Properties

Label 1.88.a
Level 1
Weight 88
Character orbit a
Rep. character \(\chi_{1}(1,\cdot)\)
Character field \(\Q\)
Dimension 7
Newform subspaces 1
Sturm bound 7
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 88 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(7\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{88}(\Gamma_0(1))\).

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

Trace form

\( 7q + 18197022042936q^{2} - 75417209598230375148q^{3} + 376404449284858469530371136q^{4} + 330373100841196567453715678850q^{5} + 16564084581260729681531185639338144q^{6} + 4559138040275533820439239270710856q^{7} + 2786143476155257309996661063804960509440q^{8} + 673901148913686380778382918990371758797539q^{9} + O(q^{10}) \) \( 7q + 18197022042936q^{2} - 75417209598230375148q^{3} + 376404449284858469530371136q^{4} + 330373100841196567453715678850q^{5} + 16564084581260729681531185639338144q^{6} + 4559138040275533820439239270710856q^{7} + 2786143476155257309996661063804960509440q^{8} + 673901148913686380778382918990371758797539q^{9} - 36837224427403653420575386409643098251772400q^{10} + 1472337847286173961997042748507295162574738204q^{11} + 201427366404167418200465939236840718875832967936q^{12} + 3014422059373249421468162881903363740495960161162q^{13} + 174857533793446829812526107753428924608746864214592q^{14} - 3625166473400973118596882684251851193011428365821800q^{15} - 15154873535380334251051728902279691626516927753351168q^{16} + 305441579408990410315615310198102211911706536410050846q^{17} + 1860981669632360899980627375655132436743258474707045912q^{18} - 23901319372326698586394239575992814866348747359503579740q^{19} - 503913338926388656356251604104339211865794221938831363200q^{20} + 9078102789159482365584926905420124740389820445221040448864q^{21} - 21663458400202993478355225948946155306682273689862970642208q^{22} - 19197455564514114566841522508484858283599987160369983612328q^{23} + 1824092148384173625358749473375767940065390921561651577825280q^{24} + 9781980869961288666104795608695289669073361244421958446550625q^{25} + 47786976140105585405534359827367610047063003260819002377752144q^{26} - 32111903914485986226800009762395451245611112029135354852933240q^{27} + 2351878931405431793473826567200385843631464643968608507561246208q^{28} - 2929622248211881589956661929145361332353613854793218003583860710q^{29} - 36957082466872023216712671594014487627603291952852808285698132800q^{30} + 6400178386642006994765449216474292517071985477354914262501322784q^{31} + 20120636584522222199515143505026752370640182957002295900464971776q^{32} - 2304099708712963231125368654513884737108580576727984624716954535856q^{33} + 8805289984040305690417218663491282456334293326041598005371418742512q^{34} + 2946164370128275916045523909273600331980424411142407876934643279600q^{35} + 98933471743766682425065068556118094922417915990574177350322935063872q^{36} - 