Properties

Label 2.88.a
Level 2
Weight 88
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 7
Newform subspaces 2
Sturm bound 22
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 88 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(22\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{88}(\Gamma_0(2))\).

Total New Old
Modular forms 23 7 16
Cusp forms 21 7 14
Eisenstein series 2 0 2

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)Dim.
\(+\)\(4\)
\(-\)\(3\)

Trace form

\( 7q - 8796093022208q^{2} + 80974745693182072212q^{3} + 541598767187353870268366848q^{4} - 2920508893103043329722952304510q^{5} - 6381576261198159352072644593713152q^{6} + 5685733200018651170793947092255103816q^{7} - 680564733841876926926749214863536422912q^{8} + 879093243194986986690731015132457684762339q^{9} + O(q^{10}) \) \( 7q - 8796093022208q^{2} + 80974745693182072212q^{3} + 541598767187353870268366848q^{4} - 2920508893103043329722952304510q^{5} - 6381576261198159352072644593713152q^{6} + 5685733200018651170793947092255103816q^{7} - 680564733841876926926749214863536422912q^{8} + 879093243194986986690731015132457684762339q^{9} + 45346814034776642044414320718694217500590080q^{10} + 1045718858653864846400722202389139493224706204q^{11} + 6265117491533843230876941959545501178520403968q^{12} - 2671899427313212817950921193897595638829922521718q^{13} - 100967058322828401682216463293916492742918286606336q^{14} + 4845183259634109809863249173665448129055865803433880q^{15} + 41904174945551648470736051523641266739574897872207872q^{16} - 911843101929464476520354640649817347730389523078087394q^{17} - 6695389209765332488782046017891867662712758472441593856q^{18} + 28924509410078824981374512978553416734376012822976487460q^{19} - 225963430866330245085937808931676908407220053845097840640q^{20} - 6592029819864249529543089444323273550331678016715264344736q^{21} - 43061396618197167843252134244216093933200210184878828814336q^{22} + 286160503294493310726157265691368539698399154450063308833112q^{23} - 493750547968143722671467529194448041338254084819611316912128q^{24} + 16773411111251446550161743540865958027446772523897993540208225q^{25} + 7590058159920279723504669770694252396564881791202346294837248q^{26} - 176541606068591278219279589633249340635185932685702156017987960q^{27} + 439912298812329995920278403244976484632759548258090763287527424q^{28} - 596105450597790504763677619332294088233825086729323088019355110q^{29} - 16684169114283099118084213828160605969035928215602441898468311040q^{30} + 203456212626134666254474285706188851196255130095897048531164029984q^{31} - 52656145834278593348959013841835216159447547700274555627155488768q^{32} + 6611172792710809377309680424239520760642205214674854003605044304q^{33} + 4335268461612023802292419723184381313338650045913048973830734741504q^{34} + 1404000317055518859792186972116908136345047884535988124862456089840q^{35} + 