Properties

Label 1.30.a
Level 1
Weight 30
Character orbit a
Rep. character \(\chi_{1}(1,\cdot)\)
Character field \(\Q\)
Dimension 2
Newforms 1
Sturm bound 2
Trace bound 0

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Defining parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 30 \)
Character orbit: \([\chi]\) = 1.a (trivial)
Character field: \(\Q\)
Newforms: \( 1 \)
Sturm bound: \(2\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{30}(\Gamma_0(1))\).

Total New Old
Modular forms 3 3 0
Cusp forms 2 2 0
Eisenstein series 1 1 0

Trace form

\(2q \) \(\mathstrut +\mathstrut 8640q^{2} \) \(\mathstrut -\mathstrut 4967640q^{3} \) \(\mathstrut -\mathstrut 89952256q^{4} \) \(\mathstrut -\mathstrut 17477788500q^{5} \) \(\mathstrut -\mathstrut 543908756736q^{6} \) \(\mathstrut -\mathstrut 3020312682800q^{7} \) \(\mathstrut +\mathstrut 3150297169920q^{8} \) \(\mathstrut +\mathstrut 163469569523706q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 8640q^{2} \) \(\mathstrut -\mathstrut 4967640q^{3} \) \(\mathstrut -\mathstrut 89952256q^{4} \) \(\mathstrut -\mathstrut 17477788500q^{5} \) \(\mathstrut -\mathstrut 543908756736q^{6} \) \(\mathstrut -\mathstrut 3020312682800q^{7} \) \(\mathstrut +\mathstrut 3150297169920q^{8} \) \(\mathstrut +\mathstrut 163469569523706q^{9} \) \(\mathstrut +\mathstrut 34285866768000q^{10} \) \(\mathstrut -\mathstrut 2055380519588616q^{11} \) \(\mathstrut -\mathstrut 4290530276229120q^{12} \) \(\mathstrut +\mathstrut 17139371179332220q^{13} \) \(\mathstrut +\mathstrut 51175123348059648q^{14} \) \(\mathstrut -\mathstrut 17192351392506000q^{15} \) \(\mathstrut -\mathstrut 453469082063208448q^{16} \) \(\mathstrut -\mathstrut 664795927907749980q^{17} \) \(\mathstrut +\mathstrut 3301524864881760960q^{18} \) \(\mathstrut +\mathstrut 1232169445452155080q^{19} \) \(\mathstrut +\mathstrut 1734668101813248000q^{20} \) \(\mathstrut -\mathstrut 27949113476248821696q^{21} \) \(\mathstrut +\mathstrut 1145797484912974080q^{22} \) \(\mathstrut -\mathstrut 18588430504374379920q^{23} \) \(\mathstrut +\mathstrut 276660084668356362240q^{24} \) \(\mathstrut -\mathstrut 207056854502617281250q^{25} \) \(\mathstrut +\mathstrut 181912132597777755264q^{26} \) \(\mathstrut -\mathstrut 1497723629507205824880q^{27} \) \(\mathstrut +\mathstrut 690727602371844997120q^{28} \) \(\mathstrut +\mathstrut 996096417765761595420q^{29} \) \(\mathstrut +\mathstrut 4218653306492474688000q^{30} \) \(\mathstrut -\mathstrut 1087258664181924030656q^{31} \) \(\mathstrut -\mathstrut 8776107971762202869760q^{32} \) \(\mathstrut -\mathstrut 428627571739162757280q^{33} \) \(\mathstrut -\mathstrut 20940413403809031000192q^{34} \) \(\mathstrut +\mathstrut 33844046536902649068000q^{35} \) \(\mathstrut +\mathstrut 15071477561016892895232q^{36} \) \(\mathstrut +\mathstrut 98878127759336138311660q^{37} \) \(\mathstrut -\mathstrut 129833487351109249470720q^{38} \) \(\mathstrut -\mathstrut 102115380027539490485328q^{39} \) \(\mathstrut -\mathstrut 87313163228088238080000q^{40} \) \(\mathstrut -\mathstrut 106699837259610025757676q^{41} \) \(\mathstrut +\mathstrut 508720764303162503608320q^{42} \) \(\mathstrut +\mathstrut 510640433114974031366200q^{43} \) \(\mathstrut +\mathstrut 179059414424913371086848q^{44} \) \(\mathstrut -\mathstrut 1127484267514124881450500q^{45} \) \(\mathstrut +\mathstrut 252841215606439776920064q^{46} \) \(\mathstrut -\mathstrut 4521107671109825795039520q^{47} \) \(\mathstrut +\mathstrut 3955787455065109907374080q^{48} \) \(\mathstrut +\mathstrut 2479210387392854586312114q^{49} \) \(\mathstrut -\mathstrut 2813370491836752543000000q^{50} \) \(\mathstrut +\mathstrut 11625042659032475755122384q^{51} \) \(\mathstrut +\mathstrut 161134672249303679180800q^{52} \) \(\mathstrut -\mathstrut 16141544629666790250663540q^{53} \) \(\mathstrut -\mathstrut 19762893017614843677688320q^{54} \) \(\mathstrut +\mathstrut 19124657798421120971058000q^{55} \) \(\mathstrut -\mathstrut 39728224533177935352299520q^{56} \) \(\mathstrut +\mathstrut 71545878452212360967003040q^{57} \) \(\mathstrut +\mathstrut 64319996083339084169992320q^{58} \) \(\mathstrut -\mathstrut 83497291515660150276062760q^{59} \) \(\mathstrut +\mathstrut 37864111042200317349888000q^{60} \) \(\mathstrut -\mathstrut 15876762943230179624113316q^{61} \) \(\mathstrut -\mathstrut 189803556071494720042936320q^{62} \) \(\mathstrut -\mathstrut 70756669640265175617870960q^{63} \) \(\mathstrut +\mathstrut 245489542813624183222697984q^{64} \) \(\mathstrut -\mathstrut 137266226551726798673511000q^{65} \) \(\mathstrut +\mathstrut 510163218803227073870066688q^{66} \) \(\mathstrut +\mathstrut 24468754434128844890782120q^{67} \) \(\mathstrut -\mathstrut 126211850011481106095677440q^{68} \) \(\mathstrut -\mathstrut 137724750477330905859409728q^{69} \) \(\mathstrut -\mathstrut 580830557953085152481664000q^{70} \) \(\mathstrut -\mathstrut 188428428253360067209577136q^{71} \) \(\mathstrut -\mathstrut 1155729487872248549372067840q^{72} \) \(\mathstrut +\mathstrut 995872721297568533052080980q^{73} \) \(\mathstrut +\mathstrut 1717790650063115374837542528q^{74} \) \(\mathstrut +\mathstrut 1573516410323456985690375000q^{75} \) \(\mathstrut -\mathstrut 1223170019527541658110935040q^{76} \) \(\mathstrut +\mathstrut 3784200478484579386131729600q^{77} \) \(\mathstrut -\mathstrut 5186288053950004913991206400q^{78} \) \(\mathstrut -\mathstrut 7218460852587735797823642080q^{79} \) \(\mathstrut +\mathstrut 3368223393577622373924864000q^{80} \) \(\mathstrut -\mathstrut 161317068789001769884865358q^{81} \) \(\mathstrut +\mathstrut 1596346485209467674952913280q^{82} \) \(\mathstrut +\mathstrut 2210846225248155523694669640q^{83} \) \(\mathstrut +\mathstrut 6695585379451908015663218688q^{84} \) \(\mathstrut +\mathstrut 3713635892367734904614943000q^{85} \) \(\mathstrut +\mathstrut 20357697303796411232450244864q^{86} \) \(\mathstrut -\mathstrut 35603430680717182696869516240q^{87} \) \(\mathstrut -\mathstrut 8696387677835767416210063360q^{88} \) \(\mathstrut +\mathstrut 5872964746600969363962031860q^{89} \) \(\mathstrut -\mathstrut 18577446803723860542221616000q^{90} \) \(\mathstrut -\mathstrut 18563550073842319069958420896q^{91} \) \(\mathstrut +\mathstrut 3714393183413159199429918720q^{92} \) \(\mathstrut +\mathstrut 104879397265778648689061310720q^{93} \) \(\mathstrut -\mathstrut 4003342107705068447401638912q^{94} \) \(\mathstrut -\mathstrut 26445947767121183571013410000q^{95} \) \(\mathstrut -\mathstrut 253006717086360117933244416q^{96} \) \(\mathstrut +\mathstrut 109649982219471997093227005380q^{97} \) \(\mathstrut -\mathstrut 183262972410594366168864121920q^{98} \) \(\mathstrut -\mathstrut 140506044809020243496724809448q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{30}^{\mathrm{new}}(\Gamma_0(1))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.30.a.a \(2\) \(5.328\) \(\Q(\sqrt{51349}) \) None \(8640\) \(-4967640\) \(-17477788500\) \(-3\!\cdots\!00\) \(+\) \(q+(4320-\beta )q^{2}+(-2483820+552\beta )q^{3}+\cdots\)