Properties

Label 20.7
Level 20
Weight 7
Dimension 34
Nonzero newspaces 3
Newform subspaces 6
Sturm bound 168
Trace bound 1

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Defining parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 7 \)
Nonzero newspaces: \( 3 \)
Newform subspaces: \( 6 \)
Sturm bound: \(168\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(\Gamma_1(20))\).

Total New Old
Modular forms 82 38 44
Cusp forms 62 34 28
Eisenstein series 20 4 16

Trace form

\( 34q - 10q^{2} + 32q^{3} + 220q^{4} - 180q^{5} - 640q^{6} - 264q^{7} + 440q^{8} + 916q^{9} + O(q^{10}) \) \( 34q - 10q^{2} + 32q^{3} + 220q^{4} - 180q^{5} - 640q^{6} - 264q^{7} + 440q^{8} + 916q^{9} + 2046q^{10} + 2200q^{11} - 440q^{12} - 4182q^{13} - 1624q^{14} - 7768q^{15} - 1232q^{16} + 3562q^{17} + 15790q^{18} - 18284q^{20} + 10816q^{21} - 26160q^{22} + 19984q^{23} + 52656q^{24} + 12382q^{25} - 30684q^{26} - 115528q^{27} + 19320q^{28} - 65080q^{29} - 17480q^{30} + 104976q^{31} - 60800q^{32} + 290200q^{33} + 149004q^{34} - 116072q^{35} - 25732q^{36} - 178914q^{37} + 74800q^{38} + 65256q^{40} + 163832q^{41} - 138360q^{42} + 60720q^{43} - 387280q^{44} - 399190q^{45} - 439960q^{46} - 355248q^{47} + 297600q^{48} + 296916q^{49} + 97486q^{50} + 641872q^{51} + 548280q^{52} + 668686q^{53} + 368272q^{54} - 310200q^{55} + 295120q^{56} - 1326256q^{57} + 350700q^{58} - 745400q^{60} + 203768q^{61} - 7120q^{62} + 2288q^{63} - 549440q^{64} + 426590q^{65} - 1468720q^{66} - 230304q^{67} - 1678280q^{68} - 294248q^{69} + 1218080q^{70} + 174128q^{71} + 2317560q^{72} - 747522q^{73} + 1917444q^{74} + 1855048q^{75} + 2723520q^{76} + 1694360q^{77} + 473200q^{78} - 1375504q^{80} - 1971498q^{81} - 3169500q^{82} - 2190936q^{83} - 4782688q^{84} - 698022q^{85} - 5052880q^{86} + 2614304q^{87} - 278880q^{88} - 287720q^{89} + 1932966q^{90} - 2186976q^{91} + 4095720q^{92} - 3403928q^{93} + 8576456q^{94} + 3484184q^{95} + 7915520q^{96} + 5683086q^{97} + 1050270q^{98} + O(q^{100}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(\Gamma_1(20))\)

We only show spaces with odd parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
20.7.b \(\chi_{20}(11, \cdot)\) 20.7.b.a 12 1
20.7.d \(\chi_{20}(19, \cdot)\) 20.7.d.a 1 1
20.7.d.b 1
20.7.d.c 2
20.7.d.d 12
20.7.f \(\chi_{20}(13, \cdot)\) 20.7.f.a 6 2

Decomposition of \(S_{7}^{\mathrm{old}}(\Gamma_1(20))\) into lower level spaces

\( S_{7}^{\mathrm{old}}(\Gamma_1(20)) \cong \) \(S_{7}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 10 T - 28 T^{2} - 760 T^{3} - 2736 T^{4} + 26240 T^{5} + 377856 T^{6} + 1679360 T^{7} - 11206656 T^{8} - 199229440 T^{9} - 469762048 T^{10} + 10737418240 T^{11} + 68719476736 T^{12} \))(\( 1 + 8 T \))(\( 1 - 8 T \))(\( 1 + 64 T^{2} \))(\( 1 - 32 T^{2} + 5744 T^{4} - 171776 T^{6} + 23527424 T^{8} - 536870912 T^{10} + 68719476736 T^{12} \))
