Properties

Label 1.90
Level 1
Weight 90
Dimension 7
Nonzero newspaces 1
Newform subspaces 1
Sturm bound 7
Trace bound 0

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 1\( 1 \) \)
Weight: \( k \) = \( 90 \)
Nonzero newspaces: \( 1 \)
Newform subspaces: \( 1 \)
Sturm bound: \(7\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{90}(\Gamma_1(1))\).

Total New Old
Modular forms 8 8 0
Cusp forms 7 7 0
Eisenstein series 1 1 0

Trace form

\( 7q - 31407330351408q^{2} - 1359636564127989407364q^{3} + 2261776677705673116713576704q^{4} + 10232470009089362495458987082250q^{5} - 47224471979937892179333042051869376q^{6} + 38501341543555466088619988316514959992q^{7} - 17954070319777658933153680910224754257920q^{8} + 5642786905031106277764534052585801627237371q^{9} + O(q^{10}) \) \( 7q - 31407330351408q^{2} - 1359636564127989407364q^{3} + 2261776677705673116713576704q^{4} + 10232470009089362495458987082250q^{5} - 47224471979937892179333042051869376q^{6} + 38501341543555466088619988316514959992q^{7} - 17954070319777658933153680910224754257920q^{8} + 5642786905031106277764534052585801627237371q^{9} - 339886969508111355644737639996417314840948000q^{10} - 33185398429231418991729589216300973489966079756q^{11} - 1004632700194650448641313614558096018594697264128q^{12} - 103208963062610039235367083636366996420076737975934q^{13} - 3327254465892753266340667920543396227686510656335232q^{14} - 3928505137426182309397624104867069340549391446959000q^{15} + 1162914962909596823700454711087105707311468348200517632q^{16} + 8319478335928844788789086096491382896131016464480742142q^{17} - 147732328649241019658282821579260643023785726547655592304q^{18} + 566291740319213482378495952187194771397009633898137615980q^{19} + 17240116916193408372006373658712519601040539123805641152000q^{20} - 90598178803547150933473132922481297065025295806785914127136q^{21} - 572030171884928043470630402880877439080047260844354367465536q^{22} - 1162243907076079309616831317442805749267469267057101679089304q^{23} + 102292564315938855068370806246211339063831852161969337222512640q^{24} - 212322678457531006783078558385295515929862869778299899884234375q^{25} + 256954281475267715641749604093742676189922873609159508555728224q^{26} + 11230609565348224313895297191070036430584541766541637832713861080q^{27} + 42271634017038816624971599134243423766860094950653754695170525184q^{28} - 141971615822865442921693407341400369750322173654805889340125766030q^{29} + 685316390944098153207628047778140189726339714889914380742075312000q^{30} + 683664012826903943332640912357777981916064560235376021892996967904q^{31} - 18295616415211966307849618677858007212056173228946481610958698446848q^{32} - 86562711553720527899763440849945042911790491733183542919656093165488q^{33} + 310076023324380793221121009797602653729382139402481891183326564397728q^{34} + 330170304491374501797216156307803170564453659546779205972398673602000q^{35} - 