Properties

Label 4.37.b
Level 4
Weight 37
Character orbit b
Rep. character \(\chi_{4}(3,\cdot)\)
Character field \(\Q\)
Dimension 17
Newform subspaces 2
Sturm bound 18
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 37 \)
Character orbit: \([\chi]\) \(=\) 4.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 4 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(18\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{37}(4, [\chi])\).

Total New Old
Modular forms 19 19 0
Cusp forms 17 17 0
Eisenstein series 2 2 0

Trace form

\( 17q - 84916q^{2} + 63108769040q^{4} + 1587598983826q^{5} - 217628996575488q^{6} - 6116838281275456q^{8} - 785089825520546847q^{9} + O(q^{10}) \) \( 17q - 84916q^{2} + 63108769040q^{4} + 1587598983826q^{5} - 217628996575488q^{6} - 6116838281275456q^{8} - 785089825520546847q^{9} - 1269669491125656104q^{10} - 42882825786868930560q^{12} + 41676149406206500018q^{13} + 360192888461659120128q^{14} - 18131657039655345618688q^{16} + 18678125271466757001058q^{17} + 98858065961248026757644q^{18} - 185987631386800333033184q^{20} - 175281296641396774711296q^{21} - 4320007363229839528400640q^{22} + 5118364442017682383085568q^{24} + 64389827691705544439075331q^{25} - 30067267580294149817323688q^{26} - 303534200373547502486016000q^{28} + 83558829778969622281411570q^{29} + 361206724962468787334008320q^{30} + 3424487663300783641955892224q^{32} - 877278707167997370072668160q^{33} - 2956121337393437394265165928q^{34} + 9386969606899626916903182864q^{36} + 1028936832542213926690666258q^{37} + 78752216997100984306444174080q^{38} - 61948625233183002270749961344q^{40} + 241910655498966263829819397954q^{41} - 193406458925326074258433843200q^{42} + 303084904999663560614745692160q^{44} + 876750889440806776372165555506q^{45} - 919681205391808405630032112128q^{46} - 1511312501851283767715389931520q^{48} - 8690584837344731555055515727247q^{49} + 10185343958170807211600579472036q^{50} + 5584769803431310739681792135968q^{52} + 2864977575306763415468638557778q^{53} - 42689634574575573966970327785984q^{54} - 18346492780981997297263940026368q^{56} - 17591710635767034592583231047680q^{57} - 13872941779387924373159026137512q^{58} - 69704584574466315130190074982400q^{60} + 488044483791787508231294360980594q^{61} + 406820870675021165265665434183680q^{62} + 58407051193644631965991706562560q^{64} - 