Properties

Label 80.6
Level 80
Weight 6
Dimension 482
Nonzero newspaces 7
Newform subspaces 21
Sturm bound 2304
Trace bound 3

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Defining parameters

Level: \( N \) = \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) = \( 6 \)
Nonzero newspaces: \( 7 \)
Newform subspaces: \( 21 \)
Sturm bound: \(2304\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_1(80))\).

Total New Old
Modular forms 1016 508 508
Cusp forms 904 482 422
Eisenstein series 112 26 86

Trace form

\( 482q - 4q^{2} + 14q^{3} + 40q^{4} + 12q^{5} - 240q^{6} - 230q^{7} - 496q^{8} - 120q^{9} + O(q^{10}) \) \( 482q - 4q^{2} + 14q^{3} + 40q^{4} + 12q^{5} - 240q^{6} - 230q^{7} - 496q^{8} - 120q^{9} + 428q^{10} + 2528q^{11} - 16q^{12} - 370q^{13} + 192q^{14} - 1062q^{15} + 1728q^{16} + 390q^{17} + 6268q^{18} + 8148q^{19} - 2980q^{20} - 5420q^{21} - 8848q^{22} - 4018q^{23} - 16744q^{24} + 10338q^{25} - 14752q^{26} + 1124q^{27} + 22856q^{28} + 8164q^{29} + 35908q^{30} + 12956q^{31} - 3184q^{32} - 30224q^{33} - 45744q^{34} - 14006q^{35} + 12120q^{36} + 38874q^{37} - 31968q^{38} + 53492q^{39} + 43200q^{40} + 21876q^{41} - 9928q^{42} - 11162q^{43} + 38856q^{44} - 56618q^{45} + 41368q^{46} - 107942q^{47} + 116840q^{48} + 42604q^{49} + 38432q^{50} + 28028q^{51} - 26320q^{52} + 7522q^{53} - 355144q^{54} + 10844q^{55} - 382272q^{56} - 184776q^{57} - 213560q^{58} - 47436q^{59} + 146168q^{60} + 154504q^{61} + 547736q^{62} + 159514q^{63} + 291040q^{64} + 95894q^{65} + 341576q^{66} + 42530q^{67} + 36032q^{68} - 151476q^{69} - 12640q^{70} - 399796q^{71} - 580616q^{72} - 133454q^{73} - 342792q^{74} + 20154q^{75} - 68600q^{76} + 105752q^{77} + 374456q^{78} + 665112q^{79} + 61568q^{80} + 193282q^{81} - 180600q^{82} + 26406q^{83} - 591696q^{84} + 88878q^{85} - 758872q^{86} - 599308q^{87} - 557712q^{88} - 27920q^{89} + 911320q^{90} - 66100q^{91} + 1144544q^{92} - 210296q^{93} + 1145200q^{94} + 59268q^{95} + 586320q^{96} - 36894q^{97} + 212756q^{98} + 881304q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_1(80))\)

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
80.6.a \(\chi_{80}(1, \cdot)\) 80.6.a.a 1 1
80.6.a.b 1
80.6.a.c 1
80.6.a.d 1
80.6.a.e 1
80.6.a.f 1
80.6.a.g 1
80.6.a.h 1
80.6.a.i 2
80.6.c \(\chi_{80}(49, \cdot)\) 80.6.c.a 2 1
80.6.c.b 2
80.6.c.c 2
80.6.c.d 8
80.6.d \(\chi_{80}(41, \cdot)\) None 0 1
80.6.f \(\chi_{80}(9, \cdot)\) None 0 1
80.6.j \(\chi_{80}(43, \cdot)\) 80.6.j.a 116 2
80.6.l \(\chi_{80}(21, \cdot)\) 80.6.l.a 80 2
80.6.n \(\chi_{80}(47, \cdot)\) 80.6.n.a 2 2
80.6.n.b 4
80.6.n.c 4
80.6.n.d 20
80.6.o \(\chi_{80}(7, \cdot)\) None 0 2
80.6.q \(\chi_{80}(29, \cdot)\) 80.6.q.a 116 2
80.6.s \(\chi_{80}(3, \cdot)\) 80.6.s.a 116 2

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_1(80))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_1(80)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(40))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 24 T + 243 T^{2} \))(\( 1 + 22 T + 243 T^{2} \))(\( 1 + 6 T + 243 T^{2} \))(\( 1 - 2 T + 243 T^{2} \))(\( 1 - 4 T + 243 T^{2} \))(\( 1 - 8 T + 243 T^{2} \))(\( 1 - 18 T + 243 T^{2} \))(\( 1 - 26 T + 243 T^{2} \))(\( 1 - 12 T + 6 T^{2} - 2916 T^{3} + 59049 T^{4} \))(\( ( 1 - 24 T + 243 T^{2} )( 1 + 24 T + 243 T^{2} ) \))(\( 1 - 362 T^{2} + 59049 T^{4} \))(\( 1 - 290 T^{2} + 59049 T^{4} \))(\( 1 - 472 T^{2} + 69724 T^{4} - 20119464 T^{6} + 8439593958 T^{8} - 1188034229736 T^{10} + 243112555575324 T^{12} - 97180614348674328 T^{14} + 12157665459056928801 T^{16} \))(\( 1 + 59049 T^{4} \))(\( 1 - 108882 T^{4} + 3486784401 T^{8} \))(\( 1 - 87122 T^{4} + 3486784401 T^{8} \))(\( 1 + 4910 T^{4} + 4404762445 T^{8} + 138021311000520 T^{12} - 6849972091873491390 T^{16} + \)\(85\!\cdots\!52\)\( T^{20} - \)\(23\!\cdots\!90\)\( T^{24} + \)\(16\!\cdots\!20\)\( T^{28} + \)\(18\!\cdots\!45\)\( T^{32} + \)\(72\!\cdots\!10\)\( T^{36} + \)\(51\!\cdots\!01\)\( T^{40} \))
