Properties

Label 3.35.b
Level 3
Weight 35
Character orbit b
Rep. character \(\chi_{3}(2,\cdot)\)
Character field \(\Q\)
Dimension 10
Newform subspaces 1
Sturm bound 11
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 35 \)
Character orbit: \([\chi]\) \(=\) 3.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 3 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(11\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{35}(3, [\chi])\).

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

Trace form

\( 10q + 119369106q^{3} - 55582533344q^{4} - 9250224859872q^{6} - 123569771565772q^{7} + 4787501147541018q^{9} + O(q^{10}) \) \( 10q + 119369106q^{3} - 55582533344q^{4} - 9250224859872q^{6} - 123569771565772q^{7} + 4787501147541018q^{9} - 2611205560422720q^{10} + 1839088058193784224q^{12} + 3639874363106470052q^{13} + 73840351311049043520q^{15} - 458913568062658993664q^{16} + 4458031837811233600320q^{18} - 18128087575667617007644q^{19} + 47155216574801503272132q^{21} - 82944038352793331672640q^{22} - 409540832361157949627904q^{24} - 306157881917435527620230q^{25} - 5188422304885629117270366q^{27} + 1200257625823339935397952q^{28} + 4536084230702466968226240q^{30} - 16210124253435972809624620q^{31} + 126080615773751026110338880q^{33} - 95987113410493892068003584q^{34} - 3778740843166378351675872q^{36} + 1107877531995608327024550788q^{37} - 1478567172848906073011055372q^{39} + 1263325098820483891139005440q^{40} - 1100302459230379163303563200q^{42} - 9626706570211152095714911228q^{43} + 6764773545409610284732206720q^{45} - 27597232473865615892091763584q^{46} - 8343379363160137604055750144q^{48} + 123592722368423574378839845566q^{49} - 154249157901217791265339790592q^{51} + 364968511672502550561914224448q^{52} - 627476296349871988055293152672q^{54} + 613243149395194155294476807040q^{55} - 200956522080425270978267102412q^{57} + 911704545930429232428533728320q^{58} - 5328764870697498499759490872320q^{60} + 3721644688213000805892336114020q^{61} - 17793115140162074693090527345068q^{63} + 31237929890100178456062152876032q^{64} - 56522434211150051720831749496640q^{66} + 64610860820592975458220110976548q^{67} - 75655347603668350171840261441152q^{69} + 173956643092708582155719748059520q^{70} - 251655858465803757687636471720960q^{72} + 219744563327934154882825652170292q^{73} - 496257346175691552817323610505790q^{75} + 755138601297458545349651061598016q^{76} - 1416035973322463045417779737274560q^{78} + 1282143700776175332608732687725076q^{79} - 1752271238163022185686811202609110q^{81} + 2617087107433191584283809414490240q^{82} - 3353754502177451840461431126935232q^{84} + 3064318591304040683100089841123840q^{85} - 1378895999771415686373632115051840q^{87} + 2890193643030846777850875310341120q^{88} - 3749676158586283582801263594857280q^{90} + 3328617839789463835132355588364424q^{91} - 913294458028957034010214182487068q^{93} - 1912224205137436189468854646420224q^{94} + 5756213457708614900907762734014464q^{96} - 11994339580278056229821837667722092q^{97} + 25463971047319347192017533065521280q^{99} + O(q^{100}) \)

