Properties

Label 40.6
Level 40
Weight 6
Dimension 117
Nonzero newspaces 5
Newform subspaces 8
Sturm bound 576
Trace bound 1

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Defining parameters

Level: \( N \) = \( 40\( 40 = 2^{3} \cdot 5 \) \)
Weight: \( k \) = \( 6 \)
Nonzero newspaces: \( 5 \)
Newform subspaces: \( 8 \)
Sturm bound: \(576\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_1(40))\).

Total New Old
Modular forms 264 129 135
Cusp forms 216 117 99
Eisenstein series 48 12 36

Trace form

\( 117q - 44q^{3} - 44q^{4} + 33q^{5} + 224q^{6} - 72q^{7} + 492q^{8} - 399q^{9} + O(q^{10}) \) \( 117q - 44q^{3} - 44q^{4} + 33q^{5} + 224q^{6} - 72q^{7} + 492q^{8} - 399q^{9} - 636q^{10} - 276q^{11} - 2184q^{12} + 222q^{13} + 2016q^{14} - 580q^{15} + 5096q^{16} + 1170q^{17} - 1924q^{18} - 1756q^{19} + 2192q^{20} + 1400q^{21} + 10688q^{22} - 3600q^{23} - 1192q^{24} - 13075q^{25} - 23144q^{26} - 4952q^{27} - 7912q^{28} + 9318q^{29} + 11888q^{30} + 18144q^{31} + 14760q^{32} - 11544q^{33} + 34148q^{34} + 8552q^{35} + 7360q^{36} + 454q^{37} - 69104q^{38} - 37800q^{39} - 75276q^{40} + 43482q^{41} + 24944q^{42} + 31676q^{43} + 83008q^{44} - 13403q^{45} + 72272q^{46} + 15256q^{47} + 4920q^{48} - 63987q^{49} - 49636q^{50} - 53608q^{51} - 106860q^{52} - 17306q^{53} - 104152q^{54} - 71868q^{55} + 13936q^{56} + 8880q^{57} + 145596q^{58} + 159068q^{59} + 198568q^{60} + 102654q^{61} + 66064q^{62} + 273288q^{63} - 58208q^{64} + 31366q^{65} - 233936q^{66} - 41596q^{67} - 146588q^{68} - 226072q^{69} + 33928q^{70} - 356744q^{71} + 237092q^{72} - 221774q^{73} + 113052q^{74} + 70044q^{75} - 126224q^{76} + 228896q^{77} - 65576q^{78} + 395664q^{79} - 198424q^{80} + 601413q^{81} - 199620q^{82} - 113428q^{83} - 462816q^{84} - 279310q^{85} - 238928q^{86} - 666752q^{87} + 187072q^{88} - 568670q^{89} + 20916q^{90} - 135504q^{91} + 333936q^{92} - 4400q^{93} + 463560q^{94} + 638508q^{95} + 517168q^{96} + 649786q^{97} + 282776q^{98} + 709860q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_1(40))\)

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
40.6.a \(\chi_{40}(1, \cdot)\) 40.6.a.a 1 1
40.6.a.b 1
40.6.a.c 1
40.6.a.d 2
40.6.c \(\chi_{40}(9, \cdot)\) 40.6.c.a 8 1
40.6.d \(\chi_{40}(21, \cdot)\) 40.6.d.a 20 1
40.6.f \(\chi_{40}(29, \cdot)\) 40.6.f.a 28 1
40.6.j \(\chi_{40}(7, \cdot)\) None 0 2
40.6.k \(\chi_{40}(3, \cdot)\) 40.6.k.a 56 2

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_1(40))\) into lower level spaces

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 2 T + 18 T^{2} - 116 T^{3} - 444 T^{4} - 1584 T^{5} - 2144 T^{6} + 186496 T^{7} - 497408 T^{8} + 567296 T^{9} - 25550848 T^{10} + 18153472 T^{11} - 509345792 T^{12} + 6111100928 T^{13} - 2248146944 T^{14} - 53150220288 T^{15} - 476741369856 T^{16} - 3985729650688 T^{17} + 19791209299968 T^{18} - 70368744177664 T^{19} + 1125899906842624 T^{20} \))
$3$ (\( 1 + 18 T + 243 T^{2} \))(\( 1 + 8 T + 243 T^{2} \))(\( 1 + 2 T + 243 T^{2} \))(\( 1 + 12 T + 6 T^{2} + 2916 T^{3} + 59049 T^{4} \))(\( 1 - 472 T^{2} + 69724 T^{4} - 20119464 T^{6} + 8439593958 T^{8} - 1188034229736 T^{10} + 243112555575324 T^{12} - 97180614348674328 T^{14} + 12157665459056928801 T^{16} \))(\( 1 - 1620 T^{2} + 1394322 T^{4} - 834927892 T^{6} + 392823148221 T^{8} - 155295909553872 T^{10} + 53814329151823576 T^{12} - 16772396183204761872 T^{14} + \)\(47\!\cdots\!30\)\( T^{16} - \)\(12\!\cdots\!00\)\( T^{18} + \)\(31\!\cdots\!00\)\( T^{20} - \)\(75\!\cdots\!00\)\( T^{22} + \)\(16\!\cdots\!30\)\( T^{24} - \)\(34\!\cdots\!28\)\( T^{26} + \)\(65\!\cdots\!76\)\( T^{28} - \)\(11\!\cdots\!28\)\( T^{30} + \)\(16\!\cdots\!21\)\( T^{32} - \)\(20\!\cdots\!08\)\( T^{34} + \)\(20\!\cdots\!22\)\( T^{36} - \)\(14\!\cdots\!80\)\( T^{38} + \)\(51\!\cdots\!01\)\( T^{40} \))
