Properties

Label 3.37.b
Level 3
Weight 37
Character orbit b
Rep. character \(\chi_{3}(2,\cdot)\)
Character field \(\Q\)
Dimension 11
Newform subspaces 2
Sturm bound 12
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 37 \)
Character orbit: \([\chi]\) \(=\) 3.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 3 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(12\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{37}(3, [\chi])\).

Total New Old
Modular forms 13 13 0
Cusp forms 11 11 0
Eisenstein series 2 2 0

Trace form

\( 11q - 164736261q^{3} - 366480813328q^{4} + 77238909590832q^{6} + 1529656139588998q^{7} + 219422675471657499q^{9} + O(q^{10}) \) \( 11q - 164736261q^{3} - 366480813328q^{4} + 77238909590832q^{6} + 1529656139588998q^{7} + 219422675471657499q^{9} + 976353968400999840q^{10} + 12713663984972326704q^{12} - 26664183895525067642q^{13} + 7346306914204243680q^{15} + 12912742276961748775040q^{16} - 116380061882656121268000q^{18} + 138821577295896839366518q^{19} - 2247956081834234679653850q^{21} + 6375713195756021276887200q^{22} - 18479722715707703137259904q^{24} + 7279508127194401821131675q^{25} + 95231216966669978862527019q^{27} - 391577988767589793638714272q^{28} + 832077359475776822335514400q^{30} - 1733608557857573255379602138q^{31} - 1833168501223628340439879200q^{33} + 11044532239488942328340194176q^{34} - 21948402658393775440817520912q^{36} + 12786450877411433164392216358q^{37} - 9936827303902532409454966170q^{39} + 14850499435375431475622142720q^{40} + 551003488989169331884355647200q^{42} - 515487319179727929010406869802q^{43} - 75992211973807694886478996800q^{45} - 887182867438274648036289139776q^{46} - 4346257047411219928317289501056q^{48} + 9110404204240350847814031518529q^{49} - 2349347159740543712848217109888q^{51} - 13203626071464804773830678836512q^{52} + 35390782846752825865826083751568q^{54} - 30568574550133842572339360395200q^{55} + 96963716675033228271210975522198q^{57} - 112605799975805654976196455444000q^{58} + 19469866547367779716225803605760q^{60} + 66999347224237022458014520403782q^{61} - 20218845986102637586445162314842q^{63} - 304236018528391125615295992251392q^{64} + 1808913396822349027193824888380960q^{66} - 3004194906649294236056896403510282q^{67} + 5226032060124551681525948207907648q^{69} - 11529100185846942980259125797809600q^{70} + 19879821238248816464160255047788800q^{72} - 10390545366461872145216469125969162q^{73} + 21376990929305061495027761443252875q^{75} - 39862084315848733842646101136440224q^{76} + 66756441224105756984056960627624800q^{78} - 71385979381146382550075928433902362q^{79} + 124901631937240804827635146103482731q^{81} - 295731520386077266321378186874491200q^{82} + 565465498335095720662452392509252320q^{84} - 