Decomposition of \(S_{35}^{\mathrm{new}}(3, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
3.35.b.a \(10\) \(21.968\) \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(0\) \(119369106\) \(0\) \(-1\!\cdots\!72\) \(q+\beta _{1}q^{2}+(11936911+41\beta _{1}-\beta _{2}+\cdots)q^{3}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 58108079248 T^{2} + \)\(20\!\cdots\!92\)\( T^{4} - \)\(57\!\cdots\!28\)\( T^{6} + \)\(13\!\cdots\!12\)\( T^{8} - \)\(24\!\cdots\!08\)\( T^{10} + \)\(38\!\cdots\!72\)\( T^{12} - \)\(49\!\cdots\!08\)\( T^{14} + \)\(52\!\cdots\!72\)\( T^{16} - \)\(44\!\cdots\!08\)\( T^{18} + \)\(22\!\cdots\!76\)\( T^{20} \)
$3$ \( 1 - 119369106 T + 4730741159849109 T^{2} + \)\(17\!\cdots\!40\)\( T^{3} + \)\(22\!\cdots\!78\)\( T^{4} - \)\(49\!\cdots\!48\)\( T^{5} + \)\(37\!\cdots\!82\)\( T^{6} + \)\(48\!\cdots\!40\)\( T^{7} + \)\(21\!\cdots\!81\)\( T^{8} - \)\(92\!\cdots\!26\)\( T^{9} + \)\(12\!\cdots\!49\)\( T^{10} \)
$5$ \( 1 - \)\(27\!\cdots\!10\)\( T^{2} + \)\(33\!\cdots\!25\)\( T^{4} - \)\(25\!\cdots\!00\)\( T^{6} + \)\(16\!\cdots\!50\)\( T^{8} - \)\(98\!\cdots\!00\)\( T^{10} + \)\(55\!\cdots\!50\)\( T^{12} - \)\(29\!\cdots\!00\)\( T^{14} + \)\(12\!\cdots\!25\)\( T^{16} - \)\(36\!\cdots\!50\)\( T^{18} + \)\(44\!\cdots\!25\)\( T^{20} \)
$7$ \( ( 1 + 61784885782886 T + \)\(10\!\cdots\!29\)\( T^{2} + \)\(61\!\cdots\!80\)\( T^{3} + \)\(66\!\cdots\!58\)\( T^{4} - \)\(44\!\cdots\!32\)\( T^{5} + \)\(36\!\cdots\!42\)\( T^{6} + \)\(17\!\cdots\!80\)\( T^{7} + \)\(16\!\cdots\!21\)\( T^{8} + \)\(52\!\cdots\!86\)\( T^{9} + \)\(46\!\cdots\!49\)\( T^{10} )^{2} \)
$11$ \( 1 - \)\(13\!\cdots\!90\)\( T^{2} + \)\(99\!\cdots\!85\)\( T^{4} - \)\(47\!\cdots\!60\)\( T^{6} + \)\(17\!\cdots\!90\)\( T^{8} - \)\(48\!\cdots\!52\)\( T^{10} + \)\(11\!\cdots\!90\)\( T^{12} - \)\(20\!\cdots\!60\)\( T^{14} + \)\(27\!\cdots\!85\)\( T^{16} - \)\(25\!\cdots\!90\)\( T^{18} + \)\(11\!\cdots\!01\)\( T^{20} \)
$13$ \( ( 1 - 1819937181553235026 T + \)\(10\!\cdots\!09\)\( T^{2} + \)\(68\!\cdots\!40\)\( T^{3} - \)\(12\!\cdots\!62\)\( T^{4} + \)\(11\!\cdots\!12\)\( T^{5} - \)\(92\!\cdots\!18\)\( T^{6} + \)\(38\!\cdots\!40\)\( T^{7} + \)\(42\!\cdots\!21\)\( T^{8} - \)\(57\!\cdots\!66\)\( T^{9} + \)\(23\!\cdots\!49\)\( T^{10} )^{2} \)
$17$ \( 1 - \)\(28\!\cdots\!78\)\( T^{2} + \)\(39\!\cdots\!17\)\( T^{4} - \)\(41\!\cdots\!28\)\( T^{6} + \)\(37\!\cdots\!22\)\( T^{8} - \)\(28\!\cdots\!68\)\( T^{10} + \)\(17\!\cdots\!02\)\( T^{12} - \)\(91\!\cdots\!68\)\( T^{14} + \)\(40\!\cdots\!57\)\( T^{16} - \)\(13\!\cdots\!58\)\( T^{18} + \)\(22\!\cdots\!01\)\( T^{20} \)
