Properties

Label 2.78.a
Level 2
Weight 78
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 6
Newforms 2
Sturm bound 19
Trace bound 2

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Defining parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 78 \)
Character orbit: \([\chi]\) = 2.a (trivial)
Character field: \(\Q\)
Newforms: \( 2 \)
Sturm bound: \(19\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{78}(\Gamma_0(2))\).

Total New Old
Modular forms 20 6 14
Cusp forms 18 6 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.
\(+\)\(3\)
\(-\)\(3\)

Trace form

\(6q \) \(\mathstrut -\mathstrut 2638706479849430280q^{3} \) \(\mathstrut +\mathstrut 453347182355485940514816q^{4} \) \(\mathstrut +\mathstrut 951948703777156427890806660q^{5} \) \(\mathstrut +\mathstrut 664069599363895225939000295424q^{6} \) \(\mathstrut -\mathstrut 404664187371826723533400217750160q^{7} \) \(\mathstrut +\mathstrut 27635744030038056171543919721032526958q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut 2638706479849430280q^{3} \) \(\mathstrut +\mathstrut 453347182355485940514816q^{4} \) \(\mathstrut +\mathstrut 951948703777156427890806660q^{5} \) \(\mathstrut +\mathstrut 664069599363895225939000295424q^{6} \) \(\mathstrut -\mathstrut 404664187371826723533400217750160q^{7} \) \(\mathstrut +\mathstrut 27635744030038056171543919721032526958q^{9} \) \(\mathstrut +\mathstrut 91257727220733797922901686662159400960q^{10} \) \(\mathstrut +\mathstrut 2813285456351650460858754578674470792552q^{11} \) \(\mathstrut -\mathstrut 199375024617150342745880842780071249838080q^{12} \) \(\mathstrut -\mathstrut 5387427944763819764030050276353666486056460q^{13} \) \(\mathstrut +\mathstrut 16520452604894625405860749985152667870035968q^{14} \) \(\mathstrut -\mathstrut 1135883271557496516132416112035892348266320560q^{15} \) \(\mathstrut +\mathstrut 34253944624943037145398863266787883273185918976q^{16} \) \(\mathstrut -\mathstrut 240351405602411934790934759670260169813104869140q^{17} \) \(\mathstrut +\mathstrut 4839316985813546497882978970098276147197546332160q^{18} \) \(\mathstrut -\mathstrut 45893971269809092729734590453917163953845035256360q^{19} \) \(\mathstrut +\mathstrut 71927210434055167140260971699563051078232320245760q^{20} \) \(\mathstrut +\mathstrut 1401012816099590795429365101383802989450997291970752q^{21} \) \(\mathstrut -\mathstrut 11125248257094086183871982045171633101780341426749440q^{22} \) \(\mathstrut +\mathstrut 48640348424149570925324498153599959856137326190461520q^{23} \) \(\mathstrut +\mathstrut 50175680293259716553311739858033147080681785774833664q^{24} \) \(\mathstrut -\mathstrut 398875278193186197856744600318615444390641586944858150q^{25} \) \(\mathstrut +\mathstrut 3415403793959182394051815855026300616551980020178878464q^{26} \) \(\mathstrut +\mathstrut 19349767246913973555813361436471974968223870568539877040q^{27} \) \(\mathstrut -\mathstrut 30575561524198343424449603702271110059135213208611061760q^{28} \) \(\mathstrut +\mathstrut 177509221507210771314192900454648567151613591454452548820q^{29} \) \(\mathstrut +\mathstrut 1340514695285109270197692869004094029901201395971407216640q^{30} \) \(\mathstrut -\mathstrut 5235559682219472906690176086938683491174471358818087437888q^{31} \) \(\mathstrut +\mathstrut 40577551469543753340467332593881981549497732378992850805280q^{33} \) \(\mathstrut -\mathstrut 28728510870588968139878894736118585268489508521741345882112q^{34} \) \(\mathstrut -\mathstrut 123064352879350240342180780799622219421217658897628058701920q^{35} \) \(\mathstrut +\mathstrut 2088097781385865762745322404541080951012278757890508653068288q^{36} \) \(\mathstrut +\mathstrut 