Properties

Label 9.38
Level 9
Weight 38
Dimension 87
Nonzero newspaces 2
Newform subspaces 5
Sturm bound 228
Trace bound 1

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Defining parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 38 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 5 \)
Sturm bound: \(228\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{38}(\Gamma_1(9))\).

Total New Old
Modular forms 115 92 23
Cusp forms 107 87 20
Eisenstein series 8 5 3

Trace form

\( 87q + 329889q^{2} - 401411892q^{3} - 1619750875629q^{4} + 18308365618722q^{5} + 392737738672791q^{6} + 2742373845211008q^{7} - 154197865950929238q^{8} - 160987063735768212q^{9} + O(q^{10}) \) \( 87q + 329889q^{2} - 401411892q^{3} - 1619750875629q^{4} + 18308365618722q^{5} + 392737738672791q^{6} + 2742373845211008q^{7} - 154197865950929238q^{8} - 160987063735768212q^{9} + 701811086045477904q^{10} + 11545997776285928088q^{11} + 205931182443362228004q^{12} - 103319493076660821414q^{13} + 3067223294721143275944q^{14} + 21425611228622029637208q^{15} - 95649770669017259211249q^{16} - 8306234468091920602458q^{17} + 82533387024967768624260q^{18} + 270035797869373653325956q^{19} + 6232134420005027125786980q^{20} + 5838301440321983715229320q^{21} + 9437426363670756221528775q^{22} + 54255877390369102651683504q^{23} - 65709136210581010081717371q^{24} - 230854284144353401270225563q^{25} + 122096071950165945112393428q^{26} + 381958868294912176823572848q^{27} - 955871471938320584845940004q^{28} + 804132498646349946302415474q^{29} - 601709628543234842558918376q^{30} + 881225512446197435361370680q^{31} - 18151730804860462796680992897q^{32} - 38173301576979446099329160940q^{33} + 27262880330075765854596958869q^{34} - 200431901416098900571796822784q^{35} + 243326989965809135416568490099q^{36} - 98036062171463270199686544726q^{37} + 1151092416460908800473807979859q^{38} - 403802968711354024639660838160q^{39} - 366940471346194657953056752368q^{40} - 471706101506504473763896837398q^{41} - 3341342390312122233865455923982q^{42} - 442935772294851618625398423048q^{43} - 10421187259538494087494836525142q^{44} - 739318658750594406568464086952q^{45} - 3144821351805580732190076216888q^{46} + 25656224846167184337775202639232q^{47} - 52356572848387126657978406782365q^{48} - 61414101723917066503976153384733q^{49} + 174211379023457416042116587774289q^{50} - 203536505460286158473319755016852q^{51} + 217730479730882346485559720698682q^{52} - 562105116327675411606666144407598q^{53} + 1010866361710929990292216974150009q^{54} - 304419666763573840933982874642024q^{55} + 523609600587293391484544540172750q^{56} - 911741534435989274642508780045708q^{57} - 1434283571032664998185636963157260q^{58} + 1030987807918163364110176110195168q^{59} + 1735895800328840301692845833732468q^{60} - 2681528401088033533091914479580158q^{61} + 9845798998683299339437765035327204q^{62} - 12027424395451184167105611735489576q^{63} + 24553200787814552000236270416344130q^{64} + 