Properties

Label 105.3.n
Level 105
Weight 3
Character orbit n
Rep. character \(\chi_{105}(31,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 20
Newform subspaces 2
Sturm bound 48
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 105.n (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(48\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(105, [\chi])\).

Total New Old
Modular forms 72 20 52
Cusp forms 56 20 36
Eisenstein series 16 0 16

Trace form

\( 20q + 4q^{2} + 6q^{3} - 28q^{4} + 6q^{7} + 8q^{8} + 30q^{9} + O(q^{10}) \) \( 20q + 4q^{2} + 6q^{3} - 28q^{4} + 6q^{7} + 8q^{8} + 30q^{9} + 40q^{11} - 48q^{12} + 16q^{14} - 84q^{16} - 96q^{17} - 12q^{18} - 6q^{19} + 84q^{21} + 40q^{22} + 64q^{23} + 108q^{24} + 50q^{25} + 156q^{26} - 248q^{28} - 200q^{29} - 18q^{31} - 72q^{32} - 60q^{35} - 168q^{36} - 114q^{37} + 240q^{38} - 54q^{39} + 204q^{42} + 228q^{43} + 216q^{44} + 196q^{46} + 24q^{47} - 22q^{49} + 40q^{50} - 60q^{51} - 492q^{52} - 152q^{53} + 308q^{56} - 12q^{57} - 92q^{58} - 324q^{59} - 456q^{61} - 54q^{63} + 696q^{64} - 120q^{65} + 108q^{66} + 166q^{67} - 204q^{68} - 300q^{70} - 464q^{71} + 12q^{72} + 582q^{73} + 104q^{74} + 30q^{75} - 44q^{77} - 168q^{78} - 98q^{79} + 480q^{80} - 90q^{81} + 684q^{82} - 60q^{84} - 120q^{85} + 308q^{86} - 544q^{88} + 96q^{89} + 330q^{91} + 168q^{92} + 234q^{93} + 60q^{94} - 120q^{95} - 432q^{96} - 176q^{98} + 240q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(105, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
105.3.n.a \(8\) \(2.861\) 8.0.\(\cdots\).16 None \(2\) \(-12\) \(0\) \(-16\) \(q+\beta _{1}q^{2}+(-1+\beta _{4})q^{3}+(-1+\beta _{1}+\cdots)q^{4}+\cdots\)
105.3.n.b \(12\) \(2.861\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(2\) \(18\) \(0\) \(22\) \(q+(-\beta _{2}-\beta _{10})q^{2}+(2+\beta _{3})q^{3}+(4\beta _{3}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(105, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(105, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 2 T - 3 T^{2} + 14 T^{3} - 13 T^{4} - 18 T^{5} + 70 T^{6} - 44 T^{7} - 156 T^{8} - 176 T^{9} + 1120 T^{10} - 1152 T^{11} - 3328 T^{12} + 14336 T^{13} - 12288 T^{14} - 32768 T^{15} + 65536 T^{16} \))(\( 1 - 2 T + T^{2} - 18 T^{3} + 35 T^{4} - 10 T^{5} + 142 T^{6} - 148 T^{7} - 208 T^{8} - 424 T^{9} - 1176 T^{10} + 4080 T^{11} + 144 T^{12} + 16320 T^{13} - 18816 T^{14} - 27136 T^{15} - 53248 T^{16} - 151552 T^{17} + 581632 T^{18} - 163840 T^{19} + 2293760 T^{20} - 4718592 T^{21} + 1048576 T^{22} - 8388608 T^{23} + 16777216 T^{24} \))
$3$ (\( ( 1 + 3 T + 3 T^{2} )^{4} \))(\( ( 1 - 3 T + 3 T^{2} )^{6} \))