68016545251019659144585255350697211368608776233487887607853792362496q^{36} + 415614429276011774504094954290616884839739198968825236706124661649826q^{37} - 414981295954429455801366187408002568554185302345422214438418309447680q^{38} - 4848075917433942437390150341067585145352417589100388240306020681209352q^{39} + 3508539796729889366487207908624035246843099907589385799724199701381120q^{40} - 6418618697764445378474998422072543198713839893055268002100719980540426q^{41} - 5897709362042313573225652906823647681445624592679859944441622179086336q^{42} - 256517552495093346533878881868115114874049551075704932806208510128442948q^{43} + 80908577810214279425242339449244072997228603775418484308777319956217856q^{44} - 1008136421053650594053909266821883015329870913151316988628297230599950070q^{45} - 2187246920712917002550544397108607274991072632429297116085248698406141952q^{46} - 16733188737624868265198708846801005986602496658340465177406132772864641744q^{47} + 484739987099808063191307956121587154538073135694033668175423182368407552q^{48} - 31666208429464795334783441124742945856803380727775507539289480402564382849q^{49} - 325107647412956537138891644142795977272301081819324164650185390975405260800q^{50} - 1488436923534378108307311683424482833828549344295074188287164039294686211096q^{51} - 206728205125918983383682533740981187243326403800086050617337009128238743552q^{52} - 3873744239282753468251342153387956686347628071807147461758111421873328064718q^{53} - 3283102625509781351631282772901147721175145116931081799392401284803938222080q^{54} - 21516706422211156955719465983343803591388253093419924345604153423284373980920q^{55} - 7811947759168217064294711991727617178670217755407559258544295859774715592704q^{56} + 14364804172733276799058201186557640544958360114702437484884205381355873721520q^{57} + 11229524416113640225861387908623839197226088206078531963629322358384398172160q^{58} + 65252100640619404855593799009539940476388245196788574888464805291977191707980q^{59} + 374877897173519797145213143903455880516931011042190085628882375527733993144320q^{60} + 1457525223204824945702563864576412154462115021937726844158273060443664370877754q^{61} - 312004483985782428273513783935283410429626502239991458777327019581334005743616q^{62} + 7772486446481763873885338030021004419412092616480944679915890933796224970961512q^{63} + 3242178498644853471859987580243261419891559570637935793104812352221567629918208q^{64} + 17501981465693343534469350932038425739742668860449414641736688344740487354991180q^{65} + 17227271677828493488011609718768884360090428630829891729822058112954320086368256q^{66} - 55063608285362556627500287200293476247490508122231737660319354361952125877519404q^{67} - 70550442839041516525388945659662118556561341681799789207271681940686335970902016q^{68} - 400760622833636458358612993092732200019663896789093288336500926158559894015798752q^{69} - 235789562607816141906138248062224322967296448509229261105547772039335525451038720q^{70} - 