$5$ (\( 1 - 25 T \))(\( 1 + 25 T \))(\( 1 + 25 T \))(\( 1 + 25 T \))(\( 1 - 25 T \))(\( 1 - 25 T \))(\( 1 + 25 T \))(\( 1 + 25 T \))(\( ( 1 - 25 T )^{2} \))(\( 1 + 90 T + 3125 T^{2} \))(\( 1 + 10 T + 3125 T^{2} \))(\( 1 - 110 T + 3125 T^{2} \))(\( 1 - 8 T + 1100 T^{2} + 113000 T^{3} - 438250 T^{4} + 353125000 T^{5} + 10742187500 T^{6} - 244140625000 T^{7} + 95367431640625 T^{8} \))(\( 1 + 76 T + 3125 T^{2} \))(\( ( 1 + 50 T + 3125 T^{2} )^{2} \))(\( ( 1 - 110 T + 3125 T^{2} )^{2} \))(\( ( 1 + 22 T + 5 T^{2} + 137760 T^{3} + 11738750 T^{4} - 4607500 T^{5} + 36683593750 T^{6} + 1345312500000 T^{7} + 152587890625 T^{8} + 2098083496093750 T^{9} + 298023223876953125 T^{10} )^{2} \))
$7$ (\( 1 - 172 T + 16807 T^{2} \))(\( 1 + 218 T + 16807 T^{2} \))(\( 1 - 118 T + 16807 T^{2} \))(\( 1 - 62 T + 16807 T^{2} \))(\( 1 + 192 T + 16807 T^{2} \))(\( 1 - 108 T + 16807 T^{2} \))(\( 1 + 242 T + 16807 T^{2} \))(\( 1 - 22 T + 16807 T^{2} \))(\( 1 + 52 T + 29646 T^{2} + 873964 T^{3} + 282475249 T^{4} \))(\( 1 - 30050 T^{2} + 282475249 T^{4} \))(\( 1 - 18610 T^{2} + 282475249 T^{4} \))(\( 1 - 8650 T^{2} + 282475249 T^{4} \))(\( 1 - 44728 T^{2} + 1493128636 T^{4} - 37445733732616 T^{6} + 682894235558230726 T^{8} - \)\(10\!\cdots\!84\)\( T^{10} + \)\(11\!\cdots\!36\)\( T^{12} - \)\(10\!\cdots\!72\)\( T^{14} + \)\(63\!\cdots\!01\)\( T^{16} \))(\( 1 + 282475249 T^{4} \))(\( 1 - 560853922 T^{4} + 79792266297612001 T^{8} \))(\( 1 - 549771682 T^{4} + 79792266297612001 T^{8} \))(\( 1 - 50878370 T^{4} + 88119764714863965 T^{8} + \)\(17\!\cdots\!80\)\( T^{12} + \)\(80\!\cdots\!70\)\( T^{16} + \)\(16\!\cdots\!76\)\( T^{20} + \)\(63\!\cdots\!70\)\( T^{24} + \)\(10\!\cdots\!80\)\( T^{28} + \)\(44\!\cdots\!65\)\( T^{32} - \)\(20\!\cdots\!70\)\( T^{36} + \)\(32\!\cdots\!01\)\( T^{40} \))
$11$ (\( 1 + 132 T + 161051 T^{2} \))(\( 1 - 480 T + 161051 T^{2} \))(\( 1 + 192 T + 161051 T^{2} \))(\( 1 - 144 T + 161051 T^{2} \))(\( 1 - 148 T + 161051 T^{2} \))(\( 1 - 604 T + 161051 T^{2} \))(\( 1 + 656 T + 161051 T^{2} \))(\( 1 - 768 T + 161051 T^{2} \))(\( 1 + 560 T + 381926 T^{2} + 90188560 T^{3} + 25937424601 T^{4} \))(\( ( 1 + 252 T + 161051 T^{2} )^{2} \))(\( ( 1 - 100 T + 161051 T^{2} )^{2} \))(\( ( 1 - 148 T + 161051 T^{2} )^{2} \))(\( ( 1 - 368 T + 176204 T^{2} - 51158896 T^{3} + 42277949270 T^{4} - 8239191359696 T^{5} + 4570277964394604 T^{6} - 1537227326344959568 T^{7} + \)\(67\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 - 161051 T^{2} )^{2} \))(\( ( 1 - 146602 T^{2} + 25937424601 T^{4} )^{2} \))(\( ( 1 - 306602 T^{2} + 25937424601 T^{4} )^{2} \))(\( ( 1 - 630250 T^{2} + 230184799605 T^{4} - 63113737617167000 T^{6} + \)\(13\!\cdots\!10\)\( T^{8} - \)\(24\!\cdots\!00\)\( T^{10} + \)\(35\!\cdots\!10\)\( T^{12} - \)\(42\!\cdots\!00\)\( T^{14} + \)\(40\!\cdots\!05\)\( T^{16} - \)\(28\!\cdots\!50\)\( T^{18} + \)\(11\!\cdots\!01\)\( T^{20} )^{2} \))
$13$ (\( 1 + 946 T + 371293 T^{2} \))(\( 1 + 622 T + 371293 T^{2} \))(\( 1 - 1106 T + 371293 T^{2} \))(\( 1 + 654 T + 371293 T^{2} \))(\( 1 - 286 T + 371293 T^{2} \))(\( 1 + 306 T + 371293 T^{2} \))(\( 1 + 206 T + 371293 T^{2} \))(\( 1 + 46 T + 371293 T^{2} \))(\( 1 - 1388 T + 1149918 T^{2} - 515354684 T^{3} + 137858491849 T^{4} \))(\( 1 - 728330 T^{2} + 137858491849 T^{4} \))(\( 1 - 202442 T^{2} + 137858491849 T^{4} \))(\( 1 - 274730 T^{2} + 137858491849 T^{4} \))(\( 1 - 1578472 T^{2} + 1074189263356 T^{4} - 437549632721743384 T^{6} + \)\(15\!\cdots\!86\)\( T^{8} - \)\(60\!\cdots\!16\)\( T^{10} + \)\(20\!\cdots\!56\)\( T^{12} - \)\(41\!\cdots\!28\)\( T^{14} + \)\(36\!\cdots\!01\)\( T^{16} \))(\( ( 1 - 1194 T + 371293 T^{2} )( 1 + 244 T + 371293 T^{2} ) \))(\( ( 1 + 1390 T + 966050 T^{2} + 516097270 T^{3} + 137858491849 T^{4} )^{2} \))(\( ( 1 - 330 T + 54450 T^{2} - 122526690 T^{3} + 137858491849 T^{4} )^{2} \))(\( ( 1 - 402 T + 80802 T^{2} - 412195466 T^{3} + 274076716325 T^{4} + 98802411059208 T^{5} + 23088035017184312 T^{6} - 33522742681396790296 T^{7} - \)\(55\!\cdots\!50\)\( T^{8} + \)\(34\!\cdots\!68\)\( T^{9} + \)\(30\!\cdots\!32\)\( T^{10} + \)\(12\!\cdots\!24\)\( T^{11} - \)\(75\!\cdots\!50\)\( T^{12} - \)\(17\!\cdots\!72\)\( T^{13} + \)\(43\!\cdots\!12\)\( T^{14} + \)\(69\!\cdots\!44\)\( T^{15} + \)\(71\!\cdots\!25\)\( T^{16} - \)\(40\!\cdots\!62\)\( T^{17} + \)\(29\!\cdots\!02\)\( T^{18} - \)\(53\!\cdots\!86\)\( T^{19} + \)\(49\!\cdots\!49\)\( T^{20} )^{2} \))
