Properties

Label 37.2
Level 37
Weight 2
Dimension 40
Nonzero newspaces 6
Newform subspaces 8
Sturm bound 228
Trace bound 2

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Defining parameters

Level: \( N \) = \( 37 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 8 \)
Sturm bound: \(228\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(37))\).

Total New Old
Modular forms 75 75 0
Cusp forms 40 40 0
Eisenstein series 35 35 0

Trace form

\( 40q - 15q^{2} - 14q^{3} - 11q^{4} - 12q^{5} - 6q^{6} - 10q^{7} - 3q^{8} - 5q^{9} + O(q^{10}) \) \( 40q - 15q^{2} - 14q^{3} - 11q^{4} - 12q^{5} - 6q^{6} - 10q^{7} - 3q^{8} - 5q^{9} - 6q^{11} + 10q^{12} - 4q^{13} + 6q^{14} + 6q^{15} + 13q^{16} + 21q^{18} + 2q^{19} + 24q^{20} + 14q^{21} + 18q^{22} + 6q^{23} + 42q^{24} + 13q^{25} + 15q^{26} - 2q^{27} - 22q^{28} - 6q^{29} - 18q^{30} - 40q^{31} - 27q^{32} - 6q^{33} - 36q^{34} - 24q^{35} - 80q^{36} - 23q^{37} - 12q^{38} - 4q^{39} - 63q^{40} - 30q^{41} + 6q^{42} - 10q^{43} - 6q^{44} + 6q^{45} - 18q^{46} + 12q^{47} + 46q^{48} + 15q^{49} + 66q^{50} + 54q^{51} + 80q^{52} + 36q^{53} + 102q^{54} + 54q^{55} + 102q^{56} + 62q^{57} + 54q^{58} + 6q^{59} + 24q^{60} - q^{61} - 12q^{62} - 22q^{63} - 17q^{64} - 15q^{65} - 126q^{66} + 14q^{67} - 66q^{69} - 72q^{70} - 18q^{71} - 111q^{72} - 34q^{73} - 51q^{74} - 92q^{75} - 58q^{76} + 6q^{77} - 102q^{78} - 10q^{79} - 30q^{80} - 41q^{81} + 30q^{83} - 46q^{84} + 9q^{85} - 12q^{86} - 6q^{87} + 72q^{88} + 27q^{89} + 90q^{90} + 52q^{91} + 60q^{92} + 56q^{93} + 54q^{94} + 30q^{95} + 54q^{96} + 8q^{97} + 9q^{98} + 48q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(37))\)

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
37.2.a \(\chi_{37}(1, \cdot)\) 37.2.a.a 1 1
37.2.a.b 1
37.2.b \(\chi_{37}(36, \cdot)\) 37.2.b.a 2 1
37.2.c \(\chi_{37}(10, \cdot)\) 37.2.c.a 2 2
37.2.e \(\chi_{37}(11, \cdot)\) 37.2.e.a 4 2
37.2.f \(\chi_{37}(7, \cdot)\) 37.2.f.a 6 6
37.2.f.b 6
37.2.h \(\chi_{37}(3, \cdot)\) 37.2.h.a 18 6

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} \))(\( 1 + 2 T^{2} \))(\( ( 1 - 2 T + 2 T^{2} )( 1 + 2 T + 2 T^{2} ) \))(\( 1 + T - T^{2} + 2 T^{3} + 4 T^{4} \))(\( 1 + 3 T^{2} + 5 T^{4} + 12 T^{6} + 16 T^{8} \))(\( 1 + 6 T + 15 T^{2} + 19 T^{3} + 6 T^{4} - 27 T^{5} - 61 T^{6} - 54 T^{7} + 24 T^{8} + 152 T^{9} + 240 T^{10} + 192 T^{11} + 64 T^{12} \))(\( 1 - 3 T + 9 T^{2} - 18 T^{3} + 36 T^{4} - 57 T^{5} + 91 T^{6} - 114 T^{7} + 144 T^{8} - 144 T^{9} + 144 T^{10} - 96 T^{11} + 64 T^{12} \))(\( 1 + 9 T + 42 T^{2} + 135 T^{3} + 333 T^{4} + 657 T^{5} + 1040 T^{6} + 1260 T^{7} + 912 T^{8} - 540 T^{9} - 3576 T^{10} - 8262 T^{11} - 13695 T^{12} - 17280 T^{13} - 14235 T^{14} + 1746 T^{15} + 35823 T^{16} + 87066 T^{17} + 142519 T^{18} + 174132 T^{19} + 143292 T^{20} + 13968 T^{21} - 227760 T^{22} - 552960 T^{23} - 876480 T^{24} - 1057536 T^{25} - 915456 T^{26} - 276480 T^{27} + 933888 T^{28} + 2580480 T^{29} + 4259840 T^{30} + 5382144 T^{31} + 5455872 T^{32} + 4423680 T^{33} + 2752512 T^{34} + 1179648 T^{35} + 262144 T^{36} \))
$3$ (\( 1 + 3 T + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( ( 1 + T + 3 T^{2} )^{2} \))(\( ( 1 - 3 T + 3 T^{2} )( 1 + 3 T + 3 T^{2} ) \))(\( 1 - 2 T + 4 T^{3} - 5 T^{4} + 12 T^{5} - 54 T^{7} + 81 T^{8} \))(\( 1 - 3 T + 9 T^{2} - 15 T^{3} + 36 T^{4} - 54 T^{5} + 117 T^{6} - 162 T^{7} + 324 T^{8} - 405 T^{9} + 729 T^{10} - 729 T^{11} + 729 T^{12} \))(\( 1 + 6 T + 15 T^{2} + 19 T^{3} + 3 T^{4} - 51 T^{5} - 134 T^{6} - 153 T^{7} + 27 T^{8} + 513 T^{9} + 1215 T^{10} + 1458 T^{11} + 729 T^{12} \))(\( 1 + 9 T + 42 T^{2} + 131 T^{3} + 315 T^{4} + 636 T^{5} + 1109 T^{6} + 1620 T^{7} + 2058 T^{8} + 2984 T^{9} + 6147 T^{10} + 13506 T^{11} + 27029 T^{12} + 49122 T^{13} + 75264 T^{14} + 78863 T^{15} + 29673 T^{16} - 64083 T^{17} - 162908 T^{18} - 192249 T^{19} + 267057 T^{20} + 2129301 T^{21} + 6096384 T^{22} + 11936646 T^{23} + 19704141 T^{24} + 29537622 T^{25} + 40330467 T^{26} + 58734072 T^{27} + 121522842 T^{28} + 286978140 T^{29} + 589368069 T^{30} + 1013989428 T^{31} + 1506635235 T^{32} + 1879706817 T^{33} + 1807962282 T^{34} + 1162261467 T^{35} + 387420489 T^{36} \))