377610413326934067670648372469436q^{65} - 181835740431829108776682384900902q^{66} + 1143324921773557096066935598695024q^{67} - 11368913876393683623825562301429937q^{68} + 45651766739864415186392523477009792q^{69} - 28240947012852252451078152466078494q^{70} - 38935502535802063850315881074884232q^{71} - 41090783447791739841512902716364389q^{72} - 27692624200493840540697160411976538q^{73} + 99032525440396489596917857141509756q^{74} + 170709895857321181267161602898401628q^{75} - 66500879941432376353175950837131459q^{76} + 342780164431500479532318352818878064q^{77} - 276607517324500733525939302718153418q^{78} - 379515845090436176152142826668618640q^{79} + 1639829527648889210970557165430298368q^{80} + 331293542492574451498143148213568148q^{81} - 399372028015877543721045070361918538q^{82} + 1500432873565728695616608098848685644q^{83} + 670584552943002331558441172923019874q^{84} - 1882364319763813075799410872225538932q^{85} + 6103812648649635186903211512013123569q^{86} - 1998354544933340665792185062503836096q^{87} - 2069954387796229410093542440310184309q^{88} + 6749162443888943821756576975670522166q^{89} + 4815138213911653652259366571211256108q^{90} - 6988899202022378884176125150365744896q^{91} + 26245691438391562481391737835538463634q^{92} - 4888442836513460260337741079738979632q^{93} - 11457187503166588496184189861962510016q^{94} + 35537636811651098189915166755679569424q^{95} + 12347574341999101028342060805474522768q^{96} - 39704159689023422777885224235664057582q^{97} + 90377088367690491144263717562764476272q^{98} - 15636137210022725768195532263739660984q^{99} + O(q^{100}) \)

Decomposition of \(S_{38}^{\mathrm{new}}(\Gamma_1(9))\)

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
9.38.a \(\chi_{9}(1, \cdot)\) 9.38.a.a 2 1
9.38.a.b 3
9.38.a.c 4
9.38.a.d 6
9.38.c \(\chi_{9}(4, \cdot)\) 9.38.c.a 72 2

Decomposition of \(S_{38}^{\mathrm{old}}(\Gamma_1(9))\) into lower level spaces

\( S_{38}^{\mathrm{old}}(\Gamma_1(9)) \cong \) \(S_{38}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{38}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 194400 T + 137474498560 T^{2} - 26718132554956800 T^{3} + \)\(18\!\cdots\!84\)\( T^{4} \))(\( 1 - 310908 T + 85252022016 T^{2} - 33372034010185728 T^{3} + \)\(11\!\cdots\!52\)\( T^{4} - \)\(58\!\cdots\!72\)\( T^{5} + \)\(25\!\cdots\!48\)\( T^{6} \))(\( 1 + 437562 T + 197558681120 T^{2} + 85300907961581568 T^{3} + \)\(26\!\cdots\!16\)\( T^{4} + \)\(11\!\cdots\!96\)\( T^{5} + \)\(37\!\cdots\!80\)\( T^{6} + \)\(11\!\cdots\!76\)\( T^{7} + \)\(35\!\cdots\!56\)\( T^{8} \))(\( 1 + 380749368432 T^{2} + \)\(73\!\cdots\!60\)\( T^{4} + \)\(10\!\cdots\!60\)\( T^{6} + \)\(13\!\cdots\!40\)\( T^{8} + \)\(13\!\cdots\!92\)\( T^{10} + \)\(67\!\cdots\!04\)\( T^{12} \))
$3$ 1