$17$ (\( 1 + 222 T + 1419857 T^{2} \))(\( 1 - 186 T + 1419857 T^{2} \))(\( 1 - 762 T + 1419857 T^{2} \))(\( 1 + 1190 T + 1419857 T^{2} \))(\( 1 + 1678 T + 1419857 T^{2} \))(\( 1 - 930 T + 1419857 T^{2} \))(\( 1 - 1690 T + 1419857 T^{2} \))(\( 1 - 378 T + 1419857 T^{2} \))(\( 1 - 148 T - 795706 T^{2} - 210138836 T^{3} + 2015993900449 T^{4} \))(\( 1 - 2363810 T^{2} + 2015993900449 T^{4} \))(\( 1 - 1879458 T^{2} + 2015993900449 T^{4} \))(\( 1 + 1354590 T^{2} + 2015993900449 T^{4} \))(\( 1 - 6260872 T^{2} + 17547242668444 T^{4} - 31297293718759478968 T^{6} + \)\(45\!\cdots\!30\)\( T^{8} - \)\(63\!\cdots\!32\)\( T^{10} + \)\(71\!\cdots\!44\)\( T^{12} - \)\(51\!\cdots\!28\)\( T^{14} + \)\(16\!\cdots\!01\)\( T^{16} \))(\( ( 1 + 808 T + 1419857 T^{2} )( 1 + 2242 T + 1419857 T^{2} ) \))(\( ( 1 + 190 T + 18050 T^{2} + 269772830 T^{3} + 2015993900449 T^{4} )^{2} \))(\( ( 1 - 2530 T + 3200450 T^{2} - 3592238210 T^{3} + 2015993900449 T^{4} )^{2} \))(\( ( 1 + 1118 T + 624962 T^{2} + 1188702526 T^{3} - 1739300659939 T^{4} - 3459207301051384 T^{5} - 2073890095879259656 T^{6} - \)\(50\!\cdots\!88\)\( T^{7} + \)\(54\!\cdots\!26\)\( T^{8} + \)\(82\!\cdots\!84\)\( T^{9} + \)\(52\!\cdots\!56\)\( T^{10} + \)\(11\!\cdots\!88\)\( T^{11} + \)\(10\!\cdots\!74\)\( T^{12} - \)\(14\!\cdots\!84\)\( T^{13} - \)\(84\!\cdots\!56\)\( T^{14} - \)\(19\!\cdots\!88\)\( T^{15} - \)\(14\!\cdots\!11\)\( T^{16} + \)\(13\!\cdots\!18\)\( T^{17} + \)\(10\!\cdots\!62\)\( T^{18} + \)\(26\!\cdots\!26\)\( T^{19} + \)\(33\!\cdots\!49\)\( T^{20} )^{2} \))
$19$ (\( 1 + 500 T + 2476099 T^{2} \))(\( 1 - 1204 T + 2476099 T^{2} \))(\( 1 - 2740 T + 2476099 T^{2} \))(\( 1 + 556 T + 2476099 T^{2} \))(\( 1 + 1060 T + 2476099 T^{2} \))(\( 1 - 1324 T + 2476099 T^{2} \))(\( 1 - 1364 T + 2476099 T^{2} \))(\( 1 + 1100 T + 2476099 T^{2} \))(\( 1 - 1000 T + 4533462 T^{2} - 2476099000 T^{3} + 6131066257801 T^{4} \))(\( ( 1 - 220 T + 2476099 T^{2} )^{2} \))(\( ( 1 + 2244 T + 2476099 T^{2} )^{2} \))(\( ( 1 - 2220 T + 2476099 T^{2} )^{2} \))(\( ( 1 + 688 T + 5308396 T^{2} + 6058136368 T^{3} + 15069081422710 T^{4} + 15000545402668432 T^{5} + 32546127598645797196 T^{6} + \)\(10\!\cdots\!12\)\( T^{7} + \)\(37\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 + 2476099 T^{2} )^{2} \))(\( ( 1 + 4250198 T^{2} + 6131066257801 T^{4} )^{2} \))(\( ( 1 - 2549802 T^{2} + 6131066257801 T^{4} )^{2} \))(\( ( 1 + 8249550 T^{2} + 36376777568405 T^{4} + \)\(13\!\cdots\!00\)\( T^{6} + \)\(42\!\cdots\!10\)\( T^{8} + \)\(11\!\cdots\!00\)\( T^{10} + \)\(26\!\cdots\!10\)\( T^{12} + \)\(49\!\cdots\!00\)\( T^{14} + \)\(83\!\cdots\!05\)\( T^{16} + \)\(11\!\cdots\!50\)\( T^{18} + \)\(86\!\cdots\!01\)\( T^{20} )^{2} \))
$23$ (\( 1 + 3564 T + 6436343 T^{2} \))(\( 1 - 3186 T + 6436343 T^{2} \))(\( 1 + 1566 T + 6436343 T^{2} \))(\( 1 + 2182 T + 6436343 T^{2} \))(\( 1 + 2976 T + 6436343 T^{2} \))(\( 1 - 852 T + 6436343 T^{2} \))(\( 1 + 2198 T + 6436343 T^{2} \))(\( 1 - 1986 T + 6436343 T^{2} \))(\( 1 - 2452 T + 6569198 T^{2} - 15781913036 T^{3} + 41426511213649 T^{4} \))(\( 1 - 6946370 T^{2} + 41426511213649 T^{4} \))(\( 1 - 1185810 T^{2} + 41426511213649 T^{4} \))(\( 1 - 11320170 T^{2} + 41426511213649 T^{4} \))(\( 1 - 34675896 T^{2} + 569897415616828 T^{4} - \)\(59\!\cdots\!20\)\( T^{6} + \)\(44\!\cdots\!82\)\( T^{8} - \)\(24\!\cdots\!80\)\( T^{10} + \)\(97\!\cdots\!28\)\( T^{12} - \)\(24\!\cdots\!04\)\( T^{14} + \)\(29\!\cdots\!01\)\( T^{16} \))(\( 1 + 41426511213649 T^{4} \))(\( 1 - 82817669402722 T^{4} + \)\(17\!\cdots\!01\)\( T^{8} \))(\( 1 + 68424579426718 T^{4} + \)\(17\!\cdots\!01\)\( T^{8} \))(\( 1 - 207859623483490 T^{4} + \)\(17\!\cdots\!45\)\( T^{8} - \)\(53\!\cdots\!80\)\( T^{12} - \)\(11\!\cdots\!90\)\( T^{16} + \)\(13\!\cdots\!52\)\( T^{20} - \)\(19\!\cdots\!90\)\( T^{24} - \)\(15\!\cdots\!80\)\( T^{28} + \)\(86\!\cdots\!45\)\( T^{32} - \)\(18\!\cdots\!90\)\( T^{36} + \)\(14\!\cdots\!01\)\( T^{40} \))