$5$ (\( ( 1 - 5 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - 5 T^{2} + 25 T^{4} )^{3} \))
$7$ (\( 1 + 16 T + 109 T^{2} + 784 T^{3} + 6664 T^{4} + 38416 T^{5} + 261709 T^{6} + 1882384 T^{7} + 5764801 T^{8} \))(\( 1 - 22 T + 272 T^{2} - 2568 T^{3} + 19600 T^{4} - 138362 T^{5} + 991858 T^{6} - 6779738 T^{7} + 47059600 T^{8} - 302122632 T^{9} + 1568025872 T^{10} - 6214455478 T^{11} + 13841287201 T^{12} \))
$11$ (\( 1 - 20 T - 147 T^{2} + 2960 T^{3} + 57131 T^{4} - 519480 T^{5} - 8882912 T^{6} + 8003440 T^{7} + 1602642534 T^{8} + 968416240 T^{9} - 130054714592 T^{10} - 920290508280 T^{11} + 12246537230411 T^{12} + 76774776818960 T^{13} - 461348971377987 T^{14} - 7594996671664820 T^{15} + 45949729863572161 T^{16} \))(\( 1 - 20 T - 77 T^{2} + 7872 T^{3} - 76264 T^{4} - 640168 T^{5} + 22115011 T^{6} - 150830740 T^{7} - 1492575946 T^{8} + 41825299028 T^{9} - 261075878769 T^{10} - 2649707325072 T^{11} + 64192943543664 T^{12} - 320614586333712 T^{13} - 3822411941056929 T^{14} + 74096068571342708 T^{15} - 319946909592076426 T^{16} - 3912160946263034740 T^{17} + 69406378073897058931 T^{18} - \)\(24\!\cdots\!88\)\( T^{19} - \)\(35\!\cdots\!04\)\( T^{20} + \)\(43\!\cdots\!32\)\( T^{21} - \)\(51\!\cdots\!77\)\( T^{22} - \)\(16\!\cdots\!20\)\( T^{23} + \)\(98\!\cdots\!41\)\( T^{24} \))
$13$ (\( 1 - 188 T^{2} + 39826 T^{4} - 8798048 T^{6} + 2113175419 T^{8} - 251281048928 T^{10} + 32487291694546 T^{12} - 4380040003026428 T^{14} + 665416609183179841 T^{16} \))(\( 1 - 1050 T^{2} + 558861 T^{4} - 197481550 T^{6} + 52237691370 T^{8} - 11158031023650 T^{10} + 2025783626034165 T^{12} - 318684524066467650 T^{14} + 42611889644625577770 T^{16} - \)\(46\!\cdots\!50\)\( T^{18} + \)\(37\!\cdots\!01\)\( T^{20} - \)\(19\!\cdots\!50\)\( T^{22} + \)\(54\!\cdots\!61\)\( T^{24} \))
$17$ (\( 1 + 18 T + 766 T^{2} + 11844 T^{3} + 262438 T^{4} + 1906254 T^{5} + 22625848 T^{6} - 382636314 T^{7} - 5169040877 T^{8} - 110581894746 T^{9} + 1889733450808 T^{10} + 46012337456526 T^{11} + 1830703831301158 T^{12} + 23877431756917956 T^{13} + 446288633717996926 T^{14} + 3030800878069216722 T^{15} + 48661191875666868481 T^{16} \))(\( 1 + 78 T + 3456 T^{2} + 111384 T^{3} + 2892825 T^{4} + 64908048 T^{5} + 1326815384 T^{6} + 25384395654 T^{7} + 459528789294 T^{8} + 7866410803470 T^{9} + 127737192900600 T^{10} + 2033291698334568 T^{11} + 33507834279238509 T^{12} + 587621300818690152 T^{13} + 10668738088251012600 T^{14} + \)\(18\!\cdots\!30\)\( T^{15} + \)\(32\!\cdots\!54\)\( T^{16} + \)\(51\!\cdots\!46\)\( T^{17} + \)\(77\!\cdots\!24\)\( T^{18} + \)\(10\!\cdots\!92\)\( T^{19} + \)\(14\!\cdots\!25\)\( T^{20} + \)\(15\!\cdots\!56\)\( T^{21} + \)\(14\!\cdots\!56\)\( T^{22} + \)\(91\!\cdots\!42\)\( T^{23} + \)\(33\!\cdots\!21\)\( T^{24} \))