$5$ (\( 1 + 5529584385900 T + \)\(30\!\cdots\!50\)\( T^{2} + \)\(40\!\cdots\!00\)\( T^{3} + \)\(52\!\cdots\!25\)\( T^{4} \))(\( 1 - 9628717886790 T + \)\(16\!\cdots\!75\)\( T^{2} - \)\(14\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!75\)\( T^{4} - \)\(50\!\cdots\!50\)\( T^{5} + \)\(38\!\cdots\!25\)\( T^{6} \))(\( 1 - 4099829756904 T + \)\(10\!\cdots\!00\)\( T^{2} + \)\(66\!\cdots\!00\)\( T^{3} + \)\(20\!\cdots\!50\)\( T^{4} + \)\(48\!\cdots\!00\)\( T^{5} + \)\(56\!\cdots\!00\)\( T^{6} - \)\(15\!\cdots\!00\)\( T^{7} + \)\(28\!\cdots\!25\)\( T^{8} \))(\( 1 + \)\(21\!\cdots\!50\)\( T^{2} + \)\(20\!\cdots\!75\)\( T^{4} + \)\(14\!\cdots\!00\)\( T^{6} + \)\(10\!\cdots\!75\)\( T^{8} + \)\(59\!\cdots\!50\)\( T^{10} + \)\(14\!\cdots\!25\)\( T^{12} \))
$7$ (\( 1 + 3448443953486000 T + \)\(26\!\cdots\!50\)\( T^{2} + \)\(64\!\cdots\!00\)\( T^{3} + \)\(34\!\cdots\!49\)\( T^{4} \))(\( 1 + 4621884343701744 T + \)\(49\!\cdots\!65\)\( T^{2} + \)\(13\!\cdots\!36\)\( T^{3} + \)\(92\!\cdots\!55\)\( T^{4} + \)\(15\!\cdots\!56\)\( T^{5} + \)\(63\!\cdots\!43\)\( T^{6} \))(\( 1 - 6605809948153184 T + \)\(38\!\cdots\!32\)\( T^{2} - \)\(16\!\cdots\!44\)\( T^{3} + \)\(53\!\cdots\!50\)\( T^{4} - \)\(31\!\cdots\!08\)\( T^{5} + \)\(13\!\cdots\!68\)\( T^{6} - \)\(42\!\cdots\!12\)\( T^{7} + \)\(11\!\cdots\!01\)\( T^{8} \))(\( ( 1 - 1497594962732100 T + \)\(30\!\cdots\!21\)\( T^{2} - \)\(20\!\cdots\!00\)\( T^{3} + \)\(57\!\cdots\!47\)\( T^{4} - \)\(51\!\cdots\!00\)\( T^{5} + \)\(63\!\cdots\!43\)\( T^{6} )^{2} \))
$11$ (\( 1 - 26734036354848538056 T + \)\(70\!\cdots\!26\)\( T^{2} - \)\(90\!\cdots\!76\)\( T^{3} + \)\(11\!\cdots\!41\)\( T^{4} \))(\( 1 + 22673303357139628620 T + \)\(11\!\cdots\!01\)\( T^{2} + \)\(15\!\cdots\!44\)\( T^{3} + \)\(39\!\cdots\!71\)\( T^{4} + \)\(26\!\cdots\!20\)\( T^{5} + \)\(39\!\cdots\!11\)\( T^{6} \))(\( 1 + 20953708852195292976 T + \)\(92\!\cdots\!92\)\( T^{2} + \)\(12\!\cdots\!20\)\( T^{3} + \)\(41\!\cdots\!06\)\( T^{4} + \)\(43\!\cdots\!20\)\( T^{5} + \)\(10\!\cdots\!72\)\( T^{6} + \)\(82\!\cdots\!36\)\( T^{7} + \)\(13\!\cdots\!81\)\( T^{8} \))(\( 1 + \)\(13\!\cdots\!26\)\( T^{2} + \)\(26\!\cdots\!15\)\( T^{4} + \)\(25\!\cdots\!20\)\( T^{6} + \)\(30\!\cdots\!15\)\( T^{8} + \)\(18\!\cdots\!06\)\( T^{10} + \)\(15\!\cdots\!21\)\( T^{12} \))
$13$ (\( 1 - \)\(53\!\cdots\!00\)\( T + \)\(32\!\cdots\!90\)\( T^{2} - \)\(87\!\cdots\!00\)\( T^{3} + \)\(27\!\cdots\!89\)\( T^{4} \))(\( 1 + \)\(16\!\cdots\!10\)\( T + \)\(21\!\cdots\!91\)\( T^{2} + \)\(71\!\cdots\!68\)\( T^{3} + \)\(35\!\cdots\!03\)\( T^{4} + \)\(45\!\cdots\!90\)\( T^{5} + \)\(44\!\cdots\!37\)\( T^{6} \))(\( 1 - 51830892788989874168 T + \)\(47\!\cdots\!04\)\( T^{2} - \)\(17\!\cdots\!76\)\( T^{3} + \)\(10\!\cdots\!50\)\( T^{4} - \)\(29\!\cdots\!08\)\( T^{5} + \)\(12\!\cdots\!56\)\( T^{6} - \)\(23\!\cdots\!16\)\( T^{7} + \)\(73\!\cdots\!21\)\( T^{8} \))(\( ( 1 + \)\(18\!\cdots\!50\)\( T + \)\(14\!\cdots\!99\)\( T^{2} - \)\(49\!\cdots\!00\)\( T^{3} + \)\(23\!\cdots\!67\)\( T^{4} + \)\(48\!\cdots\!50\)\( T^{5} + \)\(44\!\cdots\!37\)\( T^{6} )^{2} \))
