215514568 T^{6} + 905454087636 T^{7} + 23811286661761 T^{8} \))(\( 1 - 92 T + 4232 T^{2} - 180800 T^{3} + 1415657 T^{4} + 321780504 T^{5} - 19250546792 T^{6} + 951928394932 T^{7} - 45512077630256 T^{8} + 2102809824404788 T^{9} - 93936527420533352 T^{10} + 3468541341290145816 T^{11} + 33708614641728591977 T^{12} - \)\(95\!\cdots\!00\)\( T^{13} + \)\(49\!\cdots\!12\)\( T^{14} - \)\(23\!\cdots\!48\)\( T^{15} + \)\(56\!\cdots\!21\)\( T^{16} \))(\( 1 - 12486 T^{2} + 79750495 T^{4} - 364855205082 T^{6} + 1367944008581893 T^{8} - 4442534973622908972 T^{10} + \)\(12\!\cdots\!58\)\( T^{12} - \)\(32\!\cdots\!20\)\( T^{14} + \)\(75\!\cdots\!62\)\( T^{16} - \)\(15\!\cdots\!20\)\( T^{18} + \)\(30\!\cdots\!38\)\( T^{20} - \)\(51\!\cdots\!52\)\( T^{22} + \)\(77\!\cdots\!53\)\( T^{24} - \)\(10\!\cdots\!82\)\( T^{26} + \)\(10\!\cdots\!95\)\( T^{28} - \)\(82\!\cdots\!46\)\( T^{30} + \)\(32\!\cdots\!41\)\( T^{32} \))(\( 1 + 192 T + 19960 T^{2} + 1473024 T^{3} + 78367758 T^{4} + 2710347264 T^{5} + 30969352832 T^{6} - 3182896406208 T^{7} - 244778009456893 T^{8} - 7031018161313472 T^{9} + 151120562596606592 T^{10} + 29215416775022009856 T^{11} + \)\(18\!\cdots\!38\)\( T^{12} + \)\(77\!\cdots\!76\)\( T^{13} + \)\(23\!\cdots\!60\)\( T^{14} + \)\(49\!\cdots\!48\)\( T^{15} + \)\(56\!\cdots\!21\)\( T^{16} \))(\( 1 - 72 T + 2592 T^{2} - 86256 T^{3} + 4419742 T^{4} - 430225128 T^{5} + 23240286720 T^{6} - 1046121049944 T^{7} + 51328414956099 T^{8} - 2310881399326296 T^{9} + 113405185542136320 T^{10} - 4637489294658587112 T^{11} + \)\(10\!\cdots\!62\)\( T^{12} - \)\(45\!\cdots\!44\)\( T^{13} + \)\(30\!\cdots\!72\)\( T^{14} - \)\(18\!\cdots\!68\)\( T^{15} + \)\(56\!\cdots\!21\)\( T^{16} \))(\( 1 - 102 T + 5202 T^{2} + 157284 T^{3} - 29409683 T^{4} + 1814223612 T^{5} - 19692509130 T^{6} - 3140329970154 T^{7} + 264570468668724 T^{8} - 6936988904070186 T^{9} - 96093162643987530 T^{10} + 19555906968704148348 T^{11} - \)\(70\!\cdots\!63\)\( T^{12} + \)\(82\!\cdots\!16\)\( T^{13} + \)\(60\!\cdots\!82\)\( T^{14} - \)\(26\!\cdots\!38\)\( T^{15} + \)\(56\!\cdots\!21\)\( T^{16} \))(\( 1 - 46 T + 1058 T^{2} + 161412 T^{3} - 22698365 T^{4} + 1010022752 T^{5} - 9419259550 T^{6} - 2715134508658 T^{7} + 271485194118333 T^{8} - 10484618609225180 T^{9} + 53077017818847872 T^{10} + 26010553538653727460 T^{11} - \)\(20\!\cdots\!46\)\( T^{12} + \)\(72\!\cdots\!12\)\( T^{13} - \)\(91\!\cdots\!12\)\( T^{14} - \)\(16\!\cdots\!00\)\( T^{15} + \)\(11\!\cdots\!70\)\( T^{16} - \)\(36\!\cdots\!00\)\( T^{17} - \)\(44\!\cdots\!72\)\( T^{18} + \)\(78\!\cdots\!48\)\( T^{19} - \)\(49\!\cdots\!06\)\( T^{20} + \)\(13\!\cdots\!40\)\( T^{21} + \)\(61\!\cdots\!52\)\( T^{22} - \)\(26\!\cdots\!20\)\( T^{23} + \)\(15\!\cdots\!93\)\( T^{24} - \)\(34\!\cdots\!62\)\( T^{25} - \)\(26\!\cdots\!50\)\( T^{26} + \)\(61\!\cdots\!68\)\( T^{27} - \)\(30\!\cdots\!65\)\( T^{28} + \)\(48\!\cdots\!48\)\( T^{29} + \)\(69\!\cdots\!38\)\( T^{30} - \)\(66\!\cdots\!54\)\( T^{31} + \)\(32\!\cdots\!41\)\( T^{32} \))