$19$ (\( 1 + 598 T^{2} + 183481 T^{4} - 682560 T^{5} - 48973562 T^{6} - 501474240 T^{7} - 32269961996 T^{8} - 181032200640 T^{9} - 6382283573402 T^{10} - 32111636535360 T^{11} + 3116161130325721 T^{12} + 1323562321601564278 T^{14} + \)\(28\!\cdots\!81\)\( T^{16} \))(\( 1 + 6 T + 1695 T^{2} + 10098 T^{3} + 1583475 T^{4} + 6257352 T^{5} + 993857576 T^{6} + 1495351404 T^{7} + 468684238005 T^{8} - 646226336898 T^{9} + 184440578012361 T^{10} - 649631264996682 T^{11} + 67037012122357518 T^{12} - 234516886663802202 T^{13} + 24036480567148897881 T^{14} - 30402287344769217138 T^{15} + \)\(79\!\cdots\!05\)\( T^{16} + \)\(91\!\cdots\!04\)\( T^{17} + \)\(21\!\cdots\!36\)\( T^{18} + \)\(49\!\cdots\!92\)\( T^{19} + \)\(45\!\cdots\!75\)\( T^{20} + \)\(10\!\cdots\!18\)\( T^{21} + \)\(63\!\cdots\!95\)\( T^{22} + \)\(81\!\cdots\!66\)\( T^{23} + \)\(48\!\cdots\!21\)\( T^{24} \))
$23$ (\( 1 - 62 T + 1497 T^{2} + 6014 T^{3} - 1196893 T^{4} + 31086552 T^{5} - 143771420 T^{6} - 13896561704 T^{7} + 513552019554 T^{8} - 7351281141416 T^{9} - 40233137944220 T^{10} + 4601925361264728 T^{11} - 93729870105931933 T^{12} + 249139038438885086 T^{13} + 32806192774734420537 T^{14} - \)\(71\!\cdots\!58\)\( T^{15} + \)\(61\!\cdots\!61\)\( T^{16} \))(\( 1 - 2 T - 1961 T^{2} - 16242 T^{3} + 1952552 T^{4} + 28088402 T^{5} - 1236738581 T^{6} - 18913143826 T^{7} + 643866095426 T^{8} + 6409263044018 T^{9} - 382808809220061 T^{10} - 936062983320174 T^{11} + 225254373554835492 T^{12} - 495177318176372046 T^{13} - \)\(10\!\cdots\!01\)\( T^{14} + \)\(94\!\cdots\!02\)\( T^{15} + \)\(50\!\cdots\!06\)\( T^{16} - \)\(78\!\cdots\!74\)\( T^{17} - \)\(27\!\cdots\!01\)\( T^{18} + \)\(32\!\cdots\!18\)\( T^{19} + \)\(11\!\cdots\!72\)\( T^{20} - \)\(52\!\cdots\!98\)\( T^{21} - \)\(33\!\cdots\!61\)\( T^{22} - \)\(18\!\cdots\!58\)\( T^{23} + \)\(48\!\cdots\!41\)\( T^{24} \))
$29$ (\( ( 1 + 50 T + 1234 T^{2} - 15850 T^{3} - 1164374 T^{4} - 13329850 T^{5} + 872784754 T^{6} + 29741166050 T^{7} + 500246412961 T^{8} )^{2} \))(\( ( 1 + 50 T + 4152 T^{2} + 144894 T^{3} + 7271923 T^{4} + 198551984 T^{5} + 7580628328 T^{6} + 166982218544 T^{7} + 5143292971363 T^{8} + 86186330272974 T^{9} + 2077023106614072 T^{10} + 21035361665010050 T^{11} + 353814783205469041 T^{12} )^{2} \))