$17$ (\( 1 - \)\(89\!\cdots\!00\)\( T + \)\(86\!\cdots\!30\)\( T^{2} - \)\(30\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!29\)\( T^{4} \))(\( 1 - \)\(27\!\cdots\!06\)\( T + \)\(72\!\cdots\!43\)\( T^{2} - \)\(11\!\cdots\!32\)\( T^{3} + \)\(24\!\cdots\!11\)\( T^{4} - \)\(30\!\cdots\!74\)\( T^{5} + \)\(38\!\cdots\!33\)\( T^{6} \))(\( 1 + \)\(81\!\cdots\!28\)\( T + \)\(10\!\cdots\!52\)\( T^{2} + \)\(71\!\cdots\!40\)\( T^{3} + \)\(44\!\cdots\!26\)\( T^{4} + \)\(24\!\cdots\!80\)\( T^{5} + \)\(11\!\cdots\!08\)\( T^{6} + \)\(30\!\cdots\!24\)\( T^{7} + \)\(12\!\cdots\!41\)\( T^{8} \))(\( 1 + \)\(58\!\cdots\!62\)\( T^{2} - \)\(25\!\cdots\!65\)\( T^{4} - \)\(74\!\cdots\!40\)\( T^{6} - \)\(28\!\cdots\!85\)\( T^{8} + \)\(75\!\cdots\!42\)\( T^{10} + \)\(14\!\cdots\!89\)\( T^{12} \))
$19$ (\( 1 - \)\(37\!\cdots\!20\)\( T + \)\(20\!\cdots\!78\)\( T^{2} - \)\(76\!\cdots\!80\)\( T^{3} + \)\(42\!\cdots\!21\)\( T^{4} \))(\( 1 - \)\(58\!\cdots\!04\)\( T + \)\(32\!\cdots\!77\)\( T^{2} - \)\(15\!\cdots\!12\)\( T^{3} + \)\(66\!\cdots\!03\)\( T^{4} - \)\(24\!\cdots\!84\)\( T^{5} + \)\(87\!\cdots\!19\)\( T^{6} \))(\( 1 + \)\(54\!\cdots\!32\)\( T + \)\(55\!\cdots\!72\)\( T^{2} + \)\(24\!\cdots\!04\)\( T^{3} + \)\(16\!\cdots\!74\)\( T^{4} + \)\(51\!\cdots\!56\)\( T^{5} + \)\(23\!\cdots\!12\)\( T^{6} + \)\(47\!\cdots\!08\)\( T^{7} + \)\(18\!\cdots\!41\)\( T^{8} \))(\( ( 1 + \)\(35\!\cdots\!48\)\( T + \)\(41\!\cdots\!85\)\( T^{2} + \)\(14\!\cdots\!40\)\( T^{3} + \)\(84\!\cdots\!15\)\( T^{4} + \)\(14\!\cdots\!08\)\( T^{5} + \)\(87\!\cdots\!19\)\( T^{6} )^{2} \))
$23$ (\( 1 - \)\(26\!\cdots\!00\)\( T + \)\(48\!\cdots\!10\)\( T^{2} - \)\(63\!\cdots\!00\)\( T^{3} + \)\(58\!\cdots\!09\)\( T^{4} \))(\( 1 - \)\(17\!\cdots\!72\)\( T + \)\(68\!\cdots\!85\)\( T^{2} - \)\(74\!\cdots\!12\)\( T^{3} + \)\(16\!\cdots\!55\)\( T^{4} - \)\(10\!\cdots\!48\)\( T^{5} + \)\(14\!\cdots\!27\)\( T^{6} \))(\( 1 - \)\(61\!\cdots\!88\)\( T + \)\(98\!\cdots\!68\)\( T^{2} - \)\(39\!\cdots\!72\)\( T^{3} + \)\(20\!\cdots\!90\)\( T^{4} - \)\(95\!\cdots\!16\)\( T^{5} + \)\(57\!\cdots\!12\)\( T^{6} - \)\(86\!\cdots\!76\)\( T^{7} + \)\(34\!\cdots\!81\)\( T^{8} \))(\( 1 + \)\(91\!\cdots\!18\)\( T^{2} + \)\(37\!\cdots\!35\)\( T^{4} + \)\(10\!\cdots\!40\)\( T^{6} + \)\(21\!\cdots\!15\)\( T^{8} + \)\(31\!\cdots\!58\)\( T^{10} + \)\(20\!\cdots\!29\)\( T^{12} \))
$29$ (\( 1 - \)\(12\!\cdots\!20\)\( T + \)\(21\!\cdots\!18\)\( T^{2} - \)\(16\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!81\)\( T^{4} \))(\( 1 + \)\(11\!\cdots\!78\)\( T + \)\(16\!\cdots\!47\)\( T^{2} - \)\(21\!\cdots\!96\)\( T^{3} + \)\(20\!\cdots\!23\)\( T^{4} + \)\(18\!\cdots\!18\)\( T^{5} + \)\(21\!\cdots\!29\)\( T^{6} \))(\( 1 + \)\(41\!\cdots\!36\)\( T + \)\(10\!\cdots\!80\)\( T^{2} + \)\(16\!\cdots\!32\)\( T^{3} + \)\(21\!\cdots\!78\)\( T^{4} + \)\(21\!\cdots\!88\)\( T^{5} + \)\(17\!\cdots\!80\)\( T^{6} + \)\(88\!\cdots\!44\)\( T^{7} + \)\(27\!\cdots\!61\)\( T^{8} \))(\( 1 + \)\(33\!\cdots\!54\)\( T^{2} + \)\(67\!\cdots\!15\)\( T^{4} + \)\(92\!\cdots\!80\)\( T^{6} + \)\(11\!\cdots\!15\)\( T^{8} + \)\(92\!\cdots\!94\)\( T^{10} + \)\(44\!\cdots\!41\)\( T^{12} \))