257009659976866776206568130155786517264021084964194253735790191892574q^{37} - 587324989591119066664077819599555299632335764115223105085385891785440q^{38} + 2177476893515746511481604952187916849953593049482066439261256257011448q^{39} - 4055946183632870235629935478991612971979135778643370607165308368256000q^{40} + 2034963152105570026635289787511170390451724027132234453389446025270774q^{41} + 89658780429355424796599740714344224628993812797700883672374246201740032q^{42} + 309840779262375549940336342200656943846568047748786979657899388184031292q^{43} + 126162431987376782669686644089341265876891443541711549282831284121563392q^{44} + 659708759880875312399988058851743701520874054284879287864164816075165450q^{45} + 8283241800873311373975299793081290815465342713652482018901675053460849344q^{46} + 11242204606334118313269051224347428508871094904239433172921637422300084016q^{47} + 87015099799477637209089580187137614904968272649313184055446066115092201472q^{48} + 145753958954116206118167199953230990336926600227864919880446772550536417151q^{49} + 566738989487532727170004425532083774727246841989597319931442293048156885000q^{50} + 1380076634530513485502205798401631551975678313044248923585562655730689936104q^{51} + 2632332061593343239516829605468680320726925121265686622876637369717466694016q^{52} + 3419870286068841140101491223891485283445183401848447984402541666833674628402q^{53} + 17795805964745240161027162809735082301228198274502437888219986191307379792960q^{54} + 26503203833163256098848318173590643400193996002260015920275807351958253192200q^{55} + 42440292502777207541999068008134143897594666406404510394531467142051565834240q^{56} + 1706414265978560716908179695812545896970951144056546955039307620736282271920q^{57} + 18180710732775946186552894846829674783208636106804512137939695866318411858640q^{58} - 203354290958599580545241392478242597911359567625033860990941191187032144762420q^{59} - 1381723905206697211646433838762880159935776676366887561984483487076698020902400q^{60} - 1736058882648294682671101194483349409609775697166397294029791585531184983170246q^{61} - 5660242463962698328485443886007015924316011394454126206801941127817551060352768q^{62} - 5522160643840737079469334400712091779972513813965210977141538032380192156663448q^{63} - 9903787115982914515069566724978909892977619061201641390731610307175318686531584q^{64} - 18811855208405109531051606793473596313229239729475312371368341814559160624293300q^{65} - 9672594632215818613139684264175317260825714985270887382392909188658672768847232q^{66} + 11442015782476036494997320312124210629147678344838534026835401111010433735123796q^{67} + 240421477317437812711737649217217088118192667783152797163721128879125405060326528q^{68} + 510354369856920078150347546469235078219936030139359645992182838476241475664022048q^{69} + 659026527132732061108417366598403984257619301353818461746506048633537586699081600q^{70} + 