$53$ (\( ( 1 + 104 T + 12156 T^{2} + 784920 T^{3} + 53277670 T^{4} + 2204840280 T^{5} + 95916687036 T^{6} + 2305093557416 T^{7} + 62259690411361 T^{8} )^{2} \))(\( ( 1 - 3908 T^{2} + 9160166 T^{4} - 30835999748 T^{6} + 62259690411361 T^{8} )^{2} \))(\( 1 - 4 T + 8 T^{2} - 8444 T^{3} + 8425438 T^{4} - 23719196 T^{5} + 63123848 T^{6} - 88657444516 T^{7} + 62259690411361 T^{8} \))(\( 1 - 120 T + 7200 T^{2} - 373816 T^{3} + 3311084 T^{4} + 839259704 T^{5} - 54681768352 T^{6} + 3024646366328 T^{7} - 164654054909466 T^{8} + 8496231643015352 T^{9} - 431465454227857312 T^{10} + 18601655160473645816 T^{11} + \)\(20\!\cdots\!24\)\( T^{12} - \)\(65\!\cdots\!84\)\( T^{13} + \)\(35\!\cdots\!00\)\( T^{14} - \)\(16\!\cdots\!80\)\( T^{15} + \)\(38\!\cdots\!21\)\( T^{16} \))(\( 1 + 18178 T^{2} + 176430419 T^{4} + 1215858482750 T^{6} + 6624667983488729 T^{8} + 30050449713194977364 T^{10} + \)\(11\!\cdots\!10\)\( T^{12} + \)\(39\!\cdots\!28\)\( T^{14} + \)\(11\!\cdots\!98\)\( T^{16} + \)\(31\!\cdots\!68\)\( T^{18} + \)\(72\!\cdots\!10\)\( T^{20} + \)\(14\!\cdots\!24\)\( T^{22} + \)\(25\!\cdots\!09\)\( T^{24} + \)\(37\!\cdots\!50\)\( T^{26} + \)\(42\!\cdots\!39\)\( T^{28} + \)\(34\!\cdots\!58\)\( T^{30} + \)\(15\!\cdots\!41\)\( T^{32} \))(\( 1 + 176 T + 11428 T^{2} + 466336 T^{3} + 30903530 T^{4} + 1742491856 T^{5} + 35120249744 T^{6} + 828142814416 T^{7} + 94137021167923 T^{8} + 2326253165694544 T^{9} + 277115663320286864 T^{10} + 38621218760725465424 T^{11} + \)\(19\!\cdots\!30\)\( T^{12} + \)\(81\!\cdots\!64\)\( T^{13} + \)\(56\!\cdots\!48\)\( T^{14} + \)\(24\!\cdots\!44\)\( T^{15} + \)\(38\!\cdots\!21\)\( T^{16} \))(\( 1 - 76 T + 2888 T^{2} + 280744 T^{3} - 32097310 T^{4} + 1832302924 T^{5} - 7149394176 T^{6} - 4718024532996 T^{7} + 411450315196723 T^{8} - 13252930913185764 T^{9} - 56412158907238656 T^{10} + 40611823705258641196 T^{11} - \)\(19\!\cdots\!10\)\( T^{12} + \)\(49\!\cdots\!56\)\( T^{13} + \)\(14\!\cdots\!08\)\( T^{14} - \)\(10\!\cdots\!44\)\( T^{15} + \)\(38\!\cdots\!21\)\( T^{16} \))(\( 1 - 98 T + 4802 T^{2} - 358484 T^{3} + 21915517 T^{4} - 1279203212 T^{5} + 84378991270 T^{6} - 4129245378846 T^{7} + 191619264999924 T^{8} - 11599050269178414 T^{9} + 665790827415100870 T^{10} - 28352721948144746348 T^{11} + \)\(13\!\cdots\!37\)\( T^{12} - \)\(62\!\cdots\!16\)\( T^{13} + \)\(23\!\cdots\!82\)\( T^{14} - \)\(13\!\cdots\!62\)\( T^{15} + \)\(38\!\cdots\!21\)\( T^{16} \))(\( 1 - 30 T + 450 T^{2} + 53260 T^{3} - 13562801 T^{4} + 264802000 T^{5} - 422485750 T^{6} - 1346619990690 T^{7} + 94385910261585 T^{8} - 3394198091700460 T^{9} + 16961261878565400 T^{10} + 11461088832223742300 T^{11} - \)\(43\!\cdots\!94\)\( T^{12} + \)\(38\!\cdots\!80\)\( T^{13} - \)\(65\!\cdots\!00\)\( T^{14} - \)\(11\!\cdots\!00\)\( T^{15} + \)\(39\!\cdots\!74\)\( T^{16} - \)\(31\!\cdots\!00\)\( T^{17} - \)\(51\!\cdots\!00\)\( T^{18} + \)\(84\!\cdots\!20\)\( T^{19} - \)\(27\!\cdots\!34\)\( T^{20} + \)\(20\!\cdots\!00\)\( T^{21} + \)\(83\!\cdots\!00\)\( T^{22} - \)\(46\!\cdots\!40\)\( T^{23} + \)\(36\!\cdots\!85\)\( T^{24} - \)\(14\!\cdots\!10\)\( T^{25} - \)\(12\!\cdots\!50\)\( T^{26} + \)\(22\!\cdots\!00\)\( T^{27} - \)\(32\!\cdots\!81\)\( T^{28} + \)\(36\!\cdots\!40\)\( T^{29} + \)\(85\!\cdots\!50\)\( T^{30} - \)\(16\!\cdots\!70\)\( T^{31} + \)\(15\!\cdots\!41\)\( T^{32} \))
