Properties

Label 75.2
Level 75
Weight 2
Dimension 119
Nonzero newspaces 6
Newform subspaces 12
Sturm bound 800
Trace bound 1

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Defining parameters

Level: \( N \) = \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 12 \)
Sturm bound: \(800\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(75))\).

Total New Old
Modular forms 256 159 97
Cusp forms 145 119 26
Eisenstein series 111 40 71

Trace form

\( 119q - q^{2} - 9q^{3} - 25q^{4} - 6q^{5} - 23q^{6} - 28q^{7} - 21q^{8} - 13q^{9} + O(q^{10}) \) \( 119q - q^{2} - 9q^{3} - 25q^{4} - 6q^{5} - 23q^{6} - 28q^{7} - 21q^{8} - 13q^{9} - 34q^{10} - 4q^{11} - 19q^{12} - 30q^{13} - 24q^{14} - 14q^{15} - 33q^{16} - 2q^{17} - q^{18} - 8q^{19} + 36q^{20} - 6q^{21} + 16q^{22} + 16q^{23} + 51q^{24} + 14q^{25} - 6q^{26} - 9q^{27} + 44q^{28} + 14q^{29} + 14q^{30} - 28q^{31} + 17q^{32} - 10q^{33} - 20q^{34} - 20q^{35} + 9q^{36} - 28q^{37} + 8q^{38} + 12q^{39} + 2q^{40} - 22q^{41} + 66q^{42} + 8q^{43} + 48q^{44} + 64q^{45} + 12q^{46} + 16q^{47} + 67q^{48} + 47q^{49} + 106q^{50} + 18q^{51} + 98q^{52} + 56q^{53} + 115q^{54} + 4q^{55} + 58q^{57} + 26q^{58} + 28q^{59} + 14q^{60} - 54q^{61} + 4q^{62} - 28q^{63} - 101q^{64} - 58q^{65} - 58q^{66} - 112q^{67} - 122q^{68} - 104q^{69} - 140q^{70} - 56q^{71} - 171q^{72} - 114q^{73} - 134q^{74} - 114q^{75} - 124q^{76} - 96q^{77} - 118q^{78} - 100q^{79} - 154q^{80} - 61q^{81} - 46q^{82} - 28q^{83} - 106q^{84} + 38q^{85} - 4q^{86} - 14q^{87} + 64q^{88} + 72q^{89} - 24q^{90} + 12q^{91} + 32q^{92} + 68q^{93} + 172q^{94} + 56q^{95} + 33q^{96} + 158q^{97} + 95q^{98} - 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(75))\)

We only show spaces with even 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
75.2.a \(\chi_{75}(1, \cdot)\) 75.2.a.a 1 1
75.2.a.b 1
75.2.a.c 1
75.2.b \(\chi_{75}(49, \cdot)\) 75.2.b.a 2 1
75.2.b.b 2
75.2.e \(\chi_{75}(32, \cdot)\) 75.2.e.a 4 2
75.2.e.b 4
75.2.g \(\chi_{75}(16, \cdot)\) 75.2.g.a 4 4
75.2.g.b 8
75.2.g.c 12
75.2.i \(\chi_{75}(4, \cdot)\) 75.2.i.a 16 4
75.2.l \(\chi_{75}(2, \cdot)\) 75.2.l.a 64 8

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(75))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(75)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} \))(\( 1 - T + 2 T^{2} \))(\( 1 - 2 T + 2 T^{2} \))(\( ( 1 - 2 T + 2 T^{2} )( 1 + 2 T + 2 T^{2} ) \))(\( 1 - 3 T^{2} + 4 T^{4} \))(\( 1 - 7 T^{4} + 16 T^{8} \))(\( ( 1 - 2 T + 2 T^{2} )^{2}( 1 + 2 T + 2 T^{2} )^{2} \))(\( 1 + T - T^{2} - 3 T^{3} - T^{4} - 6 T^{5} - 4 T^{6} + 8 T^{7} + 16 T^{8} \))(\( ( 1 - 2 T + 2 T^{2} - 5 T^{3} + 11 T^{4} - 10 T^{5} + 8 T^{6} - 16 T^{7} + 16 T^{8} )( 1 + 3 T + 2 T^{2} + T^{4} + 8 T^{6} + 24 T^{7} + 16 T^{8} ) \))(\( 1 + 2 T^{2} - 3 T^{3} - 2 T^{4} - 2 T^{5} - 5 T^{6} + 10 T^{7} - 18 T^{8} + T^{9} - 2 T^{10} + 22 T^{11} + 121 T^{12} + 44 T^{13} - 8 T^{14} + 8 T^{15} - 288 T^{16} + 320 T^{17} - 320 T^{18} - 256 T^{19} - 512 T^{20} - 1536 T^{21} + 2048 T^{22} + 4096 T^{24} \))(\( 1 + 3 T^{2} + 10 T^{3} + 6 T^{4} + 30 T^{5} + 61 T^{6} + 80 T^{7} + 165 T^{8} + 310 T^{9} + 503 T^{10} + 690 T^{11} + 1290 T^{12} + 2060 T^{13} + 2585 T^{14} + 4150 T^{15} + 6461 T^{16} + 8300 T^{17} + 10340 T^{18} + 16480 T^{19} + 20640 T^{20} + 22080 T^{21} + 32192 T^{22} + 39680 T^{23} + 42240 T^{24} + 40960 T^{25} + 62464 T^{26} + 61440 T^{27} + 24576 T^{28} + 81920 T^{29} + 49152 T^{30} + 65536 T^{32} \))
$3$ (\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + 9 T^{4} \))(\( 1 + 9 T^{4} \))(\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{3} \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))
$5$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( 1 - 5 T + 15 T^{2} - 25 T^{3} + 25 T^{4} \))(\( 1 + 5 T + 20 T^{2} + 65 T^{3} + 155 T^{4} + 325 T^{5} + 500 T^{6} + 625 T^{7} + 625 T^{8} \))(\( 1 + 6 T + 16 T^{2} + 21 T^{3} - 4 T^{4} - 100 T^{5} - 285 T^{6} - 500 T^{7} - 100 T^{8} + 2625 T^{9} + 10000 T^{10} + 18750 T^{11} + 15625 T^{12} \))(\( 1 + 5 T^{2} + 10 T^{3} + 10 T^{4} - 50 T^{5} + 25 T^{6} - 450 T^{7} - 725 T^{8} - 2250 T^{9} + 625 T^{10} - 6250 T^{11} + 6250 T^{12} + 31250 T^{13} + 78125 T^{14} + 390625 T^{16} \))
$7$ (\( 1 - 3 T + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 + 3 T + 7 T^{2} \))(\( 1 - 5 T^{2} + 49 T^{4} \))(\( ( 1 - 7 T^{2} )^{2} \))(\( ( 1 + 49 T^{4} )^{2} \))(\( 1 + 23 T^{4} + 2401 T^{8} \))(\( ( 1 - 6 T^{2} + 49 T^{4} )^{2} \))(\( ( 1 - 2 T + 12 T^{2} - 15 T^{3} + 61 T^{4} - 105 T^{5} + 588 T^{6} - 686 T^{7} + 2401 T^{8} )^{2} \))(\( ( 1 + 6 T + 46 T^{2} + 185 T^{3} + 822 T^{4} + 2435 T^{5} + 7706 T^{6} + 17045 T^{7} + 40278 T^{8} + 63455 T^{9} + 110446 T^{10} + 100842 T^{11} + 117649 T^{12} )^{2} \))(\( 1 - 56 T^{2} + 1628 T^{4} - 32234 T^{6} + 485518 T^{8} - 5899406 T^{10} + 59922825 T^{12} - 520141390 T^{14} + 3906488461 T^{16} - 25486928110 T^{18} + 143874702825 T^{20} - 694059216494 T^{22} + 2798914651918 T^{24} - 9105307176266 T^{26} + 22533615563228 T^{28} - 37980492079544 T^{30} + 33232930569601 T^{32} \))
