Properties

Label 3.70.a
Level 3
Weight 70
Character orbit a
Rep. character \(\chi_{3}(1,\cdot)\)
Character field \(\Q\)
Dimension 12
Newforms 2
Sturm bound 23
Trace bound 2

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Defining parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 70 \)
Character orbit: \([\chi]\) = 3.a (trivial)
Character field: \(\Q\)
Newforms: \( 2 \)
Sturm bound: \(23\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{70}(\Gamma_0(3))\).

Total New Old
Modular forms 24 12 12
Cusp forms 22 12 10
Eisenstein series 2 0 2

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

\(3\)Dim.
\(+\)\(6\)
\(-\)\(6\)

Trace form

\(12q \) \(\mathstrut +\mathstrut 18831599550q^{2} \) \(\mathstrut +\mathstrut 3586833257181443225076q^{4} \) \(\mathstrut +\mathstrut 1164829908436958237509656q^{5} \) \(\mathstrut +\mathstrut 343055069760136649722795494q^{6} \) \(\mathstrut -\mathstrut 117572717428940528567593369680q^{7} \) \(\mathstrut +\mathstrut 53489353249344715486151627235240q^{8} \) \(\mathstrut +\mathstrut 3337540673324322135087429314781132q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut +\mathstrut 18831599550q^{2} \) \(\mathstrut +\mathstrut 3586833257181443225076q^{4} \) \(\mathstrut +\mathstrut 1164829908436958237509656q^{5} \) \(\mathstrut +\mathstrut 343055069760136649722795494q^{6} \) \(\mathstrut -\mathstrut 117572717428940528567593369680q^{7} \) \(\mathstrut +\mathstrut 53489353249344715486151627235240q^{8} \) \(\mathstrut +\mathstrut 3337540673324322135087429314781132q^{9} \) \(\mathstrut -\mathstrut 52741947559970814829194419204056236q^{10} \) \(\mathstrut -\mathstrut 530406647773122180123323673455989152q^{11} \) \(\mathstrut +\mathstrut 14082211801671074105707944259455245940q^{12} \) \(\mathstrut +\mathstrut 86291152728961790750567856075204537480q^{13} \) \(\mathstrut +\mathstrut 4223191121926267823962773536359436009904q^{14} \) \(\mathstrut +\mathstrut 1523155420294053595981316056317118851504q^{15} \) \(\mathstrut +\mathstrut 675083968711772954054189802341660775762960q^{16} \) \(\mathstrut -\mathstrut 5332421650216324702110425506290438680776200q^{17} \) \(\mathstrut +\mathstrut 5237602453490083476930622757907431143307550q^{18} \) \(\mathstrut +\mathstrut 72610928562133462488130939542376201807753824q^{19} \) \(\mathstrut +\mathstrut 64320410721528384212279036036456814761099064q^{20} \) \(\mathstrut +\mathstrut 3558905708901023030811108688432715439206178288q^{21} \) \(\mathstrut -\mathstrut 15843625472566800950252251932340704241830127800q^{22} \) \(\mathstrut +\mathstrut 166999280108522364256492172649952251832530273120q^{23} \) \(\mathstrut -\mathstrut 76291457153672751392638504667566560447690478488q^{24} \) \(\mathstrut +\mathstrut 4510403486751456730274868202586763408749502753396q^{25} \) \(\mathstrut +\mathstrut 6806229099931558545468034619986890000694961034564q^{26} \) \(\mathstrut +\mathstrut 98769916560403802988366998554534517007485003391840q^{28} \) \(\mathstrut +\mathstrut 1172329097642639341811595535590376287758602048749528q^{29} \) \(\mathstrut +\mathstrut 139021910090742135806818831780491051635339104092036q^{30} \) \(\mathstrut +\mathstrut 6506032846922503708608548815663143512484276367458672q^{31} \) \(\mathstrut +\mathstrut 45557451010306975910168253562000647325780865087544480q^{32} \) \(\mathstrut +\mathstrut 55212603198861050317325853719985549980507568238004400q^{33} \) \(\mathstrut +\mathstrut 281940778365507979330535631962056348548770206544865372q^{34} \) \(\mathstrut +\mathstrut 