$3$ (\( 1 - 3376 T^{2} + 5490602 T^{4} - 5841263472 T^{6} + 4649776536303 T^{8} - 3147066205418592 T^{10} + 2161997866487458188 T^{12} - \)\(16\!\cdots\!72\)\( T^{14} + \)\(13\!\cdots\!43\)\( T^{16} - \)\(87\!\cdots\!12\)\( T^{18} + \)\(43\!\cdots\!22\)\( T^{20} - \)\(14\!\cdots\!76\)\( T^{22} + \)\(22\!\cdots\!41\)\( T^{24} \))(\( 1 + 44 T + 729 T^{2} \))(\( 1 - 44 T + 729 T^{2} \))(\( ( 1 + 729 T^{2} )^{2} \))(\( ( 1 + 1698 T^{2} + 1910799 T^{4} + 1695048444 T^{6} + 1015476931359 T^{8} + 479565352944738 T^{10} + 150094635296999121 T^{12} )^{2} \))(\( 1 - 32 T + 512 T^{2} + 25272 T^{3} - 507105 T^{4} - 9777528 T^{5} + 891855648 T^{6} - 7127817912 T^{7} - 269496388305 T^{8} + 9790890598008 T^{9} + 144603922678272 T^{10} - 6588516227028768 T^{11} + 150094635296999121 T^{12} \))
$5$ (\( ( 1 - 3125 T^{2} )^{6} \))(\( 1 + 125 T \))(\( 1 + 125 T \))(\( 1 + 234 T + 15625 T^{2} \))(\( ( 1 - 230 T + 40375 T^{2} - 6332500 T^{3} + 630859375 T^{4} - 56152343750 T^{5} + 3814697265625 T^{6} )^{2} \))(\( 1 + 156 T + 24255 T^{2} + 3651000 T^{3} + 378984375 T^{4} + 38085937500 T^{5} + 3814697265625 T^{6} \))
$7$ (\( 1 - 742176 T^{2} + 278743423722 T^{4} - 72266975148027552 T^{6} + \)\(14\!\cdots\!43\)\( T^{8} - \)\(22\!\cdots\!72\)\( T^{10} + \)\(29\!\cdots\!68\)\( T^{12} - \)\(31\!\cdots\!72\)\( T^{14} + \)\(27\!\cdots\!43\)\( T^{16} - \)\(19\!\cdots\!52\)\( T^{18} + \)\(10\!\cdots\!22\)\( T^{20} - \)\(37\!\cdots\!76\)\( T^{22} + \)\(70\!\cdots\!01\)\( T^{24} \))(\( 1 - 524 T + 117649 T^{2} \))(\( 1 + 524 T + 117649 T^{2} \))(\( ( 1 + 117649 T^{2} )^{2} \))(\( ( 1 + 316498 T^{2} + 63613951199 T^{4} + 9177832017015004 T^{6} + \)\(88\!\cdots\!99\)\( T^{8} + \)\(60\!\cdots\!98\)\( T^{10} + \)\(26\!\cdots\!01\)\( T^{12} )^{2} \))(\( 1 + 264 T + 34848 T^{2} - 48340864 T^{3} - 13585926801 T^{4} + 3369384829272 T^{5} + 2531379538222304 T^{6} + 396404755779021528 T^{7} - \)\(18\!\cdots\!01\)\( T^{8} - \)\(78\!\cdots\!36\)\( T^{9} + \)\(66\!\cdots\!48\)\( T^{10} + \)\(59\!\cdots\!36\)\( T^{11} + \)\(26\!\cdots\!01\)\( T^{12} \))
$11$ (\( 1 - 8097852 T^{2} + 38966214008706 T^{4} - \)\(13\!\cdots\!80\)\( T^{6} + \)\(36\!\cdots\!55\)\( T^{8} - \)\(82\!\cdots\!12\)\( T^{10} + \)\(15\!\cdots\!04\)\( T^{12} - \)\(25\!\cdots\!52\)\( T^{14} + \)\(35\!\cdots\!55\)\( T^{16} - \)\(41\!\cdots\!80\)\( T^{18} + \)\(37\!\cdots\!86\)\( T^{20} - \)\(24\!\cdots\!52\)\( T^{22} + \)\(95\!\cdots\!21\)\( T^{24} \))(\( ( 1 - 1331 T )( 1 + 1331 T ) \))(\( ( 1 - 1331 T )( 1 + 1331 T ) \))(\( ( 1 - 1331 T )^{2}( 1 + 1331 T )^{2} \))(\( ( 1 - 4230966 T^{2} + 12435345552255 T^{4} - 27133599137440282100 T^{6} + \)\(39\!\cdots\!55\)\( T^{8} - \)\(41\!\cdots\!06\)\( T^{10} + \)\(30\!\cdots\!61\)\( T^{12} )^{2} \))(\( ( 1 - 1100 T + 3038783 T^{2} - 2443014200 T^{3} + 5383389450263 T^{4} - 3452271214393100 T^{5} + 5559917313492231481 T^{6} )^{2} \))
$13$ (\( ( 1 + 2520 T + 17586762 T^{2} + 39445743800 T^{3} + 171248210145759 T^{4} + 322338377813976240 T^{5} + \)\(10\!\cdots\!76\)\( T^{6} + \)\(15\!\cdots\!60\)\( T^{7} + \)\(39\!\cdots\!79\)\( T^{8} + \)\(44\!\cdots\!00\)\( T^{9} + \)\(95\!\cdots\!82\)\( T^{10} + \)\(66\!\cdots\!80\)\( T^{11} + \)\(12\!\cdots\!41\)\( T^{12} )^{2} \))(\( ( 1 - 2197 T )( 1 + 2197 T ) \))(\( ( 1 - 2197 T )( 1 + 2197 T ) \))(\( ( 1 - 4070 T + 4826809 T^{2} )( 1 + 4070 T + 4826809 T^{2} ) \))(\( ( 1 - 6612822 T^{2} + 51291900208479 T^{4} - \)\(31\!\cdots\!16\)\( T^{6} + \)\(11\!\cdots\!99\)\( T^{8} - \)\(35\!\cdots\!42\)\( T^{10} + \)\(12\!\cdots\!41\)\( T^{12} )^{2} \))(\( 1 - 858 T + 368082 T^{2} - 2806723722 T^{3} + 30691727023695 T^{4} - 22592900379705612 T^{5} + 12026485285270478748 T^{6} - \)\(10\!\cdots\!08\)\( T^{7} + \)\(71\!\cdots\!95\)\( T^{8} - \)\(31\!\cdots\!38\)\( T^{9} + \)\(19\!\cdots\!02\)\( T^{10} - \)\(22\!\cdots\!42\)\( T^{11} + \)\(12\!\cdots\!41\)\( T^{12} \))