$5$ (\( 1 + 2 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( 1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4} \))(\( ( 1 + 2 T + 5 T^{2} )^{2}( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} ) \))(\( 1 + 6 T + 24 T^{2} + 64 T^{3} + 132 T^{4} + 252 T^{5} + 479 T^{6} + 1260 T^{7} + 3300 T^{8} + 8000 T^{9} + 15000 T^{10} + 18750 T^{11} + 15625 T^{12} \))(\( 1 - 6 T + 18 T^{2} - 45 T^{3} + 81 T^{4} - 87 T^{5} + 109 T^{6} - 435 T^{7} + 2025 T^{8} - 5625 T^{9} + 11250 T^{10} - 18750 T^{11} + 15625 T^{12} \))(\( 1 + 3 T - 6 T^{2} - 39 T^{3} + 21 T^{4} + 363 T^{5} + 74 T^{6} - 2625 T^{7} - 2205 T^{8} + 16140 T^{9} + 22536 T^{10} - 87459 T^{11} - 179943 T^{12} + 382905 T^{13} + 1204194 T^{14} - 1414497 T^{15} - 7297755 T^{16} + 2705229 T^{17} + 38992717 T^{18} + 13526145 T^{19} - 182443875 T^{20} - 176812125 T^{21} + 752621250 T^{22} + 1196578125 T^{23} - 2811609375 T^{24} - 6832734375 T^{25} + 8803125000 T^{26} + 31523437500 T^{27} - 21533203125 T^{28} - 128173828125 T^{29} + 18066406250 T^{30} + 443115234375 T^{31} + 128173828125 T^{32} - 1190185546875 T^{33} - 915527343750 T^{34} + 2288818359375 T^{35} + 3814697265625 T^{36} \))
$7$ (\( 1 + T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( ( 1 - 3 T + 7 T^{2} )^{2} \))(\( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} \))(\( ( 1 - 13 T^{2} + 49 T^{4} )( 1 + 11 T^{2} + 49 T^{4} ) \))(\( 1 - 3 T + 12 T^{2} - 4 T^{3} + 81 T^{4} - 135 T^{5} + 1023 T^{6} - 945 T^{7} + 3969 T^{8} - 1372 T^{9} + 28812 T^{10} - 50421 T^{11} + 117649 T^{12} \))(\( 1 + 12 T + 60 T^{2} + 152 T^{3} + 108 T^{4} - 702 T^{5} - 3051 T^{6} - 4914 T^{7} + 5292 T^{8} + 52136 T^{9} + 144060 T^{10} + 201684 T^{11} + 117649 T^{12} \))(\( 1 + 3 T - 6 T^{2} - 38 T^{3} + 39 T^{4} + 453 T^{5} + 295 T^{6} - 1383 T^{7} + 1569 T^{8} + 2720 T^{9} - 51054 T^{10} - 19995 T^{11} + 714112 T^{12} + 980805 T^{13} - 4889940 T^{14} - 9815546 T^{15} + 16276629 T^{16} + 26949171 T^{17} - 61052479 T^{18} + 188644197 T^{19} + 797554821 T^{20} - 3366732278 T^{21} - 11740745940 T^{22} + 16484389635 T^{23} + 84014562688 T^{24} - 16466742285 T^{25} - 294316150254 T^{26} + 109761811040 T^{27} + 443203665681 T^{28} - 2734642885569 T^{29} + 4083179724295 T^{30} + 43890721714371 T^{31} + 26450699841111 T^{32} - 180407337377834 T^{33} - 199397583417606 T^{34} + 697891541961621 T^{35} + 1628413597910449 T^{36} \))
$11$ (\( 1 + 5 T + 11 T^{2} \))(\( 1 - 3 T + 11 T^{2} \))(\( ( 1 + 3 T + 11 T^{2} )^{2} \))(\( ( 1 + 2 T + 11 T^{2} )^{2} \))(\( ( 1 + 6 T + 28 T^{2} + 66 T^{3} + 121 T^{4} )^{2} \))(\( 1 - 30 T^{2} - 2 T^{3} + 570 T^{4} + 30 T^{5} - 7237 T^{6} + 330 T^{7} + 68970 T^{8} - 2662 T^{9} - 439230 T^{10} + 1771561 T^{12} \))(\( 1 - 9 T + 30 T^{2} - 45 T^{3} + 177 T^{4} - 1548 T^{5} + 7099 T^{6} - 17028 T^{7} + 21417 T^{8} - 59895 T^{9} + 439230 T^{10} - 1449459 T^{11} + 1771561 T^{12} \))(\( 1 - 9 T - 21 T^{2} + 324 T^{3} + 723 T^{4} - 8793 T^{5} - 17256 T^{6} + 160920 T^{7} + 372795 T^{8} - 2330154 T^{9} - 5640936 T^{10} + 24459516 T^{11} + 67109511 T^{12} - 203061375 T^{13} - 516149688 T^{14} + 1130688558 T^{15} + 2653423338 T^{16} - 4112846019 T^{17} - 9318301229 T^{18} - 45241306209 T^{19} + 321064223898 T^{20} + 1504946470698 T^{21} - 7556947582008 T^{22} - 32703237505125 T^{23} + 118888592416671 T^{24} + 476646770869236 T^{25} - 1209184728752616 T^{26} - 5494381243974414 T^{27} + 9669342204129795 T^{28} + 45912354034722120 T^{29} - 54156720068697576 T^{30} - 303558207881585283 T^{31} + 274559129680683243 T^{32} + 1353428406890670924 T^{33} - 964944327135015381 T^{34} - 4549023256493643939 T^{35} + 5559917313492231481 T^{36} \))
$13$ (\( 1 + 2 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \))(\( ( 1 - 6 T + 25 T^{2} - 78 T^{3} + 169 T^{4} )^{2} \))(\( 1 + 9 T + 27 T^{2} - 11 T^{3} - 198 T^{4} + 360 T^{5} + 4161 T^{6} + 4680 T^{7} - 33462 T^{8} - 24167 T^{9} + 771147 T^{10} + 3341637 T^{11} + 4826809 T^{12} \))(\( 1 + 12 T + 78 T^{2} + 359 T^{3} + 1485 T^{4} + 6129 T^{5} + 23841 T^{6} + 79677 T^{7} + 250965 T^{8} + 788723 T^{9} + 2227758 T^{10} + 4455516 T^{11} + 4826809 T^{12} \))(\( 1 - 9 T + 63 T^{2} - 342 T^{3} + 1584 T^{4} - 6183 T^{5} + 21936 T^{6} - 75645 T^{7} + 250839 T^{8} - 875016 T^{9} + 2693997 T^{10} - 5932935 T^{11} - 2431623 T^{12} + 113560299 T^{13} - 775998432 T^{14} + 3754732338 T^{15} - 14759374887 T^{16} + 54998570169 T^{17} - 197208543557 T^{18} + 714981412197 T^{19} - 2494334355903 T^{20} + 8249146946586 T^{21} - 22163291216352 T^{22} + 42164144096607 T^{23} - 11736979781007 T^{24} - 372282872707395 T^{25} + 2197576115181837 T^{26} - 9279106623364968 T^{27} + 34580286236911311 T^{28} - 135567973006928865 T^{29} + 511066795246743216 T^{30} - 1872676784059900299 T^{31} + 6236804194947673776 T^{32} - 17505575410819038894 T^{33} + 41921246378540329983 T^{34} - 77853743274432041397 T^{35} + \)\(11\!\cdots\!29\)\( T^{36} \))