$5$ (\( 1 + 25 T \))(\( 1 - 25 T \))(\( 1 + 25 T \))(\( ( 1 - 25 T )^{2} \))(\( 1 - 8 T + 1100 T^{2} + 113000 T^{3} - 438250 T^{4} + 353125000 T^{5} + 10742187500 T^{6} - 244140625000 T^{7} + 95367431640625 T^{8} \))(\( ( 1 + 625 T^{2} )^{10} \))
$7$ (\( 1 - 242 T + 16807 T^{2} \))(\( 1 + 108 T + 16807 T^{2} \))(\( 1 + 62 T + 16807 T^{2} \))(\( 1 - 52 T + 29646 T^{2} - 873964 T^{3} + 282475249 T^{4} \))(\( 1 - 44728 T^{2} + 1493128636 T^{4} - 37445733732616 T^{6} + 682894235558230726 T^{8} - \)\(10\!\cdots\!84\)\( T^{10} + \)\(11\!\cdots\!36\)\( T^{12} - \)\(10\!\cdots\!72\)\( T^{14} + \)\(63\!\cdots\!01\)\( T^{16} \))(\( ( 1 + 98 T + 84148 T^{2} + 8017214 T^{3} + 3556570901 T^{4} + 345044190776 T^{5} + 103920201897616 T^{6} + 10159390994080936 T^{7} + 2363994840482709802 T^{8} + \)\(22\!\cdots\!76\)\( T^{9} + \)\(43\!\cdots\!72\)\( T^{10} + \)\(37\!\cdots\!32\)\( T^{11} + \)\(66\!\cdots\!98\)\( T^{12} + \)\(48\!\cdots\!48\)\( T^{13} + \)\(82\!\cdots\!16\)\( T^{14} + \)\(46\!\cdots\!32\)\( T^{15} + \)\(80\!\cdots\!49\)\( T^{16} + \)\(30\!\cdots\!02\)\( T^{17} + \)\(53\!\cdots\!48\)\( T^{18} + \)\(10\!\cdots\!86\)\( T^{19} + \)\(17\!\cdots\!49\)\( T^{20} )^{2} \))
$11$ (\( 1 - 656 T + 161051 T^{2} \))(\( 1 + 604 T + 161051 T^{2} \))(\( 1 + 144 T + 161051 T^{2} \))(\( 1 - 560 T + 381926 T^{2} - 90188560 T^{3} + 25937424601 T^{4} \))(\( ( 1 + 368 T + 176204 T^{2} + 51158896 T^{3} + 42277949270 T^{4} + 8239191359696 T^{5} + 4570277964394604 T^{6} + 1537227326344959568 T^{7} + \)\(67\!\cdots\!01\)\( T^{8} )^{2} \))(\( 1 - 1510004 T^{2} + 1155249727726 T^{4} - 592766540112799924 T^{6} + \)\(22\!\cdots\!17\)\( T^{8} - \)\(70\!\cdots\!04\)\( T^{10} + \)\(18\!\cdots\!36\)\( T^{12} - \)\(40\!\cdots\!64\)\( T^{14} + \)\(79\!\cdots\!82\)\( T^{16} - \)\(14\!\cdots\!04\)\( T^{18} + \)\(23\!\cdots\!76\)\( T^{20} - \)\(36\!\cdots\!04\)\( T^{22} + \)\(53\!\cdots\!82\)\( T^{24} - \)\(70\!\cdots\!64\)\( T^{26} + \)\(82\!\cdots\!36\)\( T^{28} - \)\(82\!\cdots\!04\)\( T^{30} + \)\(69\!\cdots\!17\)\( T^{32} - \)\(46\!\cdots\!24\)\( T^{34} + \)\(23\!\cdots\!26\)\( T^{36} - \)\(80\!\cdots\!04\)\( T^{38} + \)\(13\!\cdots\!01\)\( T^{40} \))
$13$ (\( 1 + 206 T + 371293 T^{2} \))(\( 1 + 306 T + 371293 T^{2} \))(\( 1 + 654 T + 371293 T^{2} \))(\( 1 - 1388 T + 1149918 T^{2} - 515354684 T^{3} + 137858491849 T^{4} \))(\( 1 - 1578472 T^{2} + 1074189263356 T^{4} - 437549632721743384 T^{6} + \)\(15\!\cdots\!86\)\( T^{8} - \)\(60\!\cdots\!16\)\( T^{10} + \)\(20\!\cdots\!56\)\( T^{12} - \)\(41\!\cdots\!28\)\( T^{14} + \)\(36\!\cdots\!01\)\( T^{16} \))(\( 1 - 3520332 T^{2} + 6227853839054 T^{4} - 7441447313525468748 T^{6} + \)\(67\!\cdots\!57\)\( T^{8} - \)\(50\!\cdots\!12\)\( T^{10} + \)\(32\!\cdots\!64\)\( T^{12} - \)\(17\!\cdots\!28\)\( T^{14} + \)\(87\!\cdots\!42\)\( T^{16} - \)\(38\!\cdots\!32\)\( T^{18} + \)\(14\!\cdots\!64\)\( T^{20} - \)\(52\!\cdots\!68\)\( T^{22} + \)\(16\!\cdots\!42\)\( T^{24} - \)\(46\!\cdots\!72\)\( T^{26} + \)\(11\!\cdots\!64\)\( T^{28} - \)\(25\!\cdots\!88\)\( T^{30} + \)\(46\!\cdots\!57\)\( T^{32} - \)\(70\!\cdots\!52\)\( T^{34} + \)\(81\!\cdots\!54\)\( T^{36} - \)\(63\!\cdots\!68\)\( T^{38} + \)\(24\!\cdots\!01\)\( T^{40} \))