809919070647168245287984778020801750754647514600106512071674917136154437663707256q^{71} - 518030648835487930953142112427925210444137937256121834226706661503466875318697984q^{72} - 219512310360407845351868489935679581335396475392666027259295788773156980304650218q^{73} + 452472924926239618943165637943161754452853007389822882038116546395265373814390784q^{74} + 6593685854768020524596274859859454921366588771690718440054536491922198823162938700q^{75} + 2237925519713958252330471211436059223523937348828262137788025067535321731107389440q^{76} + 48189115305190595783092790866014996034307724133112314071272331368300210410524125472q^{77} + 48081726017840595821209933479161246054796432520697414746736368982540680665841532928q^{78} + 41887116389531328958071612822794594017886544653464533298449747875153256542229116240q^{79} - 17483073655232760837896749167977098445596166906703884539729185449335209220594728960q^{80} - 248994518459692627779800898861327495684048603642164795554115700218871127194937232673q^{81} - 153202624298947612187137453318057981364754468717800354072092846651985747066681819136q^{82} + 27257017012339100300294357239702418965717947451107447905359873023757776672679112292q^{83} - 510033603385821707465274229300107668220346554281768662136841683924010076286226530304q^{84} - 2506886845979623636803991768917828986869790710671731032315853010962471330626470636060q^{85} + 519819449213973876349059841635819218020008559557659400228146512177608052150473064448q^{86} - 1922070520588291269199218450486529887337922429718809259552885686567174203015479399240q^{87} - 3331714188825896454625401182308883924115908264476133136131968077628398518736418504704q^{88} + 4440505247623517019185317469641406104946126401490640001257528431562302963958177912870q^{89} + 15241462304536744288696433335833522172114047020301618477837980810562148664194168258560q^{90} + 74555096827161452424559032067290863540960492854480108854649860697571114878578265254064q^{91} + 22140596543144327541314530498775923307971867425518811618874714621917340133633260781568q^{92} + 68748487413042570550647295232416513125721447243886369418131426808002365372654345636224q^{93} + 58658951178719098483906984727989597905547419204747217682011490043116799856978453069824q^{94} + 141850625202675413431502055471455983500859519069709632169030111296354263823346521217400q^{95} - 38202098296803867369516485901062753981801306524845541206691164975946195491456697761792q^{96} - 792509579329487478750020088969755231354891894671840856415934423025323544960504090993554q^{97} - 1080896543784503157734778360994413992107726045156619741783739180932200778075193269551104q^{98} + 89132395521541223284365576449202212809152812784979536786243127508244728259781595844108q^{99} + O(q^{100}) \)