$11$ (\( 1 - 2 T + 11 T^{2} \))(\( 1 + 4 T + 11 T^{2} \))(\( 1 - 2 T + 11 T^{2} \))(\( ( 1 - 2 T + 11 T^{2} )^{2} \))(\( ( 1 + 4 T + 11 T^{2} )^{2} \))(\( ( 1 - 11 T^{2} )^{4} \))(\( ( 1 - 11 T^{2} )^{4} \))(\( 1 + 6 T + 5 T^{2} - 6 T^{3} + 49 T^{4} - 66 T^{5} + 605 T^{6} + 7986 T^{7} + 14641 T^{8} \))(\( 1 - 16 T + 105 T^{2} - 340 T^{3} + 390 T^{4} + 1332 T^{5} - 9707 T^{6} + 42730 T^{7} - 156045 T^{8} + 470030 T^{9} - 1174547 T^{10} + 1772892 T^{11} + 5709990 T^{12} - 54757340 T^{13} + 186013905 T^{14} - 311794736 T^{15} + 214358881 T^{16} \))(\( 1 + 4 T - 22 T^{2} - 120 T^{3} + 58 T^{4} + 900 T^{5} + 1437 T^{6} + 4466 T^{7} - 10704 T^{8} - 67762 T^{9} + 285698 T^{10} + 117068 T^{11} - 5557163 T^{12} + 1287748 T^{13} + 34569458 T^{14} - 90191222 T^{15} - 156717264 T^{16} + 719253766 T^{17} + 2545733157 T^{18} + 17538453900 T^{19} + 12432815098 T^{20} - 282953722920 T^{21} - 570623341222 T^{22} + 1141246682444 T^{23} + 3138428376721 T^{24} \))(\( 1 + 6 T + 4 T^{2} - 68 T^{3} - 237 T^{4} - 38 T^{5} + 2400 T^{6} + 9328 T^{7} - 1188 T^{8} - 114992 T^{9} - 345452 T^{10} - 52314 T^{11} + 3321817 T^{12} + 16648114 T^{13} + 30762856 T^{14} - 86386670 T^{15} - 607501509 T^{16} - 950253370 T^{17} + 3722305576 T^{18} + 22158639734 T^{19} + 48634722697 T^{20} - 8425222014 T^{21} - 611989290572 T^{22} - 2240868767632 T^{23} - 254658350628 T^{24} + 21994936061648 T^{25} + 62249819042400 T^{26} - 10841843483218 T^{27} - 743807525282877 T^{28} - 2347544425787308 T^{29} + 1518999334332964 T^{30} + 25063489016493906 T^{31} + 45949729863572161 T^{32} \))
$13$ (\( 1 + T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( 1 - 25 T^{2} + 169 T^{4} \))(\( 1 - 22 T^{2} + 169 T^{4} \))(\( ( 1 + 169 T^{4} )^{2} \))(\( 1 - 337 T^{4} + 28561 T^{8} \))(\( 1 + 2 T + 11 T^{2} + 16 T^{3} + 49 T^{4} + 208 T^{5} + 1859 T^{6} + 4394 T^{7} + 28561 T^{8} \))(\( 1 + 8 T + 17 T^{2} + 14 T^{3} + 276 T^{4} + 578 T^{5} - 4815 T^{6} - 18048 T^{7} - 25061 T^{8} - 234624 T^{9} - 813735 T^{10} + 1269866 T^{11} + 7882836 T^{12} + 5198102 T^{13} + 82055753 T^{14} + 501988136 T^{15} + 815730721 T^{16} \))(\( 1 + 2 T - 20 T^{2} + 8 T^{3} + 236 T^{4} - 1188 T^{5} - 3563 T^{6} + 9192 T^{7} - 20810 T^{8} - 140028 T^{9} + 822700 T^{10} + 1862120 T^{11} - 6309151 T^{12} + 24207560 T^{13} + 139036300 T^{14} - 307641516 T^{15} - 594354410 T^{16} + 3412925256 T^{17} - 17197920467 T^{18} - 74545238196 T^{19} + 192512450156 T^{20} + 84835994984 T^{21} - 2757169836980 T^{22} + 3584320788074 T^{23} + 23298085122481 T^{24} \))(\( 1 + 22 T^{2} + 80 T^{3} + 81 T^{4} + 1760 T^{5} + 1994 T^{6} + 7400 T^{7} + 54580 T^{8} + 208560 T^{9} + 320132 T^{10} + 4629240 T^{11} + 13798055 T^{12} + 28576760 T^{13} + 253427140 T^{14} + 524829280 T^{15} + 2023398311 T^{16} + 6822780640 T^{17} + 42829186660 T^{18} + 62783141720 T^{19} + 394086248855 T^{20} + 1718804407320 T^{21} + 1545216018788 T^{22} + 13086830705520 T^{23} + 44522582752180 T^{24} + 78473295360200 T^{25} + 274889832746906 T^{26} + 3154202293505120 T^{27} + 1887144894920961 T^{28} + 24230008527380240 T^{29} + 86622280485384358 T^{30} + 665416609183179841 T^{32} \))
$17$ (\( 1 + 2 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \))(\( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \))(\( 1 - 382 T^{4} + 83521 T^{8} \))(\( ( 1 + 289 T^{4} )^{2} \))(\( 1 + 8 T + 17 T^{2} + 60 T^{3} + 461 T^{4} + 1020 T^{5} + 4913 T^{6} + 39304 T^{7} + 83521 T^{8} \))(\( 1 + T - 27 T^{2} - 38 T^{3} + 591 T^{4} + 456 T^{5} - 11720 T^{6} - 3109 T^{7} + 167979 T^{8} - 52853 T^{9} - 3387080 T^{10} + 2240328 T^{11} + 49360911 T^{12} - 53954566 T^{13} - 651714363 T^{14} + 410338673 T^{15} + 6975757441 T^{16} \))(\( 1 + T - 22 T^{2} + 27 T^{3} - 141 T^{4} - 1839 T^{5} + 11791 T^{6} - 5192 T^{7} - 114598 T^{8} + 1351839 T^{9} - 1459527 T^{10} - 15628812 T^{11} + 54801289 T^{12} - 265689804 T^{13} - 421803303 T^{14} + 6641585007 T^{15} - 9571339558 T^{16} - 7371897544 T^{17} + 284606076079 T^{18} - 754612819647 T^{19} - 983581799181 T^{20} + 3201872665419 T^{21} - 44351865809878 T^{22} + 34271896307633 T^{23} + 582622237229761 T^{24} \))(\( 1 - 10 T + 55 T^{2} - 550 T^{3} + 4137 T^{4} - 21370 T^{5} + 140590 T^{6} - 875240 T^{7} + 4267373 T^{8} - 23180400 T^{9} + 125011430 T^{10} - 567252720 T^{11} + 2721539524 T^{12} - 13138878600 T^{13} + 55061701425 T^{14} - 238439860100 T^{15} + 1051104835255 T^{16} - 4053477621700 T^{17} + 15912831711825 T^{18} - 64551310561800 T^{19} + 227305702584004 T^{20} - 805417745261040 T^{21} + 3017472017413670 T^{22} - 9511814575609200 T^{23} + 29768158958272493 T^{24} - 103792853025234280 T^{25} + 283428582464124910 T^{26} - 732390424094117210 T^{27} + 2410308195419521257 T^{28} - 5447517918098265350 T^{29} + 9260780460767051095 T^{30} - 28624230515098157930 T^{31} + 48661191875666868481 T^{32} \))
$19$ (\( 1 + 5 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 5 T + 19 T^{2} \))(\( ( 1 - 5 T + 19 T^{2} )^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( ( 1 - 22 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 37 T^{2} + 361 T^{4} )^{2} \))(\( 1 + 4 T - 3 T^{2} + 62 T^{3} + 605 T^{4} + 1178 T^{5} - 1083 T^{6} + 27436 T^{7} + 130321 T^{8} \))(\( 1 + 5 T - 33 T^{2} - 265 T^{3} - 47 T^{4} + 6980 T^{5} + 26814 T^{6} - 68150 T^{7} - 765695 T^{8} - 1294850 T^{9} + 9679854 T^{10} + 47875820 T^{11} - 6125087 T^{12} - 656166235 T^{13} - 1552514073 T^{14} + 4469358695 T^{15} + 16983563041 T^{16} \))(\( 1 - 7 T + 16 T^{2} - 122 T^{3} + 1071 T^{4} - 728 T^{5} - 5797 T^{6} - 36918 T^{7} + 65678 T^{8} + 636002 T^{9} + 4923209 T^{10} - 25899375 T^{11} + 44345259 T^{12} - 492088125 T^{13} + 1777278449 T^{14} + 4362337718 T^{15} + 8559222638 T^{16} - 91412622882 T^{17} - 272724972157 T^{18} - 650738625992 T^{19} + 18189396016911 T^{20} - 39367899129038 T^{21} + 98097060124816 T^{22} - 815431812287533 T^{23} + 2213314919066161 T^{24} \))(\( 1 + 2 T - 53 T^{2} - 216 T^{3} + 1273 T^{4} + 10232 T^{5} - 10629 T^{6} - 233030 T^{7} - 154021 T^{8} + 2536598 T^{9} + 3525662 T^{10} + 321666 T^{11} + 122263380 T^{12} - 353312856 T^{13} - 6187122082 T^{14} + 3769508888 T^{15} + 148112464363 T^{16} + 71620668872 T^{17} - 2233551071602 T^{18} - 2423372879304 T^{19} + 15933485944980 T^{20} + 796476860934 T^{21} + 165867874898222 T^{22} + 2267393265403922 T^{23} - 2615825363137861 T^{24} - 75195914213440370 T^{25} - 65167103254166829 T^{26} + 1191928329046576808 T^{27} + 2817549891971222953 T^{28} - 9083444427847524744 T^{29} - 42347354346492858413 T^{30} + 30362254059749596598 T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \))
