Properties

Label 1.90.a
Level 1
Weight 90
Character orbit a
Rep. character \(\chi_{1}(1,\cdot)\)
Character field \(\Q\)
Dimension 7
Newforms 1
Sturm bound 7
Trace bound 0

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Defining parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 90 \)
Character orbit: \([\chi]\) = 1.a (trivial)
Character field: \(\Q\)
Newforms: \( 1 \)
Sturm bound: \(7\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{90}(\Gamma_0(1))\).

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

Trace form

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

Decomposition of \(S_{90}^{\mathrm{new}}(\Gamma_0(1))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.90.a.a \(7\) \(50.162\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-3\!\cdots\!08\) \(-1\!\cdots\!64\) \(10\!\cdots\!50\) \(38\!\cdots\!92\) \(+\) \(q+(-4486761478773-\beta _{1})q^{2}+\cdots\)