$17$ (\( ( 1 - 3420 T + 76827042 T^{2} - 320227879020 T^{3} + 3420519113736879 T^{4} - 12548863826678015160 T^{5} + \)\(10\!\cdots\!96\)\( T^{6} - \)\(30\!\cdots\!40\)\( T^{7} + \)\(19\!\cdots\!19\)\( T^{8} - \)\(45\!\cdots\!80\)\( T^{9} + \)\(26\!\cdots\!82\)\( T^{10} - \)\(28\!\cdots\!80\)\( T^{11} + \)\(19\!\cdots\!81\)\( T^{12} )^{2} \))(\( ( 1 - 4913 T )( 1 + 4913 T ) \))(\( ( 1 - 4913 T )( 1 + 4913 T ) \))(\( ( 1 - 990 T + 24137569 T^{2} )( 1 + 990 T + 24137569 T^{2} ) \))(\( ( 1 - 80718662 T^{2} + 3795004300453199 T^{4} - \)\(11\!\cdots\!16\)\( T^{6} + \)\(22\!\cdots\!39\)\( T^{8} - \)\(27\!\cdots\!02\)\( T^{10} + \)\(19\!\cdots\!81\)\( T^{12} )^{2} \))(\( 1 + 3278 T + 5372642 T^{2} + 10955616782 T^{3} - 412886782828225 T^{4} - 729783346723198108 T^{5} - \)\(11\!\cdots\!12\)\( T^{6} - \)\(17\!\cdots\!52\)\( T^{7} - \)\(24\!\cdots\!25\)\( T^{8} + \)\(15\!\cdots\!38\)\( T^{9} + \)\(18\!\cdots\!82\)\( T^{10} + \)\(26\!\cdots\!22\)\( T^{11} + \)\(19\!\cdots\!81\)\( T^{12} \))
$19$ (\( 1 - 164360172 T^{2} + 18312419319379746 T^{4} - \)\(14\!\cdots\!80\)\( T^{6} + \)\(99\!\cdots\!55\)\( T^{8} - \)\(56\!\cdots\!12\)\( T^{10} + \)\(28\!\cdots\!44\)\( T^{12} - \)\(12\!\cdots\!32\)\( T^{14} + \)\(48\!\cdots\!55\)\( T^{16} - \)\(16\!\cdots\!80\)\( T^{18} + \)\(43\!\cdots\!86\)\( T^{20} - \)\(87\!\cdots\!72\)\( T^{22} + \)\(11\!\cdots\!61\)\( T^{24} \))(\( ( 1 - 6859 T )( 1 + 6859 T ) \))(\( ( 1 - 6859 T )( 1 + 6859 T ) \))(\( ( 1 - 6859 T )^{2}( 1 + 6859 T )^{2} \))(\( ( 1 - 151791126 T^{2} + 12887986688168415 T^{4} - \)\(73\!\cdots\!40\)\( T^{6} + \)\(28\!\cdots\!15\)\( T^{8} - \)\(74\!\cdots\!46\)\( T^{10} + \)\(10\!\cdots\!81\)\( T^{12} )^{2} \))(\( 1 - 167166534 T^{2} + 13715242279167135 T^{4} - \)\(74\!\cdots\!00\)\( T^{6} + \)\(30\!\cdots\!35\)\( T^{8} - \)\(81\!\cdots\!14\)\( T^{10} + \)\(10\!\cdots\!81\)\( T^{12} \))
$23$ (\( 1 - 803107776 T^{2} + 320722238817538602 T^{4} - \)\(83\!\cdots\!92\)\( T^{6} + \)\(16\!\cdots\!03\)\( T^{8} - \)\(26\!\cdots\!52\)\( T^{10} + \)\(40\!\cdots\!88\)\( T^{12} - \)\(58\!\cdots\!92\)\( T^{14} + \)\(78\!\cdots\!23\)\( T^{16} - \)\(87\!\cdots\!12\)\( T^{18} + \)\(73\!\cdots\!62\)\( T^{20} - \)\(40\!\cdots\!76\)\( T^{22} + \)\(11\!\cdots\!21\)\( T^{24} \))(\( 1 - 15356 T + 148035889 T^{2} \))(\( 1 + 15356 T + 148035889 T^{2} \))(\( ( 1 + 148035889 T^{2} )^{2} \))(\( ( 1 + 838151698 T^{2} + 299619665559292319 T^{4} + \)\(58\!\cdots\!64\)\( T^{6} + \)\(65\!\cdots\!99\)\( T^{8} + \)\(40\!\cdots\!18\)\( T^{10} + \)\(10\!\cdots\!61\)\( T^{12} )^{2} \))(\( 1 - 19984 T + 199680128 T^{2} - 3629111359576 T^{3} + 87866945710300079 T^{4} - \)\(99\!\cdots\!72\)\( T^{5} + \)\(89\!\cdots\!24\)\( T^{6} - \)\(14\!\cdots\!08\)\( T^{7} + \)\(19\!\cdots\!59\)\( T^{8} - \)\(11\!\cdots\!44\)\( T^{9} + \)\(95\!\cdots\!48\)\( T^{10} - \)\(14\!\cdots\!16\)\( T^{11} + \)\(10\!\cdots\!61\)\( T^{12} \))