1419136027205678294994787009303228q^{65} - 816784109200805225218136599695360q^{66} + 1913850173378374444103604070923808q^{68} + 3923547517779842564469076002041856q^{69} + 312094823977274793710809307489280q^{70} - 4579322712415668078348091815468096q^{72} - 7016735948804628395640934715523902q^{73} - 670997304617426449840386839200808q^{74} - 15736875982808342938482824299100160q^{76} + 22639774155764461736245527294259200q^{77} - 18389491597233550087389536189698560q^{78} + 47267506511318977155908021422813696q^{80} - 18188477562036937666899669720078159q^{81} + 5807936682837544119287618687716888q^{82} - 37119022564017857733030089777086464q^{84} - 109102745881421109039130518648221788q^{85} + 102108696026625816655543232864792832q^{86} + 39749581676781586144951142033940480q^{88} + 536580108278993677338741367468283650q^{89} - 195283296567279720904119484240636584q^{90} - 17511543929609412714530045318983680q^{92} - 231978818535002459544922699145134080q^{93} - 696830931593880576023589572144120832q^{94} + 1050073689946737522972596709346639872q^{96} - 467501081650930946766716978739129182q^{97} + 340279939831197597514188700404039884q^{98} + O(q^{100}) \)

Decomposition of \(S_{37}^{\mathrm{new}}(4, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
4.37.b.a \(1\) \(32.837\) \(\Q\) \(\Q(\sqrt{-1}) \) \(-262144\) \(0\) \(-4\!\cdots\!34\) \(0\) \(q-2^{18}q^{2}+2^{36}q^{4}-4228490555534q^{5}+\cdots\)
4.37.b.b \(16\) \(32.837\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(177228\) \(0\) \(58\!\cdots\!60\) \(0\) \(q+(11077+\beta _{1})q^{2}+(50+198\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 262144 T \))(\( 1 - 177228 T + 18510235840 T^{2} - 5390822269734912 T^{3} + \)\(65\!\cdots\!36\)\( T^{4} - \)\(20\!\cdots\!76\)\( T^{5} + \)\(32\!\cdots\!20\)\( T^{6} - \)\(12\!\cdots\!24\)\( T^{7} + \)\(23\!\cdots\!76\)\( T^{8} - \)\(83\!\cdots\!64\)\( T^{9} + \)\(15\!\cdots\!20\)\( T^{10} - \)\(65\!\cdots\!56\)\( T^{11} + \)\(14\!\cdots\!76\)\( T^{12} - \)\(82\!\cdots\!12\)\( T^{13} + \)\(19\!\cdots\!40\)\( T^{14} - \)\(12\!\cdots\!88\)\( T^{15} + \)\(49\!\cdots\!56\)\( T^{16} \))