$17$ (\( 1 + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 - 6 T + 45 T^{2} - 198 T^{3} + 1004 T^{4} - 3366 T^{5} + 13005 T^{6} - 29478 T^{7} + 83521 T^{8} \))(\( 1 - 3 T + 36 T^{2} + 90 T^{4} + 3489 T^{5} - 8513 T^{6} + 59313 T^{7} + 26010 T^{8} + 3006756 T^{10} - 4259571 T^{11} + 24137569 T^{12} \))(\( 1 - 3 T - 72 T^{3} - 117 T^{4} + 1275 T^{5} + 1369 T^{6} + 21675 T^{7} - 33813 T^{8} - 353736 T^{9} - 4259571 T^{11} + 24137569 T^{12} \))(\( 1 + 15 T + 93 T^{2} + 327 T^{3} + 1110 T^{4} + 4911 T^{5} + 12017 T^{6} - 22476 T^{7} - 317250 T^{8} - 2116920 T^{9} - 11478213 T^{10} - 43896324 T^{11} - 137044248 T^{12} - 392964957 T^{13} + 17753187 T^{14} + 9472909323 T^{15} + 61715137998 T^{16} + 273141458883 T^{17} + 1125832673449 T^{18} + 4643404801011 T^{19} + 17835674881422 T^{20} + 46540403503899 T^{21} + 1482763931427 T^{22} - 557954044951149 T^{23} - 3307914992153112 T^{24} - 18012359339738052 T^{25} - 80069229744132933 T^{26} - 251041047514029240 T^{27} - 639574064917445250 T^{28} - 770295141410359308 T^{29} + 7001371424790037937 T^{30} + 48641382719601056607 T^{31} + \)\(18\!\cdots\!90\)\( T^{32} + \)\(93\!\cdots\!11\)\( T^{33} + \)\(45\!\cdots\!33\)\( T^{34} + \)\(12\!\cdots\!55\)\( T^{35} + \)\(14\!\cdots\!09\)\( T^{36} \))
$19$ (\( 1 + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 - 2 T^{2} + 361 T^{4} \))(\( 1 - 6 T + 17 T^{2} - 114 T^{3} + 361 T^{4} \))(\( 1 + 6 T + 44 T^{2} + 192 T^{3} + 891 T^{4} + 3648 T^{5} + 15884 T^{6} + 41154 T^{7} + 130321 T^{8} \))(\( 1 - 3 T - 36 T^{2} + 150 T^{3} + 45 T^{4} - 1767 T^{5} + 10583 T^{6} - 33573 T^{7} + 16245 T^{8} + 1028850 T^{9} - 4691556 T^{10} - 7428297 T^{11} + 47045881 T^{12} \))(\( 1 + 9 T + 18 T^{2} - 142 T^{3} - 837 T^{4} + 513 T^{5} + 15087 T^{6} + 9747 T^{7} - 302157 T^{8} - 973978 T^{9} + 2345778 T^{10} + 22284891 T^{11} + 47045881 T^{12} \))(\( 1 - 6 T - 33 T^{2} + 78 T^{3} + 1896 T^{4} + 654 T^{5} - 58725 T^{6} - 161610 T^{7} + 1407036 T^{8} + 6730218 T^{9} - 17617647 T^{10} - 214251762 T^{11} - 20455398 T^{12} + 4608278994 T^{13} + 10783094349 T^{14} - 70455746166 T^{15} - 376064748966 T^{16} + 478730321718 T^{17} + 8719338321655 T^{18} + 9095876112642 T^{19} - 135759374376726 T^{20} - 483255962952594 T^{21} + 1405263638656029 T^{22} + 11410555008764406 T^{23} - 962342220115638 T^{24} - 191513595082754118 T^{25} - 299210418458584527 T^{26} + 2171758551970785822 T^{27} + 8626630943111287836 T^{28} - 18825990740541172590 T^{29} - \)\(12\!\cdots\!25\)\( T^{30} + 27502651184316116586 T^{31} + \)\(15\!\cdots\!16\)\( T^{32} + \)\(11\!\cdots\!22\)\( T^{33} - \)\(95\!\cdots\!73\)\( T^{34} - \)\(32\!\cdots\!34\)\( T^{35} + \)\(10\!\cdots\!41\)\( T^{36} \))
$23$ (\( 1 - 2 T + 23 T^{2} \))(\( 1 - 6 T + 23 T^{2} \))(\( 1 - 30 T^{2} + 529 T^{4} \))(\( ( 1 + 4 T + 23 T^{2} )^{2} \))(\( 1 - 84 T^{2} + 2810 T^{4} - 44436 T^{6} + 279841 T^{8} \))(\( 1 + 3 T - 60 T^{2} - 67 T^{3} + 2763 T^{4} + 1518 T^{5} - 71161 T^{6} + 34914 T^{7} + 1461627 T^{8} - 815189 T^{9} - 16790460 T^{10} + 19309029 T^{11} + 148035889 T^{12} \))(\( ( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} )^{3} \))(\( 1 + 9 T + 138 T^{2} + 999 T^{3} + 8682 T^{4} + 52029 T^{5} + 352655 T^{6} + 1797453 T^{7} + 10464228 T^{8} + 44883369 T^{9} + 209564967 T^{10} + 629074395 T^{11} + 1414830477 T^{12} - 6565154292 T^{13} - 83434452042 T^{14} - 707663562552 T^{15} - 4363985947026 T^{16} - 24737828956902 T^{17} - 123933615376247 T^{18} - 568970066008746 T^{19} - 2308548565976754 T^{20} - 8610142565570184 T^{21} - 23348380493885322 T^{22} - 42255584871234156 T^{23} + 209445687446989053 T^{24} + 2141888508152129565 T^{25} + 16411239046150250727 T^{26} + 80841799529775908847 T^{27} + \)\(43\!\cdots\!72\)\( T^{28} + \)\(17\!\cdots\!31\)\( T^{29} + \)\(77\!\cdots\!55\)\( T^{30} + \)\(26\!\cdots\!07\)\( T^{31} + \)\(10\!\cdots\!38\)\( T^{32} + \)\(26\!\cdots\!93\)\( T^{33} + \)\(84\!\cdots\!18\)\( T^{34} + \)\(12\!\cdots\!27\)\( T^{35} + \)\(32\!\cdots\!69\)\( T^{36} \))