$29$ (\( 1 - 2190 T + 20511149 T^{2} \))(\( 1 - 5526 T + 20511149 T^{2} \))(\( 1 - 5910 T + 20511149 T^{2} \))(\( 1 + 1578 T + 20511149 T^{2} \))(\( 1 + 3410 T + 20511149 T^{2} \))(\( 1 - 5902 T + 20511149 T^{2} \))(\( 1 + 2218 T + 20511149 T^{2} \))(\( 1 + 5610 T + 20511149 T^{2} \))(\( 1 - 1340 T - 8758306 T^{2} - 27484939660 T^{3} + 420707233300201 T^{4} \))(\( ( 1 + 6930 T + 20511149 T^{2} )^{2} \))(\( ( 1 - 7854 T + 20511149 T^{2} )^{2} \))(\( ( 1 - 270 T + 20511149 T^{2} )^{2} \))(\( ( 1 - 2936 T + 58625996 T^{2} - 77951973928 T^{3} + 1466411094282230 T^{4} - 1598884552081323272 T^{5} + \)\(24\!\cdots\!96\)\( T^{6} - \)\(25\!\cdots\!64\)\( T^{7} + \)\(17\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 - 2950 T + 20511149 T^{2} )( 1 + 2950 T + 20511149 T^{2} ) \))(\( ( 1 - 32750922 T^{2} + 420707233300201 T^{4} )^{2} \))(\( ( 1 - 34283082 T^{2} + 420707233300201 T^{4} )^{2} \))(\( ( 1 - 99799410 T^{2} + 5177263888867445 T^{4} - \)\(18\!\cdots\!20\)\( T^{6} + \)\(49\!\cdots\!10\)\( T^{8} - \)\(10\!\cdots\!52\)\( T^{10} + \)\(20\!\cdots\!10\)\( T^{12} - \)\(32\!\cdots\!20\)\( T^{14} + \)\(38\!\cdots\!45\)\( T^{16} - \)\(31\!\cdots\!10\)\( T^{18} + \)\(13\!\cdots\!01\)\( T^{20} )^{2} \))
$31$ (\( 1 + 2312 T + 28629151 T^{2} \))(\( 1 + 9356 T + 28629151 T^{2} \))(\( 1 - 6868 T + 28629151 T^{2} \))(\( 1 + 9660 T + 28629151 T^{2} \))(\( 1 - 2448 T + 28629151 T^{2} \))(\( 1 - 3320 T + 28629151 T^{2} \))(\( 1 - 1700 T + 28629151 T^{2} \))(\( 1 - 3988 T + 28629151 T^{2} \))(\( 1 - 2248 T + 57017022 T^{2} - 64358331448 T^{3} + 819628286980801 T^{4} \))(\( ( 1 + 6752 T + 28629151 T^{2} )^{2} \))(\( ( 1 - 2144 T + 28629151 T^{2} )^{2} \))(\( ( 1 - 2048 T + 28629151 T^{2} )^{2} \))(\( ( 1 + 2112 T + 80187004 T^{2} + 163080265536 T^{3} + 3196344720873606 T^{4} + 4668849547150239936 T^{5} + \)\(65\!\cdots\!04\)\( T^{6} + \)\(49\!\cdots\!12\)\( T^{7} + \)\(67\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 - 28629151 T^{2} )^{2} \))(\( ( 1 + 35581198 T^{2} + 819628286980801 T^{4} )^{2} \))(\( ( 1 - 6898802 T^{2} + 819628286980801 T^{4} )^{2} \))(\( ( 1 - 101180050 T^{2} + 5842226806223805 T^{4} - \)\(25\!\cdots\!00\)\( T^{6} + \)\(95\!\cdots\!10\)\( T^{8} - \)\(29\!\cdots\!00\)\( T^{10} + \)\(77\!\cdots\!10\)\( T^{12} - \)\(17\!\cdots\!00\)\( T^{14} + \)\(32\!\cdots\!05\)\( T^{16} - \)\(45\!\cdots\!50\)\( T^{18} + \)\(36\!\cdots\!01\)\( T^{20} )^{2} \))
$37$ (\( 1 + 11242 T + 69343957 T^{2} \))(\( 1 - 5618 T + 69343957 T^{2} \))(\( 1 + 5518 T + 69343957 T^{2} \))(\( 1 + 3534 T + 69343957 T^{2} \))(\( 1 - 182 T + 69343957 T^{2} \))(\( 1 - 10774 T + 69343957 T^{2} \))(\( 1 + 846 T + 69343957 T^{2} \))(\( 1 + 142 T + 69343957 T^{2} \))(\( 1 + 5940 T + 123434318 T^{2} + 411903104580 T^{3} + 4808584372417849 T^{4} \))(\( 1 + 56462470 T^{2} + 4808584372417849 T^{4} \))(\( 1 - 30515770 T^{2} + 4808584372417849 T^{4} \))(\( 1 - 119573530 T^{2} + 4808584372417849 T^{4} \))(\( 1 - 251774632 T^{2} + 38631208311838780 T^{4} - \)\(41\!\cdots\!52\)\( T^{6} + \)\(32\!\cdots\!34\)\( T^{8} - \)\(19\!\cdots\!48\)\( T^{10} + \)\(89\!\cdots\!80\)\( T^{12} - \)\(27\!\cdots\!68\)\( T^{14} + \)\(53\!\cdots\!01\)\( T^{16} \))(\( ( 1 - 11292 T + 69343957 T^{2} )( 1 + 12242 T + 69343957 T^{2} ) \))(\( ( 1 + 10310 T + 53148050 T^{2} + 714936196670 T^{3} + 4808584372417849 T^{4} )^{2} \))(\( ( 1 - 2770 T + 3836450 T^{2} - 192082760890 T^{3} + 4808584372417849 T^{4} )^{2} \))(\( ( 1 - 22130 T + 244868450 T^{2} - 1308308732410 T^{3} + 13809729158765845 T^{4} - \)\(24\!\cdots\!80\)\( T^{5} + \)\(28\!\cdots\!00\)\( T^{6} - \)\(15\!\cdots\!60\)\( T^{7} + \)\(10\!\cdots\!10\)\( T^{8} - \)\(15\!\cdots\!80\)\( T^{9} + \)\(18\!\cdots\!00\)\( T^{10} - \)\(10\!\cdots\!60\)\( T^{11} + \)\(50\!\cdots\!90\)\( T^{12} - \)\(51\!\cdots\!80\)\( T^{13} + \)\(64\!\cdots\!00\)\( T^{14} - \)\(38\!\cdots\!60\)\( T^{15} + \)\(15\!\cdots\!05\)\( T^{16} - \)\(10\!\cdots\!30\)\( T^{17} + \)\(13\!\cdots\!50\)\( T^{18} - \)\(82\!\cdots\!10\)\( T^{19} + \)\(25\!\cdots\!49\)\( T^{20} )^{2} \))
$41$ (\( 1 - 1242 T + 115856201 T^{2} \))(\( 1 + 14394 T + 115856201 T^{2} \))(\( 1 + 378 T + 115856201 T^{2} \))(\( 1 - 7462 T + 115856201 T^{2} \))(\( 1 + 9398 T + 115856201 T^{2} \))(\( 1 + 17958 T + 115856201 T^{2} \))(\( 1 + 1818 T + 115856201 T^{2} \))(\( 1 - 1542 T + 115856201 T^{2} \))(\( 1 - 23076 T + 352280470 T^{2} - 2673497694276 T^{3} + 13422659310152401 T^{4} \))(\( ( 1 + 198 T + 115856201 T^{2} )^{2} \))(\( ( 1 + 7414 T + 115856201 T^{2} )^{2} \))(\( ( 1 + 2398 T + 115856201 T^{2} )^{2} \))(\( ( 1 - 11800 T + 337909340 T^{2} - 2943020124776 T^{3} + 51155654972384870 T^{4} - \)\(34\!\cdots\!76\)\( T^{5} + \)\(45\!\cdots\!40\)\( T^{6} - \)\(18\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 + 4952 T + 115856201 T^{2} )^{2} \))(\( ( 1 - 9218 T + 115856201 T^{2} )^{4} \))(\( ( 1 - 2178 T + 115856201 T^{2} )^{4} \))(\( ( 1 + 1690 T + 164119545 T^{2} - 1028151329120 T^{3} + 25291870422416110 T^{4} - 93379183076463398452 T^{5} + \)\(29\!\cdots\!10\)\( T^{6} - \)\(13\!\cdots\!20\)\( T^{7} + \)\(25\!\cdots\!45\)\( T^{8} + \)\(30\!\cdots\!90\)\( T^{9} + \)\(20\!\cdots\!01\)\( T^{10} )^{4} \))