416995278796335191384264774379644160q^{85} + 502161755301600073278417594406864800q^{87} - 1043368725545479380062123692743456000q^{88} + 1148239664725140620121403824942828960q^{90} - 676075249402808126607982688855079700q^{91} + 1136822552118204808049516782296691398q^{93} - 1775468098694354855547362712124166784q^{94} + 2402404280227868861578148783194622976q^{96} - 1638469121614694224091416117067972522q^{97} + 2995711137935834844910948248309577920q^{99} + O(q^{100}) \)

Decomposition of \(S_{37}^{\mathrm{new}}(3, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
3.37.b.a \(1\) \(24.627\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(387420489\) \(0\) \(27\!\cdots\!98\) \(q+3^{18}q^{3}+2^{36}q^{4}+2757049053441698q^{7}+\cdots\)
3.37.b.b \(10\) \(24.627\) \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(0\) \(-552156750\) \(0\) \(-1\!\cdots\!00\) \(q+\beta _{1}q^{2}+(-55215675-69\beta _{1}-\beta _{2}+\cdots)q^{3}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - 262144 T )( 1 + 262144 T ) \))(\( 1 - 125997238648 T^{2} + \)\(13\!\cdots\!92\)\( T^{4} - \)\(10\!\cdots\!08\)\( T^{6} + \)\(91\!\cdots\!52\)\( T^{8} - \)\(60\!\cdots\!28\)\( T^{10} + \)\(43\!\cdots\!92\)\( T^{12} - \)\(22\!\cdots\!28\)\( T^{14} + \)\(14\!\cdots\!12\)\( T^{16} - \)\(62\!\cdots\!88\)\( T^{18} + \)\(23\!\cdots\!76\)\( T^{20} \))
$3$ (\( 1 - 387420489 T \))(\( 1 + 552156750 T + 117774518197952061 T^{2} - \)\(34\!\cdots\!00\)\( T^{3} - \)\(33\!\cdots\!62\)\( T^{4} - \)\(20\!\cdots\!00\)\( T^{5} - \)\(49\!\cdots\!02\)\( T^{6} - \)\(77\!\cdots\!00\)\( T^{7} + \)\(39\!\cdots\!21\)\( T^{8} + \)\(28\!\cdots\!50\)\( T^{9} + \)\(76\!\cdots\!01\)\( T^{10} \))
$5$ (\( ( 1 - 3814697265625 T )( 1 + 3814697265625 T ) \))(\( 1 - \)\(69\!\cdots\!50\)\( T^{2} + \)\(24\!\cdots\!25\)\( T^{4} - \)\(57\!\cdots\!00\)\( T^{6} + \)\(10\!\cdots\!50\)\( T^{8} - \)\(16\!\cdots\!00\)\( T^{10} + \)\(22\!\cdots\!50\)\( T^{12} - \)\(25\!\cdots\!00\)\( T^{14} + \)\(23\!\cdots\!25\)\( T^{16} - \)\(13\!\cdots\!50\)\( T^{18} + \)\(42\!\cdots\!25\)\( T^{20} \))
$7$ (\( 1 - 2757049053441698 T + \)\(26\!\cdots\!01\)\( T^{2} \))(\( ( 1 + 613696456926350 T + \)\(57\!\cdots\!21\)\( T^{2} - \)\(18\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!58\)\( T^{4} - \)\(17\!\cdots\!00\)\( T^{5} + \)\(30\!\cdots\!58\)\( T^{6} - \)\(13\!\cdots\!00\)\( T^{7} + \)\(10\!\cdots\!21\)\( T^{8} + \)\(30\!\cdots\!50\)\( T^{9} + \)\(13\!\cdots\!01\)\( T^{10} )^{2} \))
$11$ (\( ( 1 - 5559917313492231481 T )( 1 + 5559917313492231481 T ) \))(\( 1 - \)\(18\!\cdots\!90\)\( T^{2} + \)\(18\!\cdots\!85\)\( T^{4} - \)\(11\!\cdots\!60\)\( T^{6} + \)\(52\!\cdots\!90\)\( T^{8} - \)\(18\!\cdots\!52\)\( T^{10} + \)\(50\!\cdots\!90\)\( T^{12} - \)\(10\!\cdots\!60\)\( T^{14} + \)\(15\!\cdots\!85\)\( T^{16} - \)\(15\!\cdots\!90\)\( T^{18} + \)\(79\!\cdots\!01\)\( T^{20} \))