$59$ (\( 1 - 16128 T^{2} + 123340988 T^{4} - 622901719296 T^{6} + 2414919503628870 T^{8} - 7547925000230297856 T^{10} + \)\(18\!\cdots\!48\)\( T^{12} - \)\(28\!\cdots\!68\)\( T^{14} + \)\(21\!\cdots\!41\)\( T^{16} \))(\( ( 1 - 3924 T^{2} + 14604166 T^{4} - 47548524564 T^{6} + 146830437604321 T^{8} )^{2} \))(\( 1 - 7592 T^{2} + 38442738 T^{4} - 91995004712 T^{6} + 146830437604321 T^{8} \))(\( 1 - 10604 T^{2} + 51410168 T^{4} - 210609725156 T^{6} + 822315101354862 T^{8} - 2552034069826033316 T^{10} + \)\(75\!\cdots\!28\)\( T^{12} - \)\(18\!\cdots\!24\)\( T^{14} + \)\(21\!\cdots\!41\)\( T^{16} \))(\( ( 1 + 186 T + 27633 T^{2} + 2994786 T^{3} + 291274901 T^{4} + 24395393880 T^{5} + 1852144355430 T^{6} + 125623993690740 T^{7} + 7786872546817290 T^{8} + 437297122037465940 T^{9} + 22443101778857620230 T^{10} + \)\(10\!\cdots\!80\)\( T^{11} + \)\(42\!\cdots\!21\)\( T^{12} + \)\(15\!\cdots\!86\)\( T^{13} + \)\(49\!\cdots\!73\)\( T^{14} + \)\(11\!\cdots\!46\)\( T^{15} + \)\(21\!\cdots\!41\)\( T^{16} )^{2} \))(\( 1 - 48 T + 8612 T^{2} - 376512 T^{3} + 30329050 T^{4} - 1530261168 T^{5} + 120119465936 T^{6} - 6323967202032 T^{7} + 540612754864243 T^{8} - 22013729830273392 T^{9} + 1455530931873714896 T^{10} - 64547232676339952688 T^{11} + \)\(44\!\cdots\!50\)\( T^{12} - \)\(19\!\cdots\!12\)\( T^{13} + \)\(15\!\cdots\!72\)\( T^{14} - \)\(29\!\cdots\!28\)\( T^{15} + \)\(21\!\cdots\!41\)\( T^{16} \))(\( 1 + 4064 T^{2} - 11259650 T^{4} + 14390721536 T^{6} + 424530130709539 T^{8} + 174377567902186496 T^{10} - \)\(16\!\cdots\!50\)\( T^{12} + \)\(72\!\cdots\!84\)\( T^{14} + \)\(21\!\cdots\!41\)\( T^{16} \))(\( 1 + 9434 T^{2} + 45444361 T^{4} + 182276889482 T^{6} + 671844820055188 T^{8} + 2208714871810497002 T^{10} + \)\(66\!\cdots\!81\)\( T^{12} + \)\(16\!\cdots\!54\)\( T^{14} + \)\(21\!\cdots\!41\)\( T^{16} \))(\( 1 + 6674 T^{2} - 7729537 T^{4} - 69884124210 T^{6} + 388809824859333 T^{8} + 851796830409747108 T^{10} - \)\(72\!\cdots\!98\)\( T^{12} - \)\(40\!\cdots\!08\)\( T^{14} + \)\(99\!\cdots\!50\)\( T^{16} - \)\(48\!\cdots\!88\)\( T^{18} - \)\(10\!\cdots\!58\)\( T^{20} + \)\(15\!\cdots\!48\)\( T^{22} + \)\(83\!\cdots\!53\)\( T^{24} - \)\(18\!\cdots\!10\)\( T^{26} - \)\(24\!\cdots\!57\)\( T^{28} + \)\(25\!\cdots\!54\)\( T^{30} + \)\(46\!\cdots\!81\)\( T^{32} \))
$61$ (\( 1 - 27152 T^{2} + 331146332 T^{4} - 2369254902768 T^{6} + 10876351870920838 T^{8} - 32804326672196187888 T^{10} + \)\(63\!\cdots\!92\)\( T^{12} - \)\(72\!\cdots\!92\)\( T^{14} + \)\(36\!\cdots\!61\)\( T^{16} \))(\( ( 1 - 10456 T^{2} + 54914478 T^{4} - 144772113496 T^{6} + 191707312997281 T^{8} )^{2} \))(\( ( 1 + 144 T + 12402 T^{2} + 535824 T^{3} + 13845841 T^{4} )^{2} \))(\( ( 1 - 48 T + 14980 T^{2} - 522512 T^{3} + 83748518 T^{4} - 1944267152 T^{5} + 207410698180 T^{6} - 2472977969328 T^{7} + 191707312997281 T^{8} )^{2} \))(\( ( 1 - 174 T + 16762 T^{2} - 1160580 T^{3} + 59379469 T^{4} - 799521084 T^{5} - 213150096146 T^{6} + 24807597435150 T^{7} - 1755715644266732 T^{8} + 92309070056193150 T^{9} - 2951242340372228786 T^{10} - 41191625557192527324 T^{11} + \)\(11\!\cdots\!89\)\( T^{12} - \)\(82\!\cdots\!80\)\( T^{13} + \)\(44\!\cdots\!02\)\( T^{14} - \)\(17\!\cdots\!34\)\( T^{15} + \)\(36\!\cdots\!61\)\( T^{16} )^{2} \))(\( 1 - 72 T + 14578 T^{2} - 925200 T^{3} + 115490913 T^{4} - 7140338640 T^{5} + 657220661810 T^{6} - 36982785415128 T^{7} + 2781811232132132 T^{8} - 137612944529691288 T^{9} + 9099772785336032210 T^{10} - \)\(36\!\cdots\!40\)\( T^{11} + \)\(22\!\cdots\!53\)\( T^{12} - \)\(65\!\cdots\!00\)\( T^{13} + \)\(38\!\cdots\!38\)\( T^{14} - \)\(71\!\cdots\!52\)\( T^{15} + \)\(36\!\cdots\!61\)\( T^{16} \))(\( ( 1 + 162 T + 12721 T^{2} + 985122 T^{3} + 71371764 T^{4} + 3665638962 T^{5} + 176132943361 T^{6} + 8346300646482 T^{7} + 191707312997281 T^{8} )^{2} \))(\( ( 1 + 42 T - 5019 T^{2} - 27678 T^{3} + 25599404 T^{4} - 102989838 T^{5} - 69492275979 T^{6} + 2163855723162 T^{7} + 191707312997281 T^{8} )^{2} \))(\( ( 1 + 60 T - 8926 T^{2} - 393256 T^{3} + 57653141 T^{4} + 1511546176 T^{5} - 266254549814 T^{6} - 2710779938676 T^{7} + 1000035050441884 T^{8} - 10086812151813396 T^{9} - 3686518162251223574 T^{10} + 77875424851457993536 T^{11} + \)\(11\!\cdots\!21\)\( T^{12} - \)\(28\!\cdots\!56\)\( T^{13} - \)\(23\!\cdots\!46\)\( T^{14} + \)\(59\!\cdots\!60\)\( T^{15} + \)\(36\!\cdots\!61\)\( T^{16} )^{2} \))