$29$ (\( 1 - 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( ( 1 - 10 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} ) \))(\( ( 1 - 9 T + 29 T^{2} )^{2} \))(\( 1 - 102 T^{2} + 4235 T^{4} - 85782 T^{6} + 707281 T^{8} \))(\( 1 + 15 T + 75 T^{2} + 396 T^{3} + 4245 T^{4} + 21165 T^{5} + 66382 T^{6} + 613785 T^{7} + 3570045 T^{8} + 9658044 T^{9} + 53046075 T^{10} + 307667235 T^{11} + 594823321 T^{12} \))(\( 1 - 9 T - 6 T^{2} + 207 T^{3} + 591 T^{4} - 5220 T^{5} + 6181 T^{6} - 151380 T^{7} + 497031 T^{8} + 5048523 T^{9} - 4243686 T^{10} - 184600341 T^{11} + 594823321 T^{12} \))(\( 1 + 18 T + 312 T^{2} + 3672 T^{3} + 40389 T^{4} + 362070 T^{5} + 3062450 T^{6} + 22252347 T^{7} + 152888334 T^{8} + 908162820 T^{9} + 5038749006 T^{10} + 22819847625 T^{11} + 87319007118 T^{12} + 129894685614 T^{13} - 1141544301585 T^{14} - 18012215688336 T^{15} - 142406720761344 T^{16} - 980000355531294 T^{17} - 5469243199756418 T^{18} - 28420010310407526 T^{19} - 119764052160290304 T^{20} - 439299928422826704 T^{21} - 807392595169340385 T^{22} + 2664289250936910486 T^{23} + 51939381800351398878 T^{24} + \)\(39\!\cdots\!25\)\( T^{25} + \)\(25\!\cdots\!66\)\( T^{26} + \)\(13\!\cdots\!80\)\( T^{27} + \)\(64\!\cdots\!34\)\( T^{28} + \)\(27\!\cdots\!63\)\( T^{29} + \)\(10\!\cdots\!50\)\( T^{30} + \)\(37\!\cdots\!30\)\( T^{31} + \)\(12\!\cdots\!09\)\( T^{32} + \)\(31\!\cdots\!28\)\( T^{33} + \)\(78\!\cdots\!52\)\( T^{34} + \)\(13\!\cdots\!62\)\( T^{35} + \)\(21\!\cdots\!61\)\( T^{36} \))
$31$ (\( 1 + 4 T + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))(\( ( 1 - 31 T^{2} )^{2} \))(\( ( 1 + 10 T + 31 T^{2} )^{2} \))(\( 1 - 52 T^{2} + 1626 T^{4} - 49972 T^{6} + 923521 T^{8} \))(\( ( 1 - 3 T + 84 T^{2} - 167 T^{3} + 2604 T^{4} - 2883 T^{5} + 29791 T^{6} )^{2} \))(\( ( 1 + 9 T + 99 T^{2} + 505 T^{3} + 3069 T^{4} + 8649 T^{5} + 29791 T^{6} )^{2} \))(\( 1 - 354 T^{2} + 60609 T^{4} - 6698055 T^{6} + 538766142 T^{8} - 33773223975 T^{10} + 1727635654875 T^{12} - 74618020136379 T^{14} + 2791322672536389 T^{16} - 91945388261561834 T^{18} + 2682461088307469829 T^{20} - 68911308574368870459 T^{22} + \)\(15\!\cdots\!75\)\( T^{24} - \)\(28\!\cdots\!75\)\( T^{26} + \)\(44\!\cdots\!42\)\( T^{28} - \)\(52\!\cdots\!55\)\( T^{30} + \)\(45\!\cdots\!89\)\( T^{32} - \)\(25\!\cdots\!74\)\( T^{34} + \)\(69\!\cdots\!41\)\( T^{36} \))
$37$ (\( 1 + T \))(\( 1 - T \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 11 T + 37 T^{2} \))(\( ( 1 - T + 37 T^{2} )^{2} \))(\( 1 + 6 T + 96 T^{2} + 371 T^{3} + 3552 T^{4} + 8214 T^{5} + 50653 T^{6} \))(\( 1 + 12 T - 30 T^{2} - 803 T^{3} - 1110 T^{4} + 16428 T^{5} + 50653 T^{6} \))(\( 1 - 6 T - 12 T^{2} + 491 T^{3} - 1692 T^{4} - 1746 T^{5} + 123974 T^{6} - 493410 T^{7} - 1147992 T^{8} + 35821187 T^{9} - 42475704 T^{10} - 675478290 T^{11} + 6279655022 T^{12} - 3272285106 T^{13} - 117329975244 T^{14} + 1259771666819 T^{15} - 1139182525596 T^{16} - 21074876723526 T^{17} + 129961739795077 T^{18} \))
$41$ (\( 1 + 9 T + 41 T^{2} \))(\( 1 + 9 T + 41 T^{2} \))(\( ( 1 + 3 T + 41 T^{2} )^{2} \))(\( 1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4} \))(\( ( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} )^{2} \))(\( 1 + 9 T^{2} - 9 T^{3} + 369 T^{4} + 11889 T^{5} - 26774 T^{6} + 487449 T^{7} + 620289 T^{8} - 620289 T^{9} + 25431849 T^{10} + 4750104241 T^{12} \))(\( 1 - 15 T + 72 T^{2} + 162 T^{3} - 2889 T^{4} + 6123 T^{5} + 21493 T^{6} + 251043 T^{7} - 4856409 T^{8} + 11165202 T^{9} + 203454792 T^{10} - 1737843015 T^{11} + 4750104241 T^{12} \))(\( 1 + 24 T + 378 T^{2} + 4905 T^{3} + 53532 T^{4} + 522402 T^{5} + 4581471 T^{6} + 36407742 T^{7} + 267792066 T^{8} + 1811269044 T^{9} + 11242071306 T^{10} + 63155161194 T^{11} + 307558451013 T^{12} + 1173628216842 T^{13} + 1745304231024 T^{14} - 25837262900685 T^{15} - 365342263122258 T^{16} - 3192138971221548 T^{17} - 22357237329705962 T^{18} - 130877697820083468 T^{19} - 614140344308515698 T^{20} - 1780729996378110885 T^{21} + 4931812629162609264 T^{22} + \)\(13\!\cdots\!42\)\( T^{23} + \)\(14\!\cdots\!33\)\( T^{24} + \)\(12\!\cdots\!14\)\( T^{25} + \)\(89\!\cdots\!26\)\( T^{26} + \)\(59\!\cdots\!84\)\( T^{27} + \)\(35\!\cdots\!66\)\( T^{28} + \)\(20\!\cdots\!22\)\( T^{29} + \)\(10\!\cdots\!51\)\( T^{30} + \)\(48\!\cdots\!42\)\( T^{31} + \)\(20\!\cdots\!52\)\( T^{32} + \)\(76\!\cdots\!05\)\( T^{33} + \)\(24\!\cdots\!98\)\( T^{34} + \)\(62\!\cdots\!44\)\( T^{35} + \)\(10\!\cdots\!21\)\( T^{36} \))