$13$ (\( 1 + \)\(17\!\cdots\!42\)\( T + \)\(12\!\cdots\!41\)\( T^{2} \))(\( ( 1 - 73221010691734315450 T + \)\(36\!\cdots\!81\)\( T^{2} - \)\(37\!\cdots\!00\)\( T^{3} + \)\(66\!\cdots\!58\)\( T^{4} - \)\(70\!\cdots\!00\)\( T^{5} + \)\(83\!\cdots\!78\)\( T^{6} - \)\(59\!\cdots\!00\)\( T^{7} + \)\(74\!\cdots\!01\)\( T^{8} - \)\(18\!\cdots\!50\)\( T^{9} + \)\(32\!\cdots\!01\)\( T^{10} )^{2} \))
$17$ (\( ( 1 - \)\(14\!\cdots\!09\)\( T )( 1 + \)\(14\!\cdots\!09\)\( T ) \))(\( 1 - \)\(12\!\cdots\!18\)\( T^{2} + \)\(80\!\cdots\!37\)\( T^{4} - \)\(33\!\cdots\!68\)\( T^{6} + \)\(10\!\cdots\!22\)\( T^{8} - \)\(23\!\cdots\!48\)\( T^{10} + \)\(40\!\cdots\!42\)\( T^{12} - \)\(51\!\cdots\!28\)\( T^{14} + \)\(48\!\cdots\!97\)\( T^{16} - \)\(29\!\cdots\!38\)\( T^{18} + \)\(91\!\cdots\!01\)\( T^{20} \))
$19$ (\( 1 + \)\(11\!\cdots\!18\)\( T + \)\(10\!\cdots\!81\)\( T^{2} \))(\( ( 1 - \)\(12\!\cdots\!18\)\( T + \)\(33\!\cdots\!97\)\( T^{2} - \)\(45\!\cdots\!88\)\( T^{3} + \)\(64\!\cdots\!82\)\( T^{4} - \)\(65\!\cdots\!48\)\( T^{5} + \)\(70\!\cdots\!42\)\( T^{6} - \)\(53\!\cdots\!68\)\( T^{7} + \)\(43\!\cdots\!77\)\( T^{8} - \)\(17\!\cdots\!78\)\( T^{9} + \)\(14\!\cdots\!01\)\( T^{10} )^{2} \))
$23$ (\( ( 1 - \)\(32\!\cdots\!69\)\( T )( 1 + \)\(32\!\cdots\!69\)\( T ) \))(\( 1 - \)\(44\!\cdots\!18\)\( T^{2} + \)\(12\!\cdots\!17\)\( T^{4} - \)\(23\!\cdots\!28\)\( T^{6} + \)\(35\!\cdots\!02\)\( T^{8} - \)\(41\!\cdots\!48\)\( T^{10} + \)\(38\!\cdots\!42\)\( T^{12} - \)\(28\!\cdots\!48\)\( T^{14} + \)\(16\!\cdots\!37\)\( T^{16} - \)\(66\!\cdots\!58\)\( T^{18} + \)\(16\!\cdots\!01\)\( T^{20} \))
$29$ (\( ( 1 - \)\(21\!\cdots\!61\)\( T )( 1 + \)\(21\!\cdots\!61\)\( T ) \))(\( 1 - \)\(15\!\cdots\!30\)\( T^{2} + \)\(13\!\cdots\!05\)\( T^{4} - \)\(91\!\cdots\!80\)\( T^{6} + \)\(51\!\cdots\!30\)\( T^{8} - \)\(24\!\cdots\!52\)\( T^{10} + \)\(10\!\cdots\!30\)\( T^{12} - \)\(35\!\cdots\!80\)\( T^{14} + \)\(10\!\cdots\!05\)\( T^{16} - \)\(22\!\cdots\!30\)\( T^{18} + \)\(29\!\cdots\!01\)\( T^{20} \))
$31$ (\( 1 - \)\(11\!\cdots\!82\)\( T + \)\(48\!\cdots\!81\)\( T^{2} \))(\( ( 1 + \)\(14\!\cdots\!10\)\( T + \)\(20\!\cdots\!85\)\( T^{2} + \)\(19\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!90\)\( T^{4} + \)\(12\!\cdots\!48\)\( T^{5} + \)\(82\!\cdots\!90\)\( T^{6} + \)\(46\!\cdots\!40\)\( T^{7} + \)\(23\!\cdots\!85\)\( T^{8} + \)\(80\!\cdots\!10\)\( T^{9} + \)\(27\!\cdots\!01\)\( T^{10} )^{2} \))
$37$ (\( 1 + \)\(33\!\cdots\!42\)\( T + \)\(28\!\cdots\!41\)\( T^{2} \))(\( ( 1 - \)\(23\!\cdots\!50\)\( T + \)\(98\!\cdots\!81\)\( T^{2} - \)\(21\!\cdots\!00\)\( T^{3} + \)\(45\!\cdots\!58\)\( T^{4} - \)\(82\!\cdots\!00\)\( T^{5} + \)\(13\!\cdots\!78\)\( T^{6} - \)\(17\!\cdots\!00\)\( T^{7} + \)\(22\!\cdots\!01\)\( T^{8} - \)\(15\!\cdots\!50\)\( T^{9} + \)\(18\!\cdots\!01\)\( T^{10} )^{2} \))