$67$ (\( ( 1 + 56 T + 12026 T^{2} + 624480 T^{3} + 76112570 T^{4} + 2803290720 T^{5} + 242337381146 T^{6} + 5065669401464 T^{7} + 406067677556641 T^{8} )^{2} \))(\( ( 1 - 17618 T^{2} + 117900386 T^{4} - 355022449778 T^{6} + 406067677556641 T^{8} )^{2} \))(\( 1 + 92 T + 4232 T^{2} + 445924 T^{3} + 46858654 T^{4} + 2001752836 T^{5} + 85279544072 T^{6} + 8322171159548 T^{7} + 406067677556641 T^{8} \))(\( 1 + 172 T + 14792 T^{2} + 964692 T^{3} + 48981824 T^{4} + 1222830796 T^{5} - 48896916264 T^{6} - 15034768401932 T^{7} - 1486231771221186 T^{8} - 67491075356272748 T^{9} - 985327676162731944 T^{10} + \)\(11\!\cdots\!24\)\( T^{11} + \)\(19\!\cdots\!84\)\( T^{12} + \)\(17\!\cdots\!08\)\( T^{13} + \)\(12\!\cdots\!12\)\( T^{14} + \)\(63\!\cdots\!88\)\( T^{15} + \)\(16\!\cdots\!81\)\( T^{16} \))(\( 1 + 23218 T^{2} + 270240458 T^{4} + 2203130578292 T^{6} + 14697787782708749 T^{8} + 87186549530108980628 T^{10} + \)\(47\!\cdots\!34\)\( T^{12} + \)\(24\!\cdots\!14\)\( T^{14} + \)\(11\!\cdots\!16\)\( T^{16} + \)\(48\!\cdots\!94\)\( T^{18} + \)\(19\!\cdots\!94\)\( T^{20} + \)\(71\!\cdots\!08\)\( T^{22} + \)\(24\!\cdots\!69\)\( T^{24} + \)\(73\!\cdots\!92\)\( T^{26} + \)\(18\!\cdots\!18\)\( T^{28} + \)\(31\!\cdots\!38\)\( T^{30} + \)\(27\!\cdots\!61\)\( T^{32} \))(\( 1 - 132 T + 788 T^{2} + 669504 T^{3} - 22985375 T^{4} - 1062230400 T^{5} + 33913660052 T^{6} - 692915274876 T^{7} + 191836445801104 T^{8} - 3110496668918364 T^{9} + 683398267260718292 T^{10} - 96087643474729737600 T^{11} - \)\(93\!\cdots\!75\)\( T^{12} + \)\(12\!\cdots\!96\)\( T^{13} + \)\(64\!\cdots\!68\)\( T^{14} - \)\(48\!\cdots\!28\)\( T^{15} + \)\(16\!\cdots\!81\)\( T^{16} \))(\( 1 - 124 T + 7688 T^{2} + 603136 T^{3} - 112977415 T^{4} + 9268146136 T^{5} - 98793237096 T^{6} - 35532386900364 T^{7} + 4178772156136048 T^{8} - 159504884795733996 T^{9} - 1990794474703184616 T^{10} + \)\(83\!\cdots\!84\)\( T^{11} - \)\(45\!\cdots\!15\)\( T^{12} + \)\(10\!\cdots\!64\)\( T^{13} + \)\(62\!\cdots\!68\)\( T^{14} - \)\(45\!\cdots\!96\)\( T^{15} + \)\(16\!\cdots\!81\)\( T^{16} \))(\( 1 + 130 T + 8450 T^{2} - 101660 T^{3} - 49206683 T^{4} - 3311248460 T^{5} - 9498450650 T^{6} + 13933744675110 T^{7} + 1410770562904548 T^{8} + 62548579846568790 T^{9} - 191404428360678650 T^{10} - \)\(29\!\cdots\!40\)\( T^{11} - \)\(19\!\cdots\!03\)\( T^{12} - \)\(18\!\cdots\!40\)\( T^{13} + \)\(69\!\cdots\!50\)\( T^{14} + \)\(47\!\cdots\!70\)\( T^{15} + \)\(16\!\cdots\!81\)\( T^{16} \))(\( 1 + 74 T + 2738 T^{2} - 587900 T^{3} - 50988730 T^{4} - 3339904986 T^{5} + 65267378776 T^{6} + 10012501185306 T^{7} + 918173402551977 T^{8} - 2374991498059262 T^{9} + 329325818085928764 T^{10} + 1284143109513809266 T^{11} + \)\(25\!\cdots\!90\)\( T^{12} + \)\(13\!\cdots\!44\)\( T^{13} + \)\(44\!\cdots\!54\)\( T^{14} - \)\(11\!\cdots\!78\)\( T^{15} - \)\(84\!\cdots\!12\)\( T^{16} - \)\(50\!\cdots\!42\)\( T^{17} + \)\(88\!\cdots\!34\)\( T^{18} + \)\(12\!\cdots\!36\)\( T^{19} + \)\(10\!\cdots\!90\)\( T^{20} + \)\(23\!\cdots\!34\)\( T^{21} + \)\(26\!\cdots\!04\)\( T^{22} - \)\(87\!\cdots\!98\)\( T^{23} + \)\(15\!\cdots\!37\)\( T^{24} + \)\(74\!\cdots\!54\)\( T^{25} + \)\(21\!\cdots\!76\)\( T^{26} - \)\(49\!\cdots\!54\)\( T^{27} - \)\(34\!\cdots\!30\)\( T^{28} - \)\(17\!\cdots\!00\)\( T^{29} + \)\(36\!\cdots\!58\)\( T^{30} + \)\(44\!\cdots\!26\)\( T^{31} + \)\(27\!\cdots\!61\)\( T^{32} \))