Decomposition of \(S_{88}^{\mathrm{new}}(\Gamma_0(2))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.88.a.a \(3\) \(95.867\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(26\!\cdots\!24\) \(-3\!\cdots\!16\) \(11\!\cdots\!50\) \(-2\!\cdots\!88\) \(-\) \(q+2^{43}q^{2}+(-107421079853628369972+\cdots)q^{3}+\cdots\)
2.88.a.b \(4\) \(95.867\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-3\!\cdots\!32\) \(40\!\cdots\!28\) \(-4\!\cdots\!60\) \(85\!\cdots\!04\) \(+\) \(q-2^{43}q^{2}+(100809496313516795532+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{88}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

\( S_{88}^{\mathrm{old}}(\Gamma_0(2)) \cong \) \(S_{88}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - 8796093022208 T )^{3} \))(\( ( 1 + 8796093022208 T )^{4} \))
$3$ (\( 1 + \)\(32\!\cdots\!16\)\( T + \)\(50\!\cdots\!13\)\( T^{2} + \)\(20\!\cdots\!32\)\( T^{3} + \)\(16\!\cdots\!31\)\( T^{4} + \)\(33\!\cdots\!04\)\( T^{5} + \)\(33\!\cdots\!03\)\( T^{6} \))(\( 1 - \)\(40\!\cdots\!28\)\( T + \)\(31\!\cdots\!92\)\( T^{2} - \)\(95\!\cdots\!80\)\( T^{3} + \)\(15\!\cdots\!46\)\( T^{4} - \)\(30\!\cdots\!60\)\( T^{5} + \)\(33\!\cdots\!48\)\( T^{6} - \)\(13\!\cdots\!84\)\( T^{7} + \)\(10\!\cdots\!61\)\( T^{8} \))
$5$ (\( 1 - \)\(11\!\cdots\!50\)\( T + \)\(15\!\cdots\!75\)\( T^{2} - \)\(14\!\cdots\!00\)\( T^{3} + \)\(99\!\cdots\!75\)\( T^{4} - \)\(46\!\cdots\!50\)\( T^{5} + \)\(26\!\cdots\!25\)\( T^{6} \))(\( 1 + \)\(40\!\cdots\!60\)\( T + \)\(76\!\cdots\!00\)\( T^{2} + \)\(29\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!50\)\( T^{4} + \)\(18\!\cdots\!00\)\( T^{5} + \)\(31\!\cdots\!00\)\( T^{6} + \)\(10\!\cdots\!00\)\( T^{7} + \)\(17\!\cdots\!25\)\( T^{8} \))
$7$ (\( 1 + \)\(28\!\cdots\!88\)\( T + \)\(92\!\cdots\!77\)\( T^{2} + \)\(19\!\cdots\!04\)\( T^{3} + \)\(31\!\cdots\!11\)\( T^{4} + \)\(32\!\cdots\!12\)\( T^{5} + \)\(37\!\cdots\!07\)\( T^{6} \))(\( 1 - \)\(85\!\cdots\!04\)\( T + \)\(80\!\cdots\!28\)\( T^{2} - \)\(49\!\cdots\!20\)\( T^{3} + \)\(39\!\cdots\!86\)\( T^{4} - \)\(16\!\cdots\!60\)\( T^{5} + \)\(90\!\cdots\!72\)\( T^{6} - \)\(31\!\cdots\!28\)\( T^{7} + \)\(12\!\cdots\!01\)\( T^{8} \))
$11$ (\( 1 + \)\(19\!\cdots\!44\)\( T + \)\(14\!\cdots\!25\)\( T^{2} - \)\(73\!\cdots\!60\)\( T^{3} + \)\(57\!\cdots\!75\)\( T^{4} + \)\(30\!\cdots\!04\)\( T^{5} + \)\(63\!\cdots\!11\)\( T^{6} \))(\( 1 - \)\(29\!\cdots\!48\)\( T + \)\(14\!\cdots\!48\)\( T^{2} - \)\(30\!\cdots\!36\)\( T^{3} + \)\(82\!\cdots\!70\)\( T^{4} - \)\(12\!\cdots\!56\)\( T^{5} + \)\(22\!\cdots\!68\)\( T^{6} - \)\(18\!\cdots\!28\)\( T^{7} + \)\(25\!\cdots\!81\)\( T^{8} \))
$13$ (\( 1 + \)\(90\!\cdots\!06\)\( T + \)\(21\!\cdots\!63\)\( T^{2} + \)\(15\!\cdots\!12\)\( T^{3} + \)\(17\!\cdots\!71\)\( T^{4} + \)\(60\!\cdots\!34\)\( T^{5} + \)\(54\!\cdots\!13\)\( T^{6} \))(\( 1 + \)\(17\!\cdots\!12\)\( T + \)\(97\!\cdots\!72\)\( T^{2} + \)\(46\!\cdots\!20\)\( T^{3} + \)\(65\!\cdots\!26\)\( T^{4} + \)\(37\!\cdots\!40\)\( T^{5} + \)\(65\!\cdots\!08\)\( T^{6} + \)\(96\!\cdots\!56\)\( T^{7} + \)\(44\!\cdots\!21\)\( T^{8} \))