$31$ (\( 1 - \)\(26\!\cdots\!24\)\( T + \)\(24\!\cdots\!66\)\( T^{2} - \)\(39\!\cdots\!64\)\( T^{3} + \)\(22\!\cdots\!21\)\( T^{4} \))(\( 1 + \)\(11\!\cdots\!36\)\( T + \)\(91\!\cdots\!57\)\( T^{2} + \)\(41\!\cdots\!92\)\( T^{3} + \)\(13\!\cdots\!27\)\( T^{4} + \)\(27\!\cdots\!56\)\( T^{5} + \)\(34\!\cdots\!31\)\( T^{6} \))(\( 1 - \)\(89\!\cdots\!64\)\( T + \)\(72\!\cdots\!68\)\( T^{2} - \)\(36\!\cdots\!12\)\( T^{3} + \)\(17\!\cdots\!54\)\( T^{4} - \)\(55\!\cdots\!32\)\( T^{5} + \)\(16\!\cdots\!28\)\( T^{6} - \)\(31\!\cdots\!84\)\( T^{7} + \)\(52\!\cdots\!41\)\( T^{8} \))(\( ( 1 - \)\(91\!\cdots\!64\)\( T + \)\(24\!\cdots\!65\)\( T^{2} - \)\(55\!\cdots\!80\)\( T^{3} + \)\(37\!\cdots\!15\)\( T^{4} - \)\(20\!\cdots\!44\)\( T^{5} + \)\(34\!\cdots\!31\)\( T^{6} )^{2} \))
$37$ (\( 1 + \)\(68\!\cdots\!00\)\( T + \)\(13\!\cdots\!90\)\( T^{2} + \)\(71\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!89\)\( T^{4} \))(\( 1 - \)\(60\!\cdots\!66\)\( T + \)\(29\!\cdots\!95\)\( T^{2} - \)\(11\!\cdots\!84\)\( T^{3} + \)\(30\!\cdots\!15\)\( T^{4} - \)\(67\!\cdots\!74\)\( T^{5} + \)\(11\!\cdots\!13\)\( T^{6} \))(\( 1 - \)\(55\!\cdots\!24\)\( T + \)\(22\!\cdots\!72\)\( T^{2} - \)\(20\!\cdots\!44\)\( T^{3} + \)\(28\!\cdots\!70\)\( T^{4} - \)\(21\!\cdots\!48\)\( T^{5} + \)\(25\!\cdots\!08\)\( T^{6} - \)\(65\!\cdots\!12\)\( T^{7} + \)\(12\!\cdots\!21\)\( T^{8} \))(\( ( 1 + \)\(48\!\cdots\!50\)\( T + \)\(11\!\cdots\!51\)\( T^{2} + \)\(66\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!67\)\( T^{4} + \)\(53\!\cdots\!50\)\( T^{5} + \)\(11\!\cdots\!13\)\( T^{6} )^{2} \))
$41$ (\( 1 - \)\(12\!\cdots\!36\)\( T + \)\(12\!\cdots\!86\)\( T^{2} - \)\(59\!\cdots\!16\)\( T^{3} + \)\(22\!\cdots\!61\)\( T^{4} \))(\( 1 + \)\(20\!\cdots\!34\)\( T + \)\(27\!\cdots\!47\)\( T^{2} + \)\(22\!\cdots\!88\)\( T^{3} + \)\(13\!\cdots\!07\)\( T^{4} + \)\(46\!\cdots\!74\)\( T^{5} + \)\(10\!\cdots\!41\)\( T^{6} \))(\( 1 - \)\(86\!\cdots\!76\)\( T + \)\(17\!\cdots\!88\)\( T^{2} - \)\(10\!\cdots\!28\)\( T^{3} + \)\(11\!\cdots\!34\)\( T^{4} - \)\(49\!\cdots\!68\)\( T^{5} + \)\(38\!\cdots\!68\)\( T^{6} - \)\(90\!\cdots\!16\)\( T^{7} + \)\(49\!\cdots\!21\)\( T^{8} \))(\( 1 + \)\(59\!\cdots\!86\)\( T^{2} + \)\(55\!\cdots\!15\)\( T^{4} + \)\(25\!\cdots\!20\)\( T^{6} + \)\(12\!\cdots\!15\)\( T^{8} + \)\(29\!\cdots\!06\)\( T^{10} + \)\(10\!\cdots\!81\)\( T^{12} \))
$43$ (\( 1 + \)\(25\!\cdots\!00\)\( T + \)\(44\!\cdots\!50\)\( T^{2} + \)\(70\!\cdots\!00\)\( T^{3} + \)\(75\!\cdots\!49\)\( T^{4} \))(\( 1 - \)\(24\!\cdots\!44\)\( T + \)\(28\!\cdots\!53\)\( T^{2} - \)\(29\!\cdots\!40\)\( T^{3} + \)\(78\!\cdots\!79\)\( T^{4} - \)\(18\!\cdots\!56\)\( T^{5} + \)\(20\!\cdots\!07\)\( T^{6} \))(\( 1 + \)\(50\!\cdots\!80\)\( T + \)\(85\!\cdots\!80\)\( T^{2} + \)\(56\!\cdots\!40\)\( T^{3} + \)\(31\!\cdots\!98\)\( T^{4} + \)\(15\!\cdots\!20\)\( T^{5} + \)\(64\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!60\)\( T^{7} + \)\(56\!\cdots\!01\)\( T^{8} \))(\( ( 1 - \)\(11\!\cdots\!00\)\( T + \)\(55\!\cdots\!29\)\( T^{2} - \)\(69\!\cdots\!00\)\( T^{3} + \)\(15\!\cdots\!47\)\( T^{4} - \)\(89\!\cdots\!00\)\( T^{5} + \)\(20\!\cdots\!07\)\( T^{6} )^{2} \))
