Properties

Label 1.100.a
Level 1
Weight 100
Character orbit a
Rep. character \(\chi_{1}(1,\cdot)\)
Character field \(\Q\)
Dimension 8
Newforms 1
Sturm bound 8
Trace bound 0

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Defining parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 100 \)
Character orbit: \([\chi]\) = 1.a (trivial)
Character field: \(\Q\)
Newforms: \( 1 \)
Sturm bound: \(8\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{100}(\Gamma_0(1))\).

Total New Old
Modular forms 9 9 0
Cusp forms 8 8 0
Eisenstein series 1 1 0

Trace form

\(8q \) \(\mathstrut -\mathstrut 208040616902520q^{2} \) \(\mathstrut -\mathstrut 282956306495420632223520q^{3} \) \(\mathstrut +\mathstrut 2874573971021731526732655093824q^{4} \) \(\mathstrut -\mathstrut 48829879146635109942685521105004560q^{5} \) \(\mathstrut -\mathstrut 77942153094234171155671704171199405344q^{6} \) \(\mathstrut -\mathstrut 56830940560713017104682273454607970657600q^{7} \) \(\mathstrut +\mathstrut 592603013041539183163433278232078795937538560q^{8} \) \(\mathstrut +\mathstrut 15585042525631880727258151421002821416820695976q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut -\mathstrut 208040616902520q^{2} \) \(\mathstrut -\mathstrut 282956306495420632223520q^{3} \) \(\mathstrut +\mathstrut 2874573971021731526732655093824q^{4} \) \(\mathstrut -\mathstrut 48829879146635109942685521105004560q^{5} \) \(\mathstrut -\mathstrut 77942153094234171155671704171199405344q^{6} \) \(\mathstrut -\mathstrut 56830940560713017104682273454607970657600q^{7} \) \(\mathstrut +\mathstrut 592603013041539183163433278232078795937538560q^{8} \) \(\mathstrut +\mathstrut 15585042525631880727258151421002821416820695976q^{9} \) \(\mathstrut -\mathstrut 20961738819400631174675750189416259674325116377360q^{10} \) \(\mathstrut +\mathstrut 668605620058921294359176615672652323197376366725536q^{11} \) \(\mathstrut -\mathstrut 260751428362765138428103958747533086169389679507011840q^{12} \) \(\mathstrut -\mathstrut 5313129021153651674083973849295770678848247690376073040q^{13} \) \(\mathstrut -\mathstrut 275891262636795191856274681125121829106120036919920873792q^{14} \) \(\mathstrut +\mathstrut 39601076201132911228470767893146157937452742673766616546880q^{15} \) \(\mathstrut +\mathstrut 1686978001168411473373901309719868789822314791151035737182208q^{16} \) \(\mathstrut +\mathstrut 29870581004484716934093126001838988402381517134653878061066640q^{17} \) \(\mathstrut +\mathstrut 104619592562318909205389831941745992324800916717508145002954280q^{18} \) \(\mathstrut -\mathstrut 2584541347057387315132230533073883085094549482599567538462153120q^{19} \) \(\mathstrut -\mathstrut 87020910945991395973478129409062640628123700866757624264022986880q^{20} \) \(\mathstrut +\mathstrut 720725986617461186277876616197863336884455731162862159407836416q^{21} \) \(\mathstrut +\mathstrut 6448630494375222037284713151002631218361976067562272720681828806560q^{22} \) \(\mathstrut -\mathstrut 591705231800483702926418342143069697446663268741979110571049882560q^{23} \) \(\mathstrut -\mathstrut 786094952626208255303024343193893163353569305701641632350239286773760q^{24} \) \(\mathstrut +\mathstrut 716804381062067209680195080749365299641535704320602348503400092268600q^{25} \) \(\mathstrut +\mathstrut 34654632463607194325743429354208471986051789816879940147342898057153456q^{26} \) \(\mathstrut +\mathstrut 13869683363495491009982193176608883185849831807068440771218044677979840q^{27} \) \(\mathstrut -\mathstrut 1135994339501291724748966539673012720374432727977564592829456188165859840q^{28} \) \(\mathstrut -\mathstrut 439009994693767454184499584774890376667926937141064013921655743937571280q^{29} \) \(\mathstrut +\mathstrut 23517293722912668382363618879896733566643340714874843272431473003480521280q^{30} \) \(\mathstrut -\mathstrut 42796079732631926995692344425065773646038440365668203549397431930127162624q^{31} \) \(\mathstrut -\mathstrut 284044778854068845159282113651411076631875258269403845301648101866131128320q^{32} \) \(\mathstrut -\mathstrut 1184428676639946519273358190717309258531732537332484528437733831641965671040q^{33} \) \(\mathstrut +\mathstrut 11470334814206853457708153296668381286712583035391282441563479133450208961168q^{34} \) \(\mathstrut -\mathstrut 13317873075112713679215664096634074386516466958921311305188860496632195839360q^{35} \) \(\mathstrut -\mathstrut 224326234591498863152148578877455893579609783196337897663675145770615318305472q^{36} \) \(\mathstrut +\mathstrut 7013041648409278763731742859978750449729753703775477297667583228204170683120q^{37} \) \(\mathstrut +\mathstrut 4097267709666535974097951231260487171744232792996835026005028682857459985875040q^{38} \) \(\mathstrut +\mathstrut 3086772337289173445407063166797596998267422181270693039748051240117483908526912q^{39} \) \(\mathstrut -\mathstrut 61154477845927513224255221208858320971333500267390856556166248662272460823577600q^{40} \) \(\mathstrut +\mathstrut 16836611518887404247693904661111767446154336367366992887095255319045118001938896q^{41} \) \(\mathstrut +\mathstrut 586782577486516292654880709693602500520055556295982039518980517393750164948778240q^{42} \) \(\mathstrut -\mathstrut 115479509651193412125043402629710680178430977878852566804453976757651553792322400q^{43} \) \(\mathstrut -\mathstrut 4823674089938931277708865059630293953738849935605913988972972769383583829463937792q^{44} \) \(\mathstrut -\mathstrut 4204225762790416656941382640299001599040327829208847740401216426127468587838672720q^{45} \) \(\mathstrut +\mathstrut 41220575732227465405339930795314328162570779910219577393801310430729179103724398656q^{46} \) \(\mathstrut +\mathstrut 28878226575287510694522520507929237505491250714878620920990009419290436045018975360q^{47} \) \(\mathstrut -\mathstrut 270645616492149970234041108687889002599221246648694105429287982543953957103033794560q^{48} \) \(\mathstrut -\mathstrut 14382803175556111227730750647629461517090856222687609645964147941666232976982659256q^{49} \) \(\mathstrut +\mathstrut 2538900041031429584911497444199766832786695308479217204477072876574482801521012636600q^{50} \) \(\mathstrut -\mathstrut 1711868817156841452117007804086601271536721152787522351736690290031659132307369497664q^{51} \) \(\mathstrut -\mathstrut 199691168563235615231272208715487263769590246824861808533841002609282642680495606400q^{52} \) \(\mathstrut -\mathstrut 30091820242734525213832735040802634061423162425240717202059443809754039115425881976720q^{53} \) \(\mathstrut +\mathstrut 136432908322968282680263912195201284966825652472631606906205576348979246713807692108480q^{54} \) \(\mathstrut +\mathstrut 226099583633880167546966079422139027647924161654010953637858494806538127768993139084480q^{55} \) \(\mathstrut +\mathstrut 323037227302956226932352973051969540564093713359973342312103259763994205180305378488320q^{56} \) \(\mathstrut -\mathstrut 273423384330404631781757518792671605280054294102341216713398337934356290096852147342720q^{57} \) \(\mathstrut +\mathstrut 5713973478801321937231450378714667908258440100056694714036585358012358074567853731219760q^{58} \) \(\mathstrut +\mathstrut 14607245332654695346837714400351649175544087130678579822473040904252509022932739720811040q^{59} \) \(\mathstrut +\mathstrut 53008257791762983739773607116394245193846456713938833829846630209166818928009302295114240q^{60} \) \(\mathstrut +\mathstrut 38323682921378832161349805479389921828014992489778724538462200957929376165086831637164336q^{61} \) \(\mathstrut +\mathstrut 201697385927609334439920541372343666252345396922224455072723131310512431934088015943381760q^{62} \) \(\mathstrut +\mathstrut 659857333915856699600072965534847901170546674796116288100652182111732663513752207410870720q^{63} \) \(\mathstrut +\mathstrut 2567569508313686836352267062189989701299823975229681223291764819542997452948800142018084864q^{64} \) \(\mathstrut +\mathstrut 2649617703605055039617798586548537725956903247199823944158672914229112797191302156162117280q^{65} \) \(\mathstrut +\mathstrut 7971156345834212651621212324655177899942634890685144307138793590710555397223588657590125952q^{66} \) \(\mathstrut +\mathstrut 11538024122310165476577023936338470394129524138300592332780359783942082451772953721242543840q^{67} \) \(\mathstrut +\mathstrut 56045208748154976002290739488865109707859265886693431912946708486067523902076046154608254080q^{68} \) \(\mathstrut +\mathstrut 54550768708090409722131753417710688638608175085564682877064185400954540597503860451967981312q^{69} \) \(\mathstrut +\mathstrut 