Properties

Label 2.66.a
Level 2
Weight 66
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 5
Newform subspaces 2
Sturm bound 16
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 66 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(16\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{66}(\Gamma_0(2))\).

Total New Old
Modular forms 17 5 12
Cusp forms 15 5 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 and the Fricke involution.

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

Trace form

\( 5q + 4294967296q^{2} + 4133618558416404q^{3} + 92233720368547758080q^{4} - 70844492277858340250850q^{5} + 7883406622088398551121920q^{6} + 7001440406811157707981723688q^{7} + 79228162514264337593543950336q^{8} + 4280010205873910625488759430465q^{9} + O(q^{10}) \) \( 5q + 4294967296q^{2} + 4133618558416404q^{3} + 92233720368547758080q^{4} - 70844492277858340250850q^{5} + 7883406622088398551121920q^{6} + 7001440406811157707981723688q^{7} + 79228162514264337593543950336q^{8} + 4280010205873910625488759430465q^{9} + 337930938987470828035602854707200q^{10} + 7618362444343764013557498728179260q^{11} + 76251803645443620481025441859108864q^{12} + 882232885765420334201808050793817414q^{13} + 6803913574135557746113036325790679040q^{14} - 165869399198448371842533041273942200200q^{15} + 1701411834604692317316873037158841057280q^{16} - 699431199975967287511256391699769193382q^{17} + 54688046875378802693745458742371033284608q^{18} - 215797600433727494753707613754433376524700q^{19} - 1306850218081545431136438477056118462873600q^{20} + 30443141210675727163896322101419389667840160q^{21} - 34407601299804566546070673016669303878975488q^{22} - 104566143245074550889373169195452729735243656q^{23} + 145423184386651800763257989747180776949022720q^{24} + 5222815973812689992833539864389023280667966875q^{25} + 1315764479050205968931892579612946188908625920q^{26} + 108154104528163539963028302441214172940259640520q^{27} + 129153779331774315634960680592466989365285879808q^{28} + 1171901356388482342344411971093074800906153580950q^{29} + 2636047695217789092070144412157500828736395673600q^{30} + 6330506755320242122975820336302432496801315635360q^{31} + 1461501637330902918203684832716283019655932542976q^{32} + 39879294444584220584040633282302322814198186549488q^{33} - 36021307174538746358640453802001276944144882728960q^{34} - 372218224586284300861056362847849176534279130224400q^{35} + 78952252900620958774140821728608214427666680381440q^{36} - 3796574591615529074089308713364716756356228339444802q^{37} + 267335410113780242328820161186798526325139848560640q^{38} - 5192750444320907817264786716986040411323790842623720q^{39} + 6233725546090231562181116522837441861517306966835200q^{40} + 35772385403070962335047731365996323671787664820390610q^{41} + 94867057371610595257263342953447427034225718373384192q^{42} + 350346941095941448933482524047995791803323727120787164q^{43} + 140533982271569742575260309124088171618023518670684160q^{44} - 457212883913540336931885029604185305367305418651677050q^{45} - 1669042359869728607100831802554318733465623847847854080q^{46} - 