$47$ (\( 1 + \)\(42\!\cdots\!00\)\( T + \)\(14\!\cdots\!70\)\( T^{2} + \)\(31\!\cdots\!00\)\( T^{3} + \)\(54\!\cdots\!69\)\( T^{4} \))(\( 1 - \)\(11\!\cdots\!32\)\( T + \)\(10\!\cdots\!17\)\( T^{2} - \)\(39\!\cdots\!60\)\( T^{3} + \)\(79\!\cdots\!79\)\( T^{4} - \)\(64\!\cdots\!08\)\( T^{5} + \)\(40\!\cdots\!03\)\( T^{6} \))(\( 1 + \)\(42\!\cdots\!20\)\( T + \)\(24\!\cdots\!20\)\( T^{2} + \)\(10\!\cdots\!40\)\( T^{3} + \)\(24\!\cdots\!38\)\( T^{4} + \)\(75\!\cdots\!80\)\( T^{5} + \)\(13\!\cdots\!80\)\( T^{6} + \)\(17\!\cdots\!60\)\( T^{7} + \)\(29\!\cdots\!61\)\( T^{8} \))(\( 1 + \)\(34\!\cdots\!22\)\( T^{2} + \)\(47\!\cdots\!35\)\( T^{4} + \)\(84\!\cdots\!60\)\( T^{6} + \)\(25\!\cdots\!15\)\( T^{8} + \)\(10\!\cdots\!42\)\( T^{10} + \)\(16\!\cdots\!09\)\( T^{12} \))
$53$ (\( 1 + \)\(15\!\cdots\!00\)\( T + \)\(16\!\cdots\!70\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{3} + \)\(39\!\cdots\!69\)\( T^{4} \))(\( 1 - \)\(11\!\cdots\!82\)\( T + \)\(23\!\cdots\!15\)\( T^{2} - \)\(14\!\cdots\!52\)\( T^{3} + \)\(14\!\cdots\!95\)\( T^{4} - \)\(45\!\cdots\!58\)\( T^{5} + \)\(24\!\cdots\!97\)\( T^{6} \))(\( 1 - \)\(12\!\cdots\!88\)\( T + \)\(20\!\cdots\!88\)\( T^{2} - \)\(19\!\cdots\!32\)\( T^{3} + \)\(18\!\cdots\!50\)\( T^{4} - \)\(12\!\cdots\!16\)\( T^{5} + \)\(79\!\cdots\!72\)\( T^{6} - \)\(31\!\cdots\!36\)\( T^{7} + \)\(15\!\cdots\!61\)\( T^{8} \))(\( 1 + \)\(25\!\cdots\!78\)\( T^{2} + \)\(31\!\cdots\!35\)\( T^{4} + \)\(24\!\cdots\!40\)\( T^{6} + \)\(12\!\cdots\!15\)\( T^{8} + \)\(40\!\cdots\!58\)\( T^{10} + \)\(61\!\cdots\!09\)\( T^{12} \))
$59$ (\( 1 - \)\(23\!\cdots\!40\)\( T + \)\(64\!\cdots\!38\)\( T^{2} - \)\(79\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!61\)\( T^{4} \))(\( 1 + \)\(10\!\cdots\!36\)\( T + \)\(11\!\cdots\!57\)\( T^{2} + \)\(73\!\cdots\!68\)\( T^{3} + \)\(38\!\cdots\!83\)\( T^{4} + \)\(12\!\cdots\!96\)\( T^{5} + \)\(36\!\cdots\!59\)\( T^{6} \))(\( 1 - \)\(13\!\cdots\!88\)\( T + \)\(62\!\cdots\!12\)\( T^{2} + \)\(19\!\cdots\!84\)\( T^{3} - \)\(29\!\cdots\!66\)\( T^{4} + \)\(64\!\cdots\!96\)\( T^{5} + \)\(68\!\cdots\!32\)\( T^{6} - \)\(48\!\cdots\!92\)\( T^{7} + \)\(12\!\cdots\!21\)\( T^{8} \))(\( 1 + \)\(11\!\cdots\!14\)\( T^{2} + \)\(73\!\cdots\!15\)\( T^{4} + \)\(30\!\cdots\!80\)\( T^{6} + \)\(80\!\cdots\!15\)\( T^{8} + \)\(14\!\cdots\!94\)\( T^{10} + \)\(13\!\cdots\!81\)\( T^{12} \))
$61$ (\( 1 - \)\(10\!\cdots\!44\)\( T + \)\(10\!\cdots\!26\)\( T^{2} - \)\(11\!\cdots\!24\)\( T^{3} + \)\(13\!\cdots\!41\)\( T^{4} \))(\( 1 - \)\(29\!\cdots\!02\)\( T + \)\(91\!\cdots\!19\)\( T^{2} - \)\(92\!\cdots\!96\)\( T^{3} + \)\(10\!\cdots\!99\)\( T^{4} - \)\(38\!\cdots\!82\)\( T^{5} + \)\(14\!\cdots\!61\)\( T^{6} \))(\( 1 + \)\(12\!\cdots\!60\)\( T + \)\(43\!\cdots\!16\)\( T^{2} + \)\(36\!\cdots\!40\)\( T^{3} + \)\(72\!\cdots\!46\)\( T^{4} + \)\(41\!\cdots\!40\)\( T^{5} + \)\(56\!\cdots\!56\)\( T^{6} + \)\(17\!\cdots\!60\)\( T^{7} + \)\(16\!\cdots\!81\)\( T^{8} \))(\( ( 1 + \)\(35\!\cdots\!34\)\( T + \)\(34\!\cdots\!15\)\( T^{2} + \)\(79\!\cdots\!80\)\( T^{3} + \)\(38\!\cdots\!15\)\( T^{4} + \)\(45\!\cdots\!94\)\( T^{5} + \)\(14\!\cdots\!61\)\( T^{6} )^{2} \))