$3$ (\( ( 1 - 387420489 T )( 1 + 387420489 T ) \))(\( 1 - 733164851967219984 T^{2} + \)\(31\!\cdots\!60\)\( T^{4} - \)\(96\!\cdots\!00\)\( T^{6} + \)\(23\!\cdots\!60\)\( T^{8} - \)\(50\!\cdots\!76\)\( T^{10} + \)\(92\!\cdots\!84\)\( T^{12} - \)\(15\!\cdots\!20\)\( T^{14} + \)\(24\!\cdots\!50\)\( T^{16} - \)\(35\!\cdots\!20\)\( T^{18} + \)\(46\!\cdots\!04\)\( T^{20} - \)\(57\!\cdots\!96\)\( T^{22} + \)\(61\!\cdots\!60\)\( T^{24} - \)\(56\!\cdots\!00\)\( T^{26} + \)\(41\!\cdots\!60\)\( T^{28} - \)\(21\!\cdots\!04\)\( T^{30} + \)\(66\!\cdots\!21\)\( T^{32} \))
$5$ (\( 1 + 4228490555534 T + \)\(14\!\cdots\!25\)\( T^{2} \))(\( ( 1 - 2908044769680 T + \)\(47\!\cdots\!00\)\( T^{2} - \)\(86\!\cdots\!00\)\( T^{3} + \)\(72\!\cdots\!00\)\( T^{4} + \)\(60\!\cdots\!00\)\( T^{5} + \)\(90\!\cdots\!00\)\( T^{6} + \)\(56\!\cdots\!00\)\( T^{7} - \)\(80\!\cdots\!50\)\( T^{8} + \)\(82\!\cdots\!00\)\( T^{9} + \)\(19\!\cdots\!00\)\( T^{10} + \)\(18\!\cdots\!00\)\( T^{11} + \)\(32\!\cdots\!00\)\( T^{12} - \)\(56\!\cdots\!00\)\( T^{13} + \)\(44\!\cdots\!00\)\( T^{14} - \)\(40\!\cdots\!00\)\( T^{15} + \)\(20\!\cdots\!25\)\( T^{16} )^{2} \))
$7$ (\( ( 1 - 1628413597910449 T )( 1 + 1628413597910449 T ) \))(\( 1 - \)\(15\!\cdots\!84\)\( T^{2} + \)\(12\!\cdots\!80\)\( T^{4} - \)\(78\!\cdots\!60\)\( T^{6} + \)\(38\!\cdots\!60\)\( T^{8} - \)\(16\!\cdots\!16\)\( T^{10} + \)\(58\!\cdots\!24\)\( T^{12} - \)\(18\!\cdots\!40\)\( T^{14} + \)\(52\!\cdots\!70\)\( T^{16} - \)\(13\!\cdots\!40\)\( T^{18} + \)\(28\!\cdots\!24\)\( T^{20} - \)\(56\!\cdots\!16\)\( T^{22} + \)\(94\!\cdots\!60\)\( T^{24} - \)\(13\!\cdots\!60\)\( T^{26} + \)\(15\!\cdots\!80\)\( T^{28} - \)\(13\!\cdots\!84\)\( T^{30} + \)\(59\!\cdots\!01\)\( T^{32} \))
$11$ (\( ( 1 - 5559917313492231481 T )( 1 + 5559917313492231481 T ) \))(\( 1 - \)\(20\!\cdots\!16\)\( T^{2} + \)\(21\!\cdots\!60\)\( T^{4} - \)\(15\!\cdots\!60\)\( T^{6} + \)\(81\!\cdots\!60\)\( T^{8} - \)\(36\!\cdots\!08\)\( T^{10} + \)\(14\!\cdots\!68\)\( T^{12} - \)\(51\!\cdots\!20\)\( T^{14} + \)\(16\!\cdots\!70\)\( T^{16} - \)\(49\!\cdots\!20\)\( T^{18} + \)\(13\!\cdots\!88\)\( T^{20} - \)\(32\!\cdots\!88\)\( T^{22} + \)\(68\!\cdots\!60\)\( T^{24} - \)\(12\!\cdots\!60\)\( T^{26} + \)\(16\!\cdots\!60\)\( T^{28} - \)\(14\!\cdots\!56\)\( T^{30} + \)\(69\!\cdots\!61\)\( T^{32} \))