$71$ (\( ( 1 + 188 T + 25704 T^{2} + 2576820 T^{3} + 208896910 T^{4} + 12989749620 T^{5} + 653181848424 T^{6} + 24082853377148 T^{7} + 645753531245761 T^{8} )^{2} \))(\( ( 1 - 126 T + 13714 T^{2} - 635166 T^{3} + 25411681 T^{4} )^{4} \))(\( ( 1 + 80 T + 11626 T^{2} + 403280 T^{3} + 25411681 T^{4} )^{2} \))(\( ( 1 - 88 T + 11996 T^{2} - 959944 T^{3} + 72709110 T^{4} - 4839077704 T^{5} + 304838525276 T^{6} - 11272824985048 T^{7} + 645753531245761 T^{8} )^{2} \))(\( ( 1 + 96 T + 20504 T^{2} + 1403664 T^{3} + 156181902 T^{4} + 7075870224 T^{5} + 521041107224 T^{6} + 12297627256416 T^{7} + 645753531245761 T^{8} )^{4} \))(\( ( 1 - 32 T + 8292 T^{2} + 81632 T^{3} + 28299158 T^{4} + 411506912 T^{5} + 210713658852 T^{6} - 4099209085472 T^{7} + 645753531245761 T^{8} )^{2} \))(\( ( 1 - 4 T + 8616 T^{2} - 20164 T^{3} + 25411681 T^{4} )^{4} \))(\( ( 1 - 100 T + 12406 T^{2} - 504100 T^{3} + 25411681 T^{4} )^{4} \))(\( ( 1 - 4 T + 10238 T^{2} + 37292 T^{3} + 54945234 T^{4} + 187988972 T^{5} + 260164790078 T^{6} - 512401135684 T^{7} + 645753531245761 T^{8} )^{4} \))
$73$ (\( 1 - 34552 T^{2} + 545494172 T^{4} - 5215338412488 T^{6} + 33447987891134918 T^{8} - \)\(14\!\cdots\!08\)\( T^{10} + \)\(43\!\cdots\!32\)\( T^{12} - \)\(79\!\cdots\!92\)\( T^{14} + \)\(65\!\cdots\!61\)\( T^{16} \))(\( ( 1 + 9216 T^{2} + 58294078 T^{4} + 261718189056 T^{6} + 806460091894081 T^{8} )^{2} \))(\( 1 + 72 T + 2592 T^{2} + 35208 T^{3} - 22947358 T^{4} + 187623432 T^{5} + 73608240672 T^{6} + 10896064292808 T^{7} + 806460091894081 T^{8} \))(\( 1 - 60 T + 1800 T^{2} - 51652 T^{3} + 31101104 T^{4} - 3155702428 T^{5} + 134694123032 T^{6} - 10014018827044 T^{7} + 380079606390366 T^{8} - 53364706329317476 T^{9} + 3825076167146386712 T^{10} - \)\(47\!\cdots\!92\)\( T^{11} + \)\(25\!\cdots\!24\)\( T^{12} - \)\(22\!\cdots\!48\)\( T^{13} + \)\(41\!\cdots\!00\)\( T^{14} - \)\(73\!\cdots\!40\)\( T^{15} + \)\(65\!\cdots\!61\)\( T^{16} \))(\( 1 - 27942 T^{2} + 406639267 T^{4} - 4007953920714 T^{6} + 29963815062731785 T^{8} - \)\(18\!\cdots\!64\)\( T^{10} + \)\(96\!\cdots\!46\)\( T^{12} - \)\(48\!\cdots\!76\)\( T^{14} + \)\(25\!\cdots\!94\)\( T^{16} - \)\(13\!\cdots\!16\)\( T^{18} + \)\(78\!\cdots\!26\)\( T^{20} - \)\(41\!\cdots\!44\)\( T^{22} + \)\(19\!\cdots\!85\)\( T^{24} - \)\(74\!\cdots\!14\)\( T^{26} + \)\(21\!\cdots\!47\)\( T^{28} - \)\(41\!\cdots\!02\)\( T^{30} + \)\(42\!\cdots\!21\)\( T^{32} \))(\( 1 - 216 T + 40020 T^{2} - 5285088 T^{3} + 638882026 T^{4} - 64569320856 T^{5} + 6069665158992 T^{6} - 501100335480312 T^{7} + 38793311793484563 T^{8} - 2670363687774582648 T^{9} + \)\(17\!\cdots\!72\)\( T^{10} - \)\(97\!\cdots\!84\)\( T^{11} + \)\(51\!\cdots\!06\)\( T^{12} - \)\(22\!\cdots\!12\)\( T^{13} + \)\(91\!\cdots\!20\)\( T^{14} - \)\(26\!\cdots\!44\)\( T^{15} + \)\(65\!\cdots\!61\)\( T^{16} \))(\( 1 - 32 T + 512 T^{2} - 51136 T^{3} + 34708862 T^{4} - 2067510688 T^{5} + 49696849920 T^{6} - 5991675606624 T^{7} + 642997612278979 T^{8} - 31929639307699296 T^{9} + 1411303120968990720 T^{10} - \)\(31\!