$23$ (\( 1 + 6 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 - 10 T^{2} + 529 T^{4} \))(\( ( 1 - 23 T^{2} )^{2} \))(\( 1 + 98 T^{4} + 279841 T^{8} \))(\( ( 1 + 529 T^{4} )^{2} \))(\( 1 - 10 T + 37 T^{2} - 200 T^{3} + 1389 T^{4} - 4600 T^{5} + 19573 T^{6} - 121670 T^{7} + 279841 T^{8} \))(\( 1 - 7 T + 12 T^{2} - 31 T^{3} + 1026 T^{4} - 672 T^{5} - 34550 T^{6} + 127672 T^{7} - 3411 T^{8} + 2936456 T^{9} - 18276950 T^{10} - 8176224 T^{11} + 287116866 T^{12} - 199526633 T^{13} + 1776430668 T^{14} - 23833778129 T^{15} + 78310985281 T^{16} \))(\( 1 - 19 T + 87 T^{2} + 664 T^{3} - 8155 T^{4} + 10802 T^{5} + 276289 T^{6} - 1712486 T^{7} - 1232028 T^{8} + 57899808 T^{9} - 214047367 T^{10} - 649484443 T^{11} + 7850412157 T^{12} - 14938142189 T^{13} - 113231057143 T^{14} + 704466963936 T^{15} - 344771947548 T^{16} - 11022147278698 T^{17} + 40900687735921 T^{18} + 36778924478494 T^{19} - 638626084966555 T^{20} + 1195965367211432 T^{21} + 3604106475587463 T^{22} - 18103385400364613 T^{23} + 21914624432020321 T^{24} \))(\( 1 + 20 T + 181 T^{2} + 1150 T^{3} + 7948 T^{4} + 59890 T^{5} + 379519 T^{6} + 2059710 T^{7} + 11607418 T^{8} + 70214570 T^{9} + 395238326 T^{10} + 1984546720 T^{11} + 9942722840 T^{12} + 53914737500 T^{13} + 283746609760 T^{14} + 1328416871210 T^{15} + 6099237415331 T^{16} + 30553588037830 T^{17} + 150101956563040 T^{18} + 655980611162500 T^{19} + 2782381502268440 T^{20} + 12773223389444960 T^{21} + 58509456956281814 T^{22} + 239068354686162790 T^{23} + 908988340148414458 T^{24} + 3709852148341955730 T^{25} + 15722148109292854831 T^{26} + 57063776401465088030 T^{27} + \)\(17\!\cdots\!08\)\( T^{28} + \)\(57\!\cdots\!50\)\( T^{29} + \)\(20\!\cdots\!29\)\( T^{30} + \)\(53\!\cdots\!40\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \))
$29$ (\( 1 - 10 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 - 10 T + 29 T^{2} \))(\( ( 1 + 10 T + 29 T^{2} )^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{4} \))(\( ( 1 + 29 T^{2} )^{4} \))(\( 1 + 8 T + 5 T^{2} - 232 T^{3} - 1711 T^{4} - 6728 T^{5} + 4205 T^{6} + 195112 T^{7} + 707281 T^{8} \))(\( 1 - 5 T + 12 T^{2} + 110 T^{3} - 322 T^{4} + 4795 T^{5} - 5456 T^{6} + 22500 T^{7} + 487955 T^{8} + 652500 T^{9} - 4588496 T^{10} + 116945255 T^{11} - 227744482 T^{12} + 2256226390 T^{13} + 7137879852 T^{14} - 86249381545 T^{15} + 500246412961 T^{16} \))(\( 1 + T - 75 T^{2} - 3 T^{3} + 2691 T^{4} + 6800 T^{5} - 35703 T^{6} - 577442 T^{7} - 700036 T^{8} + 20501600 T^{9} + 54541649 T^{10} - 290598314 T^{11} - 2133422469 T^{12} - 8427351106 T^{13} + 45869526809 T^{14} + 500013522400 T^{15} - 495122162116 T^{16} - 11843998900858 T^{17} - 21236977029663 T^{18} + 117299158901200 T^{19} + 1346163097278051 T^{20} - 43521437927607 T^{21} - 31553042497515075 T^{22} + 12200509765705829 T^{23} + 353814783205469041 T^{24} \))(\( 1 - 16 T + 101 T^{2} - 378 T^{3} + 2782 T^{4} - 28638 T^{5} + 156384 T^{6} - 372476 T^{7} + 1621718 T^{8} - 24643166 T^{9} + 158941381 T^{10} - 390275636 T^{11} + 1282706252 T^{12} - 24547929228 T^{13} + 196193283170 T^{14} - 670123125368 T^{15} + 1692236078907 T^{16} - 19433570635672 T^{17} + 164998551145970 T^{18} - 598699445941692 T^{19} + 907233760620812 T^{20} - 8005001721065764 T^{21} + 94542040090746301 T^{22} - 425091565362154294 T^{23} + 811258612334286998 T^{24} - 5403563704507781644 T^{25} + 65791879972418633184 T^{26} - \)\(34\!\cdots\!02\)\( T^{27} + \)\(98\!\cdots\!62\)\( T^{28} - \)\(38\!\cdots\!42\)\( T^{29} + \)\(30\!\cdots\!81\)\( T^{30} - \)\(13\!\cdots\!84\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \))
$31$ (\( 1 + 3 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))(\( 1 + 3 T + 31 T^{2} \))(\( ( 1 + 3 T + 31 T^{2} )^{2} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 + 8 T + 31 T^{2} )^{4} \))(\( ( 1 - 7 T + 31 T^{2} )^{4} \))(\( 1 + 9 T^{2} - 110 T^{3} + 741 T^{4} - 3410 T^{5} + 8649 T^{6} + 923521 T^{8} \))(\( 1 + 19 T + 110 T^{2} - 455 T^{3} - 10080 T^{4} - 55148 T^{5} - 24492 T^{6} + 1584870 T^{7} + 12593785 T^{8} + 49130970 T^{9} - 23536812 T^{10} - 1642914068 T^{11} - 9309091680 T^{12} - 13026263705 T^{13} + 97625404910 T^{14} + 522739668109 T^{15} + 852891037441 T^{16} \))(\( 1 - 13 T - 49 T^{2} + 1410 T^{3} - 4283 T^{4} - 42518 T^{5} + 442503 T^{6} - 1354402 T^{7} - 10366302 T^{8} + 126161252 T^{9} - 343254057 T^{10} - 2277200953 T^{11} + 23831716157 T^{12} - 70593229543 T^{13} - 329867148777 T^{14} + 3758469858332 T^{15} - 9573497589342 T^{16} - 38775379372702 T^{17} + 392723041353543 T^{18} - 1169781326771498 T^{19} - 3652932313359803 T^{20} + 37279867246546110 T^{21} - 40161786062059249 T^{22} - 330310199653262803 T^{23} + 787662783788549761 T^{24} \))(\( 1 - 6 T - 27 T^{2} + 222 T^{3} + 1398 T^{4} - 1096 T^{5} - 82911 T^{6} + 83080 T^{7} + 2495194 T^{8} + 3301786 T^{9} - 44430232 T^{10} - 311988762 T^{11} + 1830869280 T^{12} + 7523309618 T^{13} - 27963150078 T^{14} + 31095224934 T^{15} - 66108612217 T^{16} + 963951972954 T^{17} - 26872587224958 T^{18} + 224126916829838 T^{19} + 1690846228334880 T^{20} - 8931973377601062 T^{21} - 39431994447683992 T^{22} + 90840764095102246 T^{23} + 2128128599276558554 T^{24} + 2196603809108546680 T^{25} - 67956200901865191711 T^{26} - 27847690678459694776 T^{27} + \)\(11\!\cdots\!78\)\( T^{28} + \)\(54\!\cdots\!02\)\( T^{29} - \)\(20\!\cdots\!67\)\( T^{30} - \)\(14\!\cdots\!06\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \))