$17$ (\( 1 - 1690 T + 1419857 T^{2} \))(\( 1 - 930 T + 1419857 T^{2} \))(\( 1 + 1190 T + 1419857 T^{2} \))(\( 1 - 148 T - 795706 T^{2} - 210138836 T^{3} + 2015993900449 T^{4} \))(\( 1 - 6260872 T^{2} + 17547242668444 T^{4} - 31297293718759478968 T^{6} + \)\(45\!\cdots\!30\)\( T^{8} - \)\(63\!\cdots\!32\)\( T^{10} + \)\(71\!\cdots\!44\)\( T^{12} - \)\(51\!\cdots\!28\)\( T^{14} + \)\(16\!\cdots\!01\)\( T^{16} \))(\( ( 1 + 5447662 T^{2} + 1859072000 T^{3} + 15317572824845 T^{4} + 7413542230528000 T^{5} + 32518332783929091752 T^{6} + \)\(14\!\cdots\!00\)\( T^{7} + \)\(56\!\cdots\!10\)\( T^{8} + \)\(21\!\cdots\!00\)\( T^{9} + \)\(84\!\cdots\!72\)\( T^{10} + \)\(31\!\cdots\!00\)\( T^{11} + \)\(11\!\cdots\!90\)\( T^{12} + \)\(41\!\cdots\!00\)\( T^{13} + \)\(13\!\cdots\!52\)\( T^{14} + \)\(42\!\cdots\!00\)\( T^{15} + \)\(12\!\cdots\!05\)\( T^{16} + \)\(21\!\cdots\!00\)\( T^{17} + \)\(89\!\cdots\!62\)\( T^{18} + \)\(33\!\cdots\!49\)\( T^{20} )^{2} \))
$19$ (\( 1 + 1364 T + 2476099 T^{2} \))(\( 1 + 1324 T + 2476099 T^{2} \))(\( 1 - 556 T + 2476099 T^{2} \))(\( 1 + 1000 T + 4533462 T^{2} + 2476099000 T^{3} + 6131066257801 T^{4} \))(\( ( 1 - 688 T + 5308396 T^{2} - 6058136368 T^{3} + 15069081422710 T^{4} - 15000545402668432 T^{5} + 32546127598645797196 T^{6} - \)\(10\!\cdots\!12\)\( T^{7} + \)\(37\!\cdots\!01\)\( T^{8} )^{2} \))(\( 1 - 23638692 T^{2} + 296336418074734 T^{4} - \)\(25\!\cdots\!32\)\( T^{6} + \)\(17\!\cdots\!97\)\( T^{8} - \)\(93\!\cdots\!80\)\( T^{10} + \)\(42\!\cdots\!52\)\( T^{12} - \)\(16\!\cdots\!48\)\( T^{14} + \)\(56\!\cdots\!66\)\( T^{16} - \)\(16\!\cdots\!12\)\( T^{18} + \)\(44\!\cdots\!28\)\( T^{20} - \)\(10\!\cdots\!12\)\( T^{22} + \)\(21\!\cdots\!66\)\( T^{24} - \)\(38\!\cdots\!48\)\( T^{26} + \)\(60\!\cdots\!52\)\( T^{28} - \)\(80\!\cdots\!80\)\( T^{30} + \)\(91\!\cdots\!97\)\( T^{32} - \)\(83\!\cdots\!32\)\( T^{34} + \)\(59\!\cdots\!34\)\( T^{36} - \)\(28\!\cdots\!92\)\( T^{38} + \)\(75\!\cdots\!01\)\( T^{40} \))
$23$ (\( 1 - 2198 T + 6436343 T^{2} \))(\( 1 + 852 T + 6436343 T^{2} \))(\( 1 - 2182 T + 6436343 T^{2} \))(\( 1 + 2452 T + 6569198 T^{2} + 15781913036 T^{3} + 41426511213649 T^{4} \))(\( 1 - 34675896 T^{2} + 569897415616828 T^{4} - \)\(59\!\cdots\!20\)\( T^{6} + \)\(44\!\cdots\!82\)\( T^{8} - \)\(24\!\cdots\!80\)\( T^{10} + \)\(97\!\cdots\!28\)\( T^{12} - \)\(24\!\cdots\!04\)\( T^{14} + \)\(29\!\cdots\!01\)\( T^{16} \))(\( ( 1 + 2338 T + 35997660 T^{2} + 45007042654 T^{3} + 520972471777845 T^{4} + 50426407997609208 T^{5} + \)\(39\!\cdots\!60\)\( T^{6} - \)\(66\!\cdots\!96\)\( T^{7} + \)\(17\!\cdots\!10\)\( T^{8} - \)\(91\!\cdots\!52\)\( T^{9} + \)\(76\!\cdots\!60\)\( T^{10} - \)\(58\!\cdots\!36\)\( T^{11} + \)\(74\!\cdots\!90\)\( T^{12} - \)\(17\!\cdots\!72\)\( T^{13} + \)\(68\!\cdots\!60\)\( T^{14} + \)\(55\!\cdots\!44\)\( T^{15} + \)\(37\!\cdots\!05\)\( T^{16} + \)\(20\!\cdots\!78\)\( T^{17} + \)\(10\!\cdots\!60\)\( T^{18} + \)\(44\!\cdots\!34\)\( T^{19} + \)\(12\!\cdots\!49\)\( T^{20} )^{2} \))