$41$ (\( ( 1 - \)\(10\!\cdots\!21\)\( T )( 1 + \)\(10\!\cdots\!21\)\( T ) \))(\( 1 - \)\(23\!\cdots\!90\)\( T^{2} + \)\(10\!\cdots\!85\)\( T^{4} + \)\(58\!\cdots\!40\)\( T^{6} + \)\(19\!\cdots\!90\)\( T^{8} - \)\(48\!\cdots\!52\)\( T^{10} + \)\(25\!\cdots\!90\)\( T^{12} + \)\(10\!\cdots\!40\)\( T^{14} + \)\(24\!\cdots\!85\)\( T^{16} - \)\(71\!\cdots\!90\)\( T^{18} + \)\(40\!\cdots\!01\)\( T^{20} \))
$43$ (\( 1 - \)\(28\!\cdots\!98\)\( T + \)\(63\!\cdots\!01\)\( T^{2} \))(\( ( 1 + \)\(27\!\cdots\!50\)\( T + \)\(27\!\cdots\!81\)\( T^{2} + \)\(57\!\cdots\!00\)\( T^{3} + \)\(31\!\cdots\!38\)\( T^{4} + \)\(50\!\cdots\!00\)\( T^{5} + \)\(20\!\cdots\!38\)\( T^{6} + \)\(23\!\cdots\!00\)\( T^{7} + \)\(71\!\cdots\!81\)\( T^{8} + \)\(45\!\cdots\!50\)\( T^{9} + \)\(10\!\cdots\!01\)\( T^{10} )^{2} \))
$47$ (\( ( 1 - \)\(12\!\cdots\!89\)\( T )( 1 + \)\(12\!\cdots\!89\)\( T ) \))(\( 1 - \)\(90\!\cdots\!18\)\( T^{2} + \)\(34\!\cdots\!97\)\( T^{4} - \)\(76\!\cdots\!08\)\( T^{6} + \)\(10\!\cdots\!42\)\( T^{8} - \)\(14\!\cdots\!28\)\( T^{10} + \)\(26\!\cdots\!22\)\( T^{12} - \)\(46\!\cdots\!48\)\( T^{14} + \)\(52\!\cdots\!37\)\( T^{16} - \)\(33\!\cdots\!98\)\( T^{18} + \)\(90\!\cdots\!01\)\( T^{20} \))
$53$ (\( ( 1 - \)\(10\!\cdots\!89\)\( T )( 1 + \)\(10\!\cdots\!89\)\( T ) \))(\( 1 - \)\(62\!\cdots\!18\)\( T^{2} + \)\(19\!\cdots\!97\)\( T^{4} - \)\(38\!\cdots\!08\)\( T^{6} + \)\(57\!\cdots\!42\)\( T^{8} - \)\(72\!\cdots\!28\)\( T^{10} + \)\(81\!\cdots\!22\)\( T^{12} - \)\(76\!\cdots\!48\)\( T^{14} + \)\(53\!\cdots\!37\)\( T^{16} - \)\(24\!\cdots\!98\)\( T^{18} + \)\(54\!\cdots\!01\)\( T^{20} \))
$59$ (\( ( 1 - \)\(75\!\cdots\!21\)\( T )( 1 + \)\(75\!\cdots\!21\)\( T ) \))(\( 1 - \)\(26\!\cdots\!90\)\( T^{2} + \)\(39\!\cdots\!85\)\( T^{4} - \)\(40\!\cdots\!60\)\( T^{6} + \)\(31\!\cdots\!90\)\( T^{8} - \)\(19\!\cdots\!52\)\( T^{10} + \)\(10\!\cdots\!90\)\( T^{12} - \)\(40\!\cdots\!60\)\( T^{14} + \)\(12\!\cdots\!85\)\( T^{16} - \)\(27\!\cdots\!90\)\( T^{18} + \)\(32\!\cdots\!01\)\( T^{20} \))
$61$ (\( 1 + \)\(18\!\cdots\!38\)\( T + \)\(18\!\cdots\!61\)\( T^{2} \))(\( ( 1 - \)\(12\!\cdots\!10\)\( T + \)\(40\!\cdots\!45\)\( T^{2} - \)\(51\!\cdots\!20\)\( T^{3} + \)\(12\!\cdots\!10\)\( T^{4} - \)\(12\!\cdots\!52\)\( T^{5} + \)\(23\!\cdots\!10\)\( T^{6} - \)\(17\!\cdots\!20\)\( T^{7} + \)\(26\!\cdots\!45\)\( T^{8} - \)\(15\!\cdots\!10\)\( T^{9} + \)\(22\!\cdots\!01\)\( T^{10} )^{2} \))