$29$ (\( ( 1 + 37484 T + 3098539682 T^{2} + 85390956177628 T^{3} + 4145296982774939583 T^{4} + \)\(90\!\cdots\!68\)\( T^{5} + \)\(31\!\cdots\!68\)\( T^{6} + \)\(53\!\cdots\!28\)\( T^{7} + \)\(14\!\cdots\!03\)\( T^{8} + \)\(17\!\cdots\!08\)\( T^{9} + \)\(38\!\cdots\!42\)\( T^{10} + \)\(27\!\cdots\!84\)\( T^{11} + \)\(44\!\cdots\!21\)\( T^{12} )^{2} \))(\( 1 - 44858 T + 594823321 T^{2} \))(\( 1 - 44858 T + 594823321 T^{2} \))(\( ( 1 + 31878 T + 594823321 T^{2} )^{2} \))(\( ( 1 + 4018 T + 984038839 T^{2} - 4482562456036 T^{3} + 585329250206964319 T^{4} + \)\(14\!\cdots\!38\)\( T^{5} + \)\(21\!\cdots\!61\)\( T^{6} )^{4} \))(\( 1 - 1033291734 T^{2} + 1369068699154031775 T^{4} - \)\(75\!\cdots\!40\)\( T^{6} + \)\(48\!\cdots\!75\)\( T^{8} - \)\(12\!\cdots\!54\)\( T^{10} + \)\(44\!\cdots\!21\)\( T^{12} \))
$31$ (\( 1 - 6591556572 T^{2} + 20906700243515285346 T^{4} - \)\(42\!\cdots\!80\)\( T^{6} + \)\(63\!\cdots\!55\)\( T^{8} - \)\(74\!\cdots\!12\)\( T^{10} + \)\(72\!\cdots\!44\)\( T^{12} - \)\(59\!\cdots\!32\)\( T^{14} + \)\(39\!\cdots\!55\)\( T^{16} - \)\(20\!\cdots\!80\)\( T^{18} + \)\(80\!\cdots\!86\)\( T^{20} - \)\(19\!\cdots\!72\)\( T^{22} + \)\(23\!\cdots\!61\)\( T^{24} \))(\( ( 1 - 29791 T )( 1 + 29791 T ) \))(\( ( 1 - 29791 T )( 1 + 29791 T ) \))(\( ( 1 - 29791 T )^{2}( 1 + 29791 T )^{2} \))(\( ( 1 - 1079579526 T^{2} + 1559430337965859215 T^{4} - \)\(97\!\cdots\!40\)\( T^{6} + \)\(12\!\cdots\!15\)\( T^{8} - \)\(66\!\cdots\!46\)\( T^{10} + \)\(48\!\cdots\!81\)\( T^{12} )^{2} \))(\( ( 1 - 52488 T + 2507733591 T^{2} - 91420641186592 T^{3} + 2225622792979848471 T^{4} - \)\(41\!\cdots\!68\)\( T^{5} + \)\(69\!\cdots\!41\)\( T^{6} )^{2} \))
$37$ (\( ( 1 - 31320 T + 8638032522 T^{2} - 291678076865720 T^{3} + 33106290806049613119 T^{4} - \)\(11\!\cdots\!00\)\( T^{5} + \)\(89\!\cdots\!36\)\( T^{6} - \)\(30\!\cdots\!00\)\( T^{7} + \)\(21\!\cdots\!39\)\( T^{8} - \)\(49\!\cdots\!80\)\( T^{9} + \)\(37\!\cdots\!42\)\( T^{10} - \)\(34\!\cdots\!80\)\( T^{11} + \)\(28\!\cdots\!41\)\( T^{12} )^{2} \))(\( ( 1 - 50653 T )( 1 + 50653 T ) \))(\( ( 1 - 50653 T )( 1 + 50653 T ) \))(\( ( 1 - 55510 T + 2565726409 T^{2} )( 1 + 55510 T + 2565726409 T^{2} ) \))(\( ( 1 - 14610185142 T^{2} + 90894099761693350719 T^{4} - \)\(30\!\cdots\!96\)\( T^{6} + \)\(59\!\cdots\!39\)\( T^{8} - \)\(63\!\cdots\!62\)\( T^{10} + \)\(28\!\cdots\!41\)\( T^{12} )^{2} \))(\( 1 + 241554 T + 29174167458 T^{2} + 2703692719264386 T^{3} + \)\(21\!\cdots\!19\)\( T^{4} + \)\(13\!\cdots\!52\)\( T^{5} + \)\(73\!\cdots\!04\)\( T^{6} + \)\(35\!\cdots\!68\)\( T^{7} + \)\(14\!\cdots\!39\)\( T^{8} + \)\(45\!\cdots\!94\)\( T^{9} + \)\(12\!\cdots\!38\)\( T^{10} + \)\(26\!\cdots\!46\)\( T^{11} + \)\(28\!\cdots\!41\)\( T^{12} \))