$29$ (\( 1 + 2218 T + 20511149 T^{2} \))(\( 1 - 5902 T + 20511149 T^{2} \))(\( 1 + 1578 T + 20511149 T^{2} \))(\( 1 - 1340 T - 8758306 T^{2} - 27484939660 T^{3} + 420707233300201 T^{4} \))(\( ( 1 - 2936 T + 58625996 T^{2} - 77951973928 T^{3} + 1466411094282230 T^{4} - 1598884552081323272 T^{5} + \)\(24\!\cdots\!96\)\( T^{6} - \)\(25\!\cdots\!64\)\( T^{7} + \)\(17\!\cdots\!01\)\( T^{8} )^{2} \))(\( 1 - 215092900 T^{2} + 23243889276296494 T^{4} - \)\(16\!\cdots\!00\)\( T^{6} + \)\(90\!\cdots\!13\)\( T^{8} - \)\(38\!\cdots\!00\)\( T^{10} + \)\(13\!\cdots\!92\)\( T^{12} - \)\(42\!\cdots\!00\)\( T^{14} + \)\(11\!\cdots\!86\)\( T^{16} - \)\(27\!\cdots\!00\)\( T^{18} + \)\(58\!\cdots\!28\)\( T^{20} - \)\(11\!\cdots\!00\)\( T^{22} + \)\(20\!\cdots\!86\)\( T^{24} - \)\(31\!\cdots\!00\)\( T^{26} + \)\(43\!\cdots\!92\)\( T^{28} - \)\(51\!\cdots\!00\)\( T^{30} + \)\(50\!\cdots\!13\)\( T^{32} - \)\(39\!\cdots\!00\)\( T^{34} + \)\(22\!\cdots\!94\)\( T^{36} - \)\(88\!\cdots\!00\)\( T^{38} + \)\(17\!\cdots\!01\)\( T^{40} \))
$31$ (\( 1 + 1700 T + 28629151 T^{2} \))(\( 1 + 3320 T + 28629151 T^{2} \))(\( 1 - 9660 T + 28629151 T^{2} \))(\( 1 + 2248 T + 57017022 T^{2} + 64358331448 T^{3} + 819628286980801 T^{4} \))(\( ( 1 - 2112 T + 80187004 T^{2} - 163080265536 T^{3} + 3196344720873606 T^{4} - 4668849547150239936 T^{5} + \)\(65\!\cdots\!04\)\( T^{6} - \)\(49\!\cdots\!12\)\( T^{7} + \)\(67\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 - 3580 T + 132421614 T^{2} - 484936307876 T^{3} + 8983138546835629 T^{4} - 33755686429634218608 T^{5} + \)\(43\!\cdots\!88\)\( T^{6} - \)\(16\!\cdots\!96\)\( T^{7} + \)\(17\!\cdots\!38\)\( T^{8} - \)\(60\!\cdots\!32\)\( T^{9} + \)\(54\!\cdots\!44\)\( T^{10} - \)\(17\!\cdots\!32\)\( T^{11} + \)\(13\!\cdots\!38\)\( T^{12} - \)\(38\!\cdots\!96\)\( T^{13} + \)\(29\!\cdots\!88\)\( T^{14} - \)\(64\!\cdots\!08\)\( T^{15} + \)\(49\!\cdots\!29\)\( T^{16} - \)\(76\!\cdots\!76\)\( T^{17} + \)\(59\!\cdots\!14\)\( T^{18} - \)\(46\!\cdots\!80\)\( T^{19} + \)\(36\!\cdots\!01\)\( T^{20} )^{2} \))
$37$ (\( 1 + 846 T + 69343957 T^{2} \))(\( 1 - 10774 T + 69343957 T^{2} \))(\( 1 + 3534 T + 69343957 T^{2} \))(\( 1 + 5940 T + 123434318 T^{2} + 411903104580 T^{3} + 4808584372417849 T^{4} \))(\( 1 - 251774632 T^{2} + 38631208311838780 T^{4} - \)\(41\!\cdots\!52\)\( T^{6} + \)\(32\!\cdots\!34\)\( T^{8} - \)\(19\!\cdots\!48\)\( T^{10} + \)\(89\!\cdots\!80\)\( T^{12} - \)\(27\!\cdots\!68\)\( T^{14} + \)\(53\!\cdots\!01\)\( T^{16} \))(\( 1 - 565372668 T^{2} + 171401952644913934 T^{4} - \)\(36\!\cdots\!00\)\( T^{6} + \)\(59\!\cdots\!09\)\( T^{8} - \)\(80\!\cdots\!44\)\( T^{10} + \)\(93\!\cdots\!16\)\( T^{12} - \)\(93\!\cdots\!40\)\( T^{14} + \)\(83\!\cdots\!86\)\( T^{16} - \)\(67\!\cdots\!48\)\( T^{18} + \)\(48\!\cdots\!08\)\( T^{20} - \)\(32\!\cdots\!52\)\( T^{22} + \)\(19\!\cdots\!86\)\( T^{24} - \)\(10\!\cdots\!60\)\( T^{26} + \)\(49\!\cdots\!16\)\( T^{28} - \)\(20\!\cdots\!56\)\( T^{30} + \)\(73\!\cdots\!09\)\( T^{32} - \)\(21\!\cdots\!00\)\( T^{34} + \)\(48\!\cdots\!34\)\( T^{36} - \)\(77\!\cdots\!32\)\( T^{38} + \)\(66\!\cdots\!01\)\( T^{40} \))