7156514143967091986133454180146950526241890550563178852648260q^{37} \) \(\mathstrut +\mathstrut 2040897416681867651538648803119205521313170834634472538767360q^{38} \) \(\mathstrut +\mathstrut 107456379433622082812502228370241930898258059873982872409735696q^{39} \) \(\mathstrut +\mathstrut 6895238917280866375135335442099832497030641086794792760770560q^{40} \) \(\mathstrut +\mathstrut 410465139749811625309074459022728334344890109972786982566849852q^{41} \) \(\mathstrut +\mathstrut 805403647085810964014486128394288828918198462460989714800312320q^{42} \) \(\mathstrut +\mathstrut 2320058003993125449433825043551023223127010133647134942508081960q^{43} \) \(\mathstrut +\mathstrut 212565839133114693977856248066434581905269098221961512403075072q^{44} \) \(\mathstrut +\mathstrut 10203244720734129287172794367624582396187024267861702888626955380q^{45} \) \(\mathstrut +\mathstrut 2689960197998271719351769644493200790795055143703777841170612224q^{46} \) \(\mathstrut -\mathstrut 38244595912715455466995651647529711267054630500403405513549841760q^{47} \) \(\mathstrut -\mathstrut 15064350940373459149228584094629141708006792960481646191973498880q^{48} \) \(\mathstrut -\mathstrut 389094542452427259916390029434866502780300547855292070198944383978q^{49} \) \(\mathstrut -\mathstrut 831485774843342499680359664508624926672108521209001092974012006400q^{50} \) \(\mathstrut -\mathstrut 930191979149192566325986756313185136870549452751754737819177754768q^{51} \) \(\mathstrut -\mathstrut 407062546483647372579849484620264530532879283368214456976340418560q^{52} \) \(\mathstrut -\mathstrut 695797257743603428807751534595932331958483278842484417070166762460q^{53} \) \(\mathstrut +\mathstrut 1754690806597444293046334739806632909274959133479013419923611320320q^{54} \) \(\mathstrut +\mathstrut 38765245534507901360456409072653869831426630634683147406804309076720q^{55} \) \(\mathstrut +\mathstrut 1248250106611054411112330855294537584863889729514344296745059483648q^{56} \) \(\mathstrut +\mathstrut 129362944731709088997142014082973350866062059161528333699670779875040q^{57} \) \(\mathstrut +\mathstrut 50553365460753368965295588731110862830308083132310069555735581163520q^{58} \) \(\mathstrut -\mathstrut 83738260671160073339274788272749998269062795139071833447932168007160q^{59} \) \(\mathstrut -\mathstrut 85824913440887054944685196442844885241558241613465368296333414236160q^{60} \) \(\mathstrut +\mathstrut 139900325275364299372802562013721613938014575020089549956553410097812q^{61} \) \(\mathstrut -\mathstrut 2111006982576088507000264401233211102767594655185504995235583838126080q^{62} \) \(\mathstrut -\mathstrut 6126200075624497706359484909456780492729528147181203469617380029598800q^{63} \) \(\mathstrut +\mathstrut 2588154880046461420288033448353884544669165864563894958185946583924736q^{64} \) \(\mathstrut -\mathstrut 12393134856502627186159140322763470190755840041406744302014774648849160q^{65} \) \(\mathstrut +\mathstrut 15393885428335113682010960552095081769079893184032598792332776788983808q^{66} \) \(\mathstrut -\mathstrut 12854778152408665924380911481075717372896438820035823065050865579327880q^{67} \) \(\mathstrut -\mathstrut 18160438750839001419211575762557524388833159140012271964552214651863040q^{68} \) \(\mathstrut +\mathstrut 359310602285066394235528572119300341447192465781666788376235147864668736q^{69} \) \(\mathstrut +\mathstrut 103629847839080851842304847321434986509738469514573404043319760629268480q^{70} \) \(\mathstrut -\mathstrut 239348329233499159281725256062739658510043600972130704323629438986077968q^{71} \) \(\mathstrut +\mathstrut 365648453340602405398745089460256882904927686172564038282850307156213760q^{72} \) \(\mathstrut +\mathstrut 