$41$ (\( ( 1 + 8488 T + 9329931386 T^{2} - 230911529543160 T^{3} + 53548630987540978735 T^{4} - \)\(20\!\cdots\!72\)\( T^{5} + \)\(29\!\cdots\!24\)\( T^{6} - \)\(96\!\cdots\!52\)\( T^{7} + \)\(12\!\cdots\!35\)\( T^{8} - \)\(24\!\cdots\!60\)\( T^{9} + \)\(47\!\cdots\!46\)\( T^{10} + \)\(20\!\cdots\!88\)\( T^{11} + \)\(11\!\cdots\!41\)\( T^{12} )^{2} \))(\( 1 + 74338 T + 4750104241 T^{2} \))(\( 1 + 74338 T + 4750104241 T^{2} \))(\( ( 1 - 84942 T + 4750104241 T^{2} )^{2} \))(\( ( 1 + 48034 T + 14394061135 T^{2} + 453005557569020 T^{3} + 68373290842576773535 T^{4} + \)\(10\!\cdots\!54\)\( T^{5} + \)\(10\!\cdots\!21\)\( T^{6} )^{4} \))(\( ( 1 - 175868 T + 20110360631 T^{2} - 1685864350808792 T^{3} + 95526309321352536071 T^{4} - \)\(39\!\cdots\!08\)\( T^{5} + \)\(10\!\cdots\!21\)\( T^{6} )^{2} \))
$43$ (\( 1 - 19765937616 T^{2} + \)\(14\!\cdots\!82\)\( T^{4} - \)\(11\!\cdots\!72\)\( T^{6} + \)\(13\!\cdots\!23\)\( T^{8} - \)\(88\!\cdots\!12\)\( T^{10} + \)\(43\!\cdots\!88\)\( T^{12} - \)\(35\!\cdots\!12\)\( T^{14} + \)\(21\!\cdots\!23\)\( T^{16} - \)\(71\!\cdots\!72\)\( T^{18} + \)\(36\!\cdots\!82\)\( T^{20} - \)\(20\!\cdots\!16\)\( T^{22} + \)\(40\!\cdots\!01\)\( T^{24} \))(\( 1 + 17404 T + 6321363049 T^{2} \))(\( 1 - 17404 T + 6321363049 T^{2} \))(\( ( 1 + 6321363049 T^{2} )^{2} \))(\( ( 1 + 17834866498 T^{2} + \)\(21\!\cdots\!99\)\( T^{4} + \)\(15\!\cdots\!04\)\( T^{6} + \)\(85\!\cdots\!99\)\( T^{8} + \)\(28\!\cdots\!98\)\( T^{10} + \)\(63\!\cdots\!01\)\( T^{12} )^{2} \))(\( 1 - 60720 T + 1843459200 T^{2} - 170170651040280 T^{3} - 39718848676435556097 T^{4} + \)\(33\!\cdots\!60\)\( T^{5} - \)\(11\!\cdots\!00\)\( T^{6} + \)\(21\!\cdots\!40\)\( T^{7} - \)\(15\!\cdots\!97\)\( T^{8} - \)\(42\!\cdots\!20\)\( T^{9} + \)\(29\!\cdots\!00\)\( T^{10} - \)\(61\!\cdots\!80\)\( T^{11} + \)\(63\!\cdots\!01\)\( T^{12} \))
$47$ (\( 1 - 70204573056 T^{2} + \)\(25\!\cdots\!42\)\( T^{4} - \)\(65\!\cdots\!32\)\( T^{6} + \)\(12\!\cdots\!43\)\( T^{8} - \)\(18\!\cdots\!52\)\( T^{10} + \)\(21\!\cdots\!28\)\( T^{12} - \)\(21\!\cdots\!32\)\( T^{14} + \)\(16\!\cdots\!83\)\( T^{16} - \)\(10\!\cdots\!72\)\( T^{18} + \)\(47\!\cdots\!62\)\( T^{20} - \)\(14\!\cdots\!56\)\( T^{22} + \)\(24\!\cdots\!41\)\( T^{24} \))(\( 1 - 26444 T + 10779215329 T^{2} \))(\( 1 + 26444 T + 10779215329 T^{2} \))(\( ( 1 + 10779215329 T^{2} )^{2} \))(\( ( 1 + 27982006258 T^{2} + \)\(51\!\cdots\!99\)\( T^{4} + \)\(67\!\cdots\!24\)\( T^{6} + \)\(59\!\cdots\!59\)\( T^{8} + \)\(37\!\cdots\!98\)\( T^{10} + \)\(15\!\cdots\!21\)\( T^{12} )^{2} \))(\( 1 + 355248 T + 63100570752 T^{2} + 8734381777271592 T^{3} + \)\(12\!\cdots\!95\)\( T^{4} + \)\(15\!\cdots\!92\)\( T^{5} + \)\(17\!\cdots\!08\)\( T^{6} + \)\(16\!\cdots\!68\)\( T^{7} + \)\(14\!\cdots\!95\)\( T^{8} + \)\(10\!\cdots\!88\)\( T^{9} + \)\(85\!\cdots\!12\)\( T^{10} + \)\(51\!\cdots\!52\)\( T^{11} + \)\(15\!\cdots\!21\)\( T^{12} \))