$41$ (\( 1 + 1818 T + 115856201 T^{2} \))(\( 1 + 17958 T + 115856201 T^{2} \))(\( 1 - 7462 T + 115856201 T^{2} \))(\( 1 - 23076 T + 352280470 T^{2} - 2673497694276 T^{3} + 13422659310152401 T^{4} \))(\( ( 1 - 11800 T + 337909340 T^{2} - 2943020124776 T^{3} + 51155654972384870 T^{4} - \)\(34\!\cdots\!76\)\( T^{5} + \)\(45\!\cdots\!40\)\( T^{6} - \)\(18\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 - 5804 T + 580737234 T^{2} - 4176614475628 T^{3} + 181136735899530493 T^{4} - \)\(13\!\cdots\!48\)\( T^{5} + \)\(39\!\cdots\!40\)\( T^{6} - \)\(28\!\cdots\!68\)\( T^{7} + \)\(63\!\cdots\!42\)\( T^{8} - \)\(43\!\cdots\!24\)\( T^{9} + \)\(82\!\cdots\!24\)\( T^{10} - \)\(49\!\cdots\!24\)\( T^{11} + \)\(85\!\cdots\!42\)\( T^{12} - \)\(44\!\cdots\!68\)\( T^{13} + \)\(70\!\cdots\!40\)\( T^{14} - \)\(28\!\cdots\!48\)\( T^{15} + \)\(43\!\cdots\!93\)\( T^{16} - \)\(11\!\cdots\!28\)\( T^{17} + \)\(18\!\cdots\!34\)\( T^{18} - \)\(21\!\cdots\!04\)\( T^{19} + \)\(43\!\cdots\!01\)\( T^{20} )^{2} \))
$43$ (\( 1 - 10534 T + 147008443 T^{2} \))(\( 1 - 9264 T + 147008443 T^{2} \))(\( 1 + 7114 T + 147008443 T^{2} \))(\( 1 - 17684 T + 312898614 T^{2} - 2599697306012 T^{3} + 21611482313284249 T^{4} \))(\( 1 - 283211672 T^{2} + 48021567531024796 T^{4} - \)\(54\!\cdots\!84\)\( T^{6} + \)\(43\!\cdots\!06\)\( T^{8} - \)\(11\!\cdots\!16\)\( T^{10} + \)\(22\!\cdots\!96\)\( T^{12} - \)\(28\!\cdots\!28\)\( T^{14} + \)\(21\!\cdots\!01\)\( T^{16} \))(\( 1 - 1208126740 T^{2} + 787740581508660018 T^{4} - \)\(36\!\cdots\!96\)\( T^{6} + \)\(12\!\cdots\!73\)\( T^{8} - \)\(37\!\cdots\!80\)\( T^{10} + \)\(95\!\cdots\!40\)\( T^{12} - \)\(20\!\cdots\!56\)\( T^{14} + \)\(40\!\cdots\!70\)\( T^{16} - \)\(70\!\cdots\!00\)\( T^{18} + \)\(10\!\cdots\!96\)\( T^{20} - \)\(15\!\cdots\!00\)\( T^{22} + \)\(18\!\cdots\!70\)\( T^{24} - \)\(21\!\cdots\!44\)\( T^{26} + \)\(20\!\cdots\!40\)\( T^{28} - \)\(17\!\cdots\!20\)\( T^{30} + \)\(13\!\cdots\!73\)\( T^{32} - \)\(79\!\cdots\!04\)\( T^{34} + \)\(37\!\cdots\!18\)\( T^{36} - \)\(12\!\cdots\!60\)\( T^{38} + \)\(22\!\cdots\!01\)\( T^{40} \))
$47$ (\( 1 - 12074 T + 229345007 T^{2} \))(\( 1 + 9796 T + 229345007 T^{2} \))(\( 1 + 28294 T + 229345007 T^{2} \))(\( 1 + 2908 T + 56660030 T^{2} + 666935280356 T^{3} + 52599132235830049 T^{4} \))(\( 1 - 963352312 T^{2} + 473832864723586300 T^{4} - \)\(15\!\cdots\!72\)\( T^{6} + \)\(40\!\cdots\!54\)\( T^{8} - \)\(83\!\cdots\!28\)\( T^{10} + \)\(13\!\cdots\!00\)\( T^{12} - \)\(14\!\cdots\!88\)\( T^{14} + \)\(76\!\cdots\!01\)\( T^{16} \))(\( ( 1 - 22090 T + 1548510076 T^{2} - 25779450599270 T^{3} + 1065218725316011845 T^{4} - \)\(14\!\cdots\!20\)\( T^{5} + \)\(45\!\cdots\!96\)\( T^{6} - \)\(51\!\cdots\!60\)\( T^{7} + \)\(14\!\cdots\!10\)\( T^{8} - \)\(14\!\cdots\!00\)\( T^{9} + \)\(36\!\cdots\!56\)\( T^{10} - \)\(32\!\cdots\!00\)\( T^{11} + \)\(76\!\cdots\!90\)\( T^{12} - \)\(62\!\cdots\!80\)\( T^{13} + \)\(12\!\cdots\!96\)\( T^{14} - \)\(90\!\cdots\!40\)\( T^{15} + \)\(15\!\cdots\!05\)\( T^{16} - \)\(86\!\cdots\!10\)\( T^{17} + \)\(11\!\cdots\!76\)\( T^{18} - \)\(38\!\cdots\!30\)\( T^{19} + \)\(40\!\cdots\!49\)\( T^{20} )^{2} \))
$53$ (\( 1 - 32586 T + 418195493 T^{2} \))(\( 1 + 31434 T + 418195493 T^{2} \))(\( 1 + 13046 T + 418195493 T^{2} \))(\( 1 + 5412 T + 693247822 T^{2} + 2263274008116 T^{3} + 174887470365513049 T^{4} \))(\( 1 - 1183385640 T^{2} + 874785099161623996 T^{4} - \)\(49\!\cdots\!80\)\( T^{6} + \)\(23\!\cdots\!06\)\( T^{8} - \)\(86\!\cdots\!20\)\( T^{10} + \)\(26\!\cdots\!96\)\( T^{12} - \)\(63\!\cdots\!60\)\( T^{14} + \)\(93\!\cdots\!01\)\( T^{16} \))(\( 1 - 4003669356 T^{2} + 7954725909905649454 T^{4} - \)\(10\!\cdots\!60\)\( T^{6} + \)\(10\!\cdots\!77\)\( T^{8} - \)\(91\!\cdots\!76\)\( T^{10} + \)\(65\!\cdots\!16\)\( T^{12} - \)\(40\!\cdots\!80\)\( T^{14} + \)\(22\!\cdots\!54\)\( T^{16} - \)\(11\!\cdots\!56\)\( T^{18} + \)\(48\!\cdots\!96\)\( T^{20} - \)\(19\!\cdots\!44\)\( T^{22} + \)\(68\!\cdots\!54\)\( T^{24} - \)\(21\!\cdots\!20\)\( T^{26} + \)\(61\!\cdots\!16\)\( T^{28} - \)\(14\!\cdots\!24\)\( T^{30} + \)\(31\!\cdots\!77\)\( T^{32} - \)\(53\!\cdots\!40\)\( T^{34} + \)\(69\!\cdots\!54\)\( T^{36} - \)\(61\!\cdots\!44\)\( T^{38} + \)\(26\!\cdots\!01\)\( T^{40} \))