143889269423962258520958611175254488367329101991613194841590826662727327088381209016536883840q^{70} \) \(\mathstrut +\mathstrut 52677154721705052088453955375369863850612862186345822027042426779160256515633503076296638656q^{71} \) \(\mathstrut +\mathstrut 365038759307175377119135471724071125896012767600977654405238165616233444129512164397555141120q^{72} \) \(\mathstrut +\mathstrut 506688790556979736999516532365405438114465768501506115275379047447697188620686370699300754640q^{73} \) \(\mathstrut -\mathstrut 399134575097952393052355140371514479012153174980660646654807435656154590439317636650100727312q^{74} \) \(\mathstrut -\mathstrut 2324004895287993423141911565246677610480036000604538705092119528699965986086154360593593272800q^{75} \) \(\mathstrut -\mathstrut 6633935918720327036715040118664821987731317587536782451991725427750234686044104260504536239360q^{76} \) \(\mathstrut -\mathstrut 6461585481973245548189356793856436395182994379442713381123224045549509062132894968864516460800q^{77} \) \(\mathstrut -\mathstrut 28556176219359102115465270938556084350909705495286117230117205751412922131522017940137826091200q^{78} \) \(\mathstrut -\mathstrut 30611582035787076807516826687485094690084083072098723290730859803828786802497884290624307556480q^{79} \) \(\mathstrut -\mathstrut 127732467855339073698199167907855646502900419187610712488908375947950612344972404753625834332160q^{80} \) \(\mathstrut -\mathstrut 105098591020590401711840147132548556253048052082258241572206720930835350937368467193794679663032q^{81} \) \(\mathstrut -\mathstrut 68830371209738349493630432122952083611773652840970378016560437888326646942049769865516365685040q^{82} \) \(\mathstrut +\mathstrut 22938542661502254568160041367746263129225779762371870743525183586562110235427022830971680103520q^{83} \) \(\mathstrut +\mathstrut 361795660381091268015394970465025503071664436510904020432798379164529165144740663778138523854848q^{84} \) \(\mathstrut -\mathstrut 65747002822668948096103688317539312015762969670721483634214688263171509401935169398146161091360q^{85} \) \(\mathstrut +\mathstrut 2926007787385729535219437868096352598169987790407590983846487590337872781868895638215544350378656q^{86} \) \(\mathstrut +\mathstrut 4145530857628120059159343787991300967686349677839106320836691853415365479077514876503669002112320q^{87} \) \(\mathstrut +\mathstrut 13379567436970727806886279611606134975535418870618956336086465570267315661188646758052286051051520q^{88} \) \(\mathstrut +\mathstrut 13262749235321210784149676214189766161707478948372569815117556700615668825668004121877579168519760q^{89} \) \(\mathstrut +\mathstrut 18562789747759155399933926620841126049267839227249440162668332587565567838001478928890860138613680q^{90} \) \(\mathstrut -\mathstrut 6466499581384229905399040912443724309264614193875257390824803270602537852945701184143169943974784q^{91} \) \(\mathstrut -\mathstrut 3605940592933441273788896348672120936868261799818064643016984038777885904937799268971010435816960q^{92} \) \(\mathstrut -\mathstrut 6218230725998946523812046145774116878209214952116357458146814759754757343477151669356631201623040q^{93} \) \(\mathstrut -\mathstrut 111501057917354496628976353847689989005658650697758014190949880350502870630016419857891186452986752q^{94} \) \(\mathstrut -\mathstrut 310841228909329722722502224651860795134807193058850708412373586195233672473563873013656113819102400q^{95} \) \(\mathstrut -\mathstrut 967769551108604066125272773812882212494392412979843145463542262778370585211900025491604732868952064q^{96} \) \(\mathstrut -\mathstrut 870148289336425543686683480565613230312583210673688585087479722135977346731183581578056793848665840q^{97} \) \(\mathstrut -\mathstrut 1259902767579045380431363502561233259244081818944281851461225234703715688976807437199140410707221560q^{98} \) \(\mathstrut -\mathstrut 137018494226539733211453518241879004188470623376815926515879377141376994388427950391858832640432608q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{100}^{\mathrm{new}}(\Gamma_0(1))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.100.a.a \(8\) \(62.068\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-2\!\cdots\!20\) \(-2\!\cdots\!20\) \(-4\!\cdots\!60\) \(-5\!\cdots\!00\) \(+\) \(q+(-26005077112815+\beta _{1})q^{2}+\cdots\)