$53$ (\( ( 1 - 161080 T + 126322752842 T^{2} - 16418509450658840 T^{3} + \)\(67\!\cdots\!59\)\( T^{4} - \)\(69\!\cdots\!60\)\( T^{5} + \)\(19\!\cdots\!56\)\( T^{6} - \)\(15\!\cdots\!40\)\( T^{7} + \)\(33\!\cdots\!19\)\( T^{8} - \)\(17\!\cdots\!60\)\( T^{9} + \)\(30\!\cdots\!02\)\( T^{10} - \)\(86\!\cdots\!20\)\( T^{11} + \)\(11\!\cdots\!21\)\( T^{12} )^{2} \))(\( ( 1 - 148877 T )( 1 + 148877 T ) \))(\( ( 1 - 148877 T )( 1 + 148877 T ) \))(\( ( 1 - 29430 T + 22164361129 T^{2} )( 1 + 29430 T + 22164361129 T^{2} ) \))(\( ( 1 - 53634231542 T^{2} + \)\(53\!\cdots\!59\)\( T^{4} + \)\(74\!\cdots\!44\)\( T^{6} + \)\(26\!\cdots\!19\)\( T^{8} - \)\(12\!\cdots\!02\)\( T^{10} + \)\(11\!\cdots\!21\)\( T^{12} )^{2} \))(\( 1 - 346526 T + 60040134338 T^{2} - 13491114635468654 T^{3} + \)\(34\!\cdots\!59\)\( T^{4} - \)\(52\!\cdots\!68\)\( T^{5} + \)\(67\!\cdots\!84\)\( T^{6} - \)\(11\!\cdots\!72\)\( T^{7} + \)\(16\!\cdots\!19\)\( T^{8} - \)\(14\!\cdots\!06\)\( T^{9} + \)\(14\!\cdots\!78\)\( T^{10} - \)\(18\!\cdots\!74\)\( T^{11} + \)\(11\!\cdots\!21\)\( T^{12} \))
$59$ (\( 1 - 259515214572 T^{2} + \)\(35\!\cdots\!66\)\( T^{4} - \)\(33\!\cdots\!80\)\( T^{6} + \)\(23\!\cdots\!55\)\( T^{8} - \)\(13\!\cdots\!12\)\( T^{10} + \)\(63\!\cdots\!44\)\( T^{12} - \)\(24\!\cdots\!72\)\( T^{14} + \)\(75\!\cdots\!55\)\( T^{16} - \)\(18\!\cdots\!80\)\( T^{18} + \)\(35\!\cdots\!86\)\( T^{20} - \)\(46\!\cdots\!72\)\( T^{22} + \)\(31\!\cdots\!81\)\( T^{24} \))(\( ( 1 - 205379 T )( 1 + 205379 T ) \))(\( ( 1 - 205379 T )( 1 + 205379 T ) \))(\( ( 1 - 205379 T )^{2}( 1 + 205379 T )^{2} \))(\( ( 1 - 146094395766 T^{2} + \)\(10\!\cdots\!35\)\( T^{4} - \)\(50\!\cdots\!20\)\( T^{6} + \)\(18\!\cdots\!35\)\( T^{8} - \)\(46\!\cdots\!26\)\( T^{10} + \)\(56\!\cdots\!41\)\( T^{12} )^{2} \))(\( 1 - 133435411878 T^{2} + \)\(95\!\cdots\!71\)\( T^{4} - \)\(47\!\cdots\!12\)\( T^{6} + \)\(16\!\cdots\!51\)\( T^{8} - \)\(42\!\cdots\!58\)\( T^{10} + \)\(56\!\cdots\!41\)\( T^{12} \))
$61$ (\( ( 1 - 23232 T + 90461195706 T^{2} - 6669131863130560 T^{3} + \)\(44\!\cdots\!35\)\( T^{4} - \)\(87\!\cdots\!72\)\( T^{5} - \)\(16\!\cdots\!36\)\( T^{6} - \)\(44\!\cdots\!92\)\( T^{7} + \)\(11\!\cdots\!35\)\( T^{8} - \)\(91\!\cdots\!60\)\( T^{9} + \)\(63\!\cdots\!46\)\( T^{10} - \)\(84\!\cdots\!32\)\( T^{11} + \)\(18\!\cdots\!61\)\( T^{12} )^{2} \))(\( 1 - 452342 T + 51520374361 T^{2} \))(\( 1 - 452342 T + 51520374361 T^{2} \))(\( ( 1 + 234938 T + 51520374361 T^{2} )^{2} \))(\( ( 1 - 53846 T + 87983744615 T^{2} - 7715289154258100 T^{3} + \)\(45\!\cdots\!15\)\( T^{4} - \)\(14\!\cdots\!66\)\( T^{5} + \)\(13\!\cdots\!81\)\( T^{6} )^{4} \))(\( ( 1 + 246444 T + 82272859695 T^{2} + 7520154028705560 T^{3} + \)\(42\!\cdots\!95\)\( T^{4} + \)\(65\!\cdots\!24\)\( T^{5} + \)\(13\!\cdots\!81\)\( T^{6} )^{2} \))
$67$ (\( 1 - 562871355216 T^{2} + \)\(16\!\cdots\!62\)\( T^{4} - \)\(34\!\cdots\!72\)\( T^{6} + \)\(52\!\cdots\!83\)\( T^{8} - \)\(65\!\cdots\!72\)\( T^{10} + \)\(65\!\cdots\!08\)\( T^{12} - \)\(53\!\cdots\!92\)\( T^{14} + \)\(35\!\cdots\!43\)\( T^{16} - \)\(18\!\cdots\!32\)\( T^{18} + \)\(74\!\cdots\!42\)\( T^{20} - \)\(20\!\cdots\!16\)\( T^{22} + \)\(30\!\cdots\!61\)\( T^{24} \))(\( 1 + 1276 T + 90458382169 T^{2} \))(\( 1 - 1276 T + 90458382169 T^{2} \))(\( ( 1 + 90458382169 T^{2} )^{2} \))(\( ( 1 + 283076108578 T^{2} + \)\(40\!\cdots\!79\)\( T^{4} + \)\(40\!\cdots\!64\)\( T^{6} + \)\(32\!\cdots\!19\)\( T^{8} + \)\(18\!\cdots\!38\)\( T^{10} + \)\(54\!\cdots\!81\)\( T^{12} )^{2} \))(\( 1 + 230304 T + 26519966208 T^{2} - 6500435613362824 T^{3} - \)\(29\!\cdots\!61\)\( T^{4} + \)\(19\!\cdots\!12\)\( T^{5} + \)\(53\!\cdots\!24\)\( T^{6} + \)\(17\!\cdots\!28\)\( T^{7} - \)\(23\!\cdots\!21\)\( T^{8} - \)\(48\!\cdots\!16\)\( T^{9} + \)\(17\!\cdots\!68\)\( T^{10} + \)\(13\!\cdots\!96\)\( T^{11} + \)\(54\!\cdots\!81\)\( T^{12} \))
