Properties

Label 2.80.a
Level 2
Weight 80
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 7
Newforms 2
Sturm bound 20
Trace bound 1

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Defining parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 80 \)
Character orbit: \([\chi]\) = 2.a (trivial)
Character field: \(\Q\)
Newforms: \( 2 \)
Sturm bound: \(20\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{80}(\Gamma_0(2))\).

Total New Old
Modular forms 21 7 14
Cusp forms 19 7 12
Eisenstein series 2 0 2

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

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

Trace form

\(7q \) \(\mathstrut -\mathstrut 549755813888q^{2} \) \(\mathstrut -\mathstrut 187803106354177548q^{3} \) \(\mathstrut +\mathstrut 2115620184325601055735808q^{4} \) \(\mathstrut -\mathstrut 6145276452095736270220032030q^{5} \) \(\mathstrut -\mathstrut 4937981254664760836220059123712q^{6} \) \(\mathstrut +\mathstrut 4057072386346119817909861604750216q^{7} \) \(\mathstrut -\mathstrut 166153499473114484112975882535043072q^{8} \) \(\mathstrut +\mathstrut 106158954891532093765683662119882788099q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(7q \) \(\mathstrut -\mathstrut 549755813888q^{2} \) \(\mathstrut -\mathstrut 187803106354177548q^{3} \) \(\mathstrut +\mathstrut 2115620184325601055735808q^{4} \) \(\mathstrut -\mathstrut 6145276452095736270220032030q^{5} \) \(\mathstrut -\mathstrut 4937981254664760836220059123712q^{6} \) \(\mathstrut +\mathstrut 4057072386346119817909861604750216q^{7} \) \(\mathstrut -\mathstrut 166153499473114484112975882535043072q^{8} \) \(\mathstrut +\mathstrut 106158954891532093765683662119882788099q^{9} \) \(\mathstrut -\mathstrut 1084804517501313976134672355339649679360q^{10} \) \(\mathstrut -\mathstrut 321575443398600731744364478991750903164356q^{11} \) \(\mathstrut -\mathstrut 56760006068849366138259039052338259034112q^{12} \) \(\mathstrut +\mathstrut 213411284645494418776986632820026684773195562q^{13} \) \(\mathstrut -\mathstrut 599366906050755645648271828265958759303479296q^{14} \) \(\mathstrut -\mathstrut 47236557350331202300641901420895343466757909160q^{15} \) \(\mathstrut +\mathstrut 639406966332270026714112114313373821099470487552q^{16} \) \(\mathstrut +\mathstrut 3743247793127955527284345264346857360978101891486q^{17} \) \(\mathstrut -\mathstrut 65697639350913219615264006909898765724054879993856q^{18} \) \(\mathstrut +\mathstrut 704953191522321826008846597800108033002624236825860q^{19} \) \(\mathstrut -\mathstrut 1857295842902079607744164307149080077802836139704320q^{20} \) \(\mathstrut +\mathstrut 1847802831271805812450257637375456556037267988923744q^{21} \) \(\mathstrut +\mathstrut 39991540065484864542065309823861268098325288209874944q^{22} \) \(\mathstrut +\mathstrut 570542547586021985233603436982341259764668381569911832q^{23} \) \(\mathstrut -\mathstrut 1492413258884317726908907322632394572461201711408611328q^{24} \) \(\mathstrut +\mathstrut 19957769271654419938512739670219337977090400492884529025q^{25} \) \(\mathstrut -\mathstrut 70318553441869812075222937739153390710598163691944804352q^{26} \) \(\mathstrut +\mathstrut 365856946653015745317047206427167394152974657306934887240q^{27} \) \(\mathstrut +\mathstrut 1226174889974840592720485489996658830335289541249418133504q^{28} \) \(\mathstrut +\mathstrut 5184626442679594729479232177761952463897909473976388050490q^{29} \) \(\mathstrut +\mathstrut 39305862916259774066296036126452647250784633358576076718080q^{30} \) \(\mathstrut -\mathstrut 47265633703764454788346048930020178770211152273652791260896q^{31} \) \(\mathstrut -\mathstrut 50216813883093446110686315385661331328818843555712276103168q^{32} \) \(\mathstrut -\mathstrut 443508296439890381352905516054082662368523077228227357299376q^{33} \) \(\mathstrut -\mathstrut 1016951301463778536364696204661977358190236616617625582043136q^{34} \) \(\mathstrut -\mathstrut 29580493732482401782873079957065127480168545867887426839579280q^{35} \) \(\mathstrut +\mathstrut 32084575387919470863279112959593460606925275222068936998649856q^{36} \) \(\mathstrut -\mathstrut 183110545563408785476021522003985579466598732122803335845044734q^{37} \) \(\mathstrut -\mathstrut 365279086997547685070422162662919258703992343710319375370158080q^{38} \) \(\mathstrut -\mathstrut 1402240885111124150113783255259324467962586376159195982615859272q^{39} \) \(\mathstrut -\mathstrut 327862047610482084371029824873240573181824288229683525152931840q^{40} \) \(\mathstrut -\mathstrut 21569064278986051685011640474752020751200599507041865440271685066q^{41} \) \(\mathstrut -\mathstrut 48988678039070276570436382774929864624623539542039954077427695616q^{42} \) \(\mathstrut -\mathstrut 123474819603298781212804562905902539028187353682717882255638051748q^{43} \) \(\mathstrut -\mathstrut 97190214119647795614977426806223836017476827412426200078734065664q^{44} \) \(\mathstrut -\mathstrut 1436526085888697616513872506788912663604438652040404183865058407510q^{45} \) \(\mathstrut -\mathstrut 222917130179971958162793045755959914969975892162347137585739988992q^{46} \) \(\mathstrut -\mathstrut 620469883890248388028837286899680696972063204804652799021600153424q^{47} \) \(\mathstrut -\mathstrut 17154659214528761508185025776539626655167905154571579698790268928q^{48} \) \(\mathstrut +\mathstrut 9483160937021509311529957798827241848286845162471580584888478745151q^{49} \) \(\mathstrut +\mathstrut 12210123681822436913044653836820942589229872468465723155982109900800q^{50} \) \(\mathstrut +\mathstrut 104067822247885810991896145676948815995841996100753467378551129844904q^{51} \) \(\mathstrut +\mathstrut 64499603051266316658635647053582272626190469594109501939730484297728q^{52} \) \(\mathstrut +\mathstrut 230606135072497646693491502919866963474173866918967134577461667797522q^{53} \) \(\mathstrut +\mathstrut 101115128945194979022687847074859325596809456920318672471478050488320q^{54} \) \(\mathstrut +\mathstrut 1924923705417100660067664525688943628151997273898790604223532482932040q^{55} \) \(\mathstrut -\mathstrut 181147532036823552824095218682245992516768167153161852959118824833024q^{56} \) \(\mathstrut -\mathstrut 7539106847837747610947457462526926292241493999239667262638510220787280q^{57} \) \(\mathstrut -\mathstrut 1205771175030129206349110604807107040963221906376943974607229697392640q^{58} \) \(\mathstrut -\mathstrut 34967041932481408612776376262019614109870181211465148706706018896585620q^{59} \) \(\mathstrut -\mathstrut 14276373452630646230688303380217771367452257819800625568239766774743040q^{60} \) \(\mathstrut -\mathstrut 87778024853612029468672770083059023679979036091844382823436690864247206q^{61} \) \(\mathstrut -\mathstrut 78666164146592631015991057906475689118475425143309879256130837236678656q^{62} \) \(\mathstrut +\mathstrut 366103763919091517953479221942338494139020843449045270138324281072833192q^{63} \) \(\mathstrut +\mathstrut 193248897710135786048173164143756712668631051220770823544550678266380288q^{64} \) \(\mathstrut +\mathstrut 973269935970407795707127184083552322127779219817189690349021551209668940q^{65} \) \(\mathstrut +\mathstrut 1335441365255394395289431631818260356965715720023185681611536552423325696q^{66} \) \(\mathstrut +\mathstrut 4050427441314701591615347306781237590054428583083649957036351775478874676q^{67} \) \(\mathstrut +\mathstrut 1131327226581966377357188470045683687647125215224839654607987473471504384q^{68} \) \(\mathstrut -\mathstrut 7513339286426883080151503456878818672636498466381859670310371192744594912q^{69} \) \(\mathstrut -\mathstrut 7495770586000713203048170478767283518333667430601699023930133174403727360q^{70} \) \(\mathstrut -\mathstrut 28464942033305272227228958666485717522179410418777497880623206921854533176q^{71} \) \(\mathstrut -\mathstrut 