$67$ (\( 1 + \)\(10\!\cdots\!00\)\( T + \)\(94\!\cdots\!30\)\( T^{2} + \)\(39\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!29\)\( T^{4} \))(\( 1 - \)\(38\!\cdots\!44\)\( T + \)\(85\!\cdots\!93\)\( T^{2} - \)\(19\!\cdots\!68\)\( T^{3} + \)\(31\!\cdots\!11\)\( T^{4} - \)\(51\!\cdots\!76\)\( T^{5} + \)\(49\!\cdots\!83\)\( T^{6} \))(\( 1 - \)\(16\!\cdots\!48\)\( T + \)\(20\!\cdots\!72\)\( T^{2} - \)\(16\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!66\)\( T^{4} - \)\(61\!\cdots\!00\)\( T^{5} + \)\(27\!\cdots\!88\)\( T^{6} - \)\(80\!\cdots\!84\)\( T^{7} + \)\(18\!\cdots\!41\)\( T^{8} \))(\( ( 1 + \)\(47\!\cdots\!00\)\( T + \)\(10\!\cdots\!81\)\( T^{2} + \)\(31\!\cdots\!00\)\( T^{3} + \)\(37\!\cdots\!87\)\( T^{4} + \)\(63\!\cdots\!00\)\( T^{5} + \)\(49\!\cdots\!83\)\( T^{6} )^{2} \))
$71$ (\( 1 - \)\(74\!\cdots\!16\)\( T + \)\(58\!\cdots\!46\)\( T^{2} - \)\(23\!\cdots\!56\)\( T^{3} + \)\(98\!\cdots\!81\)\( T^{4} \))(\( 1 - \)\(22\!\cdots\!44\)\( T + \)\(96\!\cdots\!85\)\( T^{2} - \)\(13\!\cdots\!00\)\( T^{3} + \)\(30\!\cdots\!35\)\( T^{4} - \)\(21\!\cdots\!64\)\( T^{5} + \)\(30\!\cdots\!71\)\( T^{6} \))(\( 1 + \)\(10\!\cdots\!88\)\( T + \)\(55\!\cdots\!68\)\( T^{2} + \)\(47\!\cdots\!16\)\( T^{3} + \)\(14\!\cdots\!70\)\( T^{4} + \)\(14\!\cdots\!56\)\( T^{5} + \)\(55\!\cdots\!08\)\( T^{6} + \)\(33\!\cdots\!48\)\( T^{7} + \)\(96\!\cdots\!61\)\( T^{8} \))(\( 1 + \)\(11\!\cdots\!46\)\( T^{2} + \)\(59\!\cdots\!15\)\( T^{4} + \)\(20\!\cdots\!20\)\( T^{6} + \)\(58\!\cdots\!15\)\( T^{8} + \)\(10\!\cdots\!06\)\( T^{10} + \)\(95\!\cdots\!41\)\( T^{12} \))
$73$ (\( 1 - \)\(19\!\cdots\!00\)\( T + \)\(18\!\cdots\!10\)\( T^{2} - \)\(17\!\cdots\!00\)\( T^{3} + \)\(76\!\cdots\!09\)\( T^{4} \))(\( 1 - \)\(32\!\cdots\!78\)\( T + \)\(10\!\cdots\!55\)\( T^{2} - \)\(11\!\cdots\!68\)\( T^{3} + \)\(96\!\cdots\!15\)\( T^{4} - \)\(25\!\cdots\!02\)\( T^{5} + \)\(67\!\cdots\!77\)\( T^{6} \))(\( 1 + \)\(19\!\cdots\!48\)\( T + \)\(28\!\cdots\!68\)\( T^{2} + \)\(45\!\cdots\!52\)\( T^{3} + \)\(35\!\cdots\!90\)\( T^{4} + \)\(39\!\cdots\!56\)\( T^{5} + \)\(22\!\cdots\!12\)\( T^{6} + \)\(13\!\cdots\!96\)\( T^{7} + \)\(59\!\cdots\!81\)\( T^{8} \))(\( ( 1 + \)\(45\!\cdots\!50\)\( T + \)\(22\!\cdots\!59\)\( T^{2} + \)\(56\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!27\)\( T^{4} + \)\(34\!\cdots\!50\)\( T^{5} + \)\(67\!\cdots\!77\)\( T^{6} )^{2} \))
