Properties

Label 1.88.a
Level 1
Weight 88
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: \( k \) = \( 88 \)
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 \(M_{88}(\Gamma_0(1))\).

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

Trace form

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

Decomposition of \(S_{88}^{\mathrm{new}}(\Gamma_0(1))\) into irreducible Hecke orbits

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