1148264392602224764840958034752990388535793841983093786695632558563773541887543944q^{71} + 3501171419896287400002584502353564617118657405753157482617804067953798343009748480q^{72} - 374409897453688272052095213041058963734540054922781698723863433616033006169676778q^{73} - 6267692182697113135918363950094276057909813713595513251762135854178779844039825648q^{74} - 8476695152319207879008172548761789793080338043541943634895553197887946501021992500q^{75} - 25756991148656361027587800725601659979174599371091855384616474086918540164899774720q^{76} - 36388584729748053572991662093812077413439330540077107783935194756951898797873503968q^{77} - 64846888331810736726672544568109167776241432207116429014079230635604984028202473536q^{78} - 85259406678497763555092009258190842631846392922723869349783564051773199158532291760q^{79} - 175929871620575529814806508595522592591484087378161418574968630948298520053471846400q^{80} + 130075899482872926141800059252182559849068806790469156354739766971327125790185455327q^{81} + 418452331207151012479613113755207179560044472194983590512904029159040242435665464752q^{82} + 799436699551316625789562223449566050977301350410415390149252807991013810326220200932q^{83} + 2965971058538920161676080493537980863282051627550392135569890515226367188242918172672q^{84} + 1447960636412000317484833278982538696513805622387404516982020680022475223859101936100q^{85} + 5495402118843735819261541058852824412176052977150830855384045920547702362265426008544q^{86} + 5713391022768101576199019354064972205104781572605524049702782038175329444439055575480q^{87} - 3313551881397172977291987152605823146753541026127314958943541003010066314134793144320q^{88} - 21230493431144353192357549614759184365868949423011241361246156998259760690674646688730q^{89} - 65744952423714533714912887366831934160154558702045590193448062759143707192405601850800q^{90} - 29898056630778915601699950873820874319033299862890128755673632379318045925966038553936q^{91} - 37090709528822062152716289637586904206038391157514197340606320035430839085393574946304q^{92} - 168616378631494968479965614903805288662857594402185587160024943460939204464743190107776q^{93} - 10966792713858301785063147730704078886720309320028348378406359774378616956551513433728q^{94} + 48643414019015137232448999163272568869914681360031065070688637646060412355139258171000q^{95} + 424319017557821275979674842974911555380702469580244704972091086950734324060375424303104q^{96} + 609990462562136458041516929009042041902758328857736524008319511205831857921639181049966q^{97} + 1820101059092204919393912776274283384524227688664514279364990005487646474381614661926648q^{98} + 1562898395846749574845388338143491098526808687140358215553988854233434814380041978669708q^{99} + O(q^{100}) \)

Decomposition of \(S_{88}^{\mathrm{new}}(\Gamma_0(1))\) into newform subspaces