$17$ (\( 1 + \)\(20\!\cdots\!78\)\( T + \)\(14\!\cdots\!47\)\( T^{2} + \)\(56\!\cdots\!64\)\( T^{3} + \)\(16\!\cdots\!31\)\( T^{4} + \)\(26\!\cdots\!62\)\( T^{5} + \)\(14\!\cdots\!17\)\( T^{6} \))(\( 1 + \)\(70\!\cdots\!16\)\( T + \)\(43\!\cdots\!88\)\( T^{2} + \)\(20\!\cdots\!60\)\( T^{3} + \)\(75\!\cdots\!46\)\( T^{4} + \)\(22\!\cdots\!80\)\( T^{5} + \)\(54\!\cdots\!52\)\( T^{6} + \)\(98\!\cdots\!72\)\( T^{7} + \)\(15\!\cdots\!41\)\( T^{8} \))
$19$ (\( 1 + \)\(91\!\cdots\!00\)\( T + \)\(22\!\cdots\!17\)\( T^{2} - \)\(21\!\cdots\!00\)\( T^{3} + \)\(40\!\cdots\!63\)\( T^{4} + \)\(29\!\cdots\!00\)\( T^{5} + \)\(56\!\cdots\!19\)\( T^{6} \))(\( 1 - \)\(38\!\cdots\!60\)\( T + \)\(51\!\cdots\!56\)\( T^{2} - \)\(17\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!26\)\( T^{4} - \)\(31\!\cdots\!80\)\( T^{5} + \)\(16\!\cdots\!76\)\( T^{6} - \)\(21\!\cdots\!40\)\( T^{7} + \)\(10\!\cdots\!41\)\( T^{8} \))
$23$ (\( 1 - \)\(18\!\cdots\!84\)\( T + \)\(37\!\cdots\!93\)\( T^{2} - \)\(39\!\cdots\!48\)\( T^{3} + \)\(11\!\cdots\!71\)\( T^{4} - \)\(16\!\cdots\!56\)\( T^{5} + \)\(25\!\cdots\!23\)\( T^{6} \))(\( 1 - \)\(26\!\cdots\!28\)\( T + \)\(13\!\cdots\!32\)\( T^{2} - \)\(24\!\cdots\!20\)\( T^{3} + \)\(63\!\cdots\!66\)\( T^{4} - \)\(70\!\cdots\!40\)\( T^{5} + \)\(11\!\cdots\!88\)\( T^{6} - \)\(68\!\cdots\!44\)\( T^{7} + \)\(76\!\cdots\!81\)\( T^{8} \))
$29$ (\( 1 - \)\(34\!\cdots\!30\)\( T + \)\(42\!\cdots\!27\)\( T^{2} - \)\(56\!\cdots\!40\)\( T^{3} + \)\(71\!\cdots\!43\)\( T^{4} - \)\(97\!\cdots\!30\)\( T^{5} + \)\(48\!\cdots\!29\)\( T^{6} \))(\( 1 + \)\(93\!\cdots\!40\)\( T + \)\(53\!\cdots\!36\)\( T^{2} + \)\(33\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!86\)\( T^{4} + \)\(56\!\cdots\!20\)\( T^{5} + \)\(15\!\cdots\!16\)\( T^{6} + \)\(45\!\cdots\!60\)\( T^{7} + \)\(82\!\cdots\!61\)\( T^{8} \))
$31$ (\( 1 - \)\(83\!\cdots\!16\)\( T + \)\(17\!\cdots\!85\)\( T^{2} - \)\(94\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!35\)\( T^{4} - \)\(26\!\cdots\!36\)\( T^{5} + \)\(17\!\cdots\!31\)\( T^{6} \))(\( 1 - \)\(11\!\cdots\!68\)\( T + \)\(10\!\cdots\!28\)\( T^{2} + \)\(10\!\cdots\!04\)\( T^{3} - \)\(17\!\cdots\!30\)\( T^{4} + \)\(61\!\cdots\!44\)\( T^{5} + \)\(33\!\cdots\!88\)\( T^{6} - \)\(21\!\cdots\!08\)\( T^{7} + \)\(98\!\cdots\!41\)\( T^{8} \))
$37$ (\( 1 - \)\(23\!\cdots\!62\)\( T + \)\(62\!\cdots\!47\)\( T^{2} - \)\(12\!\cdots\!56\)\( T^{3} + \)\(16\!\cdots\!51\)\( T^{4} - \)\(17\!\cdots\!18\)\( T^{5} + \)\(19\!\cdots\!37\)\( T^{6} \))(\( 1 - \)\(18\!\cdots\!64\)\( T + \)\(10\!\cdots\!68\)\( T^{2} - \)\(14\!\cdots\!20\)\( T^{3} + \)\(40\!\cdots\!46\)\( T^{4} - \)\(38\!\cdots\!60\)\( T^{5} + \)\(75\!\cdots\!52\)\( T^{6} - \)\(36\!\cdots\!68\)\( T^{7} + \)\(54\!\cdots\!21\)\( T^{8} \))