$79$ (\( 1 - \)\(27\!\cdots\!80\)\( T + \)\(50\!\cdots\!18\)\( T^{2} - \)\(44\!\cdots\!20\)\( T^{3} + \)\(26\!\cdots\!81\)\( T^{4} \))(\( 1 - \)\(74\!\cdots\!40\)\( T + \)\(44\!\cdots\!77\)\( T^{2} - \)\(23\!\cdots\!20\)\( T^{3} + \)\(73\!\cdots\!43\)\( T^{4} - \)\(19\!\cdots\!40\)\( T^{5} + \)\(43\!\cdots\!79\)\( T^{6} \))(\( 1 - \)\(42\!\cdots\!20\)\( T + \)\(83\!\cdots\!36\)\( T^{2} - \)\(13\!\cdots\!40\)\( T^{3} - \)\(27\!\cdots\!14\)\( T^{4} - \)\(21\!\cdots\!60\)\( T^{5} + \)\(22\!\cdots\!16\)\( T^{6} - \)\(18\!\cdots\!80\)\( T^{7} + \)\(70\!\cdots\!61\)\( T^{8} \))(\( ( 1 + \)\(32\!\cdots\!32\)\( T + \)\(80\!\cdots\!85\)\( T^{2} + \)\(11\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!15\)\( T^{4} + \)\(87\!\cdots\!92\)\( T^{5} + \)\(43\!\cdots\!79\)\( T^{6} )^{2} \))
$83$ (\( 1 - \)\(47\!\cdots\!00\)\( T + \)\(24\!\cdots\!30\)\( T^{2} - \)\(47\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!29\)\( T^{4} \))(\( 1 - \)\(31\!\cdots\!04\)\( T + \)\(10\!\cdots\!49\)\( T^{2} - \)\(54\!\cdots\!36\)\( T^{3} + \)\(10\!\cdots\!27\)\( T^{4} - \)\(32\!\cdots\!16\)\( T^{5} + \)\(10\!\cdots\!67\)\( T^{6} \))(\( 1 - \)\(46\!\cdots\!24\)\( T + \)\(47\!\cdots\!60\)\( T^{2} - \)\(14\!\cdots\!44\)\( T^{3} + \)\(74\!\cdots\!46\)\( T^{4} - \)\(14\!\cdots\!12\)\( T^{5} + \)\(48\!\cdots\!40\)\( T^{6} - \)\(48\!\cdots\!08\)\( T^{7} + \)\(10\!\cdots\!41\)\( T^{8} \))(\( 1 + \)\(37\!\cdots\!38\)\( T^{2} + \)\(62\!\cdots\!35\)\( T^{4} + \)\(69\!\cdots\!40\)\( T^{6} + \)\(63\!\cdots\!15\)\( T^{8} + \)\(39\!\cdots\!58\)\( T^{10} + \)\(10\!\cdots\!89\)\( T^{12} \))
$89$ (\( 1 - \)\(13\!\cdots\!60\)\( T + \)\(23\!\cdots\!58\)\( T^{2} - \)\(17\!\cdots\!40\)\( T^{3} + \)\(17\!\cdots\!41\)\( T^{4} \))(\( 1 - \)\(16\!\cdots\!26\)\( T + \)\(69\!\cdots\!47\)\( T^{2} + \)\(72\!\cdots\!92\)\( T^{3} + \)\(93\!\cdots\!63\)\( T^{4} - \)\(28\!\cdots\!66\)\( T^{5} + \)\(24\!\cdots\!89\)\( T^{6} \))(\( 1 - \)\(31\!\cdots\!52\)\( T + \)\(35\!\cdots\!52\)\( T^{2} - \)\(89\!\cdots\!84\)\( T^{3} + \)\(60\!\cdots\!34\)\( T^{4} - \)\(12\!\cdots\!36\)\( T^{5} + \)\(64\!\cdots\!32\)\( T^{6} - \)\(75\!\cdots\!28\)\( T^{7} + \)\(32\!\cdots\!81\)\( T^{8} \))(\( 1 + \)\(51\!\cdots\!74\)\( T^{2} + \)\(13\!\cdots\!15\)\( T^{4} + \)\(22\!\cdots\!80\)\( T^{6} + \)\(24\!\cdots\!15\)\( T^{8} + \)\(16\!\cdots\!94\)\( T^{10} + \)\(58\!\cdots\!21\)\( T^{12} \))
$97$ (\( 1 - \)\(60\!\cdots\!00\)\( T + \)\(57\!\cdots\!70\)\( T^{2} - \)\(19\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!69\)\( T^{4} \))(\( 1 - \)\(26\!\cdots\!14\)\( T + \)\(74\!\cdots\!43\)\( T^{2} - \)\(13\!\cdots\!08\)\( T^{3} + \)\(24\!\cdots\!91\)\( T^{4} - \)\(28\!\cdots\!66\)\( T^{5} + \)\(34\!\cdots\!53\)\( T^{6} \))(\( 1 - \)\(44\!\cdots\!48\)\( T + \)\(13\!\cdots\!12\)\( T^{2} - \)\(24\!\cdots\!40\)\( T^{3} + \)\(18\!\cdots\!86\)\( T^{4} - \)\(80\!\cdots\!80\)\( T^{5} + \)\(13\!\cdots\!28\)\( T^{6} - \)\(15\!\cdots\!44\)\( T^{7} + \)\(11\!\cdots\!61\)\( T^{8} \))(\( ( 1 + \)\(17\!\cdots\!50\)\( T + \)\(19\!\cdots\!11\)\( T^{2} + \)\(12\!\cdots\!00\)\( T^{3} + \)\(62\!\cdots\!07\)\( T^{4} + \)\(17\!\cdots\!50\)\( T^{5} + \)\(34\!\cdots\!53\)\( T^{6} )^{2} \))
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