4646380911000170258795696401836769347105474956211412752q^{47} + 1406597507006251490063556690525924837662007810771124224q^{48} - 1492129351102935371466979010126725467565631037366182115q^{49} - 20769275918226828691437122316804487156160327057408000000q^{50} + 74851968734293288352325441533429824399212255083256806760q^{51} + 16274324257124943388074079241916687299311250450912641024q^{52} + 335677682346300958056164096291669515440665473899464980494q^{53} + 431935331265433676799136624090133427160061237003078860800q^{54} - 122303479920127600487377641644361911833275741992778946200q^{55} + 125510052401717073823312112432628728263396808710569328640q^{56} - 1435524027006552343277813430410848729092758221985585213040q^{57} - 463713973433259568312809449560182624145045665678549319680q^{58} + 3394625272218993571894661733741175653526062991894700889900q^{59} - 3059750356673741354328954615360195867853249801442505523200q^{60} + 1520772410473144755394714234498724572536428901978405638710q^{61} + 24605740180163429068226621132519593106424989224621291077632q^{62} + 135802714150119560665896310122169518736801764333945454076424q^{63} + 31385508676933403819178947116038332080511777222320172564480q^{64} + 64895485235702121487226607312575215983367937836195523451300q^{65} - 149809626692567909855591124779109907632091308722545345167360q^{66} - 147204622169015019103541524984654002213146800760232231332012q^{67} - 12902228343124234841563922514387832354408299542877214605312q^{68} - 2338952611656294569618620625821052274589789954278426765004320q^{69} - 231922377657414178457725845294759625323592133174100806860800q^{70} + 383895739208569607456368041152734954962343140940927969843560q^{71} + 1008816404601144090256994255289991135553039040467067394326528q^{72} + 1439430039400866536657771416898150420909402155311949577010034q^{73} + 2387610310641129678148832608798612527454897477792791202365440q^{74} + 9190450143105793808196455447938438580377589686680570076987500q^{75} - 3980763106921604429261768163605691925211349092580537348915200q^{76} - 26818555005238767738213214331355071802157350668242888658693664q^{77} - 46918809415480542431744096122363044050717352052893944930893824q^{78} - 66062584547778855926322557425828318229555831764447318390578800q^{79} - 24107131515621783296606039111404376381279199126192198883737600q^{80} + 60169532962747840282744178435382125692147057973395330261005405q^{81} + 161640005380680171404244248739796231721670355096896899332964352q^{82} - 11075714578911970422052892182389279259015060887262937688438876q^{83} + 561576834713135494427720737971974763446026554023921368757698560q^{84} + 1080977076209142619904291061933013670367861507974853587627651100q^{85} + 203959478417527130589662521872015124913971020759664996930027520q^{86} + 2194950060688638911356991860311064353283246752039287494417889880q^{87} - 634708215367730953097023299586309496019712292081204749934788608q^{88} - 3808726874831171149148054082536194166490796406679276483013439550q^{89} + 102744294374853525363014451995526104807175867647866785339801600q^{90} - 12918162390227545454209872616409212616925919579887907838851122640q^{91} - 