$43$ (\( 1 + 20624 T + 147008443 T^{2} \))(\( 1 - 370 T + 147008443 T^{2} \))(\( 1 - 2434 T + 147008443 T^{2} \))(\( 1 - 7114 T + 147008443 T^{2} \))(\( 1 - 1244 T + 147008443 T^{2} \))(\( 1 + 9264 T + 147008443 T^{2} \))(\( 1 + 10534 T + 147008443 T^{2} \))(\( 1 - 5026 T + 147008443 T^{2} \))(\( 1 + 17684 T + 312898614 T^{2} + 2599697306012 T^{3} + 21611482313284249 T^{4} \))(\( 1 - 293842250 T^{2} + 21611482313284249 T^{4} \))(\( 1 + 21442214 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 288754450 T^{2} + 21611482313284249 T^{4} \))(\( 1 - 283211672 T^{2} + 48021567531024796 T^{4} - \)\(54\!\cdots\!84\)\( T^{6} + \)\(43\!\cdots\!06\)\( T^{8} - \)\(11\!\cdots\!16\)\( T^{10} + \)\(22\!\cdots\!96\)\( T^{12} - \)\(28\!\cdots\!28\)\( T^{14} + \)\(21\!\cdots\!01\)\( T^{16} \))(\( 1 + 21611482313284249 T^{4} \))(\( 1 + 40420440476627918 T^{4} + \)\(46\!\cdots\!01\)\( T^{8} \))(\( 1 - 35368995141923122 T^{4} + \)\(46\!\cdots\!01\)\( T^{8} \))(\( 1 + 22292551011039310 T^{4} + \)\(40\!\cdots\!45\)\( T^{8} + \)\(23\!\cdots\!20\)\( T^{12} + \)\(67\!\cdots\!10\)\( T^{16} + \)\(21\!\cdots\!52\)\( T^{20} + \)\(31\!\cdots\!10\)\( T^{24} + \)\(51\!\cdots\!20\)\( T^{28} + \)\(41\!\cdots\!45\)\( T^{32} + \)\(10\!\cdots\!10\)\( T^{36} + \)\(22\!\cdots\!01\)\( T^{40} \))
$47$ (\( 1 + 6588 T + 229345007 T^{2} \))(\( 1 + 16146 T + 229345007 T^{2} \))(\( 1 + 13122 T + 229345007 T^{2} \))(\( 1 - 28294 T + 229345007 T^{2} \))(\( 1 - 12088 T + 229345007 T^{2} \))(\( 1 - 9796 T + 229345007 T^{2} \))(\( 1 + 12074 T + 229345007 T^{2} \))(\( 1 + 24738 T + 229345007 T^{2} \))(\( 1 - 2908 T + 56660030 T^{2} - 666935280356 T^{3} + 52599132235830049 T^{4} \))(\( 1 - 347593490 T^{2} + 52599132235830049 T^{4} \))(\( 1 - 369731298 T^{2} + 52599132235830049 T^{4} \))(\( 1 - 344584890 T^{2} + 52599132235830049 T^{4} \))(\( 1 - 963352312 T^{2} + 473832864723586300 T^{4} - \)\(15\!\cdots\!72\)\( T^{6} + \)\(40\!\cdots\!54\)\( T^{8} - \)\(83\!\cdots\!28\)\( T^{10} + \)\(13\!\cdots\!00\)\( T^{12} - \)\(14\!\cdots\!88\)\( T^{14} + \)\(76\!\cdots\!01\)\( T^{16} \))(\( 1 + 52599132235830049 T^{4} \))(\( 1 - 85407260265966082 T^{4} + \)\(27\!\cdots\!01\)\( T^{8} \))(\( 1 + 62763367693678078 T^{4} + \)\(27\!\cdots\!01\)\( T^{8} \))(\( 1 + 74279974827999230 T^{4} + \)\(48\!\cdots\!65\)\( T^{8} + \)\(19\!\cdots\!80\)\( T^{12} + \)\(14\!\cdots\!70\)\( T^{16} - \)\(25\!\cdots\!24\)\( T^{20} + \)\(39\!\cdots\!70\)\( T^{24} + \)\(15\!\cdots\!80\)\( T^{28} + \)\(10\!\cdots\!65\)\( T^{32} + \)\(43\!\cdots\!30\)\( T^{36} + \)\(16\!\cdots\!01\)\( T^{40} \))
$53$ (\( 1 + 21066 T + 418195493 T^{2} \))(\( 1 + 4374 T + 418195493 T^{2} \))(\( 1 + 9174 T + 418195493 T^{2} \))(\( 1 + 13046 T + 418195493 T^{2} \))(\( 1 - 23846 T + 418195493 T^{2} \))(\( 1 + 31434 T + 418195493 T^{2} \))(\( 1 - 32586 T + 418195493 T^{2} \))(\( 1 + 14166 T + 418195493 T^{2} \))(\( 1 + 5412 T + 693247822 T^{2} + 2263274008116 T^{3} + 174887470365513049 T^{4} \))(\( 1 - 802472090 T^{2} + 174887470365513049 T^{4} \))(\( 1 - 248174170 T^{2} + 174887470365513049 T^{4} \))(\( 1 - 827605690 T^{2} + 174887470365513049 T^{4} \))(\( 1 - 1183385640 T^{2} + 874785099161623996 T^{4} - \)\(49\!\cdots\!80\)\( T^{6} + \)\(23\!\cdots\!06\)\( T^{8} - \)\(86\!\cdots\!20\)\( T^{10} + \)\(26\!\cdots\!96\)\( T^{12} - \)\(63\!\cdots\!60\)\( T^{14} + \)\(93\!\cdots\!01\)\( T^{16} \))(\( ( 1 - 7294 T + 418195493 T^{2} )( 1 + 40244 T + 418195493 T^{2} ) \))(\( ( 1 - 22690 T + 257418050 T^{2} - 9488855736170 T^{3} + 174887470365513049 T^{4} )^{2} \))(\( ( 1 + 47830 T + 1143854450 T^{2} + 20002290430190 T^{3} + 174887470365513049 T^{4} )^{2} \))(\( ( 1 - 91226 T + 4161091538 T^{2} - 141622957033378 T^{3} + 3893612180777232821 T^{4} - \)\(81\!\cdots\!12\)\( T^{5} + \)\(12\!\cdots\!56\)\( T^{6} - \)\(10\!\cdots\!76\)\( T^{7} - \)\(19\!\cdots\!54\)\( T^{8} + \)\(10\!\cdots\!72\)\( T^{9} - \)\(26\!\cdots\!96\)\( T^{10} + \)\(45\!\cdots\!96\)\( T^{11} - \)\(34\!\cdots\!46\)\( T^{12} - \)\(80\!\cdots\!32\)\( T^{13} + \)\(39\!\cdots\!56\)\( T^{14} - \)\(10\!\cdots\!16\)\( T^{15} + \)\(20\!\cdots\!29\)\( T^{16} - \)\(31\!\cdots\!46\)\( T^{17} + \)\(38\!\cdots\!38\)\( T^{18} - \)\(35\!\cdots\!18\)\( T^{19} + \)\(16\!\cdots\!49\)\( T^{20} )^{2} \))