\cdots\!32\)\( T^{11} + \)\(27\!\cdots\!22\)\( T^{12} - \)\(21\!\cdots\!64\)\( T^{13} + \)\(11\!\cdots\!52\)\( T^{14} - \)\(39\!\cdots\!88\)\( T^{15} + \)\(65\!\cdots\!61\)\( T^{16} \))(\( 1 + 246 T + 30258 T^{2} + 1881900 T^{3} + 11613493 T^{4} - 9211461636 T^{5} - 846646828650 T^{6} - 36781012225806 T^{7} - 1359453564456732 T^{8} - 196006014151320174 T^{9} - 24043280681888404650 T^{10} - \)\(13\!\cdots\!04\)\( T^{11} + \)\(93\!\cdots\!33\)\( T^{12} + \)\(80\!\cdots\!00\)\( T^{13} + \)\(69\!\cdots\!18\)\( T^{14} + \)\(30\!\cdots\!14\)\( T^{15} + \)\(65\!\cdots\!61\)\( T^{16} \))(\( 1 + 54 T + 1458 T^{2} + 376300 T^{3} - 9032525 T^{4} + 1427883664 T^{5} + 161075984306 T^{6} + 3449490767706 T^{7} + 2006002204707837 T^{8} + 28704897132082748 T^{9} + 2397705340658794944 T^{10} + \)\(54\!\cdots\!96\)\( T^{11} - \)\(52\!\cdots\!50\)\( T^{12} + \)\(41\!\cdots\!84\)\( T^{13} + \)\(17\!\cdots\!04\)\( T^{14} + \)\(42\!\cdots\!72\)\( T^{15} + \)\(19\!\cdots\!58\)\( T^{16} + \)\(22\!\cdots\!88\)\( T^{17} + \)\(49\!\cdots\!64\)\( T^{18} + \)\(62\!\cdots\!76\)\( T^{19} - \)\(42\!\cdots\!50\)\( T^{20} + \)\(23\!\cdots\!04\)\( T^{21} + \)\(54\!\cdots\!24\)\( T^{22} + \)\(35\!\cdots\!32\)\( T^{23} + \)\(13\!\cdots\!57\)\( T^{24} + \)\(11\!\cdots\!14\)\( T^{25} + \)\(29\!\cdots\!06\)\( T^{26} + \)\(14\!\cdots\!56\)\( T^{27} - \)\(47\!\cdots\!25\)\( T^{28} + \)\(10\!\cdots\!00\)\( T^{29} + \)\(21\!\cdots\!98\)\( T^{30} + \)\(42\!\cdots\!46\)\( T^{31} + \)\(42\!\cdots\!21\)\( T^{32} \))
$79$ (\( ( 1 - 94 T + 13965 T^{2} - 1353586 T^{3} + 93524384 T^{4} - 8447730226 T^{5} + 543937881165 T^{6} - 22850220818974 T^{7} + 1517108809906561 T^{8} )^{2} \))(\( ( 1 - 173 T + 17858 T^{2} - 1079693 T^{3} + 38950081 T^{4} )^{4} \))(\( 1 - 8274 T^{2} + 26761235 T^{4} - 322272970194 T^{6} + 1517108809906561 T^{8} \))(\( 1 - 36262 T^{2} + 647137361 T^{4} - 7185915421278 T^{6} + 53992135211136996 T^{8} - \)\(27\!\cdots\!18\)\( T^{10} + \)\(98\!\cdots\!21\)\( T^{12} - \)\(21\!\cdots\!42\)\( T^{14} + \)\(23\!\cdots\!21\)\( T^{16} \))(\( ( 1 - 74 T - 15595 T^{2} + 759950 T^{3} + 174425033 T^{4} - 4475833744 T^{5} - 1421259531822 T^{6} + 13184758420596 T^{7} + 9224324946351346 T^{8} + 82286077302939636 T^{9} - 55358173886488977582 T^{10} - \)\(10\!\cdots\!24\)\( T^{11} + \)\(26\!\cdots\!13\)\( T^{12} + \)\(71\!\cdots\!50\)\( T^{13} - \)\(92\!\cdots\!95\)\( T^{14} - \)\(27\!\cdots\!94\)\( T^{15} + \)\(23\!\cdots\!21\)\( T^{16} )^{2} \))(\( 1 + 288 T + 30300 T^{2} + 2390592 T^{3} + 343035706 T^{4} + 36542007840 T^{5} + 2371926750768 T^{6} + 227910022136160 T^{7} + 23724159497509971 T^{8} + 1422386448151774560 T^{9} + 92386739068480412208 T^{10} + \)\(88\!\cdots\!40\)\( T^{11} + \)\(52\!\cdots\!66\)\( T^{12} + \)\(22\!\cdots\!92\)\( T^{13} + \)\(17\!\cdots\!00\)\( T^{14} + \)\(10\!\cdots\!28\)\( T^{15} + \)\(23\!\cdots\!21\)\( T^{16} \))(\( 1 + 13024 T^{2} + 67570270 T^{4} + 314583571456 T^{6} + 2291081588444419 T^{8} + 12253055589480487936 T^{10} + \)\(10\!\cdots\!70\)\( T^{12} + \)\(76\!\cdots\!84\)\( T^{14} + \)\(23\!\cdots\!21\)\( T^{16} \))(\( 1 + 5010 T^{2} - 52175183 T^{4} - 3130643790 T^{6} + 3476709624715428 T^{8} - 121938829202646990 T^{10} - \)\(79\!\cdots\!63\)\( T^{12} + \)\(29\!\cdots\!10\)\( T^{14} + \)\(23\!\cdots\!21\)\( T^{16} \))(\( 1 + 42538 T^{2} + 982751207 T^{4} + 15945124184102 T^{6} + 201431875816036549 T^{8} + \)\(20\!\cdots\!92\)\( T^{10} + \)\(18\!\cdots\!34\)\( T^{12} + \)\(14\!\cdots\!24\)\( T^{14} + \)\(93\!\cdots\!90\)\( T^{16} + \)\(54\!\cdots\!44\)\( T^{18} + \)\(27\!\cdots\!74\)\( T^{20} + \)\(12\!\cdots\!72\)\( T^{22} + \)\(46\!\cdots\!29\)\( T^{24} + \)\(14\!\cdots\!02\)\( T^{26} + \)\(34\!\cdots\!67\)\( T^{28} + \)\(57\!\cdots\!18\)\( T^{30} + \)\(52\!\cdots\!41\)\( T^{32} \))