1928904883216743033980111519165472790224951674764104276268548096q^{92} - 17351070997005472514228954476229969443642401928804118061384220032q^{93} + 6639539453765509358614844055808459367543338251965312321338736640q^{94} + 65642393077759635739007960160641202505805759690248833895290471000q^{95} + 2682584264764440501529820952599581505693943435030653180596715520q^{96} - 80320521866333789057339919232563862701921173611625715989905231382q^{97} + 105011570769210578923520629875788498926427144080303889020461514752q^{98} - 94799615684062844986214986394927237462820082800741591376156914420q^{99} + O(q^{100}) \)

Decomposition of \(S_{66}^{\mathrm{new}}(\Gamma_0(2))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.66.a.a \(2\) \(53.514\) \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(-8589934592\) \(11\!\cdots\!92\) \(-7\!\cdots\!00\) \(27\!\cdots\!24\) \(+\) \(q-2^{32}q^{2}+(574529980102596-111\beta )q^{3}+\cdots\)
2.66.a.b \(3\) \(53.514\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(12884901888\) \(29\!\cdots\!12\) \(39\!\cdots\!50\) \(42\!\cdots\!64\) \(-\) \(q+2^{32}q^{2}+(994852866070404-\beta _{1}+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{66}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

\( S_{66}^{\mathrm{old}}(\Gamma_0(2)) \cong \) \(S_{66}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 4294967296 T )^{2} \))(\( ( 1 - 4294967296 T )^{3} \))
$3$ (\( 1 - 1149059960205192 T + \)\(13\!\cdots\!02\)\( T^{2} - \)\(11\!\cdots\!56\)\( T^{3} + \)\(10\!\cdots\!49\)\( T^{4} \))(\( 1 - 2984558598211212 T + \)\(15\!\cdots\!77\)\( T^{2} - \)\(62\!\cdots\!96\)\( T^{3} + \)\(16\!\cdots\!11\)\( T^{4} - \)\(31\!\cdots\!88\)\( T^{5} + \)\(10\!\cdots\!07\)\( T^{6} \))
$5$ (\( 1 + \)\(74\!\cdots\!00\)\( T + \)\(29\!\cdots\!50\)\( T^{2} + \)\(20\!\cdots\!00\)\( T^{3} + \)\(73\!\cdots\!25\)\( T^{4} \))(\( 1 - \)\(39\!\cdots\!50\)\( T + \)\(39\!\cdots\!75\)\( T^{2} + \)\(15\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!75\)\( T^{4} - \)\(28\!\cdots\!50\)\( T^{5} + \)\(19\!\cdots\!25\)\( T^{6} \))
$7$ (\( 1 - \)\(27\!\cdots\!24\)\( T + \)\(18\!\cdots\!58\)\( T^{2} - \)\(23\!\cdots\!68\)\( T^{3} + \)\(72\!\cdots\!49\)\( T^{4} \))(\( 1 - \)\(42\!\cdots\!64\)\( T + \)\(16\!\cdots\!53\)\( T^{2} - \)\(52\!\cdots\!68\)\( T^{3} + \)\(13\!\cdots\!71\)\( T^{4} - \)\(31\!\cdots\!36\)\( T^{5} + \)\(62\!\cdots\!43\)\( T^{6} \))
$11$ (\( 1 - \)\(78\!\cdots\!44\)\( T + \)\(78\!\cdots\!86\)\( T^{2} - \)\(38\!\cdots\!44\)\( T^{3} + \)\(24\!\cdots\!01\)\( T^{4} \))(\( 1 + \)\(19\!\cdots\!84\)\( T + \)\(91\!\cdots\!05\)\( T^{2} - \)\(97\!\cdots\!80\)\( T^{3} + \)\(44\!\cdots\!55\)\( T^{4} + \)\(47\!\cdots\!84\)\( T^{5} + \)\(11\!\cdots\!51\)\( T^{6} \))
$13$ (\( 1 - \)\(28\!\cdots\!72\)\( T + \)\(48\!\cdots\!82\)\( T^{2} - \)\(73\!\cdots\!96\)\( T^{3} + \)\(64\!\cdots\!49\)\( T^{4} \))(\( 1 - \)\(59\!\cdots\!42\)\( T + \)\(42\!\cdots\!67\)\( T^{2} - \)\(35\!\cdots\!56\)\( T^{3} + \)\(10\!\cdots\!31\)\( T^{4} - \)\(38\!\cdots\!58\)\( T^{5} + \)\(16\!\cdots\!57\)\( T^{6} \))