1640012020416196297725034563971477848297008731151338285245876678380288q^{36} - 54562371243327727239087361205689070078467002527262667251534613287958q^{37} + 45443543517981085735407850236293406047335765118416513555057178362067520q^{38} - 52864527349249866236351055677173599634948671842685563480478962199294648q^{39} - 361606215568925170012071637858292899677067482636237269829164280222720000q^{40} - 657970372131031022656393678727303783670430723595160145307484502660397466q^{41} + 1194858754087456269163194270977035799298048792026970828715903204437563904q^{42} + 3219260988489925248073148176207907480629490254450229462745931521229023956q^{43} - 15518629834856028222698238418543583045824703222318379119582671434892588032q^{44} - 47660318642997018016862592256216157339280861352258819392244471941754530750q^{45} - 56495859283654895727924020960810320205382055229302825603343235358563739776q^{46} - 589986405817377998239864975519322982713028461832176517663311475432884156208q^{47} - 3832565523894613734560398647870601248628652435813709455578147238513735368704q^{48} - 3087918931516952942471049623523578487683423584779916315782412290559435137201q^{49} - 11186377093883721132072449077187321801629226446489467367094029042775729250000q^{50} - 44749224651513683358510561546459122133403179429459901007904418343196358896456q^{51} - 195074413541392068451275175025603658303895820240091441880921515457689944416768q^{52} - 181430063765559371241038742666701312033549399057740391278794421145415983301414q^{53} - 1069768557347791789565342861771646012899754433224109219876568781015744643201920q^{54} - 1251959124114414009910203618775792485005181014548185493458611647835407269933000q^{55} - 5833843183212315752254903583581814862872931164492369538677714690034795155128320q^{56} - 6636053151264088660994009243658731470064177381062469866816802824914691937541840q^{57} - 3085757480392710136283141086455357758895171179069060863542747741259905831138720q^{58} - 9086646688135460130541149101651496775878005651006843822379595086967226946739260q^{59} - 28597614595134268752948807453761124021285695193298938696320462632027102503168000q^{60} + 8774545844023482806788207542368099480171833682235332719617911769894989384384594q^{61} + 169526530141162019514362685680407627845528510541984421132250792903222430290676224q^{62} + 600060998677337137803649489067962875140223680130818894152235224596331850847343896q^{63} + 1452176441912468141297252840801366807510309025606268702123180748140039039923257344q^{64} + 1324597647382660271811574639166235036730362819738030027679321346411726630592583500q^{65} + 5794073040567151132693317781366382943170107912751606019648469980133340516392228608q^{66} + 5843793535214317245370457729815716742376089805910350107798697912590172409883288892q^{67} + 3383522446313435458738303622044969561038087184549368481666865721250715220292329984q^{68} - 