$31$ (\( 1 + 126 T + 9883 T^{2} + 578466 T^{3} + 27206317 T^{4} + 1079090100 T^{5} + 38160094402 T^{6} + 1243487527488 T^{7} + 38998740329170 T^{8} + 1194991513915968 T^{9} + 35241648542229442 T^{10} + 957696435880658100 T^{11} + 23204023931078714797 T^{12} + \)\(47\!\cdots\!66\)\( T^{13} + \)\(77\!\cdots\!63\)\( T^{14} + \)\(95\!\cdots\!46\)\( T^{15} + \)\(72\!\cdots\!81\)\( T^{16} \))(\( 1 - 108 T + 7296 T^{2} - 368064 T^{3} + 14679372 T^{4} - 497355660 T^{5} + 14549964692 T^{6} - 358253632404 T^{7} + 6912430087356 T^{8} - 59089176847584 T^{9} - 2401476398747496 T^{10} + 161517767136947820 T^{11} - 6040783753357557738 T^{12} + \)\(15\!\cdots\!20\)\( T^{13} - \)\(22\!\cdots\!16\)\( T^{14} - \)\(52\!\cdots\!04\)\( T^{15} + \)\(58\!\cdots\!96\)\( T^{16} - \)\(29\!\cdots\!04\)\( T^{17} + \)\(11\!\cdots\!12\)\( T^{18} - \)\(37\!\cdots\!60\)\( T^{19} + \)\(10\!\cdots\!32\)\( T^{20} - \)\(25\!\cdots\!24\)\( T^{21} + \)\(49\!\cdots\!96\)\( T^{22} - \)\(69\!\cdots\!88\)\( T^{23} + \)\(62\!\cdots\!21\)\( T^{24} \))
$37$ (\( 1 + 80 T + 1194 T^{2} + 28960 T^{3} + 3461705 T^{4} - 28416960 T^{5} - 6540374054 T^{6} - 198858748720 T^{7} - 6603314864556 T^{8} - 272237626997680 T^{9} - 12257713977418694 T^{10} - 72910144735496640 T^{11} + 12159167688035595305 T^{12} + \)\(13\!\cdots\!40\)\( T^{13} + \)\(78\!\cdots\!14\)\( T^{14} + \)\(72\!\cdots\!20\)\( T^{15} + \)\(12\!\cdots\!41\)\( T^{16} \))(\( 1 + 34 T - 4909 T^{2} - 226690 T^{3} + 12133431 T^{4} + 708409644 T^{5} - 18171294956 T^{6} - 1398986295020 T^{7} + 16325526424229 T^{8} + 1786096223883250 T^{9} - 2380658436212215 T^{10} - 1019433693058130642 T^{11} - 10072912633170313306 T^{12} - \)\(13\!\cdots\!98\)\( T^{13} - \)\(44\!\cdots\!15\)\( T^{14} + \)\(45\!\cdots\!50\)\( T^{15} + \)\(57\!\cdots\!09\)\( T^{16} - \)\(67\!\cdots\!80\)\( T^{17} - \)\(11\!\cdots\!36\)\( T^{18} + \)\(63\!\cdots\!16\)\( T^{19} + \)\(14\!\cdots\!71\)\( T^{20} - \)\(38\!\cdots\!10\)\( T^{21} - \)\(11\!\cdots\!09\)\( T^{22} + \)\(10\!\cdots\!46\)\( T^{23} + \)\(43\!\cdots\!61\)\( T^{24} \))
$41$ (\( 1 - 10106 T^{2} + 48877645 T^{4} - 146585251874 T^{6} + 296639674915264 T^{8} - 414214887920726114 T^{10} + \)\(39\!\cdots\!45\)\( T^{12} - \)\(22\!\cdots\!86\)\( T^{14} + \)\(63\!\cdots\!41\)\( T^{16} \))(\( 1 - 10494 T^{2} + 51364035 T^{4} - 152997660886 T^{6} + 310636210241223 T^{8} - 486286748354299068 T^{10} + \)\(74\!\cdots\!22\)\( T^{12} - \)\(13\!\cdots\!48\)\( T^{14} + \)\(24\!\cdots\!83\)\( T^{16} - \)\(34\!\cdots\!66\)\( T^{18} + \)\(32\!\cdots\!35\)\( T^{20} - \)\(18\!\cdots\!94\)\( T^{22} + \)\(50\!\cdots\!61\)\( T^{24} \))