$17$ (\( 1 - \)\(38\!\cdots\!64\)\( T + \)\(18\!\cdots\!38\)\( T^{2} - \)\(36\!\cdots\!48\)\( T^{3} + \)\(90\!\cdots\!49\)\( T^{4} \))(\( 1 + \)\(45\!\cdots\!46\)\( T + \)\(13\!\cdots\!43\)\( T^{2} + \)\(12\!\cdots\!12\)\( T^{3} + \)\(13\!\cdots\!51\)\( T^{4} + \)\(41\!\cdots\!54\)\( T^{5} + \)\(86\!\cdots\!93\)\( T^{6} \))
$19$ (\( 1 + \)\(13\!\cdots\!20\)\( T - \)\(46\!\cdots\!02\)\( T^{2} + \)\(18\!\cdots\!80\)\( T^{3} + \)\(17\!\cdots\!01\)\( T^{4} \))(\( 1 + \)\(76\!\cdots\!80\)\( T + \)\(31\!\cdots\!97\)\( T^{2} + \)\(14\!\cdots\!40\)\( T^{3} + \)\(42\!\cdots\!03\)\( T^{4} + \)\(13\!\cdots\!80\)\( T^{5} + \)\(22\!\cdots\!99\)\( T^{6} \))
$23$ (\( 1 - \)\(14\!\cdots\!12\)\( T + \)\(67\!\cdots\!22\)\( T^{2} - \)\(46\!\cdots\!16\)\( T^{3} + \)\(10\!\cdots\!49\)\( T^{4} \))(\( 1 + \)\(24\!\cdots\!68\)\( T + \)\(16\!\cdots\!37\)\( T^{2} + \)\(33\!\cdots\!64\)\( T^{3} + \)\(53\!\cdots\!91\)\( T^{4} + \)\(26\!\cdots\!32\)\( T^{5} + \)\(34\!\cdots\!07\)\( T^{6} \))
$29$ (\( 1 - \)\(63\!\cdots\!40\)\( T + \)\(30\!\cdots\!98\)\( T^{2} - \)\(72\!\cdots\!60\)\( T^{3} + \)\(12\!\cdots\!01\)\( T^{4} \))(\( 1 - \)\(53\!\cdots\!10\)\( T + \)\(22\!\cdots\!47\)\( T^{2} - \)\(59\!\cdots\!80\)\( T^{3} + \)\(25\!\cdots\!03\)\( T^{4} - \)\(68\!\cdots\!10\)\( T^{5} + \)\(14\!\cdots\!49\)\( T^{6} \))
$31$ (\( 1 - \)\(30\!\cdots\!84\)\( T + \)\(17\!\cdots\!66\)\( T^{2} - \)\(26\!\cdots\!84\)\( T^{3} + \)\(75\!\cdots\!01\)\( T^{4} \))(\( 1 - \)\(60\!\cdots\!76\)\( T + \)\(30\!\cdots\!45\)\( T^{2} - \)\(98\!\cdots\!40\)\( T^{3} + \)\(26\!\cdots\!95\)\( T^{4} - \)\(45\!\cdots\!76\)\( T^{5} + \)\(65\!\cdots\!51\)\( T^{6} \))
$37$ (\( 1 + \)\(21\!\cdots\!96\)\( T + \)\(24\!\cdots\!18\)\( T^{2} + \)\(18\!\cdots\!72\)\( T^{3} + \)\(73\!\cdots\!49\)\( T^{4} \))(\( 1 + \)\(16\!\cdots\!06\)\( T + \)\(22\!\cdots\!83\)\( T^{2} + \)\(27\!\cdots\!92\)\( T^{3} + \)\(19\!\cdots\!31\)\( T^{4} + \)\(11\!\cdots\!94\)\( T^{5} + \)\(62\!\cdots\!93\)\( T^{6} \))
$41$ (\( 1 + \)\(93\!\cdots\!76\)\( T + \)\(77\!\cdots\!46\)\( T^{2} + \)\(63\!\cdots\!76\)\( T^{3} + \)\(45\!\cdots\!01\)\( T^{4} \))(\( 1 - \)\(36\!\cdots\!86\)\( T + \)\(20\!\cdots\!35\)\( T^{2} - \)\(46\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!35\)\( T^{4} - \)\(16\!\cdots\!86\)\( T^{5} + \)\(31\!\cdots\!01\)\( T^{6} \))
$43$ (\( 1 - \)\(15\!\cdots\!72\)\( T + \)\(35\!\cdots\!82\)\( T^{2} - \)\(22\!\cdots\!96\)\( T^{3} + \)\(22\!\cdots\!49\)\( T^{4} \))(\( 1 - \)\(19\!\cdots\!92\)\( T + \)\(47\!\cdots\!17\)\( T^{2} - \)\(57\!\cdots\!56\)\( T^{3} + \)\(71\!\cdots\!31\)\( T^{4} - \)\(44\!\cdots\!08\)\( T^{5} + \)\(33\!\cdots\!07\)\( T^{6} \))
$47$ (\( 1 + \)\(30\!\cdots\!96\)\( T + \)\(11\!\cdots\!18\)\( T^{2} + \)\(15\!\cdots\!72\)\( T^{3} + \)\(23\!\cdots\!49\)\( T^{4} \))(\( 1 + \)\(15\!\cdots\!56\)\( T + \)\(10\!\cdots\!33\)\( T^{2} + \)\(81\!\cdots\!92\)\( T^{3} + \)\(49\!\cdots\!31\)\( T^{4} + \)\(36\!\cdots\!44\)\( T^{5} + \)\(11\!\cdots\!43\)\( T^{6} \))
$53$ (\( 1 + \)\(52\!\cdots\!88\)\( T + \)\(17\!\cdots\!22\)\( T^{2} + \)\(62\!\cdots\!84\)\( T^{3} + \)\(14\!\cdots\!49\)\( T^{4} \))(\( 1 - \)\(38\!\cdots\!82\)\( T + \)\(83\!\cdots\!87\)\( T^{2} - \)\(11\!\cdots\!36\)\( T^{3} + \)\(10\!\cdots\!91\)\( T^{4} - \)\(55\!\cdots\!18\)\( T^{5} + \)\(17\!\cdots\!57\)\( T^{6} \))