$43$ (\( 1 - 2 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 - 50 T^{2} + 1849 T^{4} \))(\( ( 1 - 2 T + 43 T^{2} )^{2} \))(\( ( 1 - 74 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 3 T + 69 T^{2} + 187 T^{3} + 2967 T^{4} + 5547 T^{5} + 79507 T^{6} )^{2} \))(\( ( 1 + 9 T + 153 T^{2} + 793 T^{3} + 6579 T^{4} + 16641 T^{5} + 79507 T^{6} )^{2} \))(\( 1 - 360 T^{2} + 65178 T^{4} - 7877220 T^{6} + 714203577 T^{8} - 51930976752 T^{10} + 3172570113552 T^{12} - 169152110585442 T^{14} + 8135756205448860 T^{16} - 361971412391702180 T^{18} + 15043013223874942140 T^{20} - \)\(57\!\cdots\!42\)\( T^{22} + \)\(20\!\cdots\!48\)\( T^{24} - \)\(60\!\cdots\!52\)\( T^{26} + \)\(15\!\cdots\!73\)\( T^{28} - \)\(31\!\cdots\!20\)\( T^{30} + \)\(48\!\cdots\!22\)\( T^{32} - \)\(49\!\cdots\!60\)\( T^{34} + \)\(25\!\cdots\!49\)\( T^{36} \))
$47$ (\( 1 + 9 T + 47 T^{2} \))(\( 1 - 3 T + 47 T^{2} \))(\( ( 1 - 3 T + 47 T^{2} )^{2} \))(\( ( 1 - 6 T + 47 T^{2} )^{2} \))(\( ( 1 - 6 T + 76 T^{2} - 282 T^{3} + 2209 T^{4} )^{2} \))(\( 1 - 15 T + 21 T^{2} - 126 T^{3} + 10023 T^{4} - 44763 T^{5} - 93242 T^{6} - 2103861 T^{7} + 22140807 T^{8} - 13081698 T^{9} + 102473301 T^{10} - 3440175105 T^{11} + 10779215329 T^{12} \))(\( 1 - 9 T - 24 T^{2} + 765 T^{3} - 1605 T^{4} - 20826 T^{5} + 241567 T^{6} - 978822 T^{7} - 3545445 T^{8} + 79424595 T^{9} - 117112344 T^{10} - 2064105063 T^{11} + 10779215329 T^{12} \))(\( 1 + 36 T + 387 T^{2} + 54 T^{3} - 3627 T^{4} + 401310 T^{5} + 3398403 T^{6} - 11982321 T^{7} - 33586911 T^{8} + 2443436091 T^{9} + 6640299531 T^{10} - 99300348351 T^{11} + 381857759151 T^{12} + 7489576689273 T^{13} - 28436798079702 T^{14} - 211416261108543 T^{15} + 2815489344002418 T^{16} + 5596907248450989 T^{17} - 134741169176022182 T^{18} + 263054640677196483 T^{19} + 6219415960901341362 T^{20} - 21949870477072259889 T^{21} - \)\(13\!\cdots\!62\)\( T^{22} + \)\(17\!\cdots\!11\)\( T^{23} + \)\(41\!\cdots\!79\)\( T^{24} - \)\(50\!\cdots\!13\)\( T^{25} + \)\(15\!\cdots\!91\)\( T^{26} + \)\(27\!\cdots\!97\)\( T^{27} - \)\(17\!\cdots\!39\)\( T^{28} - \)\(29\!\cdots\!63\)\( T^{29} + \)\(39\!\cdots\!23\)\( T^{30} + \)\(21\!\cdots\!70\)\( T^{31} - \)\(93\!\cdots\!63\)\( T^{32} + \)\(65\!\cdots\!22\)\( T^{33} + \)\(21\!\cdots\!27\)\( T^{34} + \)\(95\!\cdots\!32\)\( T^{35} + \)\(12\!\cdots\!89\)\( T^{36} \))
$53$ (\( 1 - T + 53 T^{2} \))(\( 1 + 3 T + 53 T^{2} \))(\( ( 1 - 9 T + 53 T^{2} )^{2} \))(\( 1 - 2 T - 49 T^{2} - 106 T^{3} + 2809 T^{4} \))(\( 1 - 12 T + 14 T^{2} - 288 T^{3} + 6459 T^{4} - 15264 T^{5} + 39326 T^{6} - 1786524 T^{7} + 7890481 T^{8} \))(\( 1 - 24 T + 228 T^{2} - 866 T^{3} - 3012 T^{4} + 75276 T^{5} - 701113 T^{6} + 3989628 T^{7} - 8460708 T^{8} - 128927482 T^{9} + 1799029668 T^{10} - 10036691832 T^{11} + 22164361129 T^{12} \))(\( 1 - 21 T + 180 T^{2} - 756 T^{3} + 4158 T^{4} - 67395 T^{5} + 688951 T^{6} - 3571935 T^{7} + 11679822 T^{8} - 112551012 T^{9} + 1420286580 T^{10} - 8782105353 T^{11} + 22164361129 T^{12} \))(\( 1 + 39 T + 600 T^{2} + 3348 T^{3} - 30444 T^{4} - 745503 T^{5} - 5826477 T^{6} - 371823 T^{7} + 445452720 T^{8} + 4368062646 T^{9} + 11666852094 T^{10} - 158962916373 T^{11} - 2012770140540 T^{12} - 7947243489453 T^{13} + 40655317413426 T^{14} + 686375150032296 T^{15} + 2821224335162934 T^{16} - 14061707686241979 T^{17} - 229423584704524283 T^{18} - 745270507370824887 T^{19} + 7924819157472681606 T^{20} + \)\(10\!\cdots\!92\)\( T^{21} + \)\(32\!\cdots\!06\)\( T^{22} - \)\(33\!\cdots\!29\)\( T^{23} - \)\(44\!\cdots\!60\)\( T^{24} - \)\(18\!\cdots\!01\)\( T^{25} + \)\(72\!\cdots\!34\)\( T^{26} + \)\(14\!\cdots\!18\)\( T^{27} + \)\(77\!\cdots\!80\)\( T^{28} - \)\(34\!\cdots\!31\)\( T^{29} - \)\(28\!\cdots\!57\)\( T^{30} - \)\(19\!\cdots\!19\)\( T^{31} - \)\(42\!\cdots\!36\)\( T^{32} + \)\(24\!\cdots\!36\)\( T^{33} + \)\(23\!\cdots\!00\)\( T^{34} + \)\(80\!\cdots\!07\)\( T^{35} + \)\(10\!\cdots\!89\)\( T^{36} \))