5947153100304692873901208875288315645744737627052374011935588797613066907438186848q^{69} - 48530199945646178573467957795368744286471165824229177345686884714666072932491936000q^{70} - 54831722251632385251791847370899308335600102141533329655944703784411212003030405576q^{71} - 193536676908511823968799310909661267914860871284938663775925036280217916976534056960q^{72} - 199687593578972105167207004100293505918603845674208378633352670689487827039532442554q^{73} - 965066694195901068911432718812601386557508653596585879236882657245078468484355531552q^{74} - 449845818551229824067343835155681595861236685828593692926866173145835890979882437500q^{75} - 624789819902471734428103435859366276136204176410553236414159885942801809403293936640q^{76} + 1343091849772730830408738350805190184311976794055420918621708129900429278982204783264q^{77} + 5115170224904826933556340523459220074577237696572659886093176102046008883684662673792q^{78} + 2673613187258858564803647217355170619960582874636639491339476674654505245882753222320q^{79} + 27680777280715589878137850302356927719666260300976702690203906698201250115244507136000q^{80} + 48638608516196512054280935503541026926798324553759401775728207287806340272179807855247q^{81} + 63127032979305734173962933747182283580799525530431609292523424735220314498528511785504q^{82} - 35017619719051015255233069909205249521130641969249674783596011796500257235899597331924q^{83} - 91271218351884570971920288208768314031909543306789609723100776475256076481746994388992q^{84} - 141384913067859944856588220403027346022605850144801355881993121940858052671602572135500q^{85} - 246618390628075722158819620850499384970086160242141588082920865473574265424952752922176q^{86} - 414485418912512080488627714148474005618211107267303271323056368803909094640534785472760q^{87} - 1874747156501905660418588549009712256169985369393565612731235328175018372894817920368640q^{88} - 1625431592570174192786638376427668495469444301472287736732287307560402850485922205836490q^{89} - 1500534342888898564405033173718589632643579855463696593286927469992128698108056410724000q^{90} - 304928345599594097428899578613442396706625647498145800328408392926012721126646204980336q^{91} + 7281329003117440073848907204538369781556431009214831685528490675028895216006021740500992q^{92} + 8725785078968826322471270683478576510125821912788765392594960100289595008684983594910592q^{93} + 22964567086699943860517240071514688733928422760236698641236969735081204002206026310078208q^{94} + 22546113414761055806340378014072491896390775587517280959221325649524340597082183574485000q^{95} + 84826971262863357834463681679311846519023363534588497414160609596759417378202987836473344q^{96} + 71386807957160533061753038101994485650273149078658846323482657056181047793902162983463342q^{97} - 17615586549862077419320336328445909012296979015379167958274818444332513071671081562433456q^{98} - 197573613824864988900062481565182229817848341132748736551935840297576094108952766009175068q^{99} + O(q^{100}) \)