$13$ (\( 1 + \)\(15\!\cdots\!58\)\( T + \)\(12\!\cdots\!41\)\( T^{2} \))(\( ( 1 - 97306007082829587088 T + \)\(49\!\cdots\!60\)\( T^{2} - \)\(40\!\cdots\!72\)\( T^{3} + \)\(13\!\cdots\!56\)\( T^{4} - \)\(92\!\cdots\!76\)\( T^{5} + \)\(24\!\cdots\!80\)\( T^{6} - \)\(15\!\cdots\!24\)\( T^{7} + \)\(35\!\cdots\!86\)\( T^{8} - \)\(19\!\cdots\!84\)\( T^{9} + \)\(39\!\cdots\!80\)\( T^{10} - \)\(18\!\cdots\!96\)\( T^{11} + \)\(34\!\cdots\!16\)\( T^{12} - \)\(13\!\cdots\!72\)\( T^{13} + \)\(20\!\cdots\!60\)\( T^{14} - \)\(50\!\cdots\!28\)\( T^{15} + \)\(65\!\cdots\!21\)\( T^{16} )^{2} \))
$17$ (\( 1 + \)\(23\!\cdots\!18\)\( T + \)\(19\!\cdots\!81\)\( T^{2} \))(\( ( 1 - \)\(20\!\cdots\!88\)\( T + \)\(11\!\cdots\!00\)\( T^{2} - \)\(15\!\cdots\!72\)\( T^{3} + \)\(51\!\cdots\!16\)\( T^{4} - \)\(46\!\cdots\!76\)\( T^{5} + \)\(14\!\cdots\!40\)\( T^{6} - \)\(96\!\cdots\!64\)\( T^{7} + \)\(30\!\cdots\!86\)\( T^{8} - \)\(19\!\cdots\!84\)\( T^{9} + \)\(55\!\cdots\!40\)\( T^{10} - \)\(36\!\cdots\!16\)\( T^{11} + \)\(78\!\cdots\!36\)\( T^{12} - \)\(45\!\cdots\!72\)\( T^{13} + \)\(67\!\cdots\!00\)\( T^{14} - \)\(24\!\cdots\!68\)\( T^{15} + \)\(23\!\cdots\!41\)\( T^{16} )^{2} \))
$19$ (\( ( 1 - \)\(10\!\cdots\!41\)\( T )( 1 + \)\(10\!\cdots\!41\)\( T ) \))(\( 1 - \)\(96\!\cdots\!36\)\( T^{2} + \)\(44\!\cdots\!60\)\( T^{4} - \)\(13\!\cdots\!60\)\( T^{6} + \)\(28\!\cdots\!60\)\( T^{8} - \)\(47\!\cdots\!08\)\( T^{10} + \)\(67\!\cdots\!28\)\( T^{12} - \)\(83\!\cdots\!20\)\( T^{14} + \)\(93\!\cdots\!70\)\( T^{16} - \)\(97\!\cdots\!20\)\( T^{18} + \)\(93\!\cdots\!88\)\( T^{20} - \)\(77\!\cdots\!48\)\( T^{22} + \)\(54\!\cdots\!60\)\( T^{24} - \)\(29\!\cdots\!60\)\( T^{26} + \)\(11\!\cdots\!60\)\( T^{28} - \)\(29\!\cdots\!56\)\( T^{30} + \)\(36\!\cdots\!81\)\( T^{32} \))
$23$ (\( ( 1 - \)\(32\!\cdots\!69\)\( T )( 1 + \)\(32\!\cdots\!69\)\( T ) \))(\( 1 - \)\(10\!\cdots\!24\)\( T^{2} + \)\(51\!\cdots\!60\)\( T^{4} - \)\(16\!\cdots\!60\)\( T^{6} + \)\(39\!\cdots\!80\)\( T^{8} - \)\(74\!\cdots\!16\)\( T^{10} + \)\(11\!\cdots\!64\)\( T^{12} - \)\(14\!\cdots\!40\)\( T^{14} + \)\(16\!\cdots\!70\)\( T^{16} - \)\(16\!\cdots\!40\)\( T^{18} + \)\(14\!\cdots\!24\)\( T^{20} - \)\(10\!\cdots\!76\)\( T^{22} + \)\(60\!\cdots\!80\)\( T^{24} - \)\(27\!\cdots\!60\)\( T^{26} + \)\(94\!\cdots\!60\)\( T^{28} - \)\(20\!\cdots\!84\)\( T^{30} + \)\(22\!\cdots\!61\)\( T^{32} \))