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.88.a.a \(7\) \(47.933\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(18\!\cdots\!36\) \(-7\!\cdots\!48\) \(33\!\cdots\!50\) \(45\!\cdots\!56\) \(+\) \(q+(2599574577562+\beta _{1})q^{2}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 18197022042936 T + \)\(51\!\cdots\!28\)\( T^{2} - \)\(74\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!52\)\( T^{4} - \)\(17\!\cdots\!72\)\( T^{5} + \)\(28\!\cdots\!56\)\( T^{6} - \)\(31\!\cdots\!00\)\( T^{7} + \)\(44\!\cdots\!68\)\( T^{8} - \)\(42\!\cdots\!48\)\( T^{9} + \)\(53\!\cdots\!04\)\( T^{10} - \)\(42\!\cdots\!00\)\( T^{11} + \)\(46\!\cdots\!04\)\( T^{12} - \)\(24\!\cdots\!44\)\( T^{13} + \)\(21\!\cdots\!12\)\( T^{14} \)
$3$ \( 1 + 75417209598230375148 T + \)\(79\!\cdots\!37\)\( T^{2} + \)\(62\!\cdots\!00\)\( T^{3} + \)\(33\!\cdots\!57\)\( T^{4} + \)\(89\!\cdots\!36\)\( T^{5} + \)\(11\!\cdots\!09\)\( T^{6} + \)\(43\!\cdots\!00\)\( T^{7} + \)\(36\!\cdots\!83\)\( T^{8} + \)\(93\!\cdots\!84\)\( T^{9} + \)\(11\!\cdots\!71\)\( T^{10} + \)\(68\!\cdots\!00\)\( T^{11} + \)\(28\!\cdots\!59\)\( T^{12} + \)\(86\!\cdots\!32\)\( T^{13} + \)\(36\!\cdots\!83\)\( T^{14} \)
$5$ \( 1 - \)\(33\!\cdots\!50\)\( T + \)\(17\!\cdots\!75\)\( T^{2} - \)\(25\!\cdots\!00\)\( T^{3} + \)\(16\!\cdots\!25\)\( T^{4} - \)\(44\!\cdots\!50\)\( T^{5} + \)\(10\!\cdots\!75\)\( T^{6} - \)\(39\!\cdots\!00\)\( T^{7} + \)\(69\!\cdots\!75\)\( T^{8} - \)\(18\!\cdots\!50\)\( T^{9} + \)\(44\!\cdots\!25\)\( T^{10} - \)\(43\!\cdots\!00\)\( T^{11} + \)\(20\!\cdots\!75\)\( T^{12} - \)\(24\!\cdots\!50\)\( T^{13} + \)\(47\!\cdots\!25\)\( T^{14} \)
$7$ \( 1 - \)\(45\!\cdots\!56\)\( T + \)\(43\!\cdots\!93\)\( T^{2} - \)\(20\!\cdots\!00\)\( T^{3} + \)\(22\!\cdots\!97\)\( T^{4} + \)\(14\!\cdots\!68\)\( T^{5} + \)\(61\!\cdots\!21\)\( T^{6} + \)\(56\!\cdots\!00\)\( T^{7} + \)\(20\!\cdots\!03\)\( T^{8} + \)\(16\!\cdots\!32\)\( T^{9} + \)\(82\!\cdots\!79\)\( T^{10} - \)\(26\!\cdots\!00\)\( T^{11} + \)\(18\!\cdots\!99\)\( T^{12} - \)\(63\!\cdots\!44\)\( T^{13} + \)\(46\!\cdots\!07\)\( T^{14} \)
$11$ \( 1 - \)\(14\!\cdots\!04\)\( T + \)\(19\!\cdots\!61\)\( T^{2} - \)\(23\!\cdots\!84\)\( T^{3} + \)\(18\!\cdots\!41\)\( T^{4} - \)\(17\!\cdots\!52\)\( T^{5} + \)\(10\!\cdots\!33\)\( T^{6} - \)\(85\!\cdots\!72\)\( T^{7} + \)\(43\!\cdots\!43\)\( T^{8} - \)\(28\!\cdots\!32\)\( T^{9} + \)\(11\!\cdots\!51\)\( T^{10} - \)\(59\!\cdots\!04\)\( T^{11} + \)\(20\!\cdots\!11\)\( T^{12} - \)\(59\!\cdots\!84\)\( T^{13} + \)\(16\!\cdots\!91\)\( T^{14} \)
$13$ \( 1 - \)\(30\!\cdots\!62\)\( T + \)\(26\!\cdots\!67\)\( T^{2} - \)\(84\!\cdots\!00\)\( T^{3} + \)\(44\!\cdots\!17\)\( T^{4} - \)\(12\!\cdots\!54\)\( T^{5} + \)\(50\!\cdots\!39\)\( T^{6} - \)\(12\!\cdots\!00\)\( T^{7} + \)\(41\!\cdots\!63\)\( T^{8} - \)\(86\!\cdots\!06\)\( T^{9} + \)\(24\!\cdots\!21\)\( T^{10} - \)\(37\!\cdots\!00\)\( T^{11} + \)\(98\!\cdots\!19\)\( T^{12} - \)\(90\!\cdots\!78\)\( T^{13} + \)\(24\!\cdots\!73\)\( T^{14} \)