$71$ (\( 1 - 544175734812 T^{2} + \)\(20\!\cdots\!26\)\( T^{4} - \)\(51\!\cdots\!80\)\( T^{6} + \)\(10\!\cdots\!55\)\( T^{8} - \)\(17\!\cdots\!12\)\( T^{10} + \)\(25\!\cdots\!24\)\( T^{12} - \)\(29\!\cdots\!92\)\( T^{14} + \)\(29\!\cdots\!55\)\( T^{16} - \)\(22\!\cdots\!80\)\( T^{18} + \)\(14\!\cdots\!86\)\( T^{20} - \)\(64\!\cdots\!12\)\( T^{22} + \)\(19\!\cdots\!41\)\( T^{24} \))(\( ( 1 - 357911 T )( 1 + 357911 T ) \))(\( ( 1 - 357911 T )( 1 + 357911 T ) \))(\( ( 1 - 357911 T )^{2}( 1 + 357911 T )^{2} \))(\( ( 1 - 612196656486 T^{2} + \)\(17\!\cdots\!95\)\( T^{4} - \)\(28\!\cdots\!60\)\( T^{6} + \)\(28\!\cdots\!95\)\( T^{8} - \)\(16\!\cdots\!66\)\( T^{10} + \)\(44\!\cdots\!21\)\( T^{12} )^{2} \))(\( ( 1 - 87064 T + 266865978695 T^{2} - 4191185495253760 T^{3} + \)\(34\!\cdots\!95\)\( T^{4} - \)\(14\!\cdots\!24\)\( T^{5} + \)\(21\!\cdots\!61\)\( T^{6} )^{2} \))
$73$ (\( ( 1 + 207540 T + 646443061602 T^{2} + 138932210718109860 T^{3} + \)\(20\!\cdots\!19\)\( T^{4} + \)\(38\!\cdots\!80\)\( T^{5} + \)\(39\!\cdots\!36\)\( T^{6} + \)\(58\!\cdots\!20\)\( T^{7} + \)\(46\!\cdots\!99\)\( T^{8} + \)\(48\!\cdots\!40\)\( T^{9} + \)\(33\!\cdots\!82\)\( T^{10} + \)\(16\!\cdots\!60\)\( T^{11} + \)\(12\!\cdots\!61\)\( T^{12} )^{2} \))(\( ( 1 - 389017 T )( 1 + 389017 T ) \))(\( ( 1 - 389017 T )( 1 + 389017 T ) \))(\( ( 1 - 427570 T + 151334226289 T^{2} )( 1 + 427570 T + 151334226289 T^{2} ) \))(\( ( 1 - 278917873062 T^{2} + \)\(54\!\cdots\!39\)\( T^{4} - \)\(10\!\cdots\!76\)\( T^{6} + \)\(12\!\cdots\!19\)\( T^{8} - \)\(14\!\cdots\!42\)\( T^{10} + \)\(12\!\cdots\!61\)\( T^{12} )^{2} \))(\( 1 + 332442 T + 55258841682 T^{2} - 3861289507981862 T^{3} - \)\(18\!\cdots\!25\)\( T^{4} - \)\(17\!\cdots\!32\)\( T^{5} + \)\(42\!\cdots\!28\)\( T^{6} - \)\(27\!\cdots\!48\)\( T^{7} - \)\(41\!\cdots\!25\)\( T^{8} - \)\(13\!\cdots\!78\)\( T^{9} + \)\(28\!\cdots\!62\)\( T^{10} + \)\(26\!\cdots\!58\)\( T^{11} + \)\(12\!\cdots\!61\)\( T^{12} \))
$79$ (\( 1 - 1500338197452 T^{2} + \)\(10\!\cdots\!26\)\( T^{4} - \)\(48\!\cdots\!80\)\( T^{6} + \)\(15\!\cdots\!55\)\( T^{8} - \)\(41\!\cdots\!12\)\( T^{10} + \)\(10\!\cdots\!04\)\( T^{12} - \)\(24\!\cdots\!92\)\( T^{14} + \)\(54\!\cdots\!55\)\( T^{16} - \)\(99\!\cdots\!80\)\( T^{18} + \)\(13\!\cdots\!86\)\( T^{20} - \)\(10\!\cdots\!52\)\( T^{22} + \)\(42\!\cdots\!41\)\( T^{24} \))(\( ( 1 - 493039 T )( 1 + 493039 T ) \))(\( ( 1 - 493039 T )( 1 + 493039 T ) \))(\( ( 1 - 493039 T )^{2}( 1 + 493039 T )^{2} \))(\( ( 1 - 833901716166 T^{2} + \)\(35\!\cdots\!15\)\( T^{4} - \)\(10\!\cdots\!40\)\( T^{6} + \)\(21\!\cdots\!15\)\( T^{8} - \)\(29\!\cdots\!46\)\( T^{10} + \)\(20\!\cdots\!21\)\( T^{12} )^{2} \))(\( 1 - 240117880518 T^{2} + \)\(85\!\cdots\!31\)\( T^{4} - \)\(34\!\cdots\!92\)\( T^{6} + \)\(50\!\cdots\!71\)\( T^{8} - \)\(83\!\cdots\!58\)\( T^{10} + \)\(20\!\cdots\!21\)\( T^{12} \))