$83$ (\( 1 - 28196 T^{2} + 281031704 T^{4} - 911252649996 T^{6} - 186386635165010 T^{8} - 43246520775610816716 T^{10} + \)\(63\!\cdots\!64\)\( T^{12} - \)\(30\!\cdots\!56\)\( T^{14} + \)\(50\!\cdots\!81\)\( T^{16} \))(\( ( 1 + 25678 T^{2} + 259012130 T^{4} + 1218634766638 T^{6} + 2252292232139041 T^{8} )^{2} \))(\( 1 - 256 T + 32768 T^{2} - 3746048 T^{3} + 368279842 T^{4} - 25806524672 T^{5} + 1555114262528 T^{6} - 83696735582464 T^{7} + 2252292232139041 T^{8} \))(\( 1 - 272 T + 36992 T^{2} - 4133264 T^{3} + 420964292 T^{4} - 36062285872 T^{5} + 2778566314368 T^{6} - 202040310658480 T^{7} + 15202520296678854 T^{8} - 1391855700126268720 T^{9} + \)\(13\!\cdots\!28\)\( T^{10} - \)\(11\!\cdots\!68\)\( T^{11} + \)\(94\!\cdots\!72\)\( T^{12} - \)\(64\!\cdots\!36\)\( T^{13} + \)\(39\!\cdots\!12\)\( T^{14} - \)\(20\!\cdots\!88\)\( T^{15} + \)\(50\!\cdots\!81\)\( T^{16} \))(\( ( 1 + 18260 T^{2} + 283246781 T^{4} + 2667209610360 T^{6} + 22154057717595020 T^{8} + \)\(12\!\cdots\!60\)\( T^{10} + \)\(63\!\cdots\!21\)\( T^{12} + \)\(19\!\cdots\!60\)\( T^{14} + \)\(50\!\cdots\!81\)\( T^{16} )^{2} \))(\( 1 - 49400 T^{2} + 1103892174 T^{4} - 14541036476800 T^{6} + 123368183239590851 T^{8} - \)\(69\!\cdots\!00\)\( T^{10} + \)\(24\!\cdots\!34\)\( T^{12} - \)\(52\!\cdots\!00\)\( T^{14} + \)\(50\!\cdots\!81\)\( T^{16} \))(\( ( 1 - 84 T + 3528 T^{2} - 500304 T^{3} + 70077383 T^{4} - 3446594256 T^{5} + 167432956488 T^{6} - 27462991362996 T^{7} + 2252292232139041 T^{8} )^{2} \))(\( ( 1 + 80 T + 3200 T^{2} + 317680 T^{3} + 23022958 T^{4} + 2188497520 T^{5} + 151866627200 T^{6} + 26155229869520 T^{7} + 2252292232139041 T^{8} )^{2} \))(\( ( 1 - 32 T + 512 T^{2} + 252712 T^{3} - 42882475 T^{4} - 1371597568 T^{5} + 97778626848 T^{6} - 7242029223592 T^{7} + 506959325886468 T^{8} - 49890339321325288 T^{9} + 4640409459891602208 T^{10} - \)\(44\!\cdots\!92\)\( T^{11} - \)\(96\!\cdots\!75\)\( T^{12} + \)\(39\!\cdots\!88\)\( T^{13} + \)\(54\!\cdots\!32\)\( T^{14} - \)\(23\!\cdots\!28\)\( T^{15} + \)\(50\!\cdots\!81\)\( T^{16} )^{2} \))
$89$ (\( 1 - 26360 T^{2} + 419977340 T^{4} - 4862799095880 T^{6} + 44341547037657862 T^{8} - \)\(30\!\cdots\!80\)\( T^{10} + \)\(16\!\cdots\!40\)\( T^{12} - \)\(65\!\cdots\!60\)\( T^{14} + \)\(15\!\cdots\!61\)\( T^{16} \))(\( ( 1 - 25980 T^{2} + 294219190 T^{4} - 1630043421180 T^{6} + 3936588805702081 T^{8} )^{2} \))(\( ( 1 - 9442 T^{2} + 62742241 T^{4} )^{2} \))(\( 1 - 5000 T^{2} + 161915036 T^{4} - 706085059000 T^{6} + 14582141323852486 T^{8} - 44301358938277219000 T^{10} + \)\(63\!\cdots\!16\)\( T^{12} - \)\(12\!\cdots\!00\)\( T^{14} + \)\(15\!\cdots\!61\)\( T^{16} \))(\( ( 1 + 456 T + 122262 T^{2} + 24145200 T^{3} + 3877011353 T^{4} + 528957464160 T^{5} + 62923077051270 T^{6} + 6625178757146664 T^{7} + 622894246587494484 T^{8} + 52478040935358725544 T^{9} + \)\(39\!\cdots\!70\)\( T^{10} + \)\(26\!\cdots\!60\)\( T^{11} + \)\(15\!\cdots\!93\)\( T^{12} + \)\(75\!\cdots\!00\)\( T^{13} + \)\(30\!\cdots\!02\)\( T^{14} + \)\(89\!\cdots\!96\)\( T^{15} + \)\(15\!