590732221588297211993359202991552093313332614637765009735841839598898940q^{73} \) \(\mathstrut -\mathstrut 1932561582707509952011992201876418120033979606336145283880984403039485952q^{74} \) \(\mathstrut -\mathstrut 6894358829739875418006698611431046529908046354215975823843697141950004600q^{75} \) \(\mathstrut -\mathstrut 3467650427045262567968315642170391976780739571258665204670493541489704960q^{76} \) \(\mathstrut +\mathstrut 3027746567174723274672896523599217478828065756918059258896992435435562560q^{77} \) \(\mathstrut +\mathstrut 5582523603240121029709723147076013923944333973942409491332660145569136640q^{78} \) \(\mathstrut -\mathstrut 5776495766817453464284107181954195043305348051620600864517138288432413600q^{79} \) \(\mathstrut +\mathstrut 5434666364161503150888916355684998022341974188838754175526045799006863360q^{80} \) \(\mathstrut +\mathstrut 117563120151265824119033769345677085904590527052736653327730870160256552726q^{81} \) \(\mathstrut +\mathstrut 19357494913548812795451232384100923643744067379118639730033395516288532480q^{82} \) \(\mathstrut +\mathstrut 129264760599533898767518723678436374167477584530796383608431823494550778200q^{83} \) \(\mathstrut +\mathstrut 105857535437112346177585855037195646219860979986395570285290974848349110272q^{84} \) \(\mathstrut -\mathstrut 801241925385693765103426964254546576751329262200466456455724329000162401720q^{85} \) \(\mathstrut +\mathstrut 53797180931623833539915540017975805817399081240325266040630423998645141504q^{86} \) \(\mathstrut -\mathstrut 291328796825929144330520656517167227174892291079422928234202042149107063920q^{87} \) \(\mathstrut -\mathstrut 840599991726480803409567601862008337142872621983882976102199012471173283840q^{88} \) \(\mathstrut -\mathstrut 1193812277046235700760495300028009526099839740926801826997139607286011853860q^{89} \) \(\mathstrut -\mathstrut 1609060304447922381867852111640472845776561247088529753708962342303925534720q^{90} \) \(\mathstrut +\mathstrut 2492679381298712523448873417158446985376314632434513670973361519518001603872q^{91} \) \(\mathstrut +\mathstrut 3675160817812884788672025368135229915457597531387580745772777698516239646720q^{92} \) \(\mathstrut +\mathstrut 10687213444988864959111097540950099077730437025392413394337931784475766526720q^{93} \) \(\mathstrut -\mathstrut 31395041641246475618760185950925797315969613155484425844934831292035366912q^{94} \) \(\mathstrut +\mathstrut 23652710336299468471330383881491290651477153277385046023331615359669641952400q^{95} \) \(\mathstrut +\mathstrut 3791167213953162498930463176849797580866121787939356866278082602690154594304q^{96} \) \(\mathstrut +\mathstrut 34330669035023975921697546293905865217388554866066568629115439228943107591500q^{97} \) \(\mathstrut +\mathstrut 26683820159041618494567864569402267280153427530555533481830490985535782256640q^{98} \) \(\mathstrut -\mathstrut 338117328528106478398409250709483622397469797614871765540271738811872768465464q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{78}^{\mathrm{new}}(\Gamma_0(2))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.78.a.a \(3\) \(75.096\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-824633720832\) \(-2\!\cdots\!88\) \(30\!\cdots\!10\) \(-2\!\cdots\!16\) \(+\) \(q-2^{38}q^{2}+(-842429601461527596+\cdots)q^{3}+\cdots\)
2.78.a.b \(3\) \(75.096\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(824633720832\) \(-1\!\cdots\!92\) \(64\!\cdots\!50\) \(-1\!\cdots\!44\) \(-\) \(q+2^{38}q^{2}+(-37139225154949164+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{78}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

\( S_{78}^{\mathrm{old}}(\Gamma_0(2)) \cong \) \(S_{78}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)