$43$ (\( ( 1 - 176 T + 17017 T^{2} - 1139948 T^{3} + 56853640 T^{4} - 2107763852 T^{5} + 58177736617 T^{6} - 1112559896624 T^{7} + 11688200277601 T^{8} )^{2} \))(\( ( 1 + 62 T + 7340 T^{2} + 336972 T^{3} + 27181828 T^{4} + 1019437474 T^{5} + 62011212778 T^{6} + 1884939889426 T^{7} + 92929260748228 T^{8} + 2130122349347628 T^{9} + 85791390037591340 T^{10} + 1339911903423623438 T^{11} + 39959630797262576401 T^{12} )^{2} \))
$47$ (\( 1 + 72 T + 10951 T^{2} + 664056 T^{3} + 65519473 T^{4} + 3342900456 T^{5} + 246192812578 T^{6} + 10750018584384 T^{7} + 651041931981118 T^{8} + 23746791052904256 T^{9} + 1201342389873427618 T^{10} + 36033843838636290024 T^{11} + \)\(15\!\cdots\!53\)\( T^{12} + \)\(34\!\cdots\!44\)\( T^{13} + \)\(12\!\cdots\!91\)\( T^{14} + \)\(18\!\cdots\!68\)\( T^{15} + \)\(56\!\cdots\!21\)\( T^{16} \))(\( 1 - 96 T + 10053 T^{2} - 670176 T^{3} + 37816182 T^{4} - 1580720664 T^{5} + 53466325187 T^{6} - 1308296276496 T^{7} + 51809528884176 T^{8} - 4886895580883496 T^{9} + 455990427411620625 T^{10} - 31230646812310303944 T^{11} + \)\(16\!\cdots\!56\)\( T^{12} - \)\(68\!\cdots\!96\)\( T^{13} + \)\(22\!\cdots\!25\)\( T^{14} - \)\(52\!\cdots\!84\)\( T^{15} + \)\(12\!\cdots\!36\)\( T^{16} - \)\(68\!\cdots\!04\)\( T^{17} + \)\(62\!\cdots\!67\)\( T^{18} - \)\(40\!\cdots\!16\)\( T^{19} + \)\(21\!\cdots\!22\)\( T^{20} - \)\(83\!\cdots\!64\)\( T^{21} + \)\(27\!\cdots\!53\)\( T^{22} - \)\(58\!\cdots\!64\)\( T^{23} + \)\(13\!\cdots\!81\)\( T^{24} \))
$53$ (\( 1 + 76 T - 3069 T^{2} - 443764 T^{3} + 2229785 T^{4} + 1396117872 T^{5} + 34651196266 T^{6} - 1991894657480 T^{7} - 169369280357850 T^{8} - 5595232092861320 T^{9} + 273414605764143946 T^{10} + 30944060693658997488 T^{11} + \)\(13\!\cdots\!85\)\( T^{12} - \)\(77\!\cdots\!36\)\( T^{13} - \)\(15\!\cdots\!29\)\( T^{14} + \)\(10\!\cdots\!44\)\( T^{15} + \)\(38\!\cdots\!21\)\( T^{16} \))(\( 1 + 76 T - 11231 T^{2} - 724524 T^{3} + 92836166 T^{4} + 4449767804 T^{5} - 540109351793 T^{6} - 16759328047612 T^{7} + 2565935011756424 T^{8} + 43462405312053452 T^{9} - 9681885823291571571 T^{10} - 46904208416898252492 T^{11} + \)\(30\!\cdots\!44\)\( T^{12} - \)\(13\!\cdots\!28\)\( T^{13} - \)\(76\!\cdots\!51\)\( T^{14} + \)\(96\!\cdots\!08\)\( T^{15} + \)\(15\!\cdots\!64\)\( T^{16} - \)\(29\!\cdots\!88\)\( T^{17} - \)\(26\!\cdots\!13\)\( T^{18} + \)\(61\!\cdots\!76\)\( T^{19} + \)\(35\!\cdots\!86\)\( T^{20} - \)\(78\!\cdots\!36\)\( T^{21} - \)\(34\!\cdots\!31\)\( T^{22} + \)\(65\!\cdots\!84\)\( T^{23} + \)\(24\!\cdots\!81\)\( T^{24} \))