$67$ (\( 1 - \)\(26\!\cdots\!18\)\( T + \)\(54\!\cdots\!81\)\( T^{2} \))(\( ( 1 + \)\(16\!\cdots\!50\)\( T + \)\(32\!\cdots\!41\)\( T^{2} + \)\(34\!\cdots\!00\)\( T^{3} + \)\(38\!\cdots\!58\)\( T^{4} + \)\(27\!\cdots\!00\)\( T^{5} + \)\(20\!\cdots\!98\)\( T^{6} + \)\(10\!\cdots\!00\)\( T^{7} + \)\(53\!\cdots\!81\)\( T^{8} + \)\(14\!\cdots\!50\)\( T^{9} + \)\(49\!\cdots\!01\)\( T^{10} )^{2} \))
$71$ (\( ( 1 - \)\(21\!\cdots\!61\)\( T )( 1 + \)\(21\!\cdots\!61\)\( T ) \))(\( 1 - \)\(31\!\cdots\!30\)\( T^{2} + \)\(49\!\cdots\!05\)\( T^{4} - \)\(48\!\cdots\!80\)\( T^{6} + \)\(34\!\cdots\!30\)\( T^{8} - \)\(17\!\cdots\!52\)\( T^{10} + \)\(66\!\cdots\!30\)\( T^{12} - \)\(18\!\cdots\!80\)\( T^{14} + \)\(36\!\cdots\!05\)\( T^{16} - \)\(46\!\cdots\!30\)\( T^{18} + \)\(28\!\cdots\!01\)\( T^{20} \))
$73$ (\( 1 + \)\(60\!\cdots\!62\)\( T + \)\(12\!\cdots\!61\)\( T^{2} \))(\( ( 1 + \)\(21\!\cdots\!50\)\( T + \)\(41\!\cdots\!41\)\( T^{2} + \)\(83\!\cdots\!00\)\( T^{3} + \)\(84\!\cdots\!98\)\( T^{4} + \)\(13\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!78\)\( T^{6} + \)\(12\!\cdots\!00\)\( T^{7} + \)\(72\!\cdots\!21\)\( T^{8} + \)\(45\!\cdots\!50\)\( T^{9} + \)\(25\!\cdots\!01\)\( T^{10} )^{2} \))
$79$ (\( 1 + \)\(89\!\cdots\!78\)\( T + \)\(20\!\cdots\!21\)\( T^{2} \))(\( ( 1 + \)\(31\!\cdots\!42\)\( T + \)\(86\!\cdots\!57\)\( T^{2} + \)\(16\!\cdots\!12\)\( T^{3} + \)\(29\!\cdots\!62\)\( T^{4} + \)\(42\!\cdots\!52\)\( T^{5} + \)\(60\!\cdots\!02\)\( T^{6} + \)\(70\!\cdots\!92\)\( T^{7} + \)\(75\!\cdots\!77\)\( T^{8} + \)\(56\!\cdots\!02\)\( T^{9} + \)\(37\!\cdots\!01\)\( T^{10} )^{2} \))
$83$ (\( ( 1 - \)\(34\!\cdots\!09\)\( T )( 1 + \)\(34\!\cdots\!09\)\( T ) \))(\( 1 - \)\(70\!\cdots\!98\)\( T^{2} + \)\(26\!\cdots\!37\)\( T^{4} - \)\(64\!\cdots\!48\)\( T^{6} + \)\(11\!\cdots\!22\)\( T^{8} - \)\(16\!\cdots\!28\)\( T^{10} + \)\(17\!\cdots\!42\)\( T^{12} - \)\(14\!\cdots\!08\)\( T^{14} + \)\(86\!\cdots\!97\)\( T^{16} - \)\(34\!\cdots\!18\)\( T^{18} + \)\(73\!\cdots\!01\)\( T^{20} \))
$89$ (\( ( 1 - \)\(12\!\cdots\!81\)\( T )( 1 + \)\(12\!\cdots\!81\)\( T ) \))(\( 1 - \)\(39\!\cdots\!90\)\( T^{2} + \)\(94\!\cdots\!85\)\( T^{4} - \)\(22\!\cdots\!60\)\( T^{6} + \)\(40\!\cdots\!90\)\( T^{8} - \)\(61\!\cdots\!52\)\( T^{10} + \)\(92\!\cdots\!90\)\( T^{12} - \)\(11\!\cdots\!60\)\( T^{14} + \)\(11\!\cdots\!85\)\( T^{16} - \)\(10\!\cdots\!90\)\( T^{18} + \)\(60\!\cdots\!01\)\( T^{20} \))
$97$ (\( 1 + \)\(11\!\cdots\!22\)\( T + \)\(33\!\cdots\!21\)\( T^{2} \))(\( ( 1 + \)\(25\!\cdots\!50\)\( T + \)\(11\!\cdots\!81\)\( T^{2} + \)\(43\!\cdots\!00\)\( T^{3} + \)\(60\!\cdots\!98\)\( T^{4} + \)\(23\!\cdots\!00\)\( T^{5} + \)\(20\!\cdots\!58\)\( T^{6} + \)\(48\!\cdots\!00\)\( T^{7} + \)\(42\!\cdots\!41\)\( T^{8} + \)\(31\!\cdots\!50\)\( T^{9} + \)\(41\!\cdots\!01\)\( T^{10} )^{2} \))
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