$17$ \( 1 - \)\(30\!\cdots\!46\)\( T + \)\(29\!\cdots\!23\)\( T^{2} - \)\(54\!\cdots\!00\)\( T^{3} + \)\(60\!\cdots\!37\)\( T^{4} - \)\(11\!\cdots\!02\)\( T^{5} + \)\(85\!\cdots\!51\)\( T^{6} - \)\(11\!\cdots\!00\)\( T^{7} + \)\(96\!\cdots\!23\)\( T^{8} - \)\(14\!\cdots\!58\)\( T^{9} + \)\(84\!\cdots\!29\)\( T^{10} - \)\(86\!\cdots\!00\)\( T^{11} + \)\(51\!\cdots\!39\)\( T^{12} - \)\(60\!\cdots\!94\)\( T^{13} + \)\(22\!\cdots\!97\)\( T^{14} \)
$19$ \( 1 + \)\(23\!\cdots\!40\)\( T + \)\(94\!\cdots\!73\)\( T^{2} + \)\(16\!\cdots\!60\)\( T^{3} + \)\(41\!\cdots\!41\)\( T^{4} + \)\(55\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!65\)\( T^{6} + \)\(11\!\cdots\!00\)\( T^{7} + \)\(19\!\cdots\!35\)\( T^{8} + \)\(17\!\cdots\!00\)\( T^{9} + \)\(23\!\cdots\!79\)\( T^{10} + \)\(16\!\cdots\!60\)\( T^{11} + \)\(17\!\cdots\!27\)\( T^{12} + \)\(77\!\cdots\!40\)\( T^{13} + \)\(57\!\cdots\!79\)\( T^{14} \)
$23$ \( 1 + \)\(19\!\cdots\!28\)\( T + \)\(11\!\cdots\!97\)\( T^{2} - \)\(36\!\cdots\!00\)\( T^{3} + \)\(65\!\cdots\!77\)\( T^{4} - \)\(38\!\cdots\!44\)\( T^{5} + \)\(25\!\cdots\!69\)\( T^{6} - \)\(15\!\cdots\!00\)\( T^{7} + \)\(76\!\cdots\!43\)\( T^{8} - \)\(33\!\cdots\!96\)\( T^{9} + \)\(16\!\cdots\!71\)\( T^{10} - \)\(27\!\cdots\!00\)\( T^{11} + \)\(26\!\cdots\!79\)\( T^{12} + \)\(12\!\cdots\!12\)\( T^{13} + \)\(19\!\cdots\!63\)\( T^{14} \)
$29$ \( 1 + \)\(29\!\cdots\!10\)\( T + \)\(97\!\cdots\!63\)\( T^{2} + \)\(23\!\cdots\!40\)\( T^{3} + \)\(42\!\cdots\!01\)\( T^{4} + \)\(86\!\cdots\!50\)\( T^{5} + \)\(11\!\cdots\!15\)\( T^{6} + \)\(18\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!35\)\( T^{8} + \)\(24\!\cdots\!50\)\( T^{9} + \)\(20\!\cdots\!29\)\( T^{10} + \)\(19\!\cdots\!40\)\( T^{11} + \)\(13\!\cdots\!87\)\( T^{12} + \)\(68\!\cdots\!10\)\( T^{13} + \)\(39\!\cdots\!69\)\( T^{14} \)
$31$ \( 1 - \)\(64\!\cdots\!84\)\( T + \)\(12\!\cdots\!01\)\( T^{2} + \)\(29\!\cdots\!76\)\( T^{3} + \)\(88\!\cdots\!21\)\( T^{4} + \)\(44\!\cdots\!48\)\( T^{5} + \)\(42\!\cdots\!93\)\( T^{6} + \)\(37\!\cdots\!48\)\( T^{7} + \)\(23\!\cdots\!23\)\( T^{8} + \)\(13\!\cdots\!08\)\( T^{9} + \)\(15\!\cdots\!51\)\( T^{10} + \)\(28\!\cdots\!16\)\( T^{11} + \)\(68\!\cdots\!51\)\( T^{12} - \)\(19\!\cdots\!24\)\( T^{13} + \)\(17\!\cdots\!71\)\( T^{14} \)
$37$ \( 1 + \)\(25\!\cdots\!74\)\( T + \)\(12\!\cdots\!83\)\( T^{2} + \)\(22\!\cdots\!00\)\( T^{3} + \)\(55\!\cdots\!17\)\( T^{4} + \)\(72\!\cdots\!58\)\( T^{5} + \)\(13\!\cdots\!11\)\( T^{6} + \)\(16\!\cdots\!00\)\( T^{7} + \)\(35\!\cdots\!63\)\( T^{8} + \)\(53\!\cdots\!62\)\( T^{9} + \)\(11\!\cdots\!29\)\( T^{10} + \)\(12\!\cdots\!00\)\( T^{11} + \)\(18\!\cdots\!19\)\( T^{12} + \)\(10\!\cdots\!06\)\( T^{13} + \)\(10\!\cdots\!77\)\( T^{14} \)