1167277919317003806467605526635569589433658717031483680q^{35} \) \(\mathstrut +\mathstrut 997600157022952127058593969932956937317249782262838836q^{36} \) \(\mathstrut -\mathstrut 1793993238891473327060481766868703720627341660759323320q^{37} \) \(\mathstrut -\mathstrut 3182156831449517450191211035689551578307997431817432840q^{38} \) \(\mathstrut +\mathstrut 3014018526315750621148294271559476076979191786572697696q^{39} \) \(\mathstrut -\mathstrut 114997432331491279888385296072266071239942488508557956112q^{40} \) \(\mathstrut -\mathstrut 101690940797971649675217641051808049824803223180078668328q^{41} \) \(\mathstrut -\mathstrut 165203820808862569410329123216318399737753153137396697680q^{42} \) \(\mathstrut -\mathstrut 159366284839041141434442025411722575016125749582058615520q^{43} \) \(\mathstrut +\mathstrut 873485056525078249527599829024081521381806541249174516208q^{44} \) \(\mathstrut +\mathstrut 323972266409416174750736684955389058668738029536531384216q^{45} \) \(\mathstrut +\mathstrut 13379062519102677811034662075106405720920200306661713380112q^{46} \) \(\mathstrut +\mathstrut 10051803356388297615396170066123363784791557978964101562400q^{47} \) \(\mathstrut -\mathstrut 1321347955839071916032364153357438886203386905727443090800q^{48} \) \(\mathstrut +\mathstrut 41945803295164003928393078137765476204599615638951522811820q^{49} \) \(\mathstrut +\mathstrut 9066810576976202490948256175342015914374883095961934759394q^{50} \) \(\mathstrut +\mathstrut 79785613821597081639562183356501424237759105063181357161056q^{51} \) \(\mathstrut -\mathstrut 835986717895520217098233224569221609275384221855540992184040q^{52} \) \(\mathstrut -\mathstrut 374576571253101828157737134550884704858596962504601864884040q^{53} \) \(\mathstrut +\mathstrut 95413354042880731263746555191441153280488087639498150484934q^{54} \) \(\mathstrut -\mathstrut 155106401837208677772157016915460177455759904333200044689024q^{55} \) \(\mathstrut +\mathstrut 488993929142410730815187820575763315597734720458096361933760q^{56} \) \(\mathstrut +\mathstrut 3406278691265504271604216574595652950639832368769289162922000q^{57} \) \(\mathstrut +\mathstrut 25877727213035039570410354036558827421536917935398622288469220q^{58} \) \(\mathstrut +\mathstrut 21155086795230705716179518855525237356508323599046658446134656q^{59} \) \(\mathstrut +\mathstrut 3550139634691208368928716417028215027884483899105134854250296q^{60} \) \(\mathstrut -\mathstrut 26076877786993217142672981495839488119063921232854658290338520q^{61} \) \(\mathstrut -\mathstrut 182809793688099925924143868471473826386515064385077063132582720q^{62} \) \(\mathstrut -\mathstrut 32700310541029703011420525513913607848052260891927678630406480q^{63} \) \(\mathstrut -\mathstrut 962742987658590618438943473382271772512806978171416142672238016q^{64} \) \(\mathstrut -\mathstrut 1401840139126521141612309568896020021304595354080160563971744112q^{65} \) \(\mathstrut +\mathstrut 1643674106520460586612772859333357786084941572494826379622141032q^{66} \) \(\mathstrut +\mathstrut 3114246728232518604230266590187539264027493568674209389212394880q^{67} \) \(\mathstrut +\mathstrut 424351758708200044071320732614573132810232620698037912282352040q^{68} \) \(\mathstrut +\mathstrut 4496768555334638376541272937315963548347936434460398924101666624q^{69} \) \(\mathstrut +\mathstrut 27318485572748688156205373902479618939246244132443988193283785760q^{70} \) \(\mathstrut -\mathstrut 21469819433063015331959450573544269134523554842790588147296419616q^{71} \) \(\mathstrut +\mathstrut 14876907671625039981909768926905035208501977800307445225906457640q^{72} \) \(\mathstrut +\mathstrut 13461949491598161535367788500227495912574119519273782268325991800q^{73} \) \(\mathstrut -\mathstrut 228236443879112892250304731042829834957753547253745234466153801516q^{74} \) \(\mathstrut +\mathstrut 30129054314069446359269545532225815243023222415007143812272850624q^{75} \) \(\mathstrut -\mathstrut 46046115198611244070858455855169672218209773264052666501508538224q^{76} \) \(\mathstrut -\mathstrut 413363840464986271017838789791491595145527302331818754229559252160q^{77} \) \(\mathstrut +\mathstrut 654099357358993052206094567900446932676839200404681517558383462260q^{78} \) \(\mathstrut +\mathstrut 1000872393738716879804018423834513219854462738276159891614398894640q^{79} \) \(\mathstrut -\mathstrut 292655426252051456083895116883984439096488064030338237208854495136q^{80} \) \(\mathstrut +\mathstrut 928264812174514130260181362124730419586176955083061522201421933452q^{81} \) \(\mathstrut +\mathstrut 1586126712732737743364224633707580549607415833744669370375853819500q^{82} \) \(\mathstrut -\mathstrut 2347579387394165051130287763233931339185646985858755337793066848800q^{83} \) \(\mathstrut -\mathstrut 2401696789573239691066724900686597182726363502874611534476599162784q^{84} \) \(\mathstrut -\mathstrut 5591823235676252140506584001621845687895393387234065786931272830608q^{85} \) \(\mathstrut -\mathstrut 3063629996869728466394726141974431371303435686421461193382312124152q^{86} \) \(\mathstrut -\mathstrut 6651350054692874327962789367077144023441419394961181535453528541680q^{87} \) \(\mathstrut -\mathstrut 22949942846772447944548280515369790080728593504476505562690231820320q^{88} \) \(\mathstrut +\mathstrut 57221017189652026321849663996928132337983952313344020948650410288504q^{89} \) \(\mathstrut -\mathstrut 14669032930978423518892240788052884481465330407698434816050413311596q^{90} \) \(\mathstrut -\mathstrut 37528883396746291198471872980282688396369620036465572746180454977888q^{91} \) \(\mathstrut +\mathstrut 411876800750167054838392920517639028479408520435220403610087139842400q^{92} \) \(\mathstrut -\mathstrut 122070867553908786250847199522064344319846103062115297321744985186480q^{93} \) \(\mathstrut -\mathstrut 16354380776642639890855541216901624910793114740339557666867383568864q^{94} \) \(\mathstrut -\mathstrut 187342156069685748587896986918062858683366169085989548926216904248576q^{95} \) \(\mathstrut -\mathstrut 465582458010209681938008659278486717799094520378366899856715232523872q^{96} \) \(\mathstrut -\mathstrut 245808815975386295699715603586902753094482555925543630082136700641960q^{97} \) \(\mathstrut -\mathstrut 1215582268439890856056938042502911091267814793153960536120598680303730q^{98} \) \(\mathstrut -\mathstrut 147521146695366897402530832245562206062729135440649060703735453856672q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{70}^{\mathrm{new}}(\Gamma_0(3))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3
3.70.a.a \(6\) \(90.454\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-869363388\) \(-1\!\cdots\!14\) \(53\!\cdots\!20\) \(-1\!\cdots\!16\) \(+\) \(q+(-144893898+\beta _{1})q^{2}-3^{34}q^{3}+\cdots\)
3.70.a.b \(6\) \(90.454\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(19700962938\) \(10\!\cdots\!14\) \(62\!\cdots\!36\) \(47\!\cdots\!36\) \(-\) \(q+(3283493823-\beta _{1})q^{2}+3^{34}q^{3}+\cdots\)

Decomposition of \(S_{70}^{\mathrm{old}}(\Gamma_0(3))\) into lower level spaces

\( S_{70}^{\mathrm{old}}(\Gamma_0(3)) \cong \) \(S_{70}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)