$59$ (\( 1 - 8668 T + 714924299 T^{2} \))(\( 1 - 33228 T + 714924299 T^{2} \))(\( 1 + 37092 T + 714924299 T^{2} \))(\( 1 - 62584 T + 2277965606 T^{2} - 44742822328616 T^{3} + 511116753300641401 T^{4} \))(\( ( 1 - 45840 T + 3064286732 T^{2} - 94721285480976 T^{3} + 3348109683185502486 T^{4} - \)\(67\!\cdots\!24\)\( T^{5} + \)\(15\!\cdots\!32\)\( T^{6} - \)\(16\!\cdots\!60\)\( T^{7} + \)\(26\!\cdots\!01\)\( T^{8} )^{2} \))(\( 1 - 9996949828 T^{2} + 49650800837889885966 T^{4} - \)\(16\!\cdots\!20\)\( T^{6} + \)\(39\!\cdots\!17\)\( T^{8} - \)\(73\!\cdots\!28\)\( T^{10} + \)\(11\!\cdots\!44\)\( T^{12} - \)\(14\!\cdots\!80\)\( T^{14} + \)\(15\!\cdots\!54\)\( T^{16} - \)\(13\!\cdots\!88\)\( T^{18} + \)\(10\!\cdots\!24\)\( T^{20} - \)\(70\!\cdots\!88\)\( T^{22} + \)\(39\!\cdots\!54\)\( T^{24} - \)\(19\!\cdots\!80\)\( T^{26} + \)\(77\!\cdots\!44\)\( T^{28} - \)\(25\!\cdots\!28\)\( T^{30} + \)\(69\!\cdots\!17\)\( T^{32} - \)\(14\!\cdots\!20\)\( T^{34} + \)\(23\!\cdots\!66\)\( T^{36} - \)\(23\!\cdots\!28\)\( T^{38} + \)\(12\!\cdots\!01\)\( T^{40} \))
$61$ (\( 1 + 34670 T + 844596301 T^{2} \))(\( 1 + 40210 T + 844596301 T^{2} \))(\( 1 - 39570 T + 844596301 T^{2} \))(\( 1 - 14108 T + 1110042462 T^{2} - 11915564614508 T^{3} + 713342911662882601 T^{4} \))(\( ( 1 - 61928 T + 3903014764 T^{2} - 145287706763384 T^{3} + 5198153942066716726 T^{4} - \)\(12\!\cdots\!84\)\( T^{5} + \)\(27\!\cdots\!64\)\( T^{6} - \)\(37\!\cdots\!28\)\( T^{7} + \)\(50\!\cdots\!01\)\( T^{8} )^{2} \))(\( 1 - 7152852348 T^{2} + 26523669582205677166 T^{4} - \)\(67\!\cdots\!00\)\( T^{6} + \)\(13\!\cdots\!17\)\( T^{8} - \)\(22\!\cdots\!48\)\( T^{10} + \)\(31\!\cdots\!84\)\( T^{12} - \)\(38\!\cdots\!60\)\( T^{14} + \)\(42\!\cdots\!14\)\( T^{16} - \)\(41\!\cdots\!08\)\( T^{18} + \)\(36\!\cdots\!64\)\( T^{20} - \)\(29\!\cdots\!08\)\( T^{22} + \)\(21\!\cdots\!14\)\( T^{24} - \)\(13\!\cdots\!60\)\( T^{26} + \)\(81\!\cdots\!84\)\( T^{28} - \)\(41\!\cdots\!48\)\( T^{30} + \)\(17\!\cdots\!17\)\( T^{32} - \)\(63\!\cdots\!00\)\( T^{34} + \)\(17\!\cdots\!66\)\( T^{36} - \)\(34\!\cdots\!48\)\( T^{38} + \)\(34\!\cdots\!01\)\( T^{40} \))
$67$ (\( 1 + 47566 T + 1350125107 T^{2} \))(\( 1 - 58864 T + 1350125107 T^{2} \))(\( 1 + 56734 T + 1350125107 T^{2} \))(\( 1 + 85412 T + 4371910566 T^{2} + 115316885639084 T^{3} + 1822837804551761449 T^{4} \))(\( 1 - 9281919064 T^{2} + 39492482666681482588 T^{4} - \)\(10\!\cdots\!40\)\( T^{6} + \)\(16\!\cdots\!62\)\( T^{8} - \)\(18\!\cdots\!60\)\( T^{10} + \)\(13\!\cdots\!88\)\( T^{12} - \)\(56\!\cdots\!36\)\( T^{14} + \)\(11\!\cdots\!01\)\( T^{16} \))(\( 1 - 11828518964 T^{2} + 68669252417166303634 T^{4} - \)\(26\!\cdots\!60\)\( T^{6} + \)\(76\!\cdots\!57\)\( T^{8} - \)\(18\!\cdots\!04\)\( T^{10} + \)\(38\!\cdots\!16\)\( T^{12} - \)\(70\!\cdots\!40\)\( T^{14} + \)\(11\!\cdots\!54\)\( T^{16} - \)\(18\!\cdots\!64\)\( T^{18} + \)\(25\!\cdots\!76\)\( T^{20} - \)\(33\!\cdots\!36\)\( T^{22} + \)\(39\!\cdots\!54\)\( T^{24} - \)\(42\!\cdots\!60\)\( T^{26} + \)\(42\!\cdots\!16\)\( T^{28} - \)\(36\!\cdots\!96\)\( T^{30} + \)\(28\!\cdots\!57\)\( T^{32} - \)\(17\!\cdots\!40\)\( T^{34} + \)\(83\!\cdots\!34\)\( T^{36} - \)\(26\!\cdots\!36\)\( T^{38} + \)\(40\!\cdots\!01\)\( T^{40} \))
