Properties

Label 1.68.a
Level 1
Weight 68
Character orbit a
Rep. character \(\chi_{1}(1,\cdot)\)
Character field \(\Q\)
Dimension 5
Newforms 1
Sturm bound 5
Trace bound 0

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Defining parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 68 \)
Character orbit: \([\chi]\) = 1.a (trivial)
Character field: \(\Q\)
Newforms: \( 1 \)
Sturm bound: \(5\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{68}(\Gamma_0(1))\).

Total New Old
Modular forms 6 6 0
Cusp forms 5 5 0
Eisenstein series 1 1 0

Trace form

\(5q \) \(\mathstrut +\mathstrut 5554901256q^{2} \) \(\mathstrut +\mathstrut 3443360269119372q^{3} \) \(\mathstrut +\mathstrut 350472004382296600640q^{4} \) \(\mathstrut +\mathstrut 330744836907160597528950q^{5} \) \(\mathstrut +\mathstrut 146648871573834733762548960q^{6} \) \(\mathstrut +\mathstrut 33632726520317603069152899256q^{7} \) \(\mathstrut +\mathstrut 3261706912683890613885787783680q^{8} \) \(\mathstrut +\mathstrut 27806018581845734541016911293385q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(5q \) \(\mathstrut +\mathstrut 5554901256q^{2} \) \(\mathstrut +\mathstrut 3443360269119372q^{3} \) \(\mathstrut +\mathstrut 350472004382296600640q^{4} \) \(\mathstrut +\mathstrut 330744836907160597528950q^{5} \) \(\mathstrut +\mathstrut 146648871573834733762548960q^{6} \) \(\mathstrut +\mathstrut 33632726520317603069152899256q^{7} \) \(\mathstrut +\mathstrut 3261706912683890613885787783680q^{8} \) \(\mathstrut +\mathstrut 27806018581845734541016911293385q^{9} \) \(\mathstrut -\mathstrut 5287067006725590176865792208438800q^{10} \) \(\mathstrut +\mathstrut 203006981551281631527218766884109060q^{11} \) \(\mathstrut -\mathstrut 2089518810401181456110201520986913024q^{12} \) \(\mathstrut +\mathstrut 177822884499813229570219072356766702q^{13} \) \(\mathstrut +\mathstrut 290753425875732621001638220329334092480q^{14} \) \(\mathstrut -\mathstrut 933024301033068664231979455289562936600q^{15} \) \(\mathstrut +\mathstrut 18266362102387359184851236859301091348480q^{16} \) \(\mathstrut +\mathstrut 75241497376497768433807587544150344102906q^{17} \) \(\mathstrut -\mathstrut 1011568566626978308983328666525487111002968q^{18} \) \(\mathstrut +\mathstrut 3998884102997801557241177083441346393295100q^{19} \) \(\mathstrut -\mathstrut 15074831378020267075078660994426474705846400q^{20} \) \(\mathstrut -\mathstrut 175719863949295325296418063144372762630915040q^{21} \) \(\mathstrut +\mathstrut 1841247880573095918698534631505063585473750432q^{22} \) \(\mathstrut -\mathstrut 419660874260151197180020017656265233426170968q^{23} \) \(\mathstrut -\mathstrut 13019747846573945420584224979196286639928166400q^{24} \) \(\mathstrut +\mathstrut 175279270003539476977813620578789764300129011875q^{25} \) \(\mathstrut +\mathstrut 21416337629300654169124097666599354769371062960q^{26} \) \(\mathstrut +\mathstrut 1886613800166950113417786970194782722207999659320q^{27} \) \(\mathstrut +\mathstrut 14982728306233364463143494158089725946900023694848q^{28} \) \(\mathstrut +\mathstrut 18733547497571939101380180097730717676662894233950q^{29} \) \(\mathstrut +\mathstrut 183881202860964913168534714290880945912096094414400q^{30} \) \(\mathstrut +\mathstrut 365849822471060800690376948081970588059264217288160q^{31} \) \(\mathstrut +\mathstrut 1591165751988992347021834813546084975107380979204096q^{32} \) \(\mathstrut +\mathstrut 2437038903031213839498705428654461423996034595356784q^{33} \) \(\mathstrut +\mathstrut 7817592154883765712044327776251337623941567500782480q^{34} \) \(\mathstrut +\mathstrut 4524513410554564733118320802153390591800073874285200q^{35} \) \(\mathstrut -\mathstrut 38131548457864843751915538710226876689225094472585920q^{36} \) \(\mathstrut -\mathstrut 56389392588768330454829645685931746748224606527380394q^{37} \) \(\mathstrut -\mathstrut 311451303114725973135630000941921262878276604287141280q^{38} \) \(\mathstrut -\mathstrut 717177850582388225418943791249053833224228521090828280q^{39} \) \(\mathstrut -\mathstrut 1886768345048751997029226502271137564956948018338432000q^{40} \) \(\mathstrut +\mathstrut 113555410342193387626817165052765483601006465327103010q^{41} \) \(\mathstrut +\mathstrut 2805540126450529847390961864461873572829871059166350592q^{42} \) \(\mathstrut +\mathstrut 6508304671085340439704487085091235272293711787089861092q^{43} \) \(\mathstrut +\mathstrut 49369449025916528958230735645862721892607029455332199680q^{44} \) \(\mathstrut +\mathstrut 99011086960300194550392336618799141939148872728728967150q^{45} \) \(\mathstrut +\mathstrut 79552980744915855625545425031873854934595203714399466560q^{46} \) \(\mathstrut -\mathstrut 128023594814179612806658270243785955024442659789917751344q^{47} \) \(\mathstrut -\mathstrut 198512903890152994486195174416803543869354744423782432768q^{48} \) \(\mathstrut -\mathstrut 771329931628098140389185999842408514636676231229111640435q^{49} \) \(\mathstrut -\mathstrut 3919481096957516748709549462396523762064894577356951885000q^{50} \) \(\mathstrut -\mathstrut 3397415009313164473667296477133432796692029796730856284840q^{51} \) \(\mathstrut +\mathstrut 672997930141532412373441872733517069231734104886640838016q^{52} \) \(\mathstrut +\mathstrut 1046765637663599376128164919371991945530174947374966490822q^{53} \) \(\mathstrut +\mathstrut 19066352042639687223500180230679796188217876586398799185600q^{54} \) \(\mathstrut +\mathstrut 66942753107037791618627925698973654692865254681839842449400q^{55} \) \(\mathstrut +\mathstrut 164746697572073652525116376573879338548073684495562965299200q^{56} \) \(\mathstrut -\mathstrut 9288557942524759644038647059781434529266663439727382990960q^{57} \) \(\mathstrut -\mathstrut 61137956327686914952144218440883774026433838412764071303120q^{58} \) \(\mathstrut -\mathstrut 305790738450157663862297559072875409310668010528073986627500q^{59} \) \(\mathstrut -\mathstrut 1346157500908637758724100110370785768539984220742218248588800q^{60} \) \(\mathstrut -\mathstrut 1137073938560830615075915586814815057732339483683661907261890q^{61} \) \(\mathstrut -\mathstrut 2197846158587850633685459639420082896328263471188214086352128q^{62} \) \(\mathstrut -\mathstrut 2062780018748959038039139080074136999529046449885010133153768q^{63} \) \(\mathstrut +\mathstrut 6319522016046144185339524760669817745425888834305772325437440q^{64} \) \(\mathstrut +\mathstrut 17054811715744261908524663200762497384422620874171150779332900q^{65} \) \(\mathstrut +\mathstrut 27998862803339055552343423092993933514291039899414979026779520q^{66} \) \(\mathstrut -\mathstrut 11903044989154510038048691316368615691989662189985228212145844q^{67} \) \(\mathstrut +\mathstrut 81678078346863753465296539273887923556029130514802676363013248q^{68} \) \(\mathstrut +\mathstrut 3593002077831059525793506029116145162632074080747869594484320q^{69} \) \(\mathstrut -\mathstrut 262230520857255225386866306600227822969034558447869634844476800q^{70} \) \(\mathstrut -\mathstrut 112169920751076079402369696382938772997794660792753351047772040q^{71} \) \(\mathstrut -\mathstrut 747723581155501189674233444839980291686130221322311676193789440q^{72} \) \(\mathstrut -\mathstrut 304061800126793465330511223999919410949872771539142074085150718q^{73} \) \(\mathstrut +\mathstrut 921008183492866283853128598587548033201480125306857491530532080q^{74} \) \(\mathstrut +\mathstrut 21623812027928499471467875193108522496460255319651250940492500q^{75} \) \(\mathstrut +\mathstrut 2076567511595431334156222539483464646991367936386801535927366400q^{76} \) \(\mathstrut +\mathstrut 5018131462902473316385236580394702266689060843204393245422638432q^{77} \) \(\mathstrut +\mathstrut 6187393864780941106550100388692932558738785969110711253222726464q^{78} \) \(\mathstrut +\mathstrut 2994504004186984867153587379277397704940492543518650040519815600q^{79} \) \(\mathstrut -\mathstrut 15198313539268031787508043655897930060910106707913153957798092800q^{80} \) \(\mathstrut -\mathstrut 11195187555119704986426983039082721262159900559267345597680888595q^{81} \) \(\mathstrut -\mathstrut 17699192859543411700126655967680257958811789647196784631772687408q^{82} \) \(\mathstrut -\mathstrut 47352447490969920981134910324833485175129837165394254819464203588q^{83} \) \(\mathstrut -\mathstrut 52124143836539534782259645807914007868879894236203487647926097920q^{84} \) \(\mathstrut -\mathstrut 39334683419450918052255554891934302881190270223971152032299949300q^{85} \) \(\mathstrut +\mathstrut 49699285824187070765304255878405188987409992188766856866277888160q^{86} \) \(\mathstrut +\mathstrut 357574798440063707158439877351651513807909832776407769090038489160q^{87} \) \(\mathstrut +\mathstrut 430257938373980762031490456670953174964896424231159472833711400960q^{88} \) \(\mathstrut -\mathstrut 11988583288945008097106276879882407358382120120051430459779195150q^{89} \) \(\mathstrut +\mathstrut 477054381540878916877701150770415125576892082956059127548575220400q^{90} \) \(\mathstrut +\mathstrut 232376792470844928685229521265495056030669651812230900159714321360q^{91} \) \(\mathstrut -\mathstrut 986292139147147043544160128863480288571347084605459783959415954944q^{92} \) \(\mathstrut -\mathstrut 1849280210034321990512094428557344731461785100166176155774311127936q^{93} \) \(\mathstrut -\mathstrut 4927754861315037106505826725122767884474273389521981763161929704320q^{94} \) \(\mathstrut -\mathstrut 2022377488008256909110920831467942087293430657450319294506545963000q^{95} \) \(\mathstrut -\mathstrut 2242556165580376632567217570505477297218451346654758522676359331840q^{96} \) \(\mathstrut +\mathstrut 2897556354128426205551279700626881901316694084191208635967319112106q^{97} \) \(\mathstrut +\mathstrut 12426885478574475523720488645206508880013087100212747539199689691208q^{98} \) \(\mathstrut +\mathstrut 16175146031966622867876651602334534091105243319832315048256094545620q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{68}^{\mathrm{new}}(\Gamma_0(1))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.68.a.a \(5\) \(28.429\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(5554901256\) \(34\!\cdots\!72\) \(33\!\cdots\!50\) \(33\!\cdots\!56\) \(+\) \(q+(1110980251-\beta _{1})q^{2}+(688672053797479+\cdots)q^{3}+\cdots\)