$29$ (\( 1 - \)\(17\!\cdots\!22\)\( T + \)\(44\!\cdots\!21\)\( T^{2} \))(\( ( 1 + \)\(47\!\cdots\!76\)\( T + \)\(23\!\cdots\!00\)\( T^{2} + \)\(21\!\cdots\!40\)\( T^{3} + \)\(25\!\cdots\!40\)\( T^{4} - \)\(58\!\cdots\!16\)\( T^{5} + \)\(17\!\cdots\!64\)\( T^{6} - \)\(69\!\cdots\!40\)\( T^{7} + \)\(90\!\cdots\!70\)\( T^{8} - \)\(30\!\cdots\!40\)\( T^{9} + \)\(35\!\cdots\!24\)\( T^{10} - \)\(50\!\cdots\!76\)\( T^{11} + \)\(99\!\cdots\!40\)\( T^{12} + \)\(36\!\cdots\!40\)\( T^{13} + \)\(17\!\cdots\!00\)\( T^{14} + \)\(15\!\cdots\!16\)\( T^{15} + \)\(14\!\cdots\!61\)\( T^{16} )^{2} \))
$31$ (\( ( 1 - \)\(69\!\cdots\!41\)\( T )( 1 + \)\(69\!\cdots\!41\)\( T ) \))(\( 1 - \)\(49\!\cdots\!36\)\( T^{2} + \)\(12\!\cdots\!60\)\( T^{4} - \)\(21\!\cdots\!60\)\( T^{6} + \)\(26\!\cdots\!60\)\( T^{8} - \)\(26\!\cdots\!08\)\( T^{10} + \)\(20\!\cdots\!28\)\( T^{12} - \)\(13\!\cdots\!20\)\( T^{14} + \)\(71\!\cdots\!70\)\( T^{16} - \)\(32\!\cdots\!20\)\( T^{18} + \)\(11\!\cdots\!88\)\( T^{20} - \)\(35\!\cdots\!48\)\( T^{22} + \)\(87\!\cdots\!60\)\( T^{24} - \)\(16\!\cdots\!60\)\( T^{26} + \)\(23\!\cdots\!60\)\( T^{28} - \)\(22\!\cdots\!56\)\( T^{30} + \)\(10\!\cdots\!81\)\( T^{32} \))
$37$ (\( 1 - \)\(31\!\cdots\!42\)\( T + \)\(28\!\cdots\!41\)\( T^{2} \))(\( ( 1 + \)\(15\!\cdots\!92\)\( T + \)\(13\!\cdots\!40\)\( T^{2} + \)\(17\!\cdots\!08\)\( T^{3} + \)\(90\!\cdots\!56\)\( T^{4} + \)\(91\!\cdots\!24\)\( T^{5} + \)\(38\!\cdots\!60\)\( T^{6} + \)\(32\!\cdots\!36\)\( T^{7} + \)\(12\!\cdots\!06\)\( T^{8} + \)\(92\!\cdots\!76\)\( T^{9} + \)\(31\!\cdots\!60\)\( T^{10} + \)\(21\!\cdots\!04\)\( T^{11} + \)\(59\!\cdots\!16\)\( T^{12} + \)\(32\!\cdots\!08\)\( T^{13} + \)\(74\!\cdots\!40\)\( T^{14} + \)\(23\!\cdots\!52\)\( T^{15} + \)\(43\!\cdots\!21\)\( T^{16} )^{2} \))
$41$ (\( 1 - \)\(14\!\cdots\!42\)\( T + \)\(11\!\cdots\!41\)\( T^{2} \))(\( ( 1 - \)\(49\!\cdots\!56\)\( T + \)\(54\!\cdots\!20\)\( T^{2} - \)\(19\!\cdots\!20\)\( T^{3} + \)\(13\!\cdots\!20\)\( T^{4} - \)\(37\!\cdots\!88\)\( T^{5} + \)\(21\!\cdots\!08\)\( T^{6} - \)\(56\!\cdots\!40\)\( T^{7} + \)\(26\!\cdots\!90\)\( T^{8} - \)\(64\!\cdots\!40\)\( T^{9} + \)\(28\!\cdots\!48\)\( T^{10} - \)\(56\!\cdots\!48\)\( T^{11} + \)\(23\!\cdots\!20\)\( T^{12} - \)\(38\!\cdots\!20\)\( T^{13} + \)\(12\!\cdots\!20\)\( T^{14} - \)\(13\!\cdots\!36\)\( T^{15} + \)\(30\!\cdots\!21\)\( T^{16} )^{2} \))