$37$ (\( 1 + 2 T + 37 T^{2} \))(\( 1 - 10 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( 1 + 26 T^{2} + 1369 T^{4} \))(\( ( 1 + 1369 T^{4} )^{2} \))(\( 1 - 2062 T^{4} + 1874161 T^{8} \))(\( 1 + 15 T + 63 T^{2} + 65 T^{3} + 144 T^{4} + 2405 T^{5} + 86247 T^{6} + 759795 T^{7} + 1874161 T^{8} \))(\( 1 + T + 3 T^{2} + 42 T^{3} + 2811 T^{4} + 3486 T^{5} - 59280 T^{6} + 126501 T^{7} + 3465939 T^{8} + 4680537 T^{9} - 81154320 T^{10} + 176576358 T^{11} + 5268266571 T^{12} + 2912446194 T^{13} + 7697179227 T^{14} + 94931877133 T^{15} + 3512479453921 T^{16} \))(\( 1 - 8 T - 67 T^{2} + 1147 T^{3} - 3085 T^{4} - 39031 T^{5} + 455721 T^{6} - 1535453 T^{7} - 11486553 T^{8} + 153908236 T^{9} - 464906763 T^{10} - 3103077939 T^{11} + 36758415492 T^{12} - 114813883743 T^{13} - 636457358547 T^{14} + 7795913878108 T^{15} - 21527649657033 T^{16} - 106474386807521 T^{17} + 1169255404835889 T^{18} - 3705286096378123 T^{19} - 10835999115346285 T^{20} + 149066115544953319 T^{21} - 322175152951995883 T^{22} - 1423340974235683304 T^{23} + 6582952005840035281 T^{24} \))(\( 1 + 10 T + 159 T^{2} + 1490 T^{3} + 15403 T^{4} + 128130 T^{5} + 1158066 T^{6} + 8990890 T^{7} + 72816793 T^{8} + 529727610 T^{9} + 3982453044 T^{10} + 27036193580 T^{11} + 190987118220 T^{12} + 1239202673080 T^{13} + 8238752278565 T^{14} + 50853391327250 T^{15} + 322012573505931 T^{16} + 1881575479108250 T^{17} + 11278851869355485 T^{18} + 62769332999521240 T^{19} + 357940608470313420 T^{20} + 1874796645055196060 T^{21} + 10217884947593238996 T^{22} + 50288036386477742130 T^{23} + \)\(25\!\cdots\!53\)\( T^{24} + \)\(11\!\cdots\!30\)\( T^{25} + \)\(55\!\cdots\!34\)\( T^{26} + \)\(22\!\cdots\!90\)\( T^{27} + \)\(10\!\cdots\!43\)\( T^{28} + \)\(36\!\cdots\!30\)\( T^{29} + \)\(14\!\cdots\!51\)\( T^{30} + \)\(33\!\cdots\!30\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))
$41$ (\( 1 + 8 T + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))(\( 1 + 8 T + 41 T^{2} \))(\( ( 1 + 8 T + 41 T^{2} )^{2} \))(\( ( 1 - 10 T + 41 T^{2} )^{2} \))(\( ( 1 - 41 T^{2} )^{4} \))(\( ( 1 - 41 T^{2} )^{4} \))(\( 1 - 31 T^{2} + 180 T^{3} + 1501 T^{4} + 7380 T^{5} - 52111 T^{6} + 2825761 T^{8} \))(\( ( 1 + 7 T + 28 T^{2} + 389 T^{3} + 3975 T^{4} + 15949 T^{5} + 47068 T^{6} + 482447 T^{7} + 2825761 T^{8} )^{2} \))(\( 1 - 8 T + 56 T^{2} - 840 T^{3} + 5172 T^{4} - 38968 T^{5} + 387993 T^{6} - 1909992 T^{7} + 12984488 T^{8} - 115425908 T^{9} + 544467188 T^{10} - 3312068648 T^{11} + 28013676237 T^{12} - 135794814568 T^{13} + 915249343028 T^{14} - 7955269005268 T^{15} + 36691059795368 T^{16} - 221284417060392 T^{17} + 1843007194778313 T^{18} - 7589184544594808 T^{19} + 41298033285013812 T^{20} - 275000824890927240 T^{21} + 751668921368534456 T^{22} - 4402632253729987528 T^{23} + 22563490300366186081 T^{24} \))(\( 1 + 14 T - 82 T^{2} - 2028 T^{3} - 747 T^{4} + 120014 T^{5} + 311054 T^{6} - 3602970 T^{7} - 14348876 T^{8} + 63645696 T^{9} + 522280968 T^{10} - 1039813212 T^{11} - 35365817705 T^{12} + 21504097848 T^{13} + 2388690604652 T^{14} - 220861336106 T^{15} - 115364883721937 T^{16} - 9055314780346 T^{17} + 4015388906420012 T^{18} + 1482083927782008 T^{19} - 99935348403898505 T^{20} - 120468808491927612 T^{21} + 2480889041090385288 T^{22} + 12395271310130866176 T^{23} - \)\(11\!\cdots\!96\)\( T^{24} - \)\(11\!\cdots\!70\)\( T^{25} + \)\(41\!\cdots\!54\)\( T^{26} + \)\(66\!\cdots\!74\)\( T^{27} - \)\(16\!\cdots\!07\)\( T^{28} - \)\(18\!\cdots\!88\)\( T^{29} - \)\(31\!\cdots\!02\)\( T^{30} + \)\(21\!\cdots\!14\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \))
$43$ (\( 1 + T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))(\( 1 - T + 43 T^{2} \))(\( 1 - 85 T^{2} + 1849 T^{4} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( ( 1 + 1849 T^{4} )^{2} \))(\( 1 + 23 T^{4} + 3418801 T^{8} \))(\( ( 1 - 2 T + 42 T^{2} - 86 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 16 T + 233 T^{2} - 2040 T^{3} + 16241 T^{4} - 87720 T^{5} + 430817 T^{6} - 1272112 T^{7} + 3418801 T^{8} )^{2} \))(\( ( 1 + 2 T + 167 T^{2} + 256 T^{3} + 13452 T^{4} + 15956 T^{5} + 692036 T^{6} + 686108 T^{7} + 24872748 T^{8} + 20353792 T^{9} + 570939767 T^{10} + 294016886 T^{11} + 6321363049 T^{12} )^{2} \))(\( 1 - 300 T^{2} + 44842 T^{4} - 4521820 T^{6} + 352848163 T^{8} - 23099919100 T^{10} + 1325953845604 T^{12} - 67791756187600 T^{14} + 3089570546914805 T^{16} - 125346957190872400 T^{18} + 4533172333304800804 T^{20} - \)\(14\!\cdots\!00\)\( T^{22} + \)\(41\!\cdots\!63\)\( T^{24} - \)\(97\!\cdots\!80\)\( T^{26} + \)\(17\!\cdots\!42\)\( T^{28} - \)\(22\!\cdots\!00\)\( T^{30} + \)\(13\!\cdots\!01\)\( T^{32} \))