19855893124762269568926422234712958173508054985429778343326754322371313664q^{72} \) \(\mathstrut -\mathstrut 72054718921322832062900607681868757703580929426594450916128567461868758698q^{73} \) \(\mathstrut -\mathstrut 58285966941667182290133432794451883545349516233862238743796456201037807616q^{74} \) \(\mathstrut +\mathstrut 167971908016807602597313623687204964784767823841043726104347327429005032300q^{75} \) \(\mathstrut +\mathstrut 213059028712767892150292466487328860247097601784302381426414971676094627840q^{76} \) \(\mathstrut +\mathstrut 401355015328435789957896058896899352245892288879524405045353387731141756192q^{77} \) \(\mathstrut +\mathstrut 1021787684490883563335524045565594907110884665644241914460291646996598489088q^{78} \) \(\mathstrut +\mathstrut 3040710973582394589765057501738460804911364775089555147730024472399192217040q^{79} \) \(\mathstrut -\mathstrut 561333224786810034425857319709506967164010439389991289566097604905879470080q^{80} \) \(\mathstrut +\mathstrut 5072837909823074900745211699275406404902341417793722393029156309922103027487q^{81} \) \(\mathstrut +\mathstrut 1973074838017978550756621680048244032256191877128562033686340179006549131264q^{82} \) \(\mathstrut -\mathstrut 7228846459290918089351448882727197331949066307472751880866167111571525993148q^{83} \) \(\mathstrut +\mathstrut 558464138070375045657428374799007084166603781568360322477837920763417460736q^{84} \) \(\mathstrut -\mathstrut 31244476943005476644864037046252192642363119360365574824239471754137402290780q^{85} \) \(\mathstrut -\mathstrut 37225364926982587318709858658994800858919596833790412489412404085448478031872q^{86} \) \(\mathstrut +\mathstrut 25092255393584142626638808061016866529415068292517052275231646047974687562360q^{87} \) \(\mathstrut +\mathstrut 12086701337829392691074574657643889653024480127393847024782367550228226113536q^{88} \) \(\mathstrut +\mathstrut 161183654585667545662165636598950354235384942777487698851733732514809103197670q^{89} \) \(\mathstrut +\mathstrut 458689448006897844397684541520047299850560734883113298128178674748294567034880q^{90} \) \(\mathstrut +\mathstrut 760426789894813629327825464576738708218078993889232857335934316101617122770224q^{91} \) \(\mathstrut +\mathstrut 172435904241362549152023945547292083739709018060632387615549149894625206468608q^{92} \) \(\mathstrut +\mathstrut 2804631855644147761529281182369954179015899224114816166832657367395533455882624q^{93} \) \(\mathstrut -\mathstrut 974999099921794159016154533291952205063507189826164972940500108652532996243456q^{94} \) \(\mathstrut -\mathstrut 5836982150707149730866442519963635994724665642057680479402942759990974919776200q^{95} \) \(\mathstrut -\mathstrut 451054230550115890972047978739484020128384460967505629471975092192842646290432q^{96} \) \(\mathstrut -\mathstrut 2718269228919301863076525220061127015537469327992468918107853917885631586751634q^{97} \) \(\mathstrut -\mathstrut 4136421028259613830743014560848978238441195628662449319583087334380092201631744q^{98} \) \(\mathstrut -\mathstrut 23641109639156318172231814672663782165345291836279305992323055921690539084721492q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{80}^{\mathrm{new}}(\Gamma_0(2))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.80.a.a \(3\) \(79.047\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(16\!\cdots\!64\) \(-4\!\cdots\!36\) \(-4\!\cdots\!50\) \(14\!\cdots\!12\) \(-\) \(q+2^{39}q^{2}+(-1528323113916241812+\cdots)q^{3}+\cdots\)
2.80.a.b \(4\) \(79.047\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-2\!\cdots\!52\) \(43\!\cdots\!88\) \(-2\!\cdots\!80\) \(25\!\cdots\!04\) \(+\) \(q-2^{39}q^{2}+(1099291558848636972+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{80}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

\( S_{80}^{\mathrm{old}}(\Gamma_0(2)) \cong \) \(S_{80}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)