$41$ \( 1 - \)\(20\!\cdots\!74\)\( T + \)\(82\!\cdots\!71\)\( T^{2} - \)\(32\!\cdots\!44\)\( T^{3} + \)\(34\!\cdots\!61\)\( T^{4} - \)\(17\!\cdots\!02\)\( T^{5} + \)\(96\!\cdots\!23\)\( T^{6} - \)\(49\!\cdots\!92\)\( T^{7} + \)\(19\!\cdots\!63\)\( T^{8} - \)\(75\!\cdots\!22\)\( T^{9} + \)\(29\!\cdots\!01\)\( T^{10} - \)\(58\!\cdots\!24\)\( T^{11} + \)\(30\!\cdots\!71\)\( T^{12} - \)\(15\!\cdots\!94\)\( T^{13} + \)\(15\!\cdots\!61\)\( T^{14} \)
$43$ \( 1 - \)\(30\!\cdots\!92\)\( T + \)\(10\!\cdots\!57\)\( T^{2} - \)\(20\!\cdots\!00\)\( T^{3} + \)\(41\!\cdots\!97\)\( T^{4} - \)\(59\!\cdots\!24\)\( T^{5} + \)\(87\!\cdots\!29\)\( T^{6} - \)\(98\!\cdots\!00\)\( T^{7} + \)\(11\!\cdots\!03\)\( T^{8} - \)\(99\!\cdots\!76\)\( T^{9} + \)\(89\!\cdots\!71\)\( T^{10} - \)\(57\!\cdots\!00\)\( T^{11} + \)\(38\!\cdots\!99\)\( T^{12} - \)\(14\!\cdots\!08\)\( T^{13} + \)\(60\!\cdots\!43\)\( T^{14} \)
$47$ \( 1 - \)\(11\!\cdots\!16\)\( T + \)\(15\!\cdots\!13\)\( T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(94\!\cdots\!57\)\( T^{4} - \)\(52\!\cdots\!12\)\( T^{5} + \)\(36\!\cdots\!41\)\( T^{6} - \)\(17\!\cdots\!00\)\( T^{7} + \)\(10\!\cdots\!83\)\( T^{8} - \)\(46\!\cdots\!28\)\( T^{9} + \)\(24\!\cdots\!79\)\( T^{10} - \)\(84\!\cdots\!00\)\( T^{11} + \)\(35\!\cdots\!59\)\( T^{12} - \)\(76\!\cdots\!44\)\( T^{13} + \)\(20\!\cdots\!67\)\( T^{14} \)
$53$ \( 1 - \)\(34\!\cdots\!02\)\( T + \)\(73\!\cdots\!87\)\( T^{2} - \)\(98\!\cdots\!00\)\( T^{3} + \)\(89\!\cdots\!57\)\( T^{4} - \)\(39\!\cdots\!14\)\( T^{5} - \)\(15\!\cdots\!41\)\( T^{6} + \)\(42\!\cdots\!00\)\( T^{7} - \)\(15\!\cdots\!17\)\( T^{8} - \)\(41\!\cdots\!66\)\( T^{9} + \)\(96\!\cdots\!21\)\( T^{10} - \)\(10\!\cdots\!00\)\( T^{11} + \)\(83\!\cdots\!59\)\( T^{12} - \)\(40\!\cdots\!18\)\( T^{13} + \)\(12\!\cdots\!33\)\( T^{14} \)
$59$ \( 1 + \)\(20\!\cdots\!20\)\( T + \)\(77\!\cdots\!33\)\( T^{2} + \)\(11\!\cdots\!80\)\( T^{3} + \)\(25\!\cdots\!81\)\( T^{4} + \)\(30\!\cdots\!00\)\( T^{5} + \)\(45\!\cdots\!65\)\( T^{6} + \)\(44\!\cdots\!00\)\( T^{7} + \)\(52\!\cdots\!35\)\( T^{8} + \)\(40\!\cdots\!00\)\( T^{9} + \)\(38\!\cdots\!79\)\( T^{10} + \)\(21\!\cdots\!80\)\( T^{11} + \)\(16\!\cdots\!67\)\( T^{12} + \)\(49\!\cdots\!20\)\( T^{13} + \)\(28\!\cdots\!39\)\( T^{14} \)
$61$ \( 1 + \)\(17\!\cdots\!46\)\( T + \)\(21\!\cdots\!11\)\( T^{2} + \)\(19\!\cdots\!16\)\( T^{3} + \)\(14\!\cdots\!41\)\( T^{4} + \)\(91\!\cdots\!98\)\( T^{5} + \)\(49\!\cdots\!83\)\( T^{6} + \)\(24\!\cdots\!28\)\( T^{7} + \)\(10\!\cdots\!43\)\( T^{8} + \)\(40\!\cdots\!18\)\( T^{9} + \)\(13\!\cdots\!01\)\( T^{10} + \)\(38\!\cdots\!96\)\( T^{11} + \)\(90\!\cdots\!11\)\( T^{12} + \)\(15\!\cdots\!66\)\( T^{13} + \)\(18\!\cdots\!41\)\( T^{14} \)
$67$ \( 1 - \)\(11\!\cdots\!96\)\( T + \)\(22\!\cdots\!73\)\( T^{2} - \)\(67\!\cdots\!00\)\( T^{3} + \)\(30\!\cdots\!37\)\( T^{4} - \)\(93\!\cdots\!52\)\( T^{5} + \)\(37\!\cdots\!01\)\( T^{6} - \)\(75\!\cdots\!00\)\( T^{7} + \)\(27\!\cdots\!23\)\( T^{8} - \)\(51\!\cdots\!08\)\( T^{9} + \)\(12\!\cdots\!79\)\( T^{10} - \)\(20\!\cdots\!00\)\( T^{11} + \)\(49\!\cdots\!39\)\( T^{12} - \)\(18\!\cdots\!44\)\( T^{13} + \)\(12\!\cdots\!47\)\( T^{14} \)