$59$ (\( 1 - 8 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 - 102 T^{2} + 3481 T^{4} \))(\( 1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4} \))(\( 1 + 12 T + 78 T^{2} + 360 T^{3} + 251 T^{4} + 21240 T^{5} + 271518 T^{6} + 2464548 T^{7} + 12117361 T^{8} \))(\( 1 - 6 T - 12 T^{2} + 503 T^{3} - 3207 T^{4} - 32067 T^{5} + 445277 T^{6} - 1891953 T^{7} - 11163567 T^{8} + 103305637 T^{9} - 145408332 T^{10} - 4289545794 T^{11} + 42180533641 T^{12} \))(\( 1 + 6 T + 18 T^{2} + 180 T^{3} - 972 T^{4} - 38640 T^{5} - 291923 T^{6} - 2279760 T^{7} - 3383532 T^{8} + 36968220 T^{9} + 218112498 T^{10} + 4289545794 T^{11} + 42180533641 T^{12} \))(\( 1 + 6 T + 204 T^{2} + 1749 T^{3} + 21063 T^{4} + 232917 T^{5} + 1733513 T^{6} + 19451589 T^{7} + 130830603 T^{8} + 1176560277 T^{9} + 7856124762 T^{10} + 50201213190 T^{11} + 313495504866 T^{12} + 766454784192 T^{13} + 2400376132602 T^{14} - 79631278479555 T^{15} - 866527656664779 T^{16} - 7995461698396449 T^{17} - 78364126976602151 T^{18} - 471732240205390491 T^{19} - 3016382772850095699 T^{20} - 16354592342852526345 T^{21} + 29086224134522303322 T^{22} + \)\(54\!\cdots\!08\)\( T^{23} + \)\(13\!\cdots\!06\)\( T^{24} + \)\(12\!\cdots\!10\)\( T^{25} + \)\(11\!\cdots\!02\)\( T^{26} + \)\(10\!\cdots\!03\)\( T^{27} + \)\(66\!\cdots\!03\)\( T^{28} + \)\(58\!\cdots\!51\)\( T^{29} + \)\(30\!\cdots\!53\)\( T^{30} + \)\(24\!\cdots\!43\)\( T^{31} + \)\(13\!\cdots\!43\)\( T^{32} + \)\(63\!\cdots\!51\)\( T^{33} + \)\(43\!\cdots\!64\)\( T^{34} + \)\(76\!\cdots\!14\)\( T^{35} + \)\(75\!\cdots\!21\)\( T^{36} \))
$61$ (\( 1 + 8 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))(\( ( 1 - 61 T^{2} )^{2} \))(\( ( 1 - 13 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \))(\( 1 + 6 T + 101 T^{2} + 534 T^{3} + 4932 T^{4} + 32574 T^{5} + 375821 T^{6} + 1361886 T^{7} + 13845841 T^{8} \))(\( 1 + 27 T + 324 T^{2} + 1924 T^{3} - 2538 T^{4} - 194049 T^{5} - 2164749 T^{6} - 11836989 T^{7} - 9443898 T^{8} + 436711444 T^{9} + 4486052484 T^{10} + 22804100127 T^{11} + 51520374361 T^{12} \))(\( 1 + 9 T + 45 T^{2} + 335 T^{3} + 2538 T^{4} + 11988 T^{5} - 65103 T^{6} + 731268 T^{7} + 9443898 T^{8} + 76038635 T^{9} + 623062845 T^{10} + 7601366709 T^{11} + 51520374361 T^{12} \))(\( 1 - 42 T + 885 T^{2} - 13107 T^{3} + 159972 T^{4} - 1716546 T^{5} + 16580844 T^{6} - 148842789 T^{7} + 1261767747 T^{8} - 9844414479 T^{9} + 68006901849 T^{10} - 393299629176 T^{11} + 1507960945377 T^{12} + 3164785591710 T^{13} - 145182533838693 T^{14} + 1990176992506902 T^{15} - 20957407257627255 T^{16} + 190638615040596951 T^{17} - 1558840551743433854 T^{18} + 11628955517476414011 T^{19} - 77982512405631015855 T^{20} + \)\(45\!\cdots\!62\)\( T^{21} - \)\(20\!\cdots\!13\)\( T^{22} + \)\(26\!\cdots\!10\)\( T^{23} + \)\(77\!\cdots\!97\)\( T^{24} - \)\(12\!\cdots\!96\)\( T^{25} + \)\(13\!\cdots\!69\)\( T^{26} - \)\(11\!\cdots\!39\)\( T^{27} + \)\(90\!\cdots\!47\)\( T^{28} - \)\(64\!\cdots\!29\)\( T^{29} + \)\(44\!\cdots\!24\)\( T^{30} - \)\(27\!\cdots\!26\)\( T^{31} + \)\(15\!\cdots\!52\)\( T^{32} - \)\(78\!\cdots\!07\)\( T^{33} + \)\(32\!\cdots\!85\)\( T^{34} - \)\(94\!\cdots\!82\)\( T^{35} + \)\(13\!\cdots\!81\)\( T^{36} \))
$67$ (\( 1 - 8 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( ( 1 + 12 T + 67 T^{2} )^{2} \))(\( 1 - 10 T + 33 T^{2} - 670 T^{3} + 4489 T^{4} \))(\( 1 - 18 T + 112 T^{2} - 1404 T^{3} + 18747 T^{4} - 94068 T^{5} + 502768 T^{6} - 5413734 T^{7} + 20151121 T^{8} \))(\( 1 + 18 T + 144 T^{2} + 821 T^{3} + 2133 T^{4} - 39825 T^{5} - 605283 T^{6} - 2668275 T^{7} + 9575037 T^{8} + 246926423 T^{9} + 2901761424 T^{10} + 24302251926 T^{11} + 90458382169 T^{12} \))(\( 1 - 24 T + 258 T^{2} - 1711 T^{3} + 603 T^{4} + 133227 T^{5} - 1538559 T^{6} + 8926209 T^{7} + 2706867 T^{8} - 514605493 T^{9} + 5198989218 T^{10} - 32403002568 T^{11} + 90458382169 T^{12} \))(\( 1 - 78 T^{2} - 1490 T^{3} + 4446 T^{4} + 90828 T^{5} + 1157116 T^{6} - 7343376 T^{7} - 57485709 T^{8} - 495499768 T^{9} + 7853989923 T^{10} + 17393009142 T^{11} - 19043247947 T^{12} - 6132102079608 T^{13} + 12322356993792 T^{14} + 193704058081696 T^{15} + 3350047264223664 T^{16} - 19684334390095674 T^{17} - 145296889756793845 T^{18} - 1318850404136410158 T^{19} + 15038362169100027696 T^{20} + 58259013620825134048 T^{21} + \)\(24\!\cdots\!32\)\( T^{22} - \)\(82\!\cdots\!56\)\( T^{23} - \)\(17\!\cdots\!43\)\( T^{24} + \)\(10\!\cdots\!66\)\( T^{25} + \)\(31\!\cdots\!43\)\( T^{26} - \)\(13\!\cdots\!96\)\( T^{27} - \)\(10\!\cdots\!41\)\( T^{28} - \)\(89\!\cdots\!08\)\( T^{29} + \)\(94\!\cdots\!76\)\( T^{30} + \)\(49\!\cdots\!36\)\( T^{31} + \)\(16\!\cdots\!34\)\( T^{32} - \)\(36\!\cdots\!70\)\( T^{33} - \)\(12\!\cdots\!18\)\( T^{34} + \)\(74\!\cdots\!09\)\( T^{36} \))