Decomposition of \(S_{90}^{\mathrm{new}}(\Gamma_1(1))\)

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
1.90.a \(\chi_{1}(1, \cdot)\) 1.90.a.a 7 1

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 31407330351408 T + \)\(15\!\cdots\!72\)\( T^{2} + \)\(37\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!52\)\( T^{4} + \)\(24\!\cdots\!56\)\( T^{5} + \)\(48\!\cdots\!64\)\( T^{6} + \)\(11\!\cdots\!40\)\( T^{7} + \)\(30\!\cdots\!68\)\( T^{8} + \)\(95\!\cdots\!64\)\( T^{9} + \)\(26\!\cdots\!56\)\( T^{10} + \)\(54\!\cdots\!80\)\( T^{11} + \)\(13\!\cdots\!04\)\( T^{12} + \)\(17\!\cdots\!72\)\( T^{13} + \)\(34\!\cdots\!08\)\( T^{14} \)
$3$ \( 1 + \)\(13\!\cdots\!64\)\( T + \)\(82\!\cdots\!53\)\( T^{2} + \)\(53\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!77\)\( T^{4} - \)\(42\!\cdots\!12\)\( T^{5} + \)\(67\!\cdots\!21\)\( T^{6} - \)\(45\!\cdots\!60\)\( T^{7} + \)\(19\!\cdots\!43\)\( T^{8} - \)\(35\!\cdots\!68\)\( T^{9} + \)\(47\!\cdots\!99\)\( T^{10} + \)\(38\!\cdots\!80\)\( T^{11} + \)\(17\!\cdots\!79\)\( T^{12} + \)\(82\!\cdots\!16\)\( T^{13} + \)\(17\!\cdots\!27\)\( T^{14} \)
$5$ \( 1 - \)\(10\!\cdots\!50\)\( T + \)\(72\!\cdots\!75\)\( T^{2} - \)\(67\!\cdots\!00\)\( T^{3} + \)\(23\!\cdots\!25\)\( T^{4} - \)\(20\!\cdots\!50\)\( T^{5} + \)\(50\!\cdots\!75\)\( T^{6} - \)\(38\!\cdots\!00\)\( T^{7} + \)\(81\!\cdots\!75\)\( T^{8} - \)\(52\!\cdots\!50\)\( T^{9} + \)\(10\!\cdots\!25\)\( T^{10} - \)\(46\!\cdots\!00\)\( T^{11} + \)\(79\!\cdots\!75\)\( T^{12} - \)\(18\!\cdots\!50\)\( T^{13} + \)\(28\!\cdots\!25\)\( T^{14} \)
$7$ \( 1 - \)\(38\!\cdots\!92\)\( T + \)\(80\!\cdots\!57\)\( T^{2} - \)\(27\!\cdots\!00\)\( T^{3} + \)\(31\!\cdots\!97\)\( T^{4} - \)\(93\!\cdots\!24\)\( T^{5} + \)\(76\!\cdots\!29\)\( T^{6} - \)\(19\!\cdots\!00\)\( T^{7} + \)\(12\!\cdots\!03\)\( T^{8} - \)\(25\!\cdots\!76\)\( T^{9} + \)\(13\!\cdots\!71\)\( T^{10} - \)\(19\!\cdots\!00\)\( T^{11} + \)\(93\!\cdots\!99\)\( T^{12} - \)\(73\!\cdots\!08\)\( T^{13} + \)\(31\!\cdots\!43\)\( T^{14} \)
$11$ \( 1 + \)\(33\!\cdots\!56\)\( T + \)\(18\!\cdots\!81\)\( T^{2} + \)\(51\!\cdots\!96\)\( T^{3} + \)\(19\!\cdots\!81\)\( T^{4} + \)\(43\!\cdots\!48\)\( T^{5} + \)\(13\!\cdots\!13\)\( T^{6} + \)\(25\!\cdots\!88\)\( T^{7} + \)\(62\!\cdots\!83\)\( T^{8} + \)\(10\!\cdots\!88\)\( T^{9} + \)\(21\!\cdots\!51\)\( T^{10} + \)\(27\!\cdots\!56\)\( T^{11} + \)\(47\!\cdots\!31\)\( T^{12} + \)\(42\!\cdots\!96\)\( T^{13} + \)\(61\!\cdots\!31\)\( T^{14} \)
$13$ \( 1 + \)\(10\!\cdots\!34\)\( T + \)\(79\!\cdots\!63\)\( T^{2} + \)\(36\!\cdots\!60\)\( T^{3} + \)\(12\!\cdots\!57\)\( T^{4} + \)\(24\!\cdots\!58\)\( T^{5} + \)\(13\!\cdots\!71\)\( T^{6} - \)\(13\!\cdots\!20\)\( T^{7} + \)\(19\!\cdots\!83\)\( T^{8} + \)\(47\!\cdots\!82\)\( T^{9} + \)\(34\!\cdots\!69\)\( T^{10} + \)\(13\!\cdots\!60\)\( T^{11} + \)\(40\!\cdots\!59\)\( T^{12} + \)\(72\!\cdots\!26\)\( T^{13} + \)\(96\!\cdots\!97\)\( T^{14} \)