$47$ (\( 1 + 2 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 - 2 T + 47 T^{2} \))(\( 1 - 90 T^{2} + 2209 T^{4} \))(\( 1 - 30 T^{2} + 2209 T^{4} \))(\( 1 - 4222 T^{4} + 4879681 T^{8} \))(\( ( 1 + 2209 T^{4} )^{2} \))(\( 1 + 2 T - 23 T^{2} + 250 T^{3} + 2601 T^{4} + 11750 T^{5} - 50807 T^{6} + 207646 T^{7} + 4879681 T^{8} \))(\( 1 + T - 62 T^{2} - 78 T^{3} + 3466 T^{4} + 4691 T^{5} - 155900 T^{6} - 95604 T^{7} + 3265179 T^{8} - 4493388 T^{9} - 344383100 T^{10} + 487033693 T^{11} + 16912974346 T^{12} - 17888910546 T^{13} - 668311350398 T^{14} + 506623120463 T^{15} + 23811286661761 T^{16} \))(\( 1 + 13 T + 37 T^{2} + 325 T^{3} + 7867 T^{4} + 56088 T^{5} + 222715 T^{6} + 1098334 T^{7} + 12592498 T^{8} + 108988468 T^{9} + 281826965 T^{10} + 1078581014 T^{11} + 21055334183 T^{12} + 50693307658 T^{13} + 622555765685 T^{14} + 11315509713164 T^{15} + 61447373233138 T^{16} + 251897418918338 T^{17} + 2400692941998235 T^{18} + 28415477580528744 T^{19} + 187323392168073787 T^{20} + 363717403758399275 T^{21} + 1946167892725711813 T^{22} + 32138069796092159939 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \))(\( 1 - 40 T + 865 T^{2} - 12910 T^{3} + 142622 T^{4} - 1167030 T^{5} + 6143280 T^{6} + 637840 T^{7} - 479197322 T^{8} + 6905707270 T^{9} - 64790306935 T^{10} + 444387466120 T^{11} - 1988154991916 T^{12} + 288127717900 T^{13} + 103937589685250 T^{14} - 1288860429553200 T^{15} + 10336092465289955 T^{16} - 60576440189000400 T^{17} + 229598135614717250 T^{18} + 29914284055531700 T^{19} - 9701562139107658796 T^{20} + \)\(10\!\cdots\!40\)\( T^{21} - \)\(69\!\cdots\!15\)\( T^{22} + \)\(34\!\cdots\!10\)\( T^{23} - \)\(11\!\cdots\!42\)\( T^{24} + \)\(71\!\cdots\!80\)\( T^{25} + \)\(32\!\cdots\!20\)\( T^{26} - \)\(28\!\cdots\!90\)\( T^{27} + \)\(16\!\cdots\!02\)\( T^{28} - \)\(70\!\cdots\!70\)\( T^{29} + \)\(22\!\cdots\!85\)\( T^{30} - \)\(48\!\cdots\!20\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( 1 - 4 T + 53 T^{2} \))(\( 1 - 10 T + 53 T^{2} \))(\( 1 + 4 T + 53 T^{2} \))(\( ( 1 - 14 T + 53 T^{2} )( 1 + 14 T + 53 T^{2} ) \))(\( 1 - 6 T^{2} + 2809 T^{4} \))(\( 1 + 1778 T^{4} + 7890481 T^{8} \))(\( ( 1 + 2809 T^{4} )^{2} \))(\( 1 + 5 T - 43 T^{2} - 265 T^{3} + 1244 T^{4} - 14045 T^{5} - 120787 T^{6} + 744385 T^{7} + 7890481 T^{8} \))(\( 1 + 3 T - 63 T^{2} - 6 T^{3} + 4741 T^{4} - 3522 T^{5} - 240240 T^{6} - 209553 T^{7} + 7555219 T^{8} - 11106309 T^{9} - 674834160 T^{10} - 524344794 T^{11} + 37408770421 T^{12} - 2509172958 T^{13} - 1396354751127 T^{14} + 3524133419511 T^{15} + 62259690411361 T^{16} \))(\( 1 - 44 T + 817 T^{2} - 7511 T^{3} + 18675 T^{4} + 371047 T^{5} - 5505981 T^{6} + 45196759 T^{7} - 286840983 T^{8} + 1119924348 T^{9} + 5970507613 T^{10} - 163921234483 T^{11} + 1596195784652 T^{12} - 8687825427599 T^{13} + 16771155884917 T^{14} + 166730977157196 T^{15} - 2263313326382823 T^{16} + 18901080912007187 T^{17} - 122036551253412549 T^{18} + 435873044303099339 T^{19} + 1162699718432166675 T^{20} - 24784524338025820963 T^{21} + \)\(14\!\cdots\!33\)\( T^{22} - \)\(40\!\cdots\!68\)\( T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \))(\( 1 - 10 T + 31 T^{2} + 750 T^{3} - 4567 T^{4} - 7470 T^{5} + 369374 T^{6} - 601780 T^{7} - 13186187 T^{8} + 114344200 T^{9} + 595715966 T^{10} - 8185408800 T^{11} + 16758234740 T^{12} + 506558110520 T^{13} - 2917035559495 T^{14} - 4163213158300 T^{15} + 189869197228511 T^{16} - 220650297389900 T^{17} - 8193952886621455 T^{18} + 75414851819886040 T^{19} + 132230532809509940 T^{20} - 3423101068522538400 T^{21} + 13203663800735085614 T^{22} + \)\(13\!\cdots\!00\)\( T^{23} - \)\(82\!\cdots\!07\)\( T^{24} - \)\(19\!\cdots\!40\)\( T^{25} + \)\(64\!\cdots\!26\)\( T^{26} - \)\(69\!\cdots\!90\)\( T^{27} - \)\(22\!\cdots\!47\)\( T^{28} + \)\(19\!\cdots\!50\)\( T^{29} + \)\(42\!\cdots\!39\)\( T^{30} - \)\(73\!\cdots\!70\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \))
$59$ (\( 1 + 10 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 + 10 T + 59 T^{2} \))(\( ( 1 - 10 T + 59 T^{2} )^{2} \))(\( ( 1 - 4 T + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{4} \))(\( ( 1 + 59 T^{2} )^{4} \))(\( 1 - 4 T - 43 T^{2} + 408 T^{3} + 905 T^{4} + 24072 T^{5} - 149683 T^{6} - 821516 T^{7} + 12117361 T^{8} \))(\( 1 - 30 T + 377 T^{2} - 2475 T^{3} + 8643 T^{4} - 1155 T^{5} - 582521 T^{6} + 10821600 T^{7} - 108687625 T^{8} + 638474400 T^{9} - 2027755601 T^{10} - 237212745 T^{11} + 104730351123 T^{12} - 1769437640025 T^{13} + 15902061182657 T^{14} - 74659544544570 T^{15} + 146830437604321 T^{16} \))(\( 1 + 22 T + 36 T^{2} - 2583 T^{3} - 16419 T^{4} + 137618 T^{5} + 1667358 T^{6} - 2698622 T^{7} - 107608927 T^{8} - 155893882 T^{9} + 5010588449 T^{10} + 10213783795 T^{11} - 198548784261 T^{12} + 602613243905 T^{13} + 17441858390969 T^{14} - 32017329591278 T^{15} - 1303936215281647 T^{16} - 1929310441615978 T^{17} + 70330050210590478 T^{18} + 342483240037821142 T^{19} - 2410808955025346499 T^{20} - 22376518199585707437 T^{21} + 18400203118823090436 T^{22} + \)\(66\!\cdots\!98\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \))(\( 1 - 12 T - 48 T^{2} + 796 T^{3} + 7633 T^{4} - 76112 T^{5} - 428429 T^{6} + 2657110 T^{7} + 51753489 T^{8} - 263375518 T^{9} - 2799769853 T^{10} + 8680064174 T^{11} + 262778488535 T^{12} - 989243918004 T^{13} - 11934572611572 T^{14} + 2210541870272 T^{15} + 962870787479413 T^{16} + 130421970346048 T^{17} - 41544247260882132 T^{18} - 203169926635743516 T^{19} + 3184181808612956135 T^{20} + 6205588794871964026 T^{21} - \)\(11\!