$59$ (\( 1 + \)\(65\!\cdots\!20\)\( T + \)\(28\!\cdots\!98\)\( T^{2} + \)\(83\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!01\)\( T^{4} \))(\( 1 - \)\(99\!\cdots\!20\)\( T + \)\(64\!\cdots\!97\)\( T^{2} - \)\(27\!\cdots\!60\)\( T^{3} + \)\(82\!\cdots\!03\)\( T^{4} - \)\(16\!\cdots\!20\)\( T^{5} + \)\(20\!\cdots\!99\)\( T^{6} \))
$61$ (\( 1 + \)\(29\!\cdots\!36\)\( T + \)\(43\!\cdots\!26\)\( T^{2} + \)\(32\!\cdots\!36\)\( T^{3} + \)\(12\!\cdots\!01\)\( T^{4} \))(\( 1 - \)\(30\!\cdots\!46\)\( T + \)\(56\!\cdots\!75\)\( T^{2} - \)\(67\!\cdots\!60\)\( T^{3} + \)\(62\!\cdots\!75\)\( T^{4} - \)\(38\!\cdots\!46\)\( T^{5} + \)\(13\!\cdots\!01\)\( T^{6} \))
$67$ (\( 1 + \)\(16\!\cdots\!76\)\( T + \)\(95\!\cdots\!58\)\( T^{2} + \)\(81\!\cdots\!32\)\( T^{3} + \)\(24\!\cdots\!49\)\( T^{4} \))(\( 1 - \)\(17\!\cdots\!64\)\( T + \)\(28\!\cdots\!53\)\( T^{2} + \)\(14\!\cdots\!32\)\( T^{3} + \)\(14\!\cdots\!71\)\( T^{4} - \)\(41\!\cdots\!36\)\( T^{5} + \)\(12\!\cdots\!43\)\( T^{6} \))
$71$ (\( 1 - \)\(24\!\cdots\!64\)\( T + \)\(40\!\cdots\!26\)\( T^{2} - \)\(52\!\cdots\!64\)\( T^{3} + \)\(46\!\cdots\!01\)\( T^{4} \))(\( 1 + \)\(20\!\cdots\!04\)\( T + \)\(75\!\cdots\!25\)\( T^{2} + \)\(89\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!75\)\( T^{4} + \)\(95\!\cdots\!04\)\( T^{5} + \)\(98\!\cdots\!51\)\( T^{6} \))
$73$ (\( 1 - \)\(29\!\cdots\!32\)\( T + \)\(20\!\cdots\!42\)\( T^{2} - \)\(38\!\cdots\!76\)\( T^{3} + \)\(17\!\cdots\!49\)\( T^{4} \))(\( 1 + \)\(14\!\cdots\!98\)\( T + \)\(31\!\cdots\!47\)\( T^{2} + \)\(25\!\cdots\!24\)\( T^{3} + \)\(41\!\cdots\!71\)\( T^{4} + \)\(25\!\cdots\!02\)\( T^{5} + \)\(22\!\cdots\!57\)\( T^{6} \))
$79$ (\( 1 + \)\(72\!\cdots\!00\)\( T + \)\(55\!\cdots\!98\)\( T^{2} + \)\(16\!\cdots\!00\)\( T^{3} + \)\(49\!\cdots\!01\)\( T^{4} \))(\( 1 - \)\(61\!\cdots\!00\)\( T + \)\(52\!\cdots\!97\)\( T^{2} - \)\(36\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!03\)\( T^{4} - \)\(30\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!99\)\( T^{6} \))
$83$ (\( 1 + \)\(21\!\cdots\!48\)\( T + \)\(12\!\cdots\!62\)\( T^{2} + \)\(11\!\cdots\!64\)\( T^{3} + \)\(30\!\cdots\!49\)\( T^{4} \))(\( 1 - \)\(20\!\cdots\!72\)\( T + \)\(84\!\cdots\!57\)\( T^{2} - \)\(58\!\cdots\!16\)\( T^{3} + \)\(46\!\cdots\!51\)\( T^{4} - \)\(62\!\cdots\!28\)\( T^{5} + \)\(16\!\cdots\!07\)\( T^{6} \))
$89$ (\( 1 + \)\(26\!\cdots\!20\)\( T + \)\(10\!\cdots\!98\)\( T^{2} + \)\(13\!\cdots\!80\)\( T^{3} + \)\(26\!\cdots\!01\)\( T^{4} \))(\( 1 + \)\(11\!\cdots\!30\)\( T + \)\(68\!\cdots\!47\)\( T^{2} + \)\(18\!\cdots\!40\)\( T^{3} + \)\(35\!\cdots\!03\)\( T^{4} + \)\(29\!\cdots\!30\)\( T^{5} + \)\(13\!\cdots\!49\)\( T^{6} \))
$97$ (\( 1 + \)\(59\!\cdots\!36\)\( T - \)\(13\!\cdots\!62\)\( T^{2} + \)\(82\!\cdots\!52\)\( T^{3} + \)\(19\!\cdots\!49\)\( T^{4} \))(\( 1 + \)\(74\!\cdots\!46\)\( T + \)\(59\!\cdots\!43\)\( T^{2} + \)\(21\!\cdots\!12\)\( T^{3} + \)\(81\!\cdots\!51\)\( T^{4} + \)\(14\!\cdots\!54\)\( T^{5} + \)\(26\!\cdots\!93\)\( T^{6} \))
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