$59$ (\( 1 + 7980 T + 714924299 T^{2} \))(\( 1 - 11748 T + 714924299 T^{2} \))(\( 1 - 34980 T + 714924299 T^{2} \))(\( 1 - 37092 T + 714924299 T^{2} \))(\( 1 - 20020 T + 714924299 T^{2} \))(\( 1 + 33228 T + 714924299 T^{2} \))(\( 1 + 8668 T + 714924299 T^{2} \))(\( 1 + 28380 T + 714924299 T^{2} \))(\( 1 + 62584 T + 2277965606 T^{2} + 44742822328616 T^{3} + 511116753300641401 T^{4} \))(\( ( 1 - 24660 T + 714924299 T^{2} )^{2} \))(\( ( 1 - 25972 T + 714924299 T^{2} )^{2} \))(\( ( 1 + 39740 T + 714924299 T^{2} )^{2} \))(\( ( 1 + 45840 T + 3064286732 T^{2} + 94721285480976 T^{3} + 3348109683185502486 T^{4} + \)\(67\!\cdots\!24\)\( T^{5} + \)\(15\!\cdots\!32\)\( T^{6} + \)\(16\!\cdots\!60\)\( T^{7} + \)\(26\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 + 714924299 T^{2} )^{2} \))(\( ( 1 + 1429146598 T^{2} + 511116753300641401 T^{4} )^{2} \))(\( ( 1 + 693226598 T^{2} + 511116753300641401 T^{4} )^{2} \))(\( ( 1 + 4265422750 T^{2} + 8278744468566020805 T^{4} + \)\(98\!\cdots\!00\)\( T^{6} + \)\(85\!\cdots\!10\)\( T^{8} + \)\(63\!\cdots\!00\)\( T^{10} + \)\(43\!\cdots\!10\)\( T^{12} + \)\(25\!\cdots\!00\)\( T^{14} + \)\(11\!\cdots\!05\)\( T^{16} + \)\(29\!\cdots\!50\)\( T^{18} + \)\(34\!\cdots\!01\)\( T^{20} )^{2} \))
$61$ (\( 1 - 16622 T + 844596301 T^{2} \))(\( 1 - 13202 T + 844596301 T^{2} \))(\( 1 + 9838 T + 844596301 T^{2} \))(\( 1 - 39570 T + 844596301 T^{2} \))(\( 1 - 32302 T + 844596301 T^{2} \))(\( 1 + 40210 T + 844596301 T^{2} \))(\( 1 + 34670 T + 844596301 T^{2} \))(\( 1 - 5522 T + 844596301 T^{2} \))(\( 1 - 14108 T + 1110042462 T^{2} - 11915564614508 T^{3} + 713342911662882601 T^{4} \))(\( ( 1 + 5698 T + 844596301 T^{2} )^{2} \))(\( ( 1 + 3058 T + 844596301 T^{2} )^{2} \))(\( ( 1 + 42298 T + 844596301 T^{2} )^{2} \))(\( ( 1 - 61928 T + 3903014764 T^{2} - 145287706763384 T^{3} + 5198153942066716726 T^{4} - \)\(12\!\cdots\!84\)\( T^{5} + \)\(27\!\cdots\!64\)\( T^{6} - \)\(37\!\cdots\!28\)\( T^{7} + \)\(50\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 - 54948 T + 844596301 T^{2} )^{2} \))(\( ( 1 - 18678 T + 844596301 T^{2} )^{4} \))(\( ( 1 + 35882 T + 844596301 T^{2} )^{4} \))(\( ( 1 + 10270 T + 3306972765 T^{2} + 29698468018720 T^{3} + 4981733552475787870 T^{4} + \)\(35\!\cdots\!24\)\( T^{5} + \)\(42\!\cdots\!70\)\( T^{6} + \)\(21\!\cdots\!20\)\( T^{7} + \)\(19\!\cdots\!65\)\( T^{8} + \)\(52\!\cdots\!70\)\( T^{9} + \)\(42\!\cdots\!01\)\( T^{10} )^{4} \))
$67$ (\( 1 + 1808 T + 1350125107 T^{2} \))(\( 1 - 11542 T + 1350125107 T^{2} \))(\( 1 + 33722 T + 1350125107 T^{2} \))(\( 1 - 56734 T + 1350125107 T^{2} \))(\( 1 + 60972 T + 1350125107 T^{2} \))(\( 1 + 58864 T + 1350125107 T^{2} \))(\( 1 - 47566 T + 1350125107 T^{2} \))(\( 1 - 24742 T + 1350125107 T^{2} \))(\( 1 - 85412 T + 4371910566 T^{2} - 115316885639084 T^{3} + 1822837804551761449 T^{4} \))(\( 1 - 795787610 T^{2} + 1822837804551761449 T^{4} \))(\( 1 + 755362070 T^{2} + 1822837804551761449 T^{4} \))(\( 1 - 1669968610 T^{2} + 1822837804551761449 T^{4} \))(\( 1 - 9281919064 T^{2} + 39492482666681482588 T^{4} - \)\(10\!\cdots\!40\)\( T^{6} + \)\(16\!\cdots\!62\)\( T^{8} - \)\(18\!\cdots\!60\)\( T^{10} + \)\(13\!\cdots\!88\)\( T^{12} - \)\(56\!\cdots\!36\)\( T^{14} + \)\(11\!\cdots\!01\)\( T^{16} \))(\( 1 + 1822837804551761449 T^{4} \))(\( 1 + 746452483343412398 T^{4} + \)\(33\!\cdots\!01\)\( T^{8} \))(\( 1 - 2056557036860651602 T^{4} + \)\(33\!\cdots\!01\)\( T^{8} \))(\( 1 - 2038648142805486290 T^{4} - \)\(25\!\cdots\!55\)\( T^{8} + \)\(14\!\cdots\!20\)\( T^{12} + \)\(73\!\cdots\!10\)\( T^{16} - \)\(57\!\cdots\!48\)\( T^{20} + \)\(24\!\cdots\!10\)\( T^{24} + \)\(15\!\cdots\!20\)\( T^{28} - \)\(93\!\cdots\!55\)\( T^{32} - \)\(24\!\cdots\!90\)\( T^{36} + \)\(40\!\cdots\!01\)\( T^{40} \))