$17$ \( 1 - \)\(83\!\cdots\!42\)\( T + \)\(16\!\cdots\!27\)\( T^{2} - \)\(12\!\cdots\!60\)\( T^{3} + \)\(12\!\cdots\!37\)\( T^{4} - \)\(83\!\cdots\!34\)\( T^{5} + \)\(58\!\cdots\!59\)\( T^{6} - \)\(34\!\cdots\!80\)\( T^{7} + \)\(19\!\cdots\!23\)\( T^{8} - \)\(87\!\cdots\!06\)\( T^{9} + \)\(42\!\cdots\!01\)\( T^{10} - \)\(13\!\cdots\!60\)\( T^{11} + \)\(57\!\cdots\!39\)\( T^{12} - \)\(95\!\cdots\!18\)\( T^{13} + \)\(37\!\cdots\!13\)\( T^{14} \)
$19$ \( 1 - \)\(56\!\cdots\!80\)\( T + \)\(27\!\cdots\!53\)\( T^{2} - \)\(16\!\cdots\!20\)\( T^{3} + \)\(40\!\cdots\!61\)\( T^{4} - \)\(20\!\cdots\!00\)\( T^{5} + \)\(37\!\cdots\!65\)\( T^{6} - \)\(16\!\cdots\!00\)\( T^{7} + \)\(24\!\cdots\!35\)\( T^{8} - \)\(86\!\cdots\!00\)\( T^{9} + \)\(10\!\cdots\!79\)\( T^{10} - \)\(27\!\cdots\!20\)\( T^{11} + \)\(30\!\cdots\!47\)\( T^{12} - \)\(40\!\cdots\!80\)\( T^{13} + \)\(46\!\cdots\!59\)\( T^{14} \)
$23$ \( 1 + \)\(11\!\cdots\!04\)\( T + \)\(63\!\cdots\!73\)\( T^{2} + \)\(40\!\cdots\!40\)\( T^{3} + \)\(19\!\cdots\!37\)\( T^{4} + \)\(50\!\cdots\!28\)\( T^{5} + \)\(41\!\cdots\!21\)\( T^{6} + \)\(38\!\cdots\!20\)\( T^{7} + \)\(65\!\cdots\!23\)\( T^{8} + \)\(12\!\cdots\!32\)\( T^{9} + \)\(75\!\cdots\!39\)\( T^{10} + \)\(23\!\cdots\!40\)\( T^{11} + \)\(58\!\cdots\!39\)\( T^{12} + \)\(16\!\cdots\!36\)\( T^{13} + \)\(22\!\cdots\!67\)\( T^{14} \)
$29$ \( 1 + \)\(14\!\cdots\!30\)\( T + \)\(42\!\cdots\!83\)\( T^{2} + \)\(39\!\cdots\!20\)\( T^{3} + \)\(88\!\cdots\!81\)\( T^{4} + \)\(54\!\cdots\!50\)\( T^{5} + \)\(12\!\cdots\!15\)\( T^{6} + \)\(72\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!35\)\( T^{8} + \)\(11\!\cdots\!50\)\( T^{9} + \)\(25\!\cdots\!29\)\( T^{10} + \)\(16\!\cdots\!20\)\( T^{11} + \)\(24\!\cdots\!67\)\( T^{12} + \)\(11\!\cdots\!30\)\( T^{13} + \)\(11\!\cdots\!89\)\( T^{14} \)
$31$ \( 1 - \)\(68\!\cdots\!04\)\( T + \)\(21\!\cdots\!61\)\( T^{2} - \)\(22\!\cdots\!84\)\( T^{3} + \)\(21\!\cdots\!41\)\( T^{4} - \)\(29\!\cdots\!52\)\( T^{5} + \)\(13\!\cdots\!33\)\( T^{6} - \)\(21\!\cdots\!72\)\( T^{7} + \)\(75\!\cdots\!43\)\( T^{8} - \)\(86\!\cdots\!32\)\( T^{9} + \)\(32\!\cdots\!51\)\( T^{10} - \)\(19\!\cdots\!04\)\( T^{11} + \)\(96\!\cdots\!11\)\( T^{12} - \)\(16\!\cdots\!84\)\( T^{13} + \)\(13\!\cdots\!91\)\( T^{14} \)
$37$ \( 1 + \)\(54\!\cdots\!58\)\( T + \)\(17\!\cdots\!67\)\( T^{2} + \)\(15\!\cdots\!20\)\( T^{3} + \)\(14\!\cdots\!17\)\( T^{4} + \)\(15\!\cdots\!46\)\( T^{5} + \)\(78\!\cdots\!19\)\( T^{6} + \)\(75\!\cdots\!60\)\( T^{7} + \)\(29\!\cdots\!63\)\( T^{8} + \)\(21\!\cdots\!34\)\( T^{9} + \)\(73\!\cdots\!61\)\( T^{10} + \)\(29\!\cdots\!20\)\( T^{11} + \)\(12\!\cdots\!19\)\( T^{12} + \)\(14\!\cdots\!62\)\( T^{13} + \)\(97\!\cdots\!53\)\( T^{14} \)