\cdots\!73\)\( T^{22} - \)\(65\!\cdots\!42\)\( T^{23} + \)\(75\!\cdots\!69\)\( T^{24} + \)\(23\!\cdots\!90\)\( T^{25} - \)\(21\!\cdots\!29\)\( T^{26} - \)\(22\!\cdots\!08\)\( T^{27} + \)\(13\!\cdots\!73\)\( T^{28} + \)\(83\!\cdots\!84\)\( T^{29} - \)\(29\!\cdots\!28\)\( T^{30} - \)\(43\!\cdots\!88\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( 1 - 7 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 - 7 T + 61 T^{2} \))(\( ( 1 - 7 T + 61 T^{2} )^{2} \))(\( ( 1 + 2 T + 61 T^{2} )^{2} \))(\( ( 1 - 2 T + 61 T^{2} )^{4} \))(\( ( 1 + 13 T + 61 T^{2} )^{4} \))(\( 1 - 2 T - 57 T^{2} - 64 T^{3} + 3905 T^{4} - 3904 T^{5} - 212097 T^{6} - 453962 T^{7} + 13845841 T^{8} \))(\( 1 + 14 T + 35 T^{2} + 110 T^{3} + 8070 T^{4} + 33812 T^{5} - 419217 T^{6} - 1871430 T^{7} + 12043555 T^{8} - 114157230 T^{9} - 1559906457 T^{10} + 7674681572 T^{11} + 111735936870 T^{12} + 92905593110 T^{13} + 1803213102635 T^{14} + 43998399704294 T^{15} + 191707312997281 T^{16} \))(\( 1 + 8 T - 130 T^{2} - 1358 T^{3} + 564 T^{4} + 21392 T^{5} + 487611 T^{6} + 8007850 T^{7} - 7109096 T^{8} - 542757032 T^{9} - 551554860 T^{10} + 10321250720 T^{11} + 3604709585 T^{12} + 629596293920 T^{13} - 2052335634060 T^{14} - 123195533880392 T^{15} - 98431412869736 T^{16} + 6763400488962850 T^{17} + 25121901262541571 T^{18} + 67229554748161232 T^{19} + 108122924530466484 T^{20} - 15880650394068763478 T^{21} - 92734578516174738130 T^{22} + \)\(34\!\cdots\!88\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \))(\( 1 - 66 T^{2} + 280 T^{3} + 2565 T^{4} + 73000 T^{5} - 42550 T^{6} - 4179060 T^{7} + 27062520 T^{8} + 116843800 T^{9} + 1866464452 T^{10} + 5426819300 T^{11} - 75968338617 T^{12} + 773051497020 T^{13} + 3102841052040 T^{14} + 12380110098160 T^{15} + 378207085764575 T^{16} + 755186715987760 T^{17} + 11545671554640840 T^{18} + 175468001845096620 T^{19} - 1051845537525141897 T^{20} + 4583471506975409300 T^{21} + 96160947298538715172 T^{22} + \)\(36\!\cdots\!00\)\( T^{23} + \)\(51\!\cdots\!20\)\( T^{24} - \)\(48\!\cdots\!60\)\( T^{25} - \)\(30\!\cdots\!50\)\( T^{26} + \)\(31\!\cdots\!00\)\( T^{27} + \)\(68\!\cdots\!65\)\( T^{28} + \)\(45\!\cdots\!80\)\( T^{29} - \)\(65\!\cdots\!06\)\( T^{30} + \)\(36\!\cdots\!61\)\( T^{32} \))
$67$ (\( 1 - 3 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))(\( 1 + 3 T + 67 T^{2} \))(\( 1 - 125 T^{2} + 4489 T^{4} \))(\( 1 + 10 T^{2} + 4489 T^{4} \))(\( ( 1 + 4489 T^{4} )^{2} \))(\( 1 + 2903 T^{4} + 20151121 T^{8} \))(\( 1 - 2 T - 43 T^{2} - 370 T^{3} + 5041 T^{4} - 24790 T^{5} - 193027 T^{6} - 601526 T^{7} + 20151121 T^{8} \))(\( 1 - 4 T - 97 T^{2} + 122 T^{3} + 9681 T^{4} - 8074 T^{5} - 723795 T^{6} + 600576 T^{7} + 39163204 T^{8} + 40238592 T^{9} - 3249115755 T^{10} - 2428360462 T^{11} + 195083002401 T^{12} + 164715263054 T^{13} - 8774463070393 T^{14} - 24242846421292 T^{15} + 406067677556641 T^{16} \))(\( 1 + 6 T - 172 T^{2} - 858 T^{3} + 11749 T^{4} + 57666 T^{5} - 232014 T^{6} - 4755072 T^{7} - 22912618 T^{8} + 467752054 T^{9} + 2928188158 T^{10} - 17377714672 T^{11} - 223143036976 T^{12} - 1164306883024 T^{13} + 13144636641262 T^{14} + 140682511017202 T^{15} - 461714937744778 T^{16} - 6419942092792704 T^{17} - 20987611080558366 T^{18} + 349496995432556118 T^{19} + 4770889143612975109 T^{20} - 23343206512021064526 T^{21} - \)\(31\!\cdots\!28\)\( T^{22} + \)\(73\!\cdots\!98\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \))(\( 1 + 40 T + 1030 T^{2} + 19080 T^{3} + 285687 T^{4} + 3555880 T^{5} + 38106080 T^{6} + 351599640 T^{7} + 2795567883 T^{8} + 18168145600 T^{9} + 82372777780 T^{10} - 40068357040 T^{11} - 6615976940891 T^{12} - 103670166025040 T^{13} - 1174625422928650 T^{14} - 11401205121382320 T^{15} - 98095935267474960 T^{16} - 763880743132615440 T^{17} - 5272893523526709850 T^{18} - 31180150144189105520 T^{19} - \)\(13\!\cdots\!11\)\( T^{20} - 54097294835944203280 T^{21} + \)\(74\!\cdots\!20\)\( T^{22} + \)\(11\!\cdots\!00\)\( T^{23} + \)\(11\!\cdots\!03\)\( T^{24} + \)\(95\!\cdots\!80\)\( T^{25} + \)\(69\!\cdots\!20\)\( T^{26} + \)\(43\!\cdots\!40\)\( T^{27} + \)\(23\!\cdots\!07\)\( T^{28} + \)\(10\!\cdots\!60\)\( T^{29} + \)\(37\!\cdots\!70\)\( T^{30} + \)\(98\!\cdots\!20\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \))
$71$ (\( 1 + 8 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( ( 1 + 8 T + 71 T^{2} )^{2} \))(\( ( 1 + 8 T + 71 T^{2} )^{2} \))(\( ( 1 - 71 T^{2} )^{4} \))(\( ( 1 - 71 T^{2} )^{4} \))(\( 1 - 8 T - 47 T^{2} + 434 T^{3} + 1365 T^{4} + 30814 T^{5} - 236927 T^{6} - 2863288 T^{7} + 25411681 T^{8} \))(\( 1 - 21 T + 305 T^{2} - 2480 T^{3} + 13165 T^{4} + 10682 T^{5} - 810942 T^{6} + 9320005 T^{7} - 80224815 T^{8} + 661720355 T^{9} - 4087958622 T^{10} + 3823205302 T^{11} + 334544780365 T^{12} - 4474488790480 T^{13} + 39070586595905 T^{14} - 190997523326211 T^{15} + 645753531245761 T^{16} \))(\( 1 + 21 T + 46 T^{2} - 1289 T^{3} + 2891 T^{4} + 230821 T^{5} + 1592921 T^{6} - 6954484 T^{7} - 90555524 T^{8} + 1177173511 T^{9} + 16628404231 T^{10} + 5732671986 T^{11} - 764574531549 T^{12} + 407019711006 T^{13} + 83823785728471 T^{14} + 421323348495521 T^{15} - 2301168088675844 T^{16} - 12547484153859884 T^{17} + 204053632363723241 T^{18} + 2099344730079969011 T^{19} + 1866873458831495051 T^{20} - 59098717426080800959 T^{21} + \)\(14\!\cdots\!46\)\( T^{22} + \)\(48\!\cdots\!91\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \))(\( 1 + 8 T - 249 T^{2} - 2580 T^{3} + 22325 T^{4} + 321084 T^{5} - 587178 T^{6} - 14941106 T^{7} - 19432655 T^{8} - 485658130 T^{9} - 1289469490 T^{10} + 87893386040 T^{11} + 259786638060 T^{12} - 3694736827880 T^{13} + 7661490804445 T^{14} + 51486950465450 T^{15} - 2151535697455665 T^{16} + 3655573483046950 T^{17} + 38621575145207245 T^{18} - 1322386952803358680 T^{19} + 6601615174443178860 T^{20} + \)\(15\!\cdots\!40\)\( T^{21} - \)\(16\!\cdots\!90\)\( T^{22} - \)\(44\!\cdots\!30\)\( T^{23} - \)\(12\!\cdots\!55\)\( T^{24} - \)\(68\!\cdots\!86\)\( T^{25} - \)\(19\!\cdots\!78\)\( T^{26} + \)\(74\!\cdots\!64\)\( T^{27} + \)\(36\!\cdots\!25\)\( T^{28} - \)\(30\!\cdots\!80\)\( T^{29} - \)\(20\!\cdots\!69\)\( T^{30} + \)\(46\!\cdots\!08\)\( T^{31} + \)\(41\!\cdots\!21\)\( T^{32} \))
