Properties

Label 2.68.a
Level 2
Weight 68
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 6
Newforms 2
Sturm bound 17
Trace bound 2

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Defining parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 68 \)
Character orbit: \([\chi]\) = 2.a (trivial)
Character field: \(\Q\)
Newforms: \( 2 \)
Sturm bound: \(17\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{68}(\Gamma_0(2))\).

Total New Old
Modular forms 18 6 12
Cusp forms 16 6 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.

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

Trace form

\(6q \) \(\mathstrut -\mathstrut 7608049647932760q^{3} \) \(\mathstrut +\mathstrut 442721857769029238784q^{4} \) \(\mathstrut -\mathstrut 194764873271514971106060q^{5} \) \(\mathstrut +\mathstrut 24423269365861391032909824q^{6} \) \(\mathstrut -\mathstrut 25472880664845880773818039280q^{7} \) \(\mathstrut +\mathstrut 307421164442124171189667730270142q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut 7608049647932760q^{3} \) \(\mathstrut +\mathstrut 442721857769029238784q^{4} \) \(\mathstrut -\mathstrut 194764873271514971106060q^{5} \) \(\mathstrut +\mathstrut 24423269365861391032909824q^{6} \) \(\mathstrut -\mathstrut 25472880664845880773818039280q^{7} \) \(\mathstrut +\mathstrut 307421164442124171189667730270142q^{9} \) \(\mathstrut +\mathstrut 2875535035976801610919437643284480q^{10} \) \(\mathstrut -\mathstrut 194528729227668394789524730397199048q^{11} \) \(\mathstrut -\mathstrut 561374979021966724619321603669360640q^{12} \) \(\mathstrut +\mathstrut 13426181020022526731021866395504546180q^{13} \) \(\mathstrut -\mathstrut 587441586945603476299082757570526445568q^{14} \) \(\mathstrut +\mathstrut 4122770279988267930475492594373079791280q^{15} \) \(\mathstrut +\mathstrut 32667107224410092492483962313449748299776q^{16} \) \(\mathstrut -\mathstrut 406131085963533133004709126325015926289620q^{17} \) \(\mathstrut +\mathstrut 652067067677224115958746776055852008734720q^{18} \) \(\mathstrut +\mathstrut 1713662164079298007934240347198380555420360q^{19} \) \(\mathstrut -\mathstrut 14371111087152442574205268855595512721571840q^{20} \) \(\mathstrut -\mathstrut 66346141854663534386120591270533033555664448q^{21} \) \(\mathstrut +\mathstrut 52770494517693515321135359377812025224724480q^{22} \) \(\mathstrut -\mathstrut 3769508334255602091762314982720872690295230160q^{23} \) \(\mathstrut +\mathstrut 1802119197741262615289659556219528907205902336q^{24} \) \(\mathstrut -\mathstrut 21948781054012039871502958890283570152890280150q^{25} \) \(\mathstrut +\mathstrut 44325291234558192633941056131304299851063230464q^{26} \) \(\mathstrut -\mathstrut 3047393276893726176772625798493952476190359997680q^{27} \) \(\mathstrut -\mathstrut 1879566841778225497051236475051308430804301905920q^{28} \) \(\mathstrut -\mathstrut 18975796776142919104046133207944978990412805385820q^{29} \) \(\mathstrut -\mathstrut 84109230747069657039591082214861768901104228106240q^{30} \) \(\mathstrut -\mathstrut 538210453228271688025554058616824428270408588170688q^{31} \) \(\mathstrut -\mathstrut 2230273135249039517687973903091673636121524287350240q^{33} \) \(\mathstrut -\mathstrut 4350397242280047349947151489716990554976165757452288q^{34} \) \(\mathstrut -\mathstrut 4248845690187810468893397442798655752873264304508960q^{35} \) \(\mathstrut +\mathstrut 22683678173222574361782206025361576031538389490597888q^{36} \) \(\mathstrut +\mathstrut 35559936457709464556758247952999374283062296203830580q^{37} \) \(\mathstrut +\mathstrut 107169455664530255724674464343510303460228559137669120q^{38} \) \(\mathstrut +\mathstrut 916431352383498072696586216280529700475114071206037104q^{39} \) \(\mathstrut +\mathstrut 212177035534596989653467784593161439673963097326878720q^{40} \) \(\mathstrut -\mathstrut 407329955201306049997844781648386616640068104499856548q^{41} \) \(\mathstrut +\mathstrut 2130980994385218201027756734150661109996106241603010560q^{42} \) \(\mathstrut -\mathstrut 9340848207868916101624464685354916304734010894906028680q^{43} \) \(\mathstrut -\mathstrut 14353686732186968013500031777505950907282157641528246272q^{44} \) \(\mathstrut -\mathstrut 119834981267368952169697584837637412860198078177994487420q^{45} \) \(\mathstrut +\mathstrut 11574478123089108993384700269341140506077446932554317824q^{46} \) \(\mathstrut +\mathstrut 178661178225511677548368815198287111456437152688024045920q^{47} \) \(\mathstrut -\mathstrut 41422162269609154150886786250882000265307172367195176960q^{48} \) \(\mathstrut +\mathstrut 722819010335989231697641789359655125654982642213921583478q^{49} \) \(\mathstrut +\mathstrut 592730057594650750000825972540473615868571008642528051200q^{50} \) \(\mathstrut +\mathstrut 5531238256724358302502846555847751927655956603214237953232q^{51} \) \(\mathstrut +\mathstrut 990677300654608830965708765452284494890944548069462507520q^{52} \) \(\mathstrut -\mathstrut 12241492696715674108926681175491568936325413743323667227820q^{53} \) \(\mathstrut -\mathstrut 10130271848908178375798834760765535743509814643026894520320q^{54} \) \(\mathstrut -\mathstrut 