$71$ \( 1 - \)\(11\!\cdots\!44\)\( T + \)\(98\!\cdots\!81\)\( T^{2} - \)\(62\!\cdots\!04\)\( T^{3} + \)\(32\!\cdots\!81\)\( T^{4} - \)\(14\!\cdots\!52\)\( T^{5} + \)\(57\!\cdots\!13\)\( T^{6} - \)\(20\!\cdots\!12\)\( T^{7} + \)\(66\!\cdots\!83\)\( T^{8} - \)\(19\!\cdots\!12\)\( T^{9} + \)\(48\!\cdots\!51\)\( T^{10} - \)\(10\!\cdots\!44\)\( T^{11} + \)\(19\!\cdots\!31\)\( T^{12} - \)\(26\!\cdots\!04\)\( T^{13} + \)\(26\!\cdots\!31\)\( T^{14} \)
$73$ \( 1 + \)\(37\!\cdots\!78\)\( T + \)\(49\!\cdots\!47\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!77\)\( T^{4} + \)\(21\!\cdots\!06\)\( T^{5} + \)\(17\!\cdots\!19\)\( T^{6} + \)\(36\!\cdots\!00\)\( T^{7} + \)\(22\!\cdots\!43\)\( T^{8} + \)\(35\!\cdots\!54\)\( T^{9} + \)\(23\!\cdots\!21\)\( T^{10} + \)\(28\!\cdots\!00\)\( T^{11} + \)\(17\!\cdots\!79\)\( T^{12} + \)\(16\!\cdots\!62\)\( T^{13} + \)\(58\!\cdots\!13\)\( T^{14} \)
$79$ \( 1 + \)\(85\!\cdots\!60\)\( T + \)\(10\!\cdots\!13\)\( T^{2} + \)\(62\!\cdots\!40\)\( T^{3} + \)\(43\!\cdots\!01\)\( T^{4} + \)\(19\!\cdots\!00\)\( T^{5} + \)\(93\!\cdots\!65\)\( T^{6} + \)\(31\!\cdots\!00\)\( T^{7} + \)\(11\!\cdots\!35\)\( T^{8} + \)\(29\!\cdots\!00\)\( T^{9} + \)\(82\!\cdots\!79\)\( T^{10} + \)\(14\!\cdots\!40\)\( T^{11} + \)\(30\!\cdots\!87\)\( T^{12} + \)\(31\!\cdots\!60\)\( T^{13} + \)\(45\!\cdots\!19\)\( T^{14} \)
$83$ \( 1 - \)\(79\!\cdots\!32\)\( T + \)\(70\!\cdots\!77\)\( T^{2} - \)\(36\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!37\)\( T^{4} - \)\(75\!\cdots\!84\)\( T^{5} + \)\(28\!\cdots\!49\)\( T^{6} - \)\(88\!\cdots\!00\)\( T^{7} + \)\(26\!\cdots\!23\)\( T^{8} - \)\(62\!\cdots\!36\)\( T^{9} + \)\(14\!\cdots\!71\)\( T^{10} - \)\(25\!\cdots\!00\)\( T^{11} + \)\(44\!\cdots\!39\)\( T^{12} - \)\(45\!\cdots\!48\)\( T^{13} + \)\(52\!\cdots\!03\)\( T^{14} \)
$89$ \( 1 + \)\(21\!\cdots\!30\)\( T + \)\(36\!\cdots\!03\)\( T^{2} + \)\(42\!\cdots\!20\)\( T^{3} + \)\(44\!\cdots\!61\)\( T^{4} + \)\(37\!\cdots\!50\)\( T^{5} + \)\(28\!\cdots\!15\)\( T^{6} + \)\(19\!\cdots\!00\)\( T^{7} + \)\(11\!\cdots\!35\)\( T^{8} + \)\(58\!\cdots\!50\)\( T^{9} + \)\(27\!\cdots\!29\)\( T^{10} + \)\(10\!\cdots\!20\)\( T^{11} + \)\(35\!\cdots\!47\)\( T^{12} + \)\(81\!\cdots\!30\)\( T^{13} + \)\(15\!\cdots\!09\)\( T^{14} \)
$97$ \( 1 - \)\(60\!\cdots\!66\)\( T + \)\(41\!\cdots\!63\)\( T^{2} - \)\(15\!\cdots\!00\)\( T^{3} + \)\(57\!\cdots\!57\)\( T^{4} - \)\(15\!\cdots\!62\)\( T^{5} + \)\(42\!\cdots\!91\)\( T^{6} - \)\(10\!\cdots\!00\)\( T^{7} + \)\(29\!\cdots\!83\)\( T^{8} - \)\(76\!\cdots\!78\)\( T^{9} + \)\(20\!\cdots\!29\)\( T^{10} - \)\(38\!\cdots\!00\)\( T^{11} + \)\(73\!\cdots\!59\)\( T^{12} - \)\(75\!\cdots\!94\)\( T^{13} + \)\(87\!\cdots\!17\)\( T^{14} \)
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