$41$ \( 1 + \)\(65\!\cdots\!66\)\( T + \)\(14\!\cdots\!51\)\( T^{2} + \)\(11\!\cdots\!76\)\( T^{3} + \)\(11\!\cdots\!21\)\( T^{4} + \)\(83\!\cdots\!98\)\( T^{5} + \)\(64\!\cdots\!43\)\( T^{6} + \)\(35\!\cdots\!48\)\( T^{7} + \)\(22\!\cdots\!23\)\( T^{8} + \)\(98\!\cdots\!58\)\( T^{9} + \)\(49\!\cdots\!01\)\( T^{10} + \)\(16\!\cdots\!16\)\( T^{11} + \)\(72\!\cdots\!51\)\( T^{12} + \)\(11\!\cdots\!26\)\( T^{13} + \)\(58\!\cdots\!21\)\( T^{14} \)
$43$ \( 1 - \)\(32\!\cdots\!56\)\( T + \)\(10\!\cdots\!93\)\( T^{2} - \)\(24\!\cdots\!00\)\( T^{3} + \)\(50\!\cdots\!97\)\( T^{4} - \)\(92\!\cdots\!32\)\( T^{5} + \)\(16\!\cdots\!21\)\( T^{6} - \)\(24\!\cdots\!00\)\( T^{7} + \)\(39\!\cdots\!03\)\( T^{8} - \)\(53\!\cdots\!68\)\( T^{9} + \)\(68\!\cdots\!79\)\( T^{10} - \)\(79\!\cdots\!00\)\( T^{11} + \)\(79\!\cdots\!99\)\( T^{12} - \)\(60\!\cdots\!44\)\( T^{13} + \)\(44\!\cdots\!07\)\( T^{14} \)
$47$ \( 1 + \)\(58\!\cdots\!08\)\( T + \)\(50\!\cdots\!37\)\( T^{2} + \)\(21\!\cdots\!60\)\( T^{3} + \)\(10\!\cdots\!57\)\( T^{4} + \)\(33\!\cdots\!36\)\( T^{5} + \)\(11\!\cdots\!49\)\( T^{6} + \)\(28\!\cdots\!80\)\( T^{7} + \)\(76\!\cdots\!83\)\( T^{8} + \)\(14\!\cdots\!04\)\( T^{9} + \)\(29\!\cdots\!91\)\( T^{10} + \)\(39\!\cdots\!60\)\( T^{11} + \)\(61\!\cdots\!59\)\( T^{12} + \)\(46\!\cdots\!52\)\( T^{13} + \)\(52\!\cdots\!23\)\( T^{14} \)
$53$ \( 1 + \)\(18\!\cdots\!14\)\( T + \)\(25\!\cdots\!03\)\( T^{2} + \)\(26\!\cdots\!80\)\( T^{3} + \)\(23\!\cdots\!77\)\( T^{4} + \)\(17\!\cdots\!38\)\( T^{5} + \)\(11\!\cdots\!71\)\( T^{6} + \)\(65\!\cdots\!40\)\( T^{7} + \)\(33\!\cdots\!43\)\( T^{8} + \)\(14\!\cdots\!82\)\( T^{9} + \)\(56\!\cdots\!49\)\( T^{10} + \)\(18\!\cdots\!80\)\( T^{11} + \)\(50\!\cdots\!79\)\( T^{12} + \)\(10\!\cdots\!66\)\( T^{13} + \)\(16\!\cdots\!77\)\( T^{14} \)
$59$ \( 1 + \)\(90\!\cdots\!60\)\( T + \)\(21\!\cdots\!73\)\( T^{2} + \)\(13\!\cdots\!40\)\( T^{3} + \)\(19\!\cdots\!41\)\( T^{4} + \)\(99\!\cdots\!00\)\( T^{5} + \)\(11\!\cdots\!65\)\( T^{6} + \)\(47\!\cdots\!00\)\( T^{7} + \)\(45\!\cdots\!35\)\( T^{8} + \)\(16\!\cdots\!00\)\( T^{9} + \)\(12\!\cdots\!79\)\( T^{10} + \)\(36\!\cdots\!40\)\( T^{11} + \)\(22\!\cdots\!27\)\( T^{12} + \)\(39\!\cdots\!60\)\( T^{13} + \)\(17\!\cdots\!79\)\( T^{14} \)
$61$ \( 1 - \)\(87\!\cdots\!94\)\( T + \)\(29\!\cdots\!31\)\( T^{2} - \)\(24\!\cdots\!04\)\( T^{3} + \)\(44\!\cdots\!81\)\( T^{4} - \)\(42\!\cdots\!02\)\( T^{5} + \)\(45\!\cdots\!63\)\( T^{6} - \)\(43\!\cdots\!12\)\( T^{7} + \)\(35\!\cdots\!83\)\( T^{8} - \)\(26\!\cdots\!62\)\( T^{9} + \)\(21\!\cdots\!01\)\( T^{10} - \)\(93\!\cdots\!44\)\( T^{11} + \)\(88\!\cdots\!31\)\( T^{12} - \)\(20\!\cdots\!54\)\( T^{13} + \)\(18\!\cdots\!81\)\( T^{14} \)
$67$ \( 1 - \)\(58\!\cdots\!92\)\( T + \)\(34\!\cdots\!77\)\( T^{2} - \)\(12\!\cdots\!60\)\( T^{3} + \)\(40\!\cdots\!37\)\( T^{4} - \)\(10\!\cdots\!84\)\( T^{5} + \)\(24\!\cdots\!09\)\( T^{6} - \)\(45\!\cdots\!80\)\( T^{7} + \)\(80\!\cdots\!23\)\( T^{8} - \)\(11\!\cdots\!56\)\( T^{9} + \)\(14\!\cdots\!51\)\( T^{10} - \)\(14\!\cdots\!60\)\( T^{11} + \)\(13\!\cdots\!39\)\( T^{12} - \)\(77\!\cdots\!68\)\( T^{13} + \)\(44\!\cdots\!63\)\( T^{14} \)