$71$ (\( 1 - 24528 T + 1804229351 T^{2} \))(\( 1 - 29532 T + 1804229351 T^{2} \))(\( 1 + 70212 T + 1804229351 T^{2} \))(\( 1 + 45588 T + 1804229351 T^{2} \))(\( 1 - 32648 T + 1804229351 T^{2} \))(\( 1 - 55312 T + 1804229351 T^{2} \))(\( 1 + 948 T + 1804229351 T^{2} \))(\( 1 + 42372 T + 1804229351 T^{2} \))(\( 1 + 47208 T + 4011779662 T^{2} + 85174059202008 T^{3} + 3255243551009881201 T^{4} \))(\( ( 1 + 53352 T + 1804229351 T^{2} )^{2} \))(\( ( 1 + 37608 T + 1804229351 T^{2} )^{2} \))(\( ( 1 - 4248 T + 1804229351 T^{2} )^{2} \))(\( ( 1 - 62816 T + 3398787356 T^{2} - 184024084124896 T^{3} + 8353296562609817510 T^{4} - \)\(33\!\cdots\!96\)\( T^{5} + \)\(11\!\cdots\!56\)\( T^{6} - \)\(36\!\cdots\!16\)\( T^{7} + \)\(10\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 - 1804229351 T^{2} )^{2} \))(\( ( 1 - 217623202 T^{2} + 3255243551009881201 T^{4} )^{2} \))(\( ( 1 + 894616798 T^{2} + 3255243551009881201 T^{4} )^{2} \))(\( ( 1 - 14421599170 T^{2} + 96751044929093627565 T^{4} - \)\(40\!\cdots\!20\)\( T^{6} + \)\(11\!\cdots\!70\)\( T^{8} - \)\(24\!\cdots\!24\)\( T^{10} + \)\(37\!\cdots\!70\)\( T^{12} - \)\(42\!\cdots\!20\)\( T^{14} + \)\(33\!\cdots\!65\)\( T^{16} - \)\(16\!\cdots\!70\)\( T^{18} + \)\(36\!\cdots\!01\)\( T^{20} )^{2} \))
$73$ (\( 1 - 20474 T + 2073071593 T^{2} \))(\( 1 - 33698 T + 2073071593 T^{2} \))(\( 1 - 21986 T + 2073071593 T^{2} \))(\( 1 - 11842 T + 2073071593 T^{2} \))(\( 1 + 38774 T + 2073071593 T^{2} \))(\( 1 - 27258 T + 2073071593 T^{2} \))(\( 1 + 63102 T + 2073071593 T^{2} \))(\( 1 + 52126 T + 2073071593 T^{2} \))(\( 1 + 67452 T + 4400780438 T^{2} + 139832825091036 T^{3} + 4297625829703557649 T^{4} \))(\( 1 + 883886830 T^{2} + 4297625829703557649 T^{4} \))(\( 1 - 3569749522 T^{2} + 4297625829703557649 T^{4} \))(\( 1 - 3239892370 T^{2} + 4297625829703557649 T^{4} \))(\( 1 - 9140679496 T^{2} + 46078306824990298588 T^{4} - \)\(15\!\cdots\!60\)\( T^{6} + \)\(37\!\cdots\!02\)\( T^{8} - \)\(66\!\cdots\!40\)\( T^{10} + \)\(85\!\cdots\!88\)\( T^{12} - \)\(72\!\cdots\!04\)\( T^{14} + \)\(34\!\cdots\!01\)\( T^{16} \))(\( ( 1 + 20144 T + 2073071593 T^{2} )( 1 + 88806 T + 2073071593 T^{2} ) \))(\( ( 1 + 48110 T + 1157286050 T^{2} + 99735474339230 T^{3} + 4297625829703557649 T^{4} )^{2} \))(\( ( 1 + 43230 T + 934416450 T^{2} + 89618884965390 T^{3} + 4297625829703557649 T^{4} )^{2} \))(\( ( 1 - 132186 T + 8736569298 T^{2} - 431249337673738 T^{3} + 10750357541220477581 T^{4} + \)\(12\!\cdots\!88\)\( T^{5} - \)\(17\!\cdots\!84\)\( T^{6} + \)\(58\!\cdots\!44\)\( T^{7} + \)\(47\!\cdots\!06\)\( T^{8} - \)\(60\!\cdots\!08\)\( T^{9} + \)\(32\!\cdots\!04\)\( T^{10} - \)\(12\!\cdots\!44\)\( T^{11} + \)\(20\!\cdots\!94\)\( T^{12} + \)\(52\!\cdots\!08\)\( T^{13} - \)\(32\!\cdots\!84\)\( T^{14} + \)\(47\!\cdots\!84\)\( T^{15} + \)\(85\!\cdots\!69\)\( T^{16} - \)\(70\!\cdots\!66\)\( T^{17} + \)\(29\!\cdots\!98\)\( T^{18} - \)\(93\!\cdots\!98\)\( T^{19} + \)\(14\!\cdots\!49\)\( T^{20} )^{2} \))
$79$ (\( 1 - 46240 T + 3077056399 T^{2} \))(\( 1 + 31208 T + 3077056399 T^{2} \))(\( 1 + 4520 T + 3077056399 T^{2} \))(\( 1 + 94216 T + 3077056399 T^{2} \))(\( 1 - 33360 T + 3077056399 T^{2} \))(\( 1 + 31456 T + 3077056399 T^{2} \))(\( 1 + 46536 T + 3077056399 T^{2} \))(\( 1 - 39640 T + 3077056399 T^{2} \))(\( 1 - 65904 T + 3994274078 T^{2} - 202790324919696 T^{3} + 9468276082626847201 T^{4} \))(\( ( 1 + 51920 T + 3077056399 T^{2} )^{2} \))(\( ( 1 - 79728 T + 3077056399 T^{2} )^{2} \))(\( ( 1 - 35280 T + 3077056399 T^{2} )^{2} \))(\( ( 1 + 21632 T + 7152876604 T^{2} + 332616618908288 T^{3} + 24121899620797566790 T^{4} + \)\(10\!\cdots\!12\)\( T^{5} + \)\(67\!\cdots\!04\)\( T^{6} + \)\(63\!\cdots\!68\)\( T^{7} + \)\(89\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 + 3077056399 T^{2} )^{2} \))(\( ( 1 - 2651775202 T^{2} + 9468276082626847201 T^{4} )^{2} \))(\( ( 1 + 5673984798 T^{2} + 9468276082626847201 T^{4} )^{2} \))(\( ( 1 + 18614864790 T^{2} + \)\(16\!\cdots\!45\)\( T^{4} + \)\(10\!\cdots\!80\)\( T^{6} + \)\(44\!\cdots\!10\)\( T^{8} + \)\(15\!\cdots\!48\)\( T^{10} + \)\(42\!\cdots\!10\)\( T^{12} + \)\(90\!\cdots\!80\)\( T^{14} + \)\(14\!\cdots\!45\)\( T^{16} + \)\(14\!\cdots\!90\)\( T^{18} + \)\(76\!\cdots\!01\)\( T^{20} )^{2} \))