$83$ (\( 1 - 1532795707536 T^{2} + \)\(12\!\cdots\!02\)\( T^{4} - \)\(80\!\cdots\!92\)\( T^{6} + \)\(40\!\cdots\!63\)\( T^{8} - \)\(16\!\cdots\!32\)\( T^{10} + \)\(60\!\cdots\!68\)\( T^{12} - \)\(18\!\cdots\!52\)\( T^{14} + \)\(46\!\cdots\!23\)\( T^{16} - \)\(98\!\cdots\!52\)\( T^{18} + \)\(16\!\cdots\!82\)\( T^{20} - \)\(21\!\cdots\!36\)\( T^{22} + \)\(14\!\cdots\!61\)\( T^{24} \))(\( 1 - 1131716 T + 326940373369 T^{2} \))(\( 1 + 1131716 T + 326940373369 T^{2} \))(\( ( 1 + 326940373369 T^{2} )^{2} \))(\( ( 1 + 772762953058 T^{2} + \)\(16\!\cdots\!79\)\( T^{4} + \)\(14\!\cdots\!04\)\( T^{6} + \)\(17\!\cdots\!19\)\( T^{8} + \)\(88\!\cdots\!18\)\( T^{10} + \)\(12\!\cdots\!81\)\( T^{12} )^{2} \))(\( 1 + 2190936 T + 2400100278048 T^{2} + 2073409539706878384 T^{3} + \)\(15\!\cdots\!39\)\( T^{4} + \)\(10\!\cdots\!08\)\( T^{5} + \)\(60\!\cdots\!44\)\( T^{6} + \)\(34\!\cdots\!52\)\( T^{7} + \)\(17\!\cdots\!79\)\( T^{8} + \)\(72\!\cdots\!56\)\( T^{9} + \)\(27\!\cdots\!08\)\( T^{10} + \)\(81\!\cdots\!64\)\( T^{11} + \)\(12\!\cdots\!81\)\( T^{12} \))
$89$ (\( ( 1 - 139196 T + 1314507916802 T^{2} + 63117339567205748 T^{3} + \)\(10\!\cdots\!83\)\( T^{4} + \)\(78\!\cdots\!88\)\( T^{5} + \)\(61\!\cdots\!28\)\( T^{6} + \)\(38\!\cdots\!68\)\( T^{7} + \)\(25\!\cdots\!43\)\( T^{8} + \)\(77\!\cdots\!88\)\( T^{9} + \)\(80\!\cdots\!82\)\( T^{10} - \)\(42\!\cdots\!96\)\( T^{11} + \)\(15\!\cdots\!61\)\( T^{12} )^{2} \))(\( 1 - 511058 T + 496981290961 T^{2} \))(\( 1 - 511058 T + 496981290961 T^{2} \))(\( ( 1 - 1378962 T + 496981290961 T^{2} )^{2} \))(\( ( 1 + 1086538 T + 1626117919039 T^{2} + 1075366431700477964 T^{3} + \)\(80\!\cdots\!79\)\( T^{4} + \)\(26\!\cdots\!98\)\( T^{5} + \)\(12\!\cdots\!81\)\( T^{6} )^{4} \))(\( 1 - 2595301222374 T^{2} + \)\(29\!\cdots\!55\)\( T^{4} - \)\(19\!\cdots\!20\)\( T^{6} + \)\(73\!\cdots\!55\)\( T^{8} - \)\(15\!\cdots\!34\)\( T^{10} + \)\(15\!\cdots\!61\)\( T^{12} \))
$97$ (\( ( 1 - 1172340 T + 3445180935762 T^{2} - 3393805340997736740 T^{3} + \)\(56\!\cdots\!99\)\( T^{4} - \)\(47\!\cdots\!60\)\( T^{5} + \)\(57\!\cdots\!36\)\( T^{6} - \)\(39\!\cdots\!40\)\( T^{7} + \)\(38\!\cdots\!59\)\( T^{8} - \)\(19\!\cdots\!60\)\( T^{9} + \)\(16\!\cdots\!22\)\( T^{10} - \)\(47\!\cdots\!60\)\( T^{11} + \)\(33\!\cdots\!21\)\( T^{12} )^{2} \))(\( ( 1 - 912673 T )( 1 + 912673 T ) \))(\( ( 1 - 912673 T )( 1 + 912673 T ) \))(\( ( 1 - 1472510 T + 832972004929 T^{2} )( 1 + 1472510 T + 832972004929 T^{2} ) \))(\( ( 1 - 2704280608902 T^{2} + \)\(40\!\cdots\!19\)\( T^{4} - \)\(41\!\cdots\!16\)\( T^{6} + \)\(28\!\cdots\!79\)\( T^{8} - \)\(13\!\cdots\!62\)\( T^{10} + \)\(33\!\cdots\!21\)\( T^{12} )^{2} \))(\( 1 - 3338406 T + 5572477310418 T^{2} - 7517418539616763974 T^{3} + \)\(94\!\cdots\!59\)\( T^{4} - \)\(10\!\cdots\!08\)\( T^{5} + \)\(96\!\cdots\!24\)\( T^{6} - \)\(85\!\cdots\!32\)\( T^{7} + \)\(65\!\cdots\!19\)\( T^{8} - \)\(43\!\cdots\!86\)\( T^{9} + \)\(26\!\cdots\!58\)\( T^{10} - \)\(13\!\cdots\!94\)\( T^{11} + \)\(33\!\cdots\!21\)\( T^{12} \))
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