$43$ (\( ( 1 - \)\(25\!\cdots\!49\)\( T )( 1 + \)\(25\!\cdots\!49\)\( T ) \))(\( 1 - \)\(46\!\cdots\!24\)\( T^{2} + \)\(11\!\cdots\!20\)\( T^{4} - \)\(20\!\cdots\!00\)\( T^{6} + \)\(27\!\cdots\!00\)\( T^{8} - \)\(29\!\cdots\!56\)\( T^{10} + \)\(27\!\cdots\!04\)\( T^{12} - \)\(21\!\cdots\!20\)\( T^{14} + \)\(14\!\cdots\!50\)\( T^{16} - \)\(88\!\cdots\!20\)\( T^{18} + \)\(45\!\cdots\!04\)\( T^{20} - \)\(20\!\cdots\!56\)\( T^{22} + \)\(74\!\cdots\!00\)\( T^{24} - \)\(22\!\cdots\!00\)\( T^{26} + \)\(52\!\cdots\!20\)\( T^{28} - \)\(85\!\cdots\!24\)\( T^{30} + \)\(75\!\cdots\!01\)\( T^{32} \))
$47$ (\( ( 1 - \)\(12\!\cdots\!89\)\( T )( 1 + \)\(12\!\cdots\!89\)\( T ) \))(\( 1 - \)\(20\!\cdots\!64\)\( T^{2} + \)\(20\!\cdots\!20\)\( T^{4} - \)\(13\!\cdots\!00\)\( T^{6} + \)\(60\!\cdots\!00\)\( T^{8} - \)\(21\!\cdots\!96\)\( T^{10} + \)\(57\!\cdots\!84\)\( T^{12} - \)\(12\!\cdots\!20\)\( T^{14} + \)\(21\!\cdots\!50\)\( T^{16} - \)\(30\!\cdots\!20\)\( T^{18} + \)\(34\!\cdots\!04\)\( T^{20} - \)\(31\!\cdots\!16\)\( T^{22} + \)\(22\!\cdots\!00\)\( T^{24} - \)\(12\!\cdots\!00\)\( T^{26} + \)\(46\!\cdots\!20\)\( T^{28} - \)\(11\!\cdots\!84\)\( T^{30} + \)\(13\!\cdots\!21\)\( T^{32} \))
$53$ (\( 1 + \)\(18\!\cdots\!78\)\( T + \)\(11\!\cdots\!21\)\( T^{2} \))(\( ( 1 - \)\(10\!\cdots\!28\)\( T + \)\(82\!\cdots\!00\)\( T^{2} - \)\(68\!\cdots\!12\)\( T^{3} + \)\(30\!\cdots\!36\)\( T^{4} - \)\(20\!\cdots\!56\)\( T^{5} + \)\(65\!\cdots\!00\)\( T^{6} - \)\(36\!\cdots\!24\)\( T^{7} + \)\(94\!\cdots\!86\)\( T^{8} - \)\(43\!\cdots\!04\)\( T^{9} + \)\(92\!\cdots\!00\)\( T^{10} - \)\(33\!\cdots\!16\)\( T^{11} + \)\(59\!\cdots\!16\)\( T^{12} - \)\(16\!\cdots\!12\)\( T^{13} + \)\(22\!\cdots\!00\)\( T^{14} - \)\(34\!\cdots\!48\)\( T^{15} + \)\(39\!\cdots\!61\)\( T^{16} )^{2} \))
$59$ (\( ( 1 - \)\(75\!\cdots\!21\)\( T )( 1 + \)\(75\!\cdots\!21\)\( T ) \))(\( 1 - \)\(53\!\cdots\!36\)\( T^{2} + \)\(14\!\cdots\!20\)\( T^{4} - \)\(27\!\cdots\!00\)\( T^{6} + \)\(38\!\cdots\!20\)\( T^{8} - \)\(41\!\cdots\!48\)\( T^{10} + \)\(37\!\cdots\!88\)\( T^{12} - \)\(27\!\cdots\!40\)\( T^{14} + \)\(16\!\cdots\!50\)\( T^{16} - \)\(86\!\cdots\!40\)\( T^{18} + \)\(37\!\cdots\!68\)\( T^{20} - \)\(13\!\cdots\!68\)\( T^{22} + \)\(38\!\cdots\!20\)\( T^{24} - \)\(88\!\cdots\!00\)\( T^{26} + \)\(15\!\cdots\!20\)\( T^{28} - \)\(17\!\cdots\!96\)\( T^{30} + \)\(10\!\cdots\!41\)\( T^{32} \))