$71$ (\( 1 - 9 T + 71 T^{2} \))(\( 1 + 15 T + 71 T^{2} \))(\( ( 1 + 3 T + 71 T^{2} )^{2} \))(\( 1 + 6 T - 35 T^{2} + 426 T^{3} + 5041 T^{4} \))(\( ( 1 + 6 T - 35 T^{2} + 426 T^{3} + 5041 T^{4} )^{2} \))(\( 1 - 33 T + 459 T^{2} - 3015 T^{3} + 3528 T^{4} + 104358 T^{5} - 1167947 T^{6} + 7409418 T^{7} + 17784648 T^{8} - 1079101665 T^{9} + 11663961579 T^{10} - 59539568583 T^{11} + 128100283921 T^{12} \))(\( 1 + 12 T + 126 T^{2} + 1692 T^{3} + 16038 T^{4} + 158646 T^{5} + 1508509 T^{6} + 11263866 T^{7} + 80847558 T^{8} + 605585412 T^{9} + 3201871806 T^{10} + 21650752212 T^{11} + 128100283921 T^{12} \))(\( 1 + 9 T - 63 T^{2} - 1917 T^{3} - 8496 T^{4} + 166716 T^{5} + 1884603 T^{6} + 410022 T^{7} - 152283924 T^{8} - 1131943221 T^{9} + 2498233545 T^{10} + 82136597535 T^{11} + 627503832690 T^{12} + 120906464391 T^{13} - 36145626008895 T^{14} - 418565647771353 T^{15} - 1652505020171100 T^{16} + 18990058148780418 T^{17} + 274648717254513679 T^{18} + 1348294128563409678 T^{19} - 8330277806682515100 T^{20} - \)\(14\!\cdots\!83\)\( T^{21} - \)\(91\!\cdots\!95\)\( T^{22} + \)\(21\!\cdots\!41\)\( T^{23} + \)\(80\!\cdots\!90\)\( T^{24} + \)\(74\!\cdots\!85\)\( T^{25} + \)\(16\!\cdots\!45\)\( T^{26} - \)\(51\!\cdots\!51\)\( T^{27} - \)\(49\!\cdots\!24\)\( T^{28} + \)\(94\!\cdots\!62\)\( T^{29} + \)\(30\!\cdots\!23\)\( T^{30} + \)\(19\!\cdots\!76\)\( T^{31} - \)\(70\!\cdots\!76\)\( T^{32} - \)\(11\!\cdots\!67\)\( T^{33} - \)\(26\!\cdots\!23\)\( T^{34} + \)\(26\!\cdots\!19\)\( T^{35} + \)\(21\!\cdots\!61\)\( T^{36} \))
$73$ (\( 1 + T + 73 T^{2} \))(\( 1 - 11 T + 73 T^{2} \))(\( ( 1 - 9 T + 73 T^{2} )^{2} \))(\( ( 1 + 10 T + 73 T^{2} )^{2} \))(\( ( 1 + 73 T^{2} )^{4} \))(\( ( 1 - 6 T + 168 T^{2} - 767 T^{3} + 12264 T^{4} - 31974 T^{5} + 389017 T^{6} )^{2} \))(\( ( 1 + 180 T^{2} - 89 T^{3} + 13140 T^{4} + 389017 T^{6} )^{2} \))(\( ( 1 + 27 T + 618 T^{2} + 8981 T^{3} + 119166 T^{4} + 1281135 T^{5} + 13641888 T^{6} + 132169647 T^{7} + 1272564606 T^{8} + 11104917955 T^{9} + 92897216238 T^{10} + 704332048863 T^{11} + 5306926344096 T^{12} + 36381980483535 T^{13} + 247039649451438 T^{14} + 1359132686301509 T^{15} + 6827292284801946 T^{16} + 21774422481140187 T^{17} + 58871586708267913 T^{18} )^{2} \))
$79$ (\( 1 - 4 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( 1 - 122 T^{2} + 6241 T^{4} \))(\( 1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4} \))(\( 1 + 18 T + 212 T^{2} + 1872 T^{3} + 13107 T^{4} + 147888 T^{5} + 1323092 T^{6} + 8874702 T^{7} + 38950081 T^{8} \))(\( 1 - 33 T + 516 T^{2} - 4424 T^{3} + 9234 T^{4} + 298269 T^{5} - 4310931 T^{6} + 23563251 T^{7} + 57629394 T^{8} - 2181204536 T^{9} + 20098241796 T^{10} - 101542861167 T^{11} + 243087455521 T^{12} \))(\( 1 + 3 T + 114 T^{2} + 656 T^{3} + 8793 T^{4} + 56367 T^{5} + 695565 T^{6} + 4452993 T^{7} + 54877113 T^{8} + 323433584 T^{9} + 4440309234 T^{10} + 9231169197 T^{11} + 243087455521 T^{12} \))(\( 1 + 6 T + 51 T^{2} - 744 T^{3} - 11859 T^{4} - 181887 T^{5} - 1140321 T^{6} + 4326360 T^{7} + 93553728 T^{8} + 1772350074 T^{9} + 9557226795 T^{10} + 53738786658 T^{11} - 637396371546 T^{12} - 6901346692677 T^{13} - 61226482104588 T^{14} - 392101601141514 T^{15} + 1699968786747330 T^{16} + 14774685876886602 T^{17} + 422333648288668249 T^{18} + 1167200184274041558 T^{19} + 10609505198090086530 T^{20} - \)\(19\!\cdots\!46\)\( T^{21} - \)\(23\!\cdots\!28\)\( T^{22} - \)\(21\!\cdots\!23\)\( T^{23} - \)\(15\!\cdots\!66\)\( T^{24} + \)\(10\!\cdots\!22\)\( T^{25} + \)\(14\!\cdots\!95\)\( T^{26} + \)\(21\!\cdots\!06\)\( T^{27} + \)\(88\!\cdots\!28\)\( T^{28} + \)\(32\!\cdots\!40\)\( T^{29} - \)\(67\!\cdots\!61\)\( T^{30} - \)\(84\!\cdots\!93\)\( T^{31} - \)\(43\!\cdots\!79\)\( T^{32} - \)\(21\!\cdots\!56\)\( T^{33} + \)\(11\!\cdots\!71\)\( T^{34} + \)\(10\!\cdots\!54\)\( T^{35} + \)\(14\!\cdots\!61\)\( T^{36} \))