$71$ \( 1 + \)\(54\!\cdots\!76\)\( T + \)\(38\!\cdots\!21\)\( T^{2} + \)\(14\!\cdots\!56\)\( T^{3} + \)\(62\!\cdots\!61\)\( T^{4} + \)\(18\!\cdots\!48\)\( T^{5} + \)\(57\!\cdots\!73\)\( T^{6} + \)\(13\!\cdots\!08\)\( T^{7} + \)\(33\!\cdots\!63\)\( T^{8} + \)\(61\!\cdots\!28\)\( T^{9} + \)\(12\!\cdots\!51\)\( T^{10} + \)\(16\!\cdots\!76\)\( T^{11} + \)\(24\!\cdots\!71\)\( T^{12} + \)\(20\!\cdots\!56\)\( T^{13} + \)\(21\!\cdots\!11\)\( T^{14} \)
$73$ \( 1 + \)\(19\!\cdots\!54\)\( T + \)\(53\!\cdots\!23\)\( T^{2} + \)\(79\!\cdots\!40\)\( T^{3} + \)\(11\!\cdots\!37\)\( T^{4} + \)\(13\!\cdots\!78\)\( T^{5} + \)\(13\!\cdots\!71\)\( T^{6} + \)\(12\!\cdots\!20\)\( T^{7} + \)\(94\!\cdots\!23\)\( T^{8} + \)\(62\!\cdots\!82\)\( T^{9} + \)\(37\!\cdots\!89\)\( T^{10} + \)\(17\!\cdots\!40\)\( T^{11} + \)\(81\!\cdots\!39\)\( T^{12} + \)\(20\!\cdots\!86\)\( T^{13} + \)\(70\!\cdots\!17\)\( T^{14} \)
$79$ \( 1 - \)\(26\!\cdots\!20\)\( T + \)\(19\!\cdots\!33\)\( T^{2} - \)\(22\!\cdots\!80\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{4} - \)\(15\!\cdots\!00\)\( T^{5} + \)\(14\!\cdots\!65\)\( T^{6} - \)\(24\!\cdots\!00\)\( T^{7} + \)\(10\!\cdots\!35\)\( T^{8} - \)\(93\!\cdots\!00\)\( T^{9} + \)\(64\!\cdots\!79\)\( T^{10} - \)\(79\!\cdots\!80\)\( T^{11} + \)\(54\!\cdots\!67\)\( T^{12} - \)\(57\!\cdots\!20\)\( T^{13} + \)\(16\!\cdots\!39\)\( T^{14} \)
$83$ \( 1 + \)\(35\!\cdots\!24\)\( T + \)\(34\!\cdots\!33\)\( T^{2} + \)\(80\!\cdots\!20\)\( T^{3} + \)\(46\!\cdots\!17\)\( T^{4} + \)\(78\!\cdots\!48\)\( T^{5} + \)\(37\!\cdots\!21\)\( T^{6} + \)\(51\!\cdots\!60\)\( T^{7} + \)\(23\!\cdots\!63\)\( T^{8} + \)\(30\!\cdots\!32\)\( T^{9} + \)\(11\!\cdots\!59\)\( T^{10} + \)\(12\!\cdots\!20\)\( T^{11} + \)\(33\!\cdots\!19\)\( T^{12} + \)\(21\!\cdots\!96\)\( T^{13} + \)\(38\!\cdots\!87\)\( T^{14} \)
$89$ \( 1 + \)\(16\!\cdots\!90\)\( T + \)\(25\!\cdots\!63\)\( T^{2} + \)\(22\!\cdots\!60\)\( T^{3} + \)\(20\!\cdots\!01\)\( T^{4} + \)\(13\!\cdots\!50\)\( T^{5} + \)\(88\!\cdots\!15\)\( T^{6} + \)\(47\!\cdots\!00\)\( T^{7} + \)\(27\!\cdots\!35\)\( T^{8} + \)\(12\!\cdots\!50\)\( T^{9} + \)\(62\!\cdots\!29\)\( T^{10} + \)\(22\!\cdots\!60\)\( T^{11} + \)\(75\!\cdots\!87\)\( T^{12} + \)\(15\!\cdots\!90\)\( T^{13} + \)\(29\!\cdots\!69\)\( T^{14} \)
$97$ \( 1 - \)\(71\!\cdots\!42\)\( T + \)\(48\!\cdots\!87\)\( T^{2} - \)\(20\!\cdots\!40\)\( T^{3} + \)\(79\!\cdots\!57\)\( T^{4} - \)\(24\!\cdots\!14\)\( T^{5} + \)\(72\!\cdots\!99\)\( T^{6} - \)\(19\!\cdots\!20\)\( T^{7} + \)\(47\!\cdots\!83\)\( T^{8} - \)\(11\!\cdots\!46\)\( T^{9} + \)\(23\!\cdots\!41\)\( T^{10} - \)\(40\!\cdots\!40\)\( T^{11} + \)\(62\!\cdots\!59\)\( T^{12} - \)\(61\!\cdots\!98\)\( T^{13} + \)\(57\!\cdots\!73\)\( T^{14} \)
show more
show less