$73$ (\( 1 - 14 T + 73 T^{2} \))(\( 1 + 10 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))(\( 1 + 50 T^{2} + 5329 T^{4} \))(\( 1 - 46 T^{2} + 5329 T^{4} \))(\( ( 1 + 5329 T^{4} )^{2} \))(\( 1 - 8542 T^{4} + 28398241 T^{8} \))(\( 1 + 10 T + 27 T^{2} + 740 T^{3} + 11429 T^{4} + 54020 T^{5} + 143883 T^{6} + 3890170 T^{7} + 28398241 T^{8} \))(\( 1 - 2 T - 163 T^{2} - 986 T^{3} + 8481 T^{4} + 176518 T^{5} + 938895 T^{6} - 6925578 T^{7} - 156207836 T^{8} - 505567194 T^{9} + 5003371455 T^{10} + 68668502806 T^{11} + 240845481921 T^{12} - 2044048590698 T^{13} - 24667478885107 T^{14} - 22094797038194 T^{15} + 806460091894081 T^{16} \))(\( 1 + 16 T - 28 T^{2} - 506 T^{3} + 12385 T^{4} - 2328 T^{5} - 561526 T^{6} + 14932744 T^{7} + 69930462 T^{8} - 421338712 T^{9} + 9017966158 T^{10} + 43966609042 T^{11} - 621466327768 T^{12} + 3209562460066 T^{13} + 48056741655982 T^{14} - 163907921726104 T^{15} + 1985902113117342 T^{16} + 30956647391941192 T^{17} - 84978102751157014 T^{18} - 25718343752457816 T^{19} + 9988008238108193185 T^{20} - 29789022874383563978 T^{21} - \)\(12\!\cdots\!72\)\( T^{22} + \)\(50\!\cdots\!32\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))(\( 1 + 20 T + 426 T^{2} + 5180 T^{3} + 61943 T^{4} + 512020 T^{5} + 4294444 T^{6} + 28526480 T^{7} + 237139103 T^{8} + 1934704520 T^{9} + 18247662296 T^{10} + 153434416460 T^{11} + 1126098548485 T^{12} + 8592560378500 T^{13} + 56602836572850 T^{14} + 546633756136020 T^{15} + 3853646456873416 T^{16} + 39904264197929460 T^{17} + 301636516096717650 T^{18} + 3342652060762934500 T^{19} + 31979217969627214885 T^{20} + \)\(31\!\cdots\!80\)\( T^{21} + \)\(27\!\cdots\!44\)\( T^{22} + \)\(21\!\cdots\!40\)\( T^{23} + \)\(19\!\cdots\!43\)\( T^{24} + \)\(16\!\cdots\!40\)\( T^{25} + \)\(18\!\cdots\!56\)\( T^{26} + \)\(16\!\cdots\!40\)\( T^{27} + \)\(14\!\cdots\!03\)\( T^{28} + \)\(86\!\cdots\!40\)\( T^{29} + \)\(51\!\cdots\!34\)\( T^{30} + \)\(17\!\cdots\!40\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \))
$79$ (\( 1 + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( ( 1 + 79 T^{2} )^{2} \))(\( ( 1 + 79 T^{2} )^{2} \))(\( ( 1 + 98 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 - 142 T^{2} + 6241 T^{4} )^{2} \))(\( 1 - 79 T^{2} + 6241 T^{4} - 493039 T^{6} + 38950081 T^{8} \))(\( 1 + 30 T + 447 T^{2} + 5690 T^{3} + 63618 T^{4} + 615980 T^{5} + 6355819 T^{6} + 63162750 T^{7} + 565067755 T^{8} + 4989857250 T^{9} + 39666666379 T^{10} + 303702163220 T^{11} + 2477926253058 T^{12} + 17508450910310 T^{13} + 108660092617887 T^{14} + 576117269584770 T^{15} + 1517108809906561 T^{16} \))(\( 1 - 10 T - 272 T^{2} + 2840 T^{3} + 26866 T^{4} - 289420 T^{5} - 489675 T^{6} + 1858730 T^{7} - 91007200 T^{8} + 1966674780 T^{9} + 3705819298 T^{10} - 105092388950 T^{11} + 146277106069 T^{12} - 8302298727050 T^{13} + 23128018238818 T^{14} + 969647366856420 T^{15} - 3544737811583200 T^{16} + 5719417040513270 T^{17} - 119033849782245675 T^{18} - 5557995338774137780 T^{19} + 40758645286949667826 T^{20} + \)\(34\!\cdots\!60\)\( T^{21} - \)\(25\!\cdots\!72\)\( T^{22} - \)\(74\!\cdots\!90\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \))(\( 1 + 20 T - 26 T^{2} - 3900 T^{3} - 29155 T^{4} + 347300 T^{5} + 6752970 T^{6} + 1986700 T^{7} - 796980480 T^{8} - 5624999100 T^{9} + 44951710692 T^{10} + 864380570440 T^{11} + 1361154199023 T^{12} - 70535599976700 T^{13} - 564113457968440 T^{14} + 2384852381457900 T^{15} + 60762066594476935 T^{16} + 188403338135174100 T^{17} - 3520632091181034040 T^{18} - 34776801676912191300 T^{19} + 53017066305435970863 T^{20} + \)\(26\!\cdots\!60\)\( T^{21} + \)\(10\!\cdots\!32\)\( T^{22} - \)\(10\!\cdots\!00\)\( T^{23} - \)\(12\!\cdots\!80\)\( T^{24} + \)\(23\!\cdots\!00\)\( T^{25} + \)\(63\!\cdots\!70\)\( T^{26} + \)\(25\!\cdots\!00\)\( T^{27} - \)\(17\!\cdots\!55\)\( T^{28} - \)\(18\!\cdots\!00\)\( T^{29} - \)\(95\!\cdots\!06\)\( T^{30} + \)\(58\!\cdots\!80\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \))