$41$ (\( 1 + \)\(11\!\cdots\!34\)\( T + \)\(21\!\cdots\!95\)\( T^{2} + \)\(59\!\cdots\!60\)\( T^{3} + \)\(44\!\cdots\!95\)\( T^{4} + \)\(50\!\cdots\!74\)\( T^{5} + \)\(86\!\cdots\!41\)\( T^{6} \))(\( 1 - \)\(54\!\cdots\!08\)\( T + \)\(31\!\cdots\!48\)\( T^{2} - \)\(15\!\cdots\!76\)\( T^{3} + \)\(71\!\cdots\!70\)\( T^{4} - \)\(31\!\cdots\!56\)\( T^{5} + \)\(13\!\cdots\!28\)\( T^{6} - \)\(47\!\cdots\!28\)\( T^{7} + \)\(17\!\cdots\!21\)\( T^{8} \))
$43$ (\( 1 + \)\(98\!\cdots\!96\)\( T + \)\(23\!\cdots\!93\)\( T^{2} + \)\(22\!\cdots\!12\)\( T^{3} + \)\(30\!\cdots\!51\)\( T^{4} + \)\(16\!\cdots\!04\)\( T^{5} + \)\(21\!\cdots\!43\)\( T^{6} \))(\( 1 + \)\(15\!\cdots\!52\)\( T + \)\(51\!\cdots\!92\)\( T^{2} + \)\(54\!\cdots\!80\)\( T^{3} + \)\(98\!\cdots\!26\)\( T^{4} + \)\(70\!\cdots\!60\)\( T^{5} + \)\(85\!\cdots\!08\)\( T^{6} + \)\(34\!\cdots\!36\)\( T^{7} + \)\(27\!\cdots\!01\)\( T^{8} \))
$47$ (\( 1 + \)\(50\!\cdots\!08\)\( T + \)\(49\!\cdots\!77\)\( T^{2} + \)\(21\!\cdots\!64\)\( T^{3} + \)\(14\!\cdots\!51\)\( T^{4} + \)\(44\!\cdots\!52\)\( T^{5} + \)\(26\!\cdots\!47\)\( T^{6} \))(\( 1 + \)\(11\!\cdots\!36\)\( T + \)\(12\!\cdots\!88\)\( T^{2} + \)\(81\!\cdots\!20\)\( T^{3} + \)\(53\!\cdots\!86\)\( T^{4} + \)\(24\!\cdots\!60\)\( T^{5} + \)\(11\!\cdots\!72\)\( T^{6} + \)\(30\!\cdots\!92\)\( T^{7} + \)\(77\!\cdots\!61\)\( T^{8} \))
$53$ (\( 1 + \)\(66\!\cdots\!06\)\( T + \)\(13\!\cdots\!23\)\( T^{2} + \)\(17\!\cdots\!52\)\( T^{3} + \)\(14\!\cdots\!51\)\( T^{4} + \)\(70\!\cdots\!14\)\( T^{5} + \)\(10\!\cdots\!53\)\( T^{6} \))(\( 1 + \)\(32\!\cdots\!12\)\( T + \)\(67\!\cdots\!52\)\( T^{2} + \)\(96\!\cdots\!40\)\( T^{3} + \)\(11\!\cdots\!66\)\( T^{4} + \)\(98\!\cdots\!80\)\( T^{5} + \)\(71\!\cdots\!88\)\( T^{6} + \)\(34\!\cdots\!36\)\( T^{7} + \)\(11\!\cdots\!61\)\( T^{8} \))
$59$ (\( 1 - \)\(14\!\cdots\!60\)\( T + \)\(23\!\cdots\!57\)\( T^{2} - \)\(35\!\cdots\!80\)\( T^{3} + \)\(26\!\cdots\!83\)\( T^{4} - \)\(19\!\cdots\!60\)\( T^{5} + \)\(15\!\cdots\!59\)\( T^{6} \))(\( 1 + \)\(81\!\cdots\!80\)\( T + \)\(31\!\cdots\!76\)\( T^{2} + \)\(21\!\cdots\!60\)\( T^{3} + \)\(46\!\cdots\!66\)\( T^{4} + \)\(24\!\cdots\!40\)\( T^{5} + \)\(42\!\cdots\!36\)\( T^{6} + \)\(12\!\cdots\!20\)\( T^{7} + \)\(18\!\cdots\!21\)\( T^{8} \))
$61$ (\( 1 - \)\(54\!\cdots\!26\)\( T + \)\(52\!\cdots\!55\)\( T^{2} - \)\(19\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!55\)\( T^{4} - \)\(24\!\cdots\!66\)\( T^{5} + \)\(93\!\cdots\!61\)\( T^{6} \))(\( 1 - \)\(90\!\cdots\!28\)\( T + \)\(94\!\cdots\!28\)\( T^{2} - \)\(47\!\cdots\!36\)\( T^{3} + \)\(28\!\cdots\!70\)\( T^{4} - \)\(10\!\cdots\!56\)\( T^{5} + \)\(41\!\cdots\!48\)\( T^{6} - \)\(84\!\cdots\!08\)\( T^{7} + \)\(19\!\cdots\!81\)\( T^{8} \))
$67$ (\( 1 - \)\(99\!\cdots\!92\)\( T + \)\(19\!\cdots\!57\)\( T^{2} - \)\(14\!\cdots\!76\)\( T^{3} + \)\(14\!\cdots\!11\)\( T^{4} - \)\(54\!\cdots\!68\)\( T^{5} + \)\(40\!\cdots\!67\)\( T^{6} \))(\( 1 + \)\(65\!\cdots\!96\)\( T + \)\(27\!\cdots\!48\)\( T^{2} + \)\(80\!\cdots\!20\)\( T^{3} + \)\(22\!\cdots\!26\)\( T^{4} + \)\(59\!\cdots\!60\)\( T^{5} + \)\(14\!\cdots\!92\)\( T^{6} + \)\(26\!\cdots\!32\)\( T^{7} + \)\(29\!\cdots\!41\)\( T^{8} \))