$83$ (\( 1 - 51576 T + 3939040643 T^{2} \))(\( 1 - 38466 T + 3939040643 T^{2} \))(\( 1 - 109074 T + 3939040643 T^{2} \))(\( 1 - 31482 T + 3939040643 T^{2} \))(\( 1 + 16716 T + 3939040643 T^{2} \))(\( 1 + 24552 T + 3939040643 T^{2} \))(\( 1 - 88778 T + 3939040643 T^{2} \))(\( 1 - 59826 T + 3939040643 T^{2} \))(\( 1 + 108724 T + 10572459494 T^{2} + 428268254869532 T^{3} + 15516041187205853449 T^{4} \))(\( 1 - 4053674810 T^{2} + 15516041187205853449 T^{4} \))(\( 1 - 7612675530 T^{2} + 15516041187205853449 T^{4} \))(\( 1 - 7103795010 T^{2} + 15516041187205853449 T^{4} \))(\( 1 - 6759897816 T^{2} + 40943120759345365468 T^{4} - \)\(86\!\cdots\!80\)\( T^{6} + \)\(39\!\cdots\!42\)\( T^{8} - \)\(13\!\cdots\!20\)\( T^{10} + \)\(98\!\cdots\!68\)\( T^{12} - \)\(25\!\cdots\!84\)\( T^{14} + \)\(57\!\cdots\!01\)\( T^{16} \))(\( 1 + 15516041187205853449 T^{4} \))(\( 1 - 2557737354504584722 T^{4} + \)\(24\!\cdots\!01\)\( T^{8} \))(\( 1 - 7276763020942840082 T^{4} + \)\(24\!\cdots\!01\)\( T^{8} \))(\( 1 - 18609684090616711570 T^{4} - \)\(46\!\cdots\!35\)\( T^{8} + \)\(34\!\cdots\!80\)\( T^{12} + \)\(25\!\cdots\!70\)\( T^{16} - \)\(17\!\cdots\!24\)\( T^{20} + \)\(60\!\cdots\!70\)\( T^{24} + \)\(20\!\cdots\!80\)\( T^{28} - \)\(64\!\cdots\!35\)\( T^{32} - \)\(62\!\cdots\!70\)\( T^{36} + \)\(80\!\cdots\!01\)\( T^{40} \))
$89$ (\( 1 + 110310 T + 5584059449 T^{2} \))(\( 1 - 119514 T + 5584059449 T^{2} \))(\( 1 - 38490 T + 5584059449 T^{2} \))(\( 1 + 94054 T + 5584059449 T^{2} \))(\( 1 - 101370 T + 5584059449 T^{2} \))(\( 1 + 90854 T + 5584059449 T^{2} \))(\( 1 + 104934 T + 5584059449 T^{2} \))(\( 1 - 57690 T + 5584059449 T^{2} \))(\( 1 + 55020 T + 10818978262 T^{2} + 307234950883980 T^{3} + 31181719929966183601 T^{4} \))(\( ( 1 + 9990 T + 5584059449 T^{2} )^{2} \))(\( ( 1 - 826 T + 5584059449 T^{2} )^{2} \))(\( ( 1 - 85210 T + 5584059449 T^{2} )^{2} \))(\( ( 1 + 20952 T + 16118164796 T^{2} + 497915996461992 T^{3} + \)\(11\!\cdots\!30\)\( T^{4} + \)\(27\!\cdots\!08\)\( T^{5} + \)\(50\!\cdots\!96\)\( T^{6} + \)\(36\!\cdots\!48\)\( T^{7} + \)\(97\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 - 51050 T + 5584059449 T^{2} )( 1 + 51050 T + 5584059449 T^{2} ) \))(\( ( 1 - 5648223282 T^{2} + 31181719929966183601 T^{4} )^{2} \))(\( ( 1 + 1924732878 T^{2} + 31181719929966183601 T^{4} )^{2} \))(\( ( 1 - 33120686970 T^{2} + \)\(41\!\cdots\!65\)\( T^{4} - \)\(20\!\cdots\!20\)\( T^{6} - \)\(32\!\cdots\!30\)\( T^{8} + \)\(73\!\cdots\!76\)\( T^{10} - \)\(10\!\cdots\!30\)\( T^{12} - \)\(20\!\cdots\!20\)\( T^{14} + \)\(12\!\cdots\!65\)\( T^{16} - \)\(31\!\cdots\!70\)\( T^{18} + \)\(29\!\cdots\!01\)\( T^{20} )^{2} \))
$97$ (\( 1 + 78382 T + 8587340257 T^{2} \))(\( 1 - 94658 T + 8587340257 T^{2} \))(\( 1 + 1918 T + 8587340257 T^{2} \))(\( 1 - 23714 T + 8587340257 T^{2} \))(\( 1 + 119038 T + 8587340257 T^{2} \))(\( 1 - 154706 T + 8587340257 T^{2} \))(\( 1 + 36254 T + 8587340257 T^{2} \))(\( 1 + 144382 T + 8587340257 T^{2} \))(\( 1 - 147668 T + 11612429670 T^{2} - 1268075361070676 T^{3} + 73742412689492826049 T^{4} \))(\( 1 - 6923133890 T^{2} + 73742412689492826049 T^{4} \))(\( 1 - 15761405890 T^{2} + 73742412689492826049 T^{4} \))(\( 1 - 7720618690 T^{2} + 73742412689492826049 T^{4} \))(\( 1 - 45263915272 T^{2} + \)\(10\!\cdots\!40\)\( T^{4} - \)\(14\!\cdots\!12\)\( T^{6} + \)\(15\!\cdots\!94\)\( T^{8} - \)\(11\!\cdots\!88\)\( T^{10} + \)\(55\!\cdots\!40\)\( T^{12} - \)\(18\!\cdots\!28\)\( T^{14} + \)\(29\!\cdots\!01\)\( T^{16} \))(\( ( 1 + 92142 T + 8587340257 T^{2} )( 1 + 160808 T + 8587340257 T^{2} ) \))(\( ( 1 + 98510 T + 4852110050 T^{2} + 845938888717070 T^{3} + 73742412689492826049 T^{4} )^{2} \))(\( ( 1 + 1230 T + 756450 T^{2} + 10562428516110 T^{3} + 73742412689492826049 T^{4} )^{2} \))(\( ( 1 - 187386 T + 17556756498 T^{2} - 497237634071002 T^{3} - \)\(23\!\cdots\!35\)\( T^{4} + \)\(42\!\cdots\!24\)\( T^{5} - \)\(37\!\cdots\!32\)\( T^{6} + \)\(18\!\cdots\!68\)\( T^{7} + \)\(11\!\cdots\!50\)\( T^{8} - \)\(33\!\cdots\!76\)\( T^{9} + \)\(36\!\cdots\!68\)\( T^{10} - \)\(29\!\cdots\!32\)\( T^{11} + \)\(87\!\cdots\!50\)\( T^{12} + \)\(11\!\cdots\!24\)\( T^{13} - \)\(20\!\cdots\!32\)\( T^{14} + \)\(19\!\cdots\!68\)\( T^{15} - \)\(93\!\cdots\!15\)\( T^{16} - \)\(17\!\cdots\!86\)\( T^{17} + \)\(51\!\cdots\!98\)\( T^{18} - \)\(47\!\cdots\!02\)\( T^{19} + \)\(21\!\cdots\!49\)\( T^{20} )^{2} \))
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