$59$ (\( 1 + 54 T + 7198 T^{2} + 336204 T^{3} + 19932742 T^{4} - 202333950 T^{5} - 35478676088 T^{6} - 7641841019598 T^{7} - 385856896323245 T^{8} - 26601248589220638 T^{9} - 429907925960363768 T^{10} - 8534553984691411950 T^{11} + \)\(29\!\cdots\!82\)\( T^{12} + \)\(17\!\cdots\!04\)\( T^{13} + \)\(12\!\cdots\!38\)\( T^{14} + \)\(33\!\cdots\!94\)\( T^{15} + \)\(21\!\cdots\!41\)\( T^{16} \))(\( 1 + 270 T + 48372 T^{2} + 6499440 T^{3} + 729674841 T^{4} + 72198474768 T^{5} + 6488082937652 T^{6} + 539451035302806 T^{7} + 41805568251303822 T^{8} + 3030722232785131878 T^{9} + \)\(20\!\cdots\!28\)\( T^{10} + \)\(13\!\cdots\!68\)\( T^{11} + \)\(80\!\cdots\!17\)\( T^{12} + \)\(46\!\cdots\!08\)\( T^{13} + \)\(25\!\cdots\!08\)\( T^{14} + \)\(12\!\cdots\!98\)\( T^{15} + \)\(61\!\cdots\!62\)\( T^{16} + \)\(27\!\cdots\!06\)\( T^{17} + \)\(11\!\cdots\!12\)\( T^{18} + \)\(44\!\cdots\!48\)\( T^{19} + \)\(15\!\cdots\!81\)\( T^{20} + \)\(48\!\cdots\!40\)\( T^{21} + \)\(12\!\cdots\!72\)\( T^{22} + \)\(24\!\cdots\!70\)\( T^{23} + \)\(31\!\cdots\!61\)\( T^{24} \))
$61$ (\( 1 + 396 T + 83164 T^{2} + 12233232 T^{3} + 1413738778 T^{4} + 136250283708 T^{5} + 11318984386192 T^{6} + 825650586150588 T^{7} + 53403008176121923 T^{8} + 3072245831066337948 T^{9} + \)\(15\!\cdots\!72\)\( T^{10} + \)\(70\!\cdots\!88\)\( T^{11} + \)\(27\!\cdots\!18\)\( T^{12} + \)\(87\!\cdots\!32\)\( T^{13} + \)\(22\!\cdots\!44\)\( T^{14} + \)\(39\!\cdots\!36\)\( T^{15} + \)\(36\!\cdots\!61\)\( T^{16} \))(\( 1 + 60 T + 13086 T^{2} + 713160 T^{3} + 79444341 T^{4} + 3022528800 T^{5} + 251078089370 T^{6} + 4114433299500 T^{7} + 360604571968890 T^{8} - 6628548155521500 T^{9} + 198447695188114806 T^{10} - 25628280588886519440 T^{11} + \)\(58\!\cdots\!41\)\( T^{12} - \)\(95\!\cdots\!40\)\( T^{13} + \)\(27\!\cdots\!46\)\( T^{14} - \)\(34\!\cdots\!00\)\( T^{15} + \)\(69\!\cdots\!90\)\( T^{16} + \)\(29\!\cdots\!00\)\( T^{17} + \)\(66\!\cdots\!70\)\( T^{18} + \)\(29\!\cdots\!00\)\( T^{19} + \)\(29\!\cdots\!01\)\( T^{20} + \)\(97\!\cdots\!60\)\( T^{21} + \)\(66\!\cdots\!86\)\( T^{22} + \)\(11\!\cdots\!60\)\( T^{23} + \)\(70\!\cdots\!41\)\( T^{24} \))
$67$ (\( 1 - 184 T + 6231 T^{2} + 140176 T^{3} + 74370665 T^{4} - 7237038408 T^{5} + 24063966106 T^{6} - 15184872524680 T^{7} + 3087140085953070 T^{8} - 68164892763288520 T^{9} + 484915892741904826 T^{10} - \)\(65\!\cdots\!52\)\( T^{11} + \)\(30\!\cdots\!65\)\( T^{12} + \)\(25\!\cdots\!24\)\( T^{13} + \)\(50\!\cdots\!91\)\( T^{14} - \)\(67\!\cdots\!36\)\( T^{15} + \)\(16\!\cdots\!81\)\( T^{16} \))(\( 1 + 18 T - 15576 T^{2} + 716224 T^{3} + 161647428 T^{4} - 11551101282 T^{5} - 593219478248 T^{6} + 115316141768802 T^{7} - 912928968126828 T^{8} - 541451373822575840 T^{9} + 35055175141983029184 T^{10} + \)\(12\!\cdots\!10\)\( T^{11} - \)\(20\!\cdots\!66\)\( T^{12} + \)\(55\!\cdots\!90\)\( T^{13} + \)\(70\!\cdots\!64\)\( T^{14} - \)\(48\!\cdots\!60\)\( T^{15} - \)\(37\!\cdots\!48\)\( T^{16} + \)\(21\!\cdots\!98\)\( T^{17} - \)\(48\!\cdots\!28\)\( T^{18} - \)\(42\!\cdots\!78\)\( T^{19} + \)\(26\!\cdots\!68\)\( T^{20} + \)\(53\!\cdots\!16\)\( T^{21} - \)\(51\!\cdots\!76\)\( T^{22} + \)\(26\!\cdots\!02\)\( T^{23} + \)\(66\!\cdots\!21\)\( T^{24} \))