$61$ (\( 1 - \)\(27\!\cdots\!62\)\( T + \)\(18\!\cdots\!61\)\( T^{2} \))(\( ( 1 - \)\(10\!\cdots\!16\)\( T + \)\(99\!\cdots\!20\)\( T^{2} - \)\(11\!\cdots\!20\)\( T^{3} + \)\(47\!\cdots\!20\)\( T^{4} - \)\(55\!\cdots\!88\)\( T^{5} + \)\(14\!\cdots\!88\)\( T^{6} - \)\(15\!\cdots\!40\)\( T^{7} + \)\(30\!\cdots\!90\)\( T^{8} - \)\(29\!\cdots\!40\)\( T^{9} + \)\(49\!\cdots\!48\)\( T^{10} - \)\(36\!\cdots\!28\)\( T^{11} + \)\(57\!\cdots\!20\)\( T^{12} - \)\(26\!\cdots\!20\)\( T^{13} + \)\(42\!\cdots\!20\)\( T^{14} - \)\(86\!\cdots\!36\)\( T^{15} + \)\(14\!\cdots\!81\)\( T^{16} )^{2} \))
$67$ (\( ( 1 - \)\(74\!\cdots\!09\)\( T )( 1 + \)\(74\!\cdots\!09\)\( T ) \))(\( 1 - \)\(44\!\cdots\!04\)\( T^{2} + \)\(10\!\cdots\!40\)\( T^{4} - \)\(17\!\cdots\!80\)\( T^{6} + \)\(22\!\cdots\!40\)\( T^{8} - \)\(22\!\cdots\!56\)\( T^{10} + \)\(18\!\cdots\!84\)\( T^{12} - \)\(13\!\cdots\!80\)\( T^{14} + \)\(77\!\cdots\!10\)\( T^{16} - \)\(39\!\cdots\!80\)\( T^{18} + \)\(16\!\cdots\!64\)\( T^{20} - \)\(60\!\cdots\!36\)\( T^{22} + \)\(17\!\cdots\!40\)\( T^{24} - \)\(42\!\cdots\!80\)\( T^{26} + \)\(77\!\cdots\!40\)\( T^{28} - \)\(97\!\cdots\!84\)\( T^{30} + \)\(65\!\cdots\!81\)\( T^{32} \))
$71$ (\( ( 1 - \)\(21\!\cdots\!61\)\( T )( 1 + \)\(21\!\cdots\!61\)\( T ) \))(\( 1 - \)\(33\!\cdots\!16\)\( T^{2} + \)\(57\!\cdots\!20\)\( T^{4} - \)\(70\!\cdots\!00\)\( T^{6} + \)\(66\!\cdots\!20\)\( T^{8} - \)\(50\!\cdots\!48\)\( T^{10} + \)\(32\!\cdots\!28\)\( T^{12} - \)\(18\!\cdots\!40\)\( T^{14} + \)\(86\!\cdots\!50\)\( T^{16} - \)\(35\!\cdots\!40\)\( T^{18} + \)\(12\!\cdots\!68\)\( T^{20} - \)\(37\!\cdots\!08\)\( T^{22} + \)\(96\!\cdots\!20\)\( T^{24} - \)\(19\!\cdots\!00\)\( T^{26} + \)\(32\!\cdots\!20\)\( T^{28} - \)\(35\!\cdots\!96\)\( T^{30} + \)\(21\!\cdots\!21\)\( T^{32} \))
$73$ (\( 1 - \)\(65\!\cdots\!62\)\( T + \)\(12\!\cdots\!61\)\( T^{2} \))(\( ( 1 + \)\(67\!\cdots\!32\)\( T + \)\(51\!\cdots\!60\)\( T^{2} + \)\(28\!\cdots\!68\)\( T^{3} + \)\(17\!\cdots\!96\)\( T^{4} + \)\(75\!\cdots\!24\)\( T^{5} + \)\(33\!\cdots\!80\)\( T^{6} + \)\(12\!\cdots\!96\)\( T^{7} + \)\(48\!\cdots\!66\)\( T^{8} + \)\(15\!\cdots\!56\)\( T^{9} + \)\(48\!\cdots\!80\)\( T^{10} + \)\(13\!\cdots\!44\)\( T^{11} + \)\(35\!\cdots\!36\)\( T^{12} + \)\(72\!\cdots\!68\)\( T^{13} + \)\(15\!\cdots\!60\)\( T^{14} + \)\(24\!\cdots\!72\)\( T^{15} + \)\(43\!\cdots\!81\)\( T^{16} )^{2} \))