$83$ (\( 1 + 6 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))(\( 1 - 130 T^{2} + 6889 T^{4} \))(\( 1 - 22 T^{2} + 6889 T^{4} \))(\( 1 + 9938 T^{4} + 47458321 T^{8} \))(\( ( 1 + 6889 T^{4} )^{2} \))(\( 1 + 18 T + 41 T^{2} - 1416 T^{3} - 18731 T^{4} - 117528 T^{5} + 282449 T^{6} + 10292166 T^{7} + 47458321 T^{8} \))(\( 1 - 2 T - 123 T^{2} - 626 T^{3} + 1416 T^{4} + 170328 T^{5} + 1146775 T^{6} - 9202888 T^{7} - 131919081 T^{8} - 763839704 T^{9} + 7900132975 T^{10} + 97391336136 T^{11} + 67200982536 T^{12} - 2465839442518 T^{13} - 40213665924387 T^{14} - 54272101979254 T^{15} + 2252292232139041 T^{16} \))(\( 1 + 10 T - 32 T^{2} - 586 T^{3} + 7028 T^{4} + 84104 T^{5} - 240235 T^{6} - 5165580 T^{7} + 16282512 T^{8} + 55940648 T^{9} - 5964955348 T^{10} - 18358343086 T^{11} + 292581600901 T^{12} - 1523742476138 T^{13} - 41092577392372 T^{14} + 31986135297976 T^{15} + 772740681182352 T^{16} - 20347429564667940 T^{17} - 78542520596301715 T^{18} + 2282250432431589208 T^{19} + 15829109807473180148 T^{20} - \)\(10\!\cdots\!58\)\( T^{21} - \)\(49\!\cdots\!68\)\( T^{22} + \)\(12\!\cdots\!70\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))(\( 1 - 10 T + 202 T^{2} + 1560 T^{3} - 11179 T^{4} + 496950 T^{5} + 1018034 T^{6} + 15276030 T^{7} + 550764400 T^{8} + 1724892540 T^{9} + 40801859152 T^{10} + 455708099560 T^{11} + 3046921596515 T^{12} + 45917600363320 T^{13} + 368604392570920 T^{14} + 3224407333551250 T^{15} + 38409379558310191 T^{16} + 267625808684753750 T^{17} + 2539315660421067880 T^{18} + 26255086958941652840 T^{19} + \)\(14\!\cdots\!15\)\( T^{20} + \)\(17\!\cdots\!80\)\( T^{21} + \)\(13\!\cdots\!88\)\( T^{22} + \)\(46\!\cdots\!80\)\( T^{23} + \)\(12\!\cdots\!00\)\( T^{24} + \)\(28\!\cdots\!90\)\( T^{25} + \)\(15\!\cdots\!66\)\( T^{26} + \)\(63\!\cdots\!50\)\( T^{27} - \)\(11\!\cdots\!19\)\( T^{28} + \)\(13\!\cdots\!80\)\( T^{29} + \)\(14\!\cdots\!58\)\( T^{30} - \)\(61\!\cdots\!70\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \))
$89$ (\( 1 + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( ( 1 + 89 T^{2} )^{2} \))(\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 + 89 T^{2} )^{4} \))(\( ( 1 + 89 T^{2} )^{4} \))(\( 1 + 9 T - 43 T^{2} - 993 T^{3} - 4460 T^{4} - 88377 T^{5} - 340603 T^{6} + 6344721 T^{7} + 62742241 T^{8} \))(\( 1 + 77 T^{2} + 700 T^{3} + 12658 T^{4} + 121100 T^{5} + 1037049 T^{6} + 10883950 T^{7} + 150329955 T^{8} + 968671550 T^{9} + 8214465129 T^{10} + 85371745900 T^{11} + 794191286578 T^{12} + 3908841614300 T^{13} + 38267559403997 T^{14} + 3936588805702081 T^{16} \))(\( 1 - 57 T + 1416 T^{2} - 19412 T^{3} + 146681 T^{4} - 436608 T^{5} + 1888 T^{6} - 50698623 T^{7} + 1130538888 T^{8} - 10860338043 T^{9} + 55652919649 T^{10} - 136415194025 T^{11} + 148330818154 T^{12} - 12140952268225 T^{13} + 440826776539729 T^{14} - 7656201649835667 T^{15} + 70932543370768008 T^{16} - 283104124814438727 T^{17} + 938300677334368 T^{18} - 19311754666067125632 T^{19} + \)\(57\!\cdots\!61\)\( T^{20} - \)\(68\!\cdots\!08\)\( T^{21} + \)\(44\!\cdots\!16\)\( T^{22} - \)\(15\!\cdots\!73\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \))(\( 1 - 18 T - 288 T^{2} + 7264 T^{3} + 14783 T^{4} - 1100918 T^{5} + 4764016 T^{6} + 40139780 T^{7} - 746571076 T^{8} + 9051008048 T^{9} + 13821315872 T^{10} - 1302515170094 T^{11} + 4134501893785 T^{12} + 64272919989494 T^{13} - 170401937947852 T^{14} - 989839747535702 T^{15} - 10684112304900357 T^{16} - 88095737530677478 T^{17} - 1349753750484935692 T^{18} + 45310416132073595686 T^{19} + \)\(25\!\cdots\!85\)\( T^{20} - \)\(72\!\cdots\!06\)\( T^{21} + \)\(68\!\cdots\!92\)\( T^{22} + \)\(40\!\cdots\!92\)\( T^{23} - \)\(29\!\cdots\!56\)\( T^{24} + \)\(14\!\cdots\!20\)\( T^{25} + \)\(14\!\cdots\!16\)\( T^{26} - \)\(30\!\cdots\!02\)\( T^{27} + \)\(36\!\cdots\!43\)\( T^{28} + \)\(15\!\cdots\!16\)\( T^{29} - \)\(56\!\cdots\!08\)\( T^{30} - \)\(31\!\cdots\!82\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} \))
$97$ (\( 1 + 17 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 17 T + 97 T^{2} \))(\( 1 + 95 T^{2} + 9409 T^{4} \))(\( 1 - 190 T^{2} + 9409 T^{4} \))(\( ( 1 + 9409 T^{4} )^{2} \))(\( 1 + 9743 T^{4} + 88529281 T^{8} \))(\( 1 - 2 T - 33 T^{2} - 820 T^{3} + 10061 T^{4} - 79540 T^{5} - 310497 T^{6} - 1825346 T^{7} + 88529281 T^{8} \))(\( 1 + 6 T - 187 T^{2} - 1878 T^{3} + 3816 T^{4} + 286836 T^{5} + 2735335 T^{6} - 15219804 T^{7} - 421030321 T^{8} - 1476320988 T^{9} + 25736767015 T^{10} + 261787472628 T^{11} + 337827736296 T^{12} - 16127025002646 T^{13} - 155765764921723 T^{14} + 484789706868678 T^{15} + 7837433594376961 T^{16} \))(\( 1 - 4 T - 152 T^{2} - 258 T^{3} + 15244 T^{4} + 16576 T^{5} - 519489 T^{6} - 1347552 T^{7} - 32374768 T^{8} + 207718764 T^{9} + 20086873388 T^{10} - 47224426932 T^{11} - 2473927275691 T^{12} - 4580769412404 T^{13} + 188997391707692 T^{14} + 189579307496172 T^{15} - 2866114933581808 T^{16} - 11571887538000864 T^{17} - 432719793868561281 T^{18} + 1339312363509201088 T^{19} + \)\(11\!\cdots\!84\)\( T^{20} - \)\(19\!\cdots\!86\)\( T^{21} - \)\(11\!\cdots\!48\)\( T^{22} - \)\(28\!\cdots\!12\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \))(\( 1 - 40 T + 1170 T^{2} - 28840 T^{3} + 597037 T^{4} - 11124280 T^{5} + 188742510 T^{6} - 2956583740 T^{7} + 43370532748 T^{8} - 598437364480 T^{9} + 7830452653220 T^{10} - 97502185336460 T^{11} + 1158651974029219 T^{12} - 13184566116562660 T^{13} + 143770204308646900 T^{14} - 1504531362263433880 T^{15} + 15122407401217954415 T^{16} - \)\(14\!\cdots\!60\)\( T^{17} + \)\(13\!\cdots\!00\)\( T^{18} - \)\(12\!\cdots\!80\)\( T^{19} + \)\(10\!\cdots\!39\)\( T^{20} - \)\(83\!\cdots\!20\)\( T^{21} + \)\(65\!\cdots\!80\)\( T^{22} - \)\(48\!\cdots\!40\)\( T^{23} + \)\(33\!\cdots\!28\)\( T^{24} - \)\(22\!\cdots\!80\)\( T^{25} + \)\(13\!\cdots\!90\)\( T^{26} - \)\(79\!\cdots\!40\)\( T^{27} + \)\(41\!\cdots\!17\)\( T^{28} - \)\(19\!\cdots\!80\)\( T^{29} + \)\(76\!\cdots\!30\)\( T^{30} - \)\(25\!\cdots\!20\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \))
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