$71$ (\( ( 1 - 82 T + 12166 T^{2} - 846262 T^{3} + 94594474 T^{4} - 4266006742 T^{5} + 309158511046 T^{6} - 10504223281522 T^{7} + 645753531245761 T^{8} )^{2} \))(\( ( 1 + 314 T + 58704 T^{2} + 7369374 T^{3} + 727286203 T^{4} + 59185452824 T^{5} + 4395000293464 T^{6} + 298353867685784 T^{7} + 18481564986337243 T^{8} + 944018901720035454 T^{9} + 37908315298251153744 T^{10} + \)\(10\!\cdots\!14\)\( T^{11} + \)\(16\!\cdots\!41\)\( T^{12} )^{2} \))
$73$ (\( 1 - 348 T + 68263 T^{2} - 9707460 T^{3} + 1072498525 T^{4} - 96253557984 T^{5} + 7434307414846 T^{6} - 526544361727584 T^{7} + 37365046682274814 T^{8} - 2805954903646295136 T^{9} + \)\(21\!\cdots\!86\)\( T^{10} - \)\(14\!\cdots\!76\)\( T^{11} + \)\(86\!\cdots\!25\)\( T^{12} - \)\(41\!\cdots\!40\)\( T^{13} + \)\(15\!\cdots\!23\)\( T^{14} - \)\(42\!\cdots\!32\)\( T^{15} + \)\(65\!\cdots\!61\)\( T^{16} \))(\( 1 - 234 T + 50280 T^{2} - 7494552 T^{3} + 1045125336 T^{4} - 124045277670 T^{5} + 13798749511760 T^{6} - 1393240344068466 T^{7} + 132320540590792632 T^{8} - 11749990134656584872 T^{9} + \)\(98\!\cdots\!48\)\( T^{10} - \)\(77\!\cdots\!58\)\( T^{11} + \)\(58\!\cdots\!42\)\( T^{12} - \)\(41\!\cdots\!82\)\( T^{13} + \)\(27\!\cdots\!68\)\( T^{14} - \)\(17\!\cdots\!08\)\( T^{15} + \)\(10\!\cdots\!92\)\( T^{16} - \)\(59\!\cdots\!34\)\( T^{17} + \)\(31\!\cdots\!60\)\( T^{18} - \)\(15\!\cdots\!30\)\( T^{19} + \)\(67\!\cdots\!96\)\( T^{20} - \)\(25\!\cdots\!88\)\( T^{21} + \)\(92\!\cdots\!80\)\( T^{22} - \)\(23\!\cdots\!86\)\( T^{23} + \)\(52\!\cdots\!41\)\( T^{24} \))
$79$ (\( 1 + 206 T + 5583 T^{2} - 659438 T^{3} + 124066817 T^{4} + 19494076044 T^{5} + 428008398310 T^{6} + 33090623674568 T^{7} + 7605703397631354 T^{8} + 206518582352978888 T^{9} + 16670961782854763110 T^{10} + \)\(47\!\cdots\!24\)\( T^{11} + \)\(18\!\cdots\!37\)\( T^{12} - \)\(62\!\cdots\!38\)\( T^{13} + \)\(32\!\cdots\!03\)\( T^{14} + \)\(75\!\cdots\!86\)\( T^{15} + \)\(23\!\cdots\!21\)\( T^{16} \))(\( 1 - 108 T - 6108 T^{2} + 1752736 T^{3} - 90950172 T^{4} - 3971475972 T^{5} + 1036682005132 T^{6} - 72467068572108 T^{7} - 540849229955460 T^{8} + 549266052938802784 T^{9} - 33282724854561757956 T^{10} - \)\(11\!\cdots\!32\)\( T^{11} + \)\(26\!\cdots\!02\)\( T^{12} - \)\(70\!\cdots\!12\)\( T^{13} - \)\(12\!\cdots\!36\)\( T^{14} + \)\(13\!\cdots\!64\)\( T^{15} - \)\(82\!\cdots\!60\)\( T^{16} - \)\(68\!\cdots\!08\)\( T^{17} + \)\(61\!\cdots\!12\)\( T^{18} - \)\(14\!\cdots\!32\)\( T^{19} - \)\(20\!\cdots\!12\)\( T^{20} + \)\(25\!\cdots\!96\)\( T^{21} - \)\(54\!\cdots\!08\)\( T^{22} - \)\(60\!\cdots\!28\)\( T^{23} + \)\(34\!\cdots\!81\)\( T^{24} \))