$71$ (\( 1 + \)\(62\!\cdots\!64\)\( T + \)\(19\!\cdots\!05\)\( T^{2} + \)\(36\!\cdots\!20\)\( T^{3} + \)\(21\!\cdots\!55\)\( T^{4} + \)\(82\!\cdots\!84\)\( T^{5} + \)\(15\!\cdots\!71\)\( T^{6} \))(\( 1 + \)\(18\!\cdots\!92\)\( T + \)\(37\!\cdots\!88\)\( T^{2} + \)\(47\!\cdots\!84\)\( T^{3} + \)\(59\!\cdots\!70\)\( T^{4} + \)\(54\!\cdots\!44\)\( T^{5} + \)\(49\!\cdots\!28\)\( T^{6} + \)\(27\!\cdots\!32\)\( T^{7} + \)\(17\!\cdots\!61\)\( T^{8} \))
$73$ (\( 1 + \)\(57\!\cdots\!06\)\( T + \)\(19\!\cdots\!03\)\( T^{2} + \)\(56\!\cdots\!72\)\( T^{3} + \)\(24\!\cdots\!91\)\( T^{4} + \)\(94\!\cdots\!54\)\( T^{5} + \)\(21\!\cdots\!73\)\( T^{6} \))(\( 1 - \)\(35\!\cdots\!88\)\( T + \)\(28\!\cdots\!92\)\( T^{2} - \)\(92\!\cdots\!00\)\( T^{3} + \)\(52\!\cdots\!86\)\( T^{4} - \)\(11\!\cdots\!00\)\( T^{5} + \)\(47\!\cdots\!28\)\( T^{6} - \)\(75\!\cdots\!24\)\( T^{7} + \)\(27\!\cdots\!81\)\( T^{8} \))
$79$ (\( 1 - \)\(26\!\cdots\!20\)\( T + \)\(37\!\cdots\!77\)\( T^{2} - \)\(63\!\cdots\!60\)\( T^{3} + \)\(46\!\cdots\!43\)\( T^{4} - \)\(40\!\cdots\!20\)\( T^{5} + \)\(19\!\cdots\!79\)\( T^{6} \))(\( 1 - \)\(15\!\cdots\!20\)\( T + \)\(22\!\cdots\!36\)\( T^{2} - \)\(33\!\cdots\!40\)\( T^{3} + \)\(41\!\cdots\!86\)\( T^{4} - \)\(41\!\cdots\!60\)\( T^{5} + \)\(33\!\cdots\!16\)\( T^{6} - \)\(29\!\cdots\!80\)\( T^{7} + \)\(23\!\cdots\!61\)\( T^{8} \))
$83$ (\( 1 + \)\(84\!\cdots\!76\)\( T + \)\(43\!\cdots\!73\)\( T^{2} - \)\(12\!\cdots\!08\)\( T^{3} + \)\(39\!\cdots\!71\)\( T^{4} + \)\(70\!\cdots\!04\)\( T^{5} + \)\(75\!\cdots\!83\)\( T^{6} \))(\( 1 - \)\(11\!\cdots\!68\)\( T + \)\(23\!\cdots\!92\)\( T^{2} - \)\(32\!\cdots\!60\)\( T^{3} + \)\(27\!\cdots\!06\)\( T^{4} - \)\(29\!\cdots\!20\)\( T^{5} + \)\(19\!\cdots\!68\)\( T^{6} - \)\(84\!\cdots\!44\)\( T^{7} + \)\(69\!\cdots\!41\)\( T^{8} \))
$89$ (\( 1 - \)\(55\!\cdots\!30\)\( T + \)\(73\!\cdots\!87\)\( T^{2} - \)\(49\!\cdots\!40\)\( T^{3} + \)\(29\!\cdots\!23\)\( T^{4} - \)\(86\!\cdots\!30\)\( T^{5} + \)\(61\!\cdots\!89\)\( T^{6} \))(\( 1 + \)\(11\!\cdots\!60\)\( T + \)\(10\!\cdots\!16\)\( T^{2} + \)\(12\!\cdots\!20\)\( T^{3} + \)\(48\!\cdots\!46\)\( T^{4} + \)\(50\!\cdots\!80\)\( T^{5} + \)\(15\!\cdots\!56\)\( T^{6} + \)\(68\!\cdots\!40\)\( T^{7} + \)\(24\!\cdots\!81\)\( T^{8} \))
$97$ (\( 1 + \)\(87\!\cdots\!38\)\( T + \)\(46\!\cdots\!87\)\( T^{2} + \)\(14\!\cdots\!24\)\( T^{3} + \)\(32\!\cdots\!31\)\( T^{4} + \)\(43\!\cdots\!22\)\( T^{5} + \)\(35\!\cdots\!97\)\( T^{6} \))(\( 1 - \)\(84\!\cdots\!84\)\( T + \)\(14\!\cdots\!48\)\( T^{2} - \)\(30\!\cdots\!20\)\( T^{3} + \)\(97\!\cdots\!66\)\( T^{4} - \)\(21\!\cdots\!60\)\( T^{5} + \)\(70\!\cdots\!12\)\( T^{6} - \)\(29\!\cdots\!48\)\( T^{7} + \)\(24\!\cdots\!61\)\( T^{8} \))
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