23004965892657671573434211243380648737175644041642955089520q^{55} \) \(\mathstrut -\mathstrut 43345538450557380908604633137015345836892657567550041751552q^{56} \) \(\mathstrut -\mathstrut 166694765891788067318477166628066058095660030743552928987680q^{57} \) \(\mathstrut +\mathstrut 111599735655643602542412280887578869388802258819321000099840q^{58} \) \(\mathstrut +\mathstrut 441265192596176373107574341794337060718049626078708679691160q^{59} \) \(\mathstrut +\mathstrut 304206752918557801082388868232467456837135473424958666833920q^{60} \) \(\mathstrut +\mathstrut 1190999677431039748060049041408186173577768157943715984330212q^{61} \) \(\mathstrut +\mathstrut 177579177194603693817537403470957123090920383246954250895360q^{62} \) \(\mathstrut +\mathstrut 1781707618725880102267065796065958417932331622894448119960400q^{63} \) \(\mathstrut +\mathstrut 2410407066388485413312943138511743903783304490674189252952064q^{64} \) \(\mathstrut -\mathstrut 17014249594153100176941848930235219643875560772265066518598920q^{65} \) \(\mathstrut -\mathstrut 18323907210228963216244380862396726996820812018726146033057792q^{66} \) \(\mathstrut -\mathstrut 14130250287061935214434024229279303628947271905621898845646040q^{67} \) \(\mathstrut -\mathstrut 29967184812588117134987994434872120424430063232587187020103680q^{68} \) \(\mathstrut +\mathstrut 59397393705678114949513905408973237584363242832892113003065664q^{69} \) \(\mathstrut +\mathstrut 151359529314070680116471083768031130830953932273582601855303680q^{70} \) \(\mathstrut +\mathstrut 41340774020983520169015276829401277717319910799136988136562832q^{71} \) \(\mathstrut +\mathstrut 48114057265343996319061730166355386093510241153476506765230080q^{72} \) \(\mathstrut +\mathstrut 618709200056416488517131921856719346264203807424896740476909980q^{73} \) \(\mathstrut -\mathstrut 196714776259890201586090221611088065667543298215726786489090048q^{74} \) \(\mathstrut -\mathstrut 460131762587231510466574375997947411790386156041963769297981800q^{75} \) \(\mathstrut +\mathstrut 126445949478280303150648712608458038903943943396831095989207040q^{76} \) \(\mathstrut -\mathstrut 7651452259806639980201822914203088779370146448853331630607715520q^{77} \) \(\mathstrut -\mathstrut 2267492623252981251020624280255026653491793912853595600055173120q^{78} \) \(\mathstrut +\mathstrut 7720715696945865590463871323925920624391457706165908903663901600q^{79} \) \(\mathstrut -\mathstrut 1060400833118203806395193753656810262457014821830576708528373760q^{80} \) \(\mathstrut +\mathstrut 7365026793095143817274580003278450604511726994241946863537222326q^{81} \) \(\mathstrut +\mathstrut 15012798411264507710787646918011484815898800989896769888153763840q^{82} \) \(\mathstrut +\mathstrut 31489915361822334560188711693604949352841211701312605165190339400q^{83} \) \(\mathstrut -\mathstrut 4895481196284031170058973833771616839202181915446482556128591872q^{84} \) \(\mathstrut +\mathstrut 78142426591912234333681756244948230271920702940004457876989641640q^{85} \) \(\mathstrut -\mathstrut 32986497567414733106619261411311304087985073091878715612721053696q^{86} \) \(\mathstrut -\mathstrut 351985235467347889798399589356480953343808396604484464579292665360q^{87} \) \(\mathstrut +\mathstrut 3893775228043940948171347826488845188894320286161772231755038720q^{88} \) \(\mathstrut -\mathstrut 123426521949058543473593445867186345816512087857189099156785955140q^{89} \) \(\mathstrut -\mathstrut 76464107345529467755356214071532252283658907962764212180761968640q^{90} \) \(\mathstrut -\mathstrut 564824914214332057553970702582901165038636076565440326180832890528q^{91} \) \(\mathstrut -\mathstrut 278140622102913165933437666614113146387926835473920359028479754240q^{92} \) \(\mathstrut +\mathstrut 3331276272901403046526870062417073206614851388308318555780388970240q^{93} \) \(\mathstrut +\mathstrut 913387847886008990512847111276821560789069894876286047013790810112q^{94} \) \(\mathstrut +\mathstrut 4337582135928558136377686930964417676836263985858732036555055052400q^{95} \) \(\mathstrut +\mathstrut 132972926524207390900451772141758277608232221976366066300987899904q^{96} \) \(\mathstrut -\mathstrut 6824871461188772314037540224803732928542096436791132468788037640500q^{97} \) \(\mathstrut -\mathstrut 9653718711210471788616546940897111465443860582012248308386491269120q^{98} \) \(\mathstrut +\mathstrut 4165067909549884750024286102826729628775548538024955012922814797464q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{68}^{\mathrm{new}}(\Gamma_0(2))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.68.a.a \(3\) \(56.858\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-25769803776\) \(-5\!\cdots\!16\) \(-2\!\cdots\!50\) \(21\!\cdots\!12\) \(+\) \(q-2^{33}q^{2}+(-1741882068537572+\cdots)q^{3}+\cdots\)
2.68.a.b \(3\) \(56.858\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(25769803776\) \(-2\!\cdots\!44\) \(69\!\cdots\!90\) \(-4\!\cdots\!92\) \(-\) \(q+2^{33}q^{2}+(-794134480773348+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{68}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

\( S_{68}^{\mathrm{old}}(\Gamma_0(2)) \cong \) \(S_{68}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)