$79$ (\( ( 1 - \)\(14\!\cdots\!61\)\( T )( 1 + \)\(14\!\cdots\!61\)\( T ) \))(\( 1 - \)\(10\!\cdots\!76\)\( T^{2} + \)\(62\!\cdots\!60\)\( T^{4} - \)\(26\!\cdots\!60\)\( T^{6} + \)\(90\!\cdots\!60\)\( T^{8} - \)\(26\!\cdots\!08\)\( T^{10} + \)\(71\!\cdots\!48\)\( T^{12} - \)\(16\!\cdots\!20\)\( T^{14} + \)\(36\!\cdots\!70\)\( T^{16} - \)\(71\!\cdots\!20\)\( T^{18} + \)\(12\!\cdots\!88\)\( T^{20} - \)\(20\!\cdots\!68\)\( T^{22} + \)\(29\!\cdots\!60\)\( T^{24} - \)\(36\!\cdots\!60\)\( T^{26} + \)\(37\!\cdots\!60\)\( T^{28} - \)\(27\!\cdots\!56\)\( T^{30} + \)\(10\!\cdots\!21\)\( T^{32} \))
$83$ (\( ( 1 - \)\(34\!\cdots\!09\)\( T )( 1 + \)\(34\!\cdots\!09\)\( T ) \))(\( 1 - \)\(73\!\cdots\!24\)\( T^{2} + \)\(27\!\cdots\!00\)\( T^{4} - \)\(66\!\cdots\!60\)\( T^{6} + \)\(12\!\cdots\!40\)\( T^{8} - \)\(20\!\cdots\!96\)\( T^{10} + \)\(30\!\cdots\!44\)\( T^{12} - \)\(43\!\cdots\!40\)\( T^{14} + \)\(57\!\cdots\!70\)\( T^{16} - \)\(65\!\cdots\!40\)\( T^{18} + \)\(67\!\cdots\!24\)\( T^{20} - \)\(66\!\cdots\!76\)\( T^{22} + \)\(61\!\cdots\!40\)\( T^{24} - \)\(49\!\cdots\!60\)\( T^{26} + \)\(30\!\cdots\!00\)\( T^{28} - \)\(12\!\cdots\!04\)\( T^{30} + \)\(24\!\cdots\!81\)\( T^{32} \))
$89$ (\( 1 - \)\(75\!\cdots\!62\)\( T + \)\(15\!\cdots\!61\)\( T^{2} \))(\( ( 1 - \)\(23\!\cdots\!44\)\( T + \)\(10\!\cdots\!20\)\( T^{2} - \)\(16\!\cdots\!00\)\( T^{3} + \)\(42\!\cdots\!00\)\( T^{4} - \)\(55\!\cdots\!56\)\( T^{5} + \)\(10\!\cdots\!24\)\( T^{6} - \)\(11\!\cdots\!20\)\( T^{7} + \)\(19\!\cdots\!50\)\( T^{8} - \)\(17\!\cdots\!20\)\( T^{9} + \)\(24\!\cdots\!04\)\( T^{10} - \)\(19\!\cdots\!36\)\( T^{11} + \)\(22\!\cdots\!00\)\( T^{12} - \)\(12\!\cdots\!00\)\( T^{13} + \)\(11\!\cdots\!20\)\( T^{14} - \)\(40\!\cdots\!24\)\( T^{15} + \)\(26\!\cdots\!81\)\( T^{16} )^{2} \))
$97$ (\( 1 + \)\(91\!\cdots\!78\)\( T + \)\(33\!\cdots\!21\)\( T^{2} \))(\( ( 1 - \)\(22\!\cdots\!48\)\( T + \)\(14\!\cdots\!40\)\( T^{2} - \)\(61\!\cdots\!12\)\( T^{3} + \)\(83\!\cdots\!76\)\( T^{4} + \)\(19\!\cdots\!04\)\( T^{5} + \)\(26\!\cdots\!00\)\( T^{6} + \)\(16\!\cdots\!16\)\( T^{7} + \)\(74\!\cdots\!86\)\( T^{8} + \)\(56\!\cdots\!36\)\( T^{9} + \)\(29\!\cdots\!00\)\( T^{10} + \)\(71\!\cdots\!44\)\( T^{11} + \)\(10\!\cdots\!56\)\( T^{12} - \)\(25\!\cdots\!12\)\( T^{13} + \)\(19\!\cdots\!40\)\( T^{14} - \)\(10\!\cdots\!68\)\( T^{15} + \)\(15\!\cdots\!61\)\( T^{16} )^{2} \))
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