$83$ (\( 1 - 20672 T^{2} + 223804480 T^{4} - 2182268545136 T^{6} + 17948924233578718 T^{8} - \)\(10\!\cdots\!56\)\( T^{10} + \)\(50\!\cdots\!80\)\( T^{12} - \)\(22\!\cdots\!92\)\( T^{14} + \)\(50\!\cdots\!81\)\( T^{16} \))(\( 1 - 52956 T^{2} + 1343643534 T^{4} - 21859222482508 T^{6} + 258399723525197679 T^{8} - \)\(23\!\cdots\!76\)\( T^{10} + \)\(18\!\cdots\!16\)\( T^{12} - \)\(11\!\cdots\!96\)\( T^{14} + \)\(58\!\cdots\!39\)\( T^{16} - \)\(23\!\cdots\!88\)\( T^{18} + \)\(68\!\cdots\!54\)\( T^{20} - \)\(12\!\cdots\!56\)\( T^{22} + \)\(11\!\cdots\!21\)\( T^{24} \))
$89$ (\( 1 - 282 T + 59686 T^{2} - 9356196 T^{3} + 1240796086 T^{4} - 138656838366 T^{5} + 14271061565800 T^{6} - 1337157406377822 T^{7} + 121622616146107507 T^{8} - 10591623815918728062 T^{9} + \)\(89\!\cdots\!00\)\( T^{10} - \)\(68\!\cdots\!26\)\( T^{11} + \)\(48\!\cdots\!66\)\( T^{12} - \)\(29\!\cdots\!96\)\( T^{13} + \)\(14\!\cdots\!06\)\( T^{14} - \)\(55\!\cdots\!62\)\( T^{15} + \)\(15\!\cdots\!61\)\( T^{16} \))(\( 1 + 186 T + 44532 T^{2} + 6138000 T^{3} + 889877673 T^{4} + 104389916664 T^{5} + 11976622429892 T^{6} + 1266314492339586 T^{7} + 128180246542580718 T^{8} + 12521360973919319370 T^{9} + \)\(11\!\cdots\!68\)\( T^{10} + \)\(10\!\cdots\!44\)\( T^{11} + \)\(99\!\cdots\!61\)\( T^{12} + \)\(86\!\cdots\!24\)\( T^{13} + \)\(74\!\cdots\!88\)\( T^{14} + \)\(62\!\cdots\!70\)\( T^{15} + \)\(50\!\cdots\!58\)\( T^{16} + \)\(39\!\cdots\!86\)\( T^{17} + \)\(29\!\cdots\!32\)\( T^{18} + \)\(20\!\cdots\!24\)\( T^{19} + \)\(13\!\cdots\!53\)\( T^{20} + \)\(75\!\cdots\!00\)\( T^{21} + \)\(43\!\cdots\!32\)\( T^{22} + \)\(14\!\cdots\!06\)\( T^{23} + \)\(61\!\cdots\!41\)\( T^{24} \))
$97$ (\( 1 - 44576 T^{2} + 925514428 T^{4} - 12414040936928 T^{6} + 128325632901816454 T^{8} - \)\(10\!\cdots\!68\)\( T^{10} + \)\(72\!\cdots\!08\)\( T^{12} - \)\(30\!\cdots\!16\)\( T^{14} + \)\(61\!\cdots\!21\)\( T^{16} \))(\( 1 - 64740 T^{2} + 2084566242 T^{4} - 44252577868756 T^{6} + 697286703828717423 T^{8} - \)\(87\!\cdots\!56\)\( T^{10} + \)\(89\!\cdots\!48\)\( T^{12} - \)\(77\!\cdots\!36\)\( T^{14} + \)\(54\!\cdots\!03\)\( T^{16} - \)\(30\!\cdots\!96\)\( T^{18} + \)\(12\!\cdots\!82\)\( T^{20} - \)\(35\!\cdots\!40\)\( T^{22} + \)\(48\!\cdots\!81\)\( T^{24} \))
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