$71$ (\( 1 - 948 T + 1804229351 T^{2} \))(\( 1 + 55312 T + 1804229351 T^{2} \))(\( 1 - 45588 T + 1804229351 T^{2} \))(\( 1 - 47208 T + 4011779662 T^{2} - 85174059202008 T^{3} + 3255243551009881201 T^{4} \))(\( ( 1 + 62816 T + 3398787356 T^{2} + 184024084124896 T^{3} + 8353296562609817510 T^{4} + \)\(33\!\cdots\!96\)\( T^{5} + \)\(11\!\cdots\!56\)\( T^{6} + \)\(36\!\cdots\!16\)\( T^{7} + \)\(10\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 + 100156 T + 13448448446 T^{2} + 957445823975748 T^{3} + 76873855601451932317 T^{4} + \)\(43\!\cdots\!20\)\( T^{5} + \)\(26\!\cdots\!28\)\( T^{6} + \)\(12\!\cdots\!92\)\( T^{7} + \)\(67\!\cdots\!74\)\( T^{8} + \)\(28\!\cdots\!84\)\( T^{9} + \)\(13\!\cdots\!68\)\( T^{10} + \)\(51\!\cdots\!84\)\( T^{11} + \)\(21\!\cdots\!74\)\( T^{12} + \)\(74\!\cdots\!92\)\( T^{13} + \)\(28\!\cdots\!28\)\( T^{14} + \)\(82\!\cdots\!20\)\( T^{15} + \)\(26\!\cdots\!17\)\( T^{16} + \)\(59\!\cdots\!48\)\( T^{17} + \)\(15\!\cdots\!46\)\( T^{18} + \)\(20\!\cdots\!56\)\( T^{19} + \)\(36\!\cdots\!01\)\( T^{20} )^{2} \))
$73$ (\( 1 + 63102 T + 2073071593 T^{2} \))(\( 1 - 27258 T + 2073071593 T^{2} \))(\( 1 - 11842 T + 2073071593 T^{2} \))(\( 1 + 67452 T + 4400780438 T^{2} + 139832825091036 T^{3} + 4297625829703557649 T^{4} \))(\( 1 - 9140679496 T^{2} + 46078306824990298588 T^{4} - \)\(15\!\cdots\!60\)\( T^{6} + \)\(37\!\cdots\!02\)\( T^{8} - \)\(66\!\cdots\!40\)\( T^{10} + \)\(85\!\cdots\!88\)\( T^{12} - \)\(72\!\cdots\!04\)\( T^{14} + \)\(34\!\cdots\!01\)\( T^{16} \))(\( ( 1 + 52568 T + 9379342894 T^{2} + 374528688123736 T^{3} + 37721970904921608509 T^{4} + \)\(11\!\cdots\!56\)\( T^{5} + \)\(82\!\cdots\!04\)\( T^{6} + \)\(19\!\cdots\!04\)\( T^{7} + \)\(11\!\cdots\!58\)\( T^{8} + \)\(24\!\cdots\!96\)\( T^{9} + \)\(16\!\cdots\!04\)\( T^{10} + \)\(51\!\cdots\!28\)\( T^{11} + \)\(49\!\cdots\!42\)\( T^{12} + \)\(17\!\cdots\!28\)\( T^{13} + \)\(15\!\cdots\!04\)\( T^{14} + \)\(44\!\cdots\!08\)\( T^{15} + \)\(29\!\cdots\!41\)\( T^{16} + \)\(61\!\cdots\!52\)\( T^{17} + \)\(31\!\cdots\!94\)\( T^{18} + \)\(37\!\cdots\!24\)\( T^{19} + \)\(14\!\cdots\!49\)\( T^{20} )^{2} \))
$79$ (\( 1 - 46536 T + 3077056399 T^{2} \))(\( 1 - 31456 T + 3077056399 T^{2} \))(\( 1 - 94216 T + 3077056399 T^{2} \))(\( 1 + 65904 T + 3994274078 T^{2} + 202790324919696 T^{3} + 9468276082626847201 T^{4} \))(\( ( 1 - 21632 T + 7152876604 T^{2} - 332616618908288 T^{3} + 24121899620797566790 T^{4} - \)\(10\!\cdots\!12\)\( T^{5} + \)\(67\!\cdots\!04\)\( T^{6} - \)\(63\!\cdots\!68\)\( T^{7} + \)\(89\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 - 141040 T + 30508439526 T^{2} - 3027236690742416 T^{3} + \)\(38\!\cdots\!93\)\( T^{4} - \)\(29\!\cdots\!88\)\( T^{5} + \)\(27\!\cdots\!48\)\( T^{6} - \)\(17\!\cdots\!16\)\( T^{7} + \)\(13\!\cdots\!30\)\( T^{8} - \)\(73\!\cdots\!40\)\( T^{9} + \)\(47\!\cdots\!04\)\( T^{10} - \)\(22\!\cdots\!60\)\( T^{11} + \)\(12\!\cdots\!30\)\( T^{12} - \)\(51\!\cdots\!84\)\( T^{13} + \)\(24\!\cdots\!48\)\( T^{14} - \)\(81\!\cdots\!12\)\( T^{15} + \)\(32\!\cdots\!93\)\( T^{16} - \)\(79\!\cdots\!84\)\( T^{17} + \)\(24\!\cdots\!26\)\( T^{18} - \)\(34\!\cdots\!60\)\( T^{19} + \)\(76\!\cdots\!01\)\( T^{20} )^{2} \))