$83$ (\( 1 + 15 T + 83 T^{2} \))(\( 1 - 9 T + 83 T^{2} \))(\( ( 1 - 9 T + 83 T^{2} )^{2} \))(\( 1 - 12 T + 61 T^{2} - 996 T^{3} + 6889 T^{4} \))(\( 1 - 6 T - 112 T^{2} + 108 T^{3} + 12027 T^{4} + 8964 T^{5} - 771568 T^{6} - 3430722 T^{7} + 47458321 T^{8} \))(\( 1 - 21 T + 144 T^{2} + 936 T^{3} - 15894 T^{4} - 5853 T^{5} + 1245277 T^{6} - 485799 T^{7} - 109493766 T^{8} + 535192632 T^{9} + 6833998224 T^{10} - 82719853503 T^{11} + 326940373369 T^{12} \))(\( 1 - 3 T - 117 T^{2} + 639 T^{3} - 1080 T^{4} - 30306 T^{5} + 685621 T^{6} - 2515398 T^{7} - 7440120 T^{8} + 365371893 T^{9} - 5552623557 T^{10} - 11817121929 T^{11} + 326940373369 T^{12} \))(\( 1 + 24 T + 306 T^{2} + 2196 T^{3} + 9297 T^{4} + 6297 T^{5} - 170718 T^{6} - 5406636 T^{7} - 114672105 T^{8} - 1257754167 T^{9} - 5405518044 T^{10} + 9382388541 T^{11} + 123718017525 T^{12} - 108892198758 T^{13} + 7519210640532 T^{14} + 265085046923256 T^{15} + 186458381755263 T^{16} - 58788027469865559 T^{17} - 891894459605788325 T^{18} - 4879406279998841397 T^{19} + 1284511791912006807 T^{20} + \)\(15\!\cdots\!72\)\( T^{21} + \)\(35\!\cdots\!72\)\( T^{22} - \)\(42\!\cdots\!94\)\( T^{23} + \)\(40\!\cdots\!25\)\( T^{24} + \)\(25\!\cdots\!07\)\( T^{25} - \)\(12\!\cdots\!04\)\( T^{26} - \)\(23\!\cdots\!01\)\( T^{27} - \)\(17\!\cdots\!45\)\( T^{28} - \)\(69\!\cdots\!12\)\( T^{29} - \)\(18\!\cdots\!98\)\( T^{30} + \)\(55\!\cdots\!11\)\( T^{31} + \)\(68\!\cdots\!13\)\( T^{32} + \)\(13\!\cdots\!72\)\( T^{33} + \)\(15\!\cdots\!86\)\( T^{34} + \)\(10\!\cdots\!52\)\( T^{35} + \)\(34\!\cdots\!09\)\( T^{36} \))
$89$ (\( 1 - 4 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 18 T^{2} + 7921 T^{4} \))(\( 1 + 7 T - 40 T^{2} + 623 T^{3} + 7921 T^{4} \))(\( 1 - 18 T + 297 T^{2} - 3402 T^{3} + 37412 T^{4} - 302778 T^{5} + 2352537 T^{6} - 12689442 T^{7} + 62742241 T^{8} \))(\( 1 - 12 T + 159 T^{2} - 2483 T^{3} + 27897 T^{4} - 256761 T^{5} + 2868014 T^{6} - 22851729 T^{7} + 220972137 T^{8} - 1750438027 T^{9} + 9976016319 T^{10} - 67008713388 T^{11} + 496981290961 T^{12} \))(\( 1 + 24 T + 207 T^{2} + 225 T^{3} - 6615 T^{4} + 1455 T^{5} + 482158 T^{6} + 129495 T^{7} - 52397415 T^{8} + 158618025 T^{9} + 12987643887 T^{10} + 134017426776 T^{11} + 496981290961 T^{12} \))(\( 1 - 18 T + 387 T^{2} - 3087 T^{3} + 20286 T^{4} + 196929 T^{5} - 4226320 T^{6} + 37573632 T^{7} + 10578546 T^{8} - 6523012935 T^{9} + 98231925060 T^{10} - 887074577217 T^{11} + 2728200116556 T^{12} + 46048793439078 T^{13} - 781796060350416 T^{14} + 5860053773073657 T^{15} + 12657347148996963 T^{16} - 742825743146149311 T^{17} + 10004771758282270642 T^{18} - 66111491140007288679 T^{19} + \)\(10\!\cdots\!23\)\( T^{20} + \)\(41\!\cdots\!33\)\( T^{21} - \)\(49\!\cdots\!56\)\( T^{22} + \)\(25\!\cdots\!22\)\( T^{23} + \)\(13\!\cdots\!16\)\( T^{24} - \)\(39\!\cdots\!93\)\( T^{25} + \)\(38\!\cdots\!60\)\( T^{26} - \)\(22\!\cdots\!15\)\( T^{27} + \)\(32\!\cdots\!46\)\( T^{28} + \)\(10\!\cdots\!48\)\( T^{29} - \)\(10\!\cdots\!20\)\( T^{30} + \)\(43\!\cdots\!01\)\( T^{31} + \)\(39\!\cdots\!26\)\( T^{32} - \)\(53\!\cdots\!63\)\( T^{33} + \)\(59\!\cdots\!07\)\( T^{34} - \)\(24\!\cdots\!22\)\( T^{35} + \)\(12\!\cdots\!81\)\( T^{36} \))
$97$ (\( 1 - 4 T + 97 T^{2} \))(\( 1 - 8 T + 97 T^{2} \))(\( 1 - 50 T^{2} + 9409 T^{4} \))(\( ( 1 - 7 T + 97 T^{2} )^{2} \))(\( 1 - 310 T^{2} + 42411 T^{4} - 2916790 T^{6} + 88529281 T^{8} \))(\( 1 - 18 T - 36 T^{2} + 646 T^{3} + 30546 T^{4} - 194922 T^{5} - 1099401 T^{6} - 18907434 T^{7} + 287407314 T^{8} + 589586758 T^{9} - 3187054116 T^{10} - 154572124626 T^{11} + 832972004929 T^{12} \))(\( 1 + 27 T + 234 T^{2} + 2171 T^{3} + 45891 T^{4} + 431748 T^{5} + 2525169 T^{6} + 41879556 T^{7} + 431788419 T^{8} + 1981413083 T^{9} + 20715851754 T^{10} + 231858186939 T^{11} + 832972004929 T^{12} \))(\( 1 + 9 T + 504 T^{2} + 4293 T^{3} + 125847 T^{4} + 1080882 T^{5} + 20074899 T^{6} + 179587512 T^{7} + 2287045863 T^{8} + 21853371753 T^{9} + 211737312162 T^{10} + 2232395399421 T^{11} + 21173600117616 T^{12} + 239838383190627 T^{13} + 2724651031247154 T^{14} + 29512060670153709 T^{15} + 360648208158018129 T^{16} + 3478609963446443532 T^{17} + 39492592051828867297 T^{18} + \)\(33\!\cdots\!04\)\( T^{19} + \)\(33\!\cdots\!61\)\( T^{20} + \)\(26\!\cdots\!57\)\( T^{21} + \)\(24\!\cdots\!74\)\( T^{22} + \)\(20\!\cdots\!39\)\( T^{23} + \)\(17\!\cdots\!64\)\( T^{24} + \)\(18\!\cdots\!73\)\( T^{25} + \)\(16\!\cdots\!82\)\( T^{26} + \)\(16\!\cdots\!01\)\( T^{27} + \)\(16\!\cdots\!87\)\( T^{28} + \)\(12\!\cdots\!36\)\( T^{29} + \)\(13\!\cdots\!59\)\( T^{30} + \)\(72\!\cdots\!14\)\( T^{31} + \)\(82\!\cdots\!43\)\( T^{32} + \)\(27\!\cdots\!49\)\( T^{33} + \)\(30\!\cdots\!84\)\( T^{34} + \)\(53\!\cdots\!33\)\( T^{35} + \)\(57\!\cdots\!89\)\( T^{36} \))
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