\cdots\!61\)\( T^{16} )^{2} \))(\( 1 + 132 T + 36502 T^{2} + 4051608 T^{3} + 716519265 T^{4} + 69860823192 T^{5} + 9283830649286 T^{6} + 786348344536908 T^{7} + 86046231197013188 T^{8} + 6228665237076848268 T^{9} + \)\(58\!\cdots\!26\)\( T^{10} + \)\(34\!\cdots\!12\)\( T^{11} + \)\(28\!\cdots\!65\)\( T^{12} + \)\(12\!\cdots\!08\)\( T^{13} + \)\(90\!\cdots\!42\)\( T^{14} + \)\(25\!\cdots\!12\)\( T^{15} + \)\(15\!\cdots\!61\)\( T^{16} \))(\( 1 + 22466 T^{2} + 258050305 T^{4} + 2722572965954 T^{6} + 25242583519811524 T^{8} + \)\(17\!\cdots\!14\)\( T^{10} + \)\(10\!\cdots\!05\)\( T^{12} + \)\(55\!\cdots\!86\)\( T^{14} + \)\(15\!\cdots\!61\)\( T^{16} \))(\( 1 + 24902 T^{2} + 348507721 T^{4} + 3638615519702 T^{6} + 30923927822861044 T^{8} + \)\(22\!\cdots\!82\)\( T^{10} + \)\(13\!\cdots\!01\)\( T^{12} + \)\(61\!\cdots\!42\)\( T^{14} + \)\(15\!\cdots\!61\)\( T^{16} \))(\( 1 + 32552 T^{2} + 612475958 T^{4} + 7153003757936 T^{6} + 52449920499191449 T^{8} + \)\(12\!\cdots\!92\)\( T^{10} - \)\(24\!\cdots\!66\)\( T^{12} - \)\(42\!\cdots\!64\)\( T^{14} - \)\(40\!\cdots\!12\)\( T^{16} - \)\(26\!\cdots\!24\)\( T^{18} - \)\(95\!\cdots\!46\)\( T^{20} + \)\(31\!\cdots\!32\)\( T^{22} + \)\(81\!\cdots\!89\)\( T^{24} + \)\(69\!\cdots\!36\)\( T^{26} + \)\(37\!\cdots\!78\)\( T^{28} + \)\(12\!\cdots\!12\)\( T^{30} + \)\(24\!\cdots\!21\)\( T^{32} \))
$97$ (\( 1 - 33190 T^{2} + 746913305 T^{4} - 10749643947030 T^{6} + 119807243476794452 T^{8} - \)\(95\!\cdots\!30\)\( T^{10} + \)\(58\!\cdots\!05\)\( T^{12} - \)\(23\!\cdots\!90\)\( T^{14} + \)\(61\!\cdots\!21\)\( T^{16} \))(\( ( 1 + 22275 T^{2} + 265834660 T^{4} + 1971989734275 T^{6} + 7837433594376961 T^{8} )^{2} \))(\( 1 + 84 T + 3528 T^{2} + 652176 T^{3} + 117853367 T^{4} + 6136323984 T^{5} + 312331303368 T^{6} + 69969648414036 T^{7} + 7837433594376961 T^{8} \))(\( 1 + 304 T + 46208 T^{2} + 5567716 T^{3} + 677799269 T^{4} + 82289884992 T^{5} + 9196107143944 T^{6} + 965526846684508 T^{7} + 96868152787138756 T^{8} + 9084642100454535772 T^{9} + \)\(81\!\cdots\!64\)\( T^{10} + \)\(68\!\cdots\!68\)\( T^{11} + \)\(53\!\cdots\!09\)\( T^{12} + \)\(41\!\cdots\!84\)\( T^{13} + \)\(32\!\cdots\!28\)\( T^{14} + \)\(19\!\cdots\!76\)\( T^{15} + \)\(61\!\cdots\!21\)\( T^{16} \))(\( ( 1 + 45720 T^{2} + 1061437628 T^{4} + 16194732863400 T^{6} + 177610134878867718 T^{8} + \)\(14\!\cdots\!00\)\( T^{10} + \)\(83\!\cdots\!08\)\( T^{12} + \)\(31\!\cdots\!20\)\( T^{14} + \)\(61\!\cdots\!21\)\( T^{16} )^{2} \))(\( 1 - 5880 T^{2} - 9876452 T^{4} - 849078891720 T^{6} + 12359117115903558 T^{8} - 75168343796248453320 T^{10} - \)\(77\!\cdots\!72\)\( T^{12} - \)\(40\!\cdots\!80\)\( T^{14} + \)\(61\!\cdots\!21\)\( T^{16} \))(\( ( 1 + 36 T + 648 T^{2} + 128556 T^{3} - 8578162 T^{4} + 1209583404 T^{5} + 57366974088 T^{6} + 29986992177444 T^{7} + 7837433594376961 T^{8} )^{2} \))(\( ( 1 + 216 T + 23328 T^{2} + 3287304 T^{3} + 429479822 T^{4} + 30930243336 T^{5} + 2065211067168 T^{6} + 179921953064664 T^{7} + 7837433594376961 T^{8} )^{2} \))(\( ( 1 - 68 T + 2312 T^{2} + 258100 T^{3} - 260409124 T^{4} + 11295624444 T^{5} - 132728762504 T^{6} - 78472536305132 T^{7} + 30681693645016390 T^{8} - 738348094094986988 T^{9} - 11750381912498879624 T^{10} + \)\(94\!\cdots\!76\)\( T^{11} - \)\(20\!\cdots\!64\)\( T^{12} + \)\(19\!\cdots\!00\)\( T^{13} + \)\(16\!\cdots\!92\)\( T^{14} - \)\(44\!\cdots\!92\)\( T^{15} + \)\(61\!\cdots\!21\)\( T^{16} )^{2} \))
show more
show less