$83$ (\( 1 + 88778 T + 3939040643 T^{2} \))(\( 1 - 24552 T + 3939040643 T^{2} \))(\( 1 + 31482 T + 3939040643 T^{2} \))(\( 1 - 108724 T + 10572459494 T^{2} - 428268254869532 T^{3} + 15516041187205853449 T^{4} \))(\( 1 - 6759897816 T^{2} + 40943120759345365468 T^{4} - \)\(86\!\cdots\!80\)\( T^{6} + \)\(39\!\cdots\!42\)\( T^{8} - \)\(13\!\cdots\!20\)\( T^{10} + \)\(98\!\cdots\!68\)\( T^{12} - \)\(25\!\cdots\!84\)\( T^{14} + \)\(57\!\cdots\!01\)\( T^{16} \))(\( 1 - 34768653380 T^{2} + \)\(55\!\cdots\!58\)\( T^{4} - \)\(53\!\cdots\!56\)\( T^{6} + \)\(34\!\cdots\!33\)\( T^{8} - \)\(15\!\cdots\!20\)\( T^{10} + \)\(44\!\cdots\!40\)\( T^{12} - \)\(24\!\cdots\!36\)\( T^{14} - \)\(64\!\cdots\!10\)\( T^{16} + \)\(49\!\cdots\!20\)\( T^{18} - \)\(23\!\cdots\!44\)\( T^{20} + \)\(76\!\cdots\!80\)\( T^{22} - \)\(15\!\cdots\!10\)\( T^{24} - \)\(90\!\cdots\!64\)\( T^{26} + \)\(25\!\cdots\!40\)\( T^{28} - \)\(14\!\cdots\!80\)\( T^{30} + \)\(48\!\cdots\!33\)\( T^{32} - \)\(11\!\cdots\!44\)\( T^{34} + \)\(18\!\cdots\!58\)\( T^{36} - \)\(18\!\cdots\!20\)\( T^{38} + \)\(80\!\cdots\!01\)\( T^{40} \))
$89$ (\( 1 + 104934 T + 5584059449 T^{2} \))(\( 1 + 90854 T + 5584059449 T^{2} \))(\( 1 + 94054 T + 5584059449 T^{2} \))(\( 1 + 55020 T + 10818978262 T^{2} + 307234950883980 T^{3} + 31181719929966183601 T^{4} \))(\( ( 1 + 20952 T + 16118164796 T^{2} + 497915996461992 T^{3} + \)\(11\!\cdots\!30\)\( T^{4} + \)\(27\!\cdots\!08\)\( T^{5} + \)\(50\!\cdots\!96\)\( T^{6} + \)\(36\!\cdots\!48\)\( T^{7} + \)\(97\!\cdots\!01\)\( T^{8} )^{2} \))(\( ( 1 + 1580 T + 27398194046 T^{2} + 460040940498284 T^{3} + \)\(38\!\cdots\!93\)\( T^{4} + \)\(90\!\cdots\!72\)\( T^{5} + \)\(38\!\cdots\!08\)\( T^{6} + \)\(95\!\cdots\!24\)\( T^{7} + \)\(29\!\cdots\!70\)\( T^{8} + \)\(73\!\cdots\!60\)\( T^{9} + \)\(18\!\cdots\!64\)\( T^{10} + \)\(40\!\cdots\!40\)\( T^{11} + \)\(91\!\cdots\!70\)\( T^{12} + \)\(16\!\cdots\!76\)\( T^{13} + \)\(37\!\cdots\!08\)\( T^{14} + \)\(49\!\cdots\!28\)\( T^{15} + \)\(11\!\cdots\!93\)\( T^{16} + \)\(77\!\cdots\!16\)\( T^{17} + \)\(25\!\cdots\!46\)\( T^{18} + \)\(83\!\cdots\!20\)\( T^{19} + \)\(29\!\cdots\!01\)\( T^{20} )^{2} \))
$97$ (\( 1 + 36254 T + 8587340257 T^{2} \))(\( 1 - 154706 T + 8587340257 T^{2} \))(\( 1 - 23714 T + 8587340257 T^{2} \))(\( 1 - 147668 T + 11612429670 T^{2} - 1268075361070676 T^{3} + 73742412689492826049 T^{4} \))(\( 1 - 45263915272 T^{2} + \)\(10\!\cdots\!40\)\( T^{4} - \)\(14\!\cdots\!12\)\( T^{6} + \)\(15\!\cdots\!94\)\( T^{8} - \)\(11\!\cdots\!88\)\( T^{10} + \)\(55\!\cdots\!40\)\( T^{12} - \)\(18\!\cdots\!28\)\( T^{14} + \)\(29\!\cdots\!01\)\( T^{16} \))(\( ( 1 - 73688 T + 43672916862 T^{2} - 2817316775448856 T^{3} + \)\(98\!\cdots\!25\)\( T^{4} - \)\(59\!\cdots\!28\)\( T^{5} + \)\(15\!\cdots\!52\)\( T^{6} - \)\(87\!\cdots\!16\)\( T^{7} + \)\(18\!\cdots\!70\)\( T^{8} - \)\(96\!\cdots\!88\)\( T^{9} + \)\(17\!\cdots\!72\)\( T^{10} - \)\(82\!\cdots\!16\)\( T^{11} + \)\(13\!\cdots\!30\)\( T^{12} - \)\(55\!\cdots\!88\)\( T^{13} + \)\(84\!\cdots\!52\)\( T^{14} - \)\(27\!\cdots\!96\)\( T^{15} + \)\(39\!\cdots\!25\)\( T^{16} - \)\(97\!\cdots\!08\)\( T^{17} + \)\(12\!\cdots\!62\)\( T^{18} - \)\(18\!\cdots\!16\)\( T^{19} + \)\(21\!\cdots\!49\)\( T^{20} )^{2} \))
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