$19$ \( ( 1 + \)\(90\!\cdots\!22\)\( T + \)\(12\!\cdots\!77\)\( T^{2} + \)\(72\!\cdots\!32\)\( T^{3} + \)\(55\!\cdots\!42\)\( T^{4} + \)\(26\!\cdots\!52\)\( T^{5} + \)\(16\!\cdots\!82\)\( T^{6} + \)\(65\!\cdots\!12\)\( T^{7} + \)\(32\!\cdots\!97\)\( T^{8} + \)\(73\!\cdots\!82\)\( T^{9} + \)\(24\!\cdots\!01\)\( T^{10} )^{2} \)
$23$ \( 1 - \)\(12\!\cdots\!78\)\( T^{2} + \)\(69\!\cdots\!97\)\( T^{4} - \)\(23\!\cdots\!48\)\( T^{6} + \)\(56\!\cdots\!82\)\( T^{8} - \)\(11\!\cdots\!08\)\( T^{10} + \)\(22\!\cdots\!42\)\( T^{12} - \)\(36\!\cdots\!28\)\( T^{14} + \)\(42\!\cdots\!77\)\( T^{16} - \)\(30\!\cdots\!38\)\( T^{18} + \)\(97\!\cdots\!01\)\( T^{20} \)
$29$ \( 1 - \)\(33\!\cdots\!30\)\( T^{2} + \)\(48\!\cdots\!05\)\( T^{4} - \)\(41\!\cdots\!80\)\( T^{6} + \)\(24\!\cdots\!30\)\( T^{8} - \)\(12\!\cdots\!52\)\( T^{10} + \)\(68\!\cdots\!30\)\( T^{12} - \)\(31\!\cdots\!80\)\( T^{14} + \)\(10\!\cdots\!05\)\( T^{16} - \)\(19\!\cdots\!30\)\( T^{18} + \)\(16\!\cdots\!01\)\( T^{20} \)
$31$ \( ( 1 + \)\(81\!\cdots\!10\)\( T + \)\(19\!\cdots\!85\)\( T^{2} + \)\(19\!\cdots\!40\)\( T^{3} + \)\(17\!\cdots\!90\)\( T^{4} + \)\(16\!\cdots\!48\)\( T^{5} + \)\(89\!\cdots\!90\)\( T^{6} + \)\(51\!\cdots\!40\)\( T^{7} + \)\(26\!\cdots\!85\)\( T^{8} + \)\(54\!\cdots\!10\)\( T^{9} + \)\(34\!\cdots\!01\)\( T^{10} )^{2} \)
$37$ \( ( 1 - \)\(55\!\cdots\!94\)\( T + \)\(80\!\cdots\!89\)\( T^{2} - \)\(30\!\cdots\!40\)\( T^{3} + \)\(26\!\cdots\!18\)\( T^{4} - \)\(76\!\cdots\!72\)\( T^{5} + \)\(55\!\cdots\!02\)\( T^{6} - \)\(13\!\cdots\!40\)\( T^{7} + \)\(72\!\cdots\!41\)\( T^{8} - \)\(10\!\cdots\!54\)\( T^{9} + \)\(39\!\cdots\!49\)\( T^{10} )^{2} \)
$41$ \( 1 - \)\(40\!\cdots\!90\)\( T^{2} + \)\(87\!\cdots\!85\)\( T^{4} - \)\(12\!\cdots\!60\)\( T^{6} + \)\(13\!\cdots\!90\)\( T^{8} - \)\(10\!\cdots\!52\)\( T^{10} + \)\(61\!\cdots\!90\)\( T^{12} - \)\(27\!\cdots\!60\)\( T^{14} + \)\(89\!\cdots\!85\)\( T^{16} - \)\(19\!\cdots\!90\)\( T^{18} + \)\(22\!\cdots\!01\)\( T^{20} \)
$43$ \( ( 1 + \)\(48\!\cdots\!14\)\( T + \)\(12\!\cdots\!69\)\( T^{2} + \)\(51\!\cdots\!80\)\( T^{3} + \)\(71\!\cdots\!38\)\( T^{4} + \)\(25\!\cdots\!12\)\( T^{5} + \)\(24\!\cdots\!62\)\( T^{6} + \)\(61\!\cdots\!80\)\( T^{7} + \)\(49\!\cdots\!81\)\( T^{8} + \)\(68\!\cdots\!14\)\( T^{9} + \)\(48\!\cdots\!49\)\( T^{10} )^{2} \)
$47$ \( 1 - \)\(22\!\cdots\!78\)\( T^{2} + \)\(19\!\cdots\!77\)\( T^{4} - \)\(10\!\cdots\!48\)\( T^{6} + \)\(11\!\cdots\!82\)\( T^{8} - \)\(10\!\cdots\!68\)\( T^{10} + \)\(56\!\cdots\!02\)\( T^{12} - \)\(27\!\cdots\!08\)\( T^{14} + \)\(24\!\cdots\!37\)\( T^{16} - \)\(14\!\cdots\!98\)\( T^{18} + \)\(32\!\cdots\!01\)\( T^{20} \)
$53$ \( 1 - \)\(22\!\cdots\!78\)\( T^{2} + \)\(27\!\cdots\!77\)\( T^{4} - \)\(22\!\cdots\!48\)\( T^{6} + \)\(13\!\cdots\!82\)\( T^{8} - \)\(65\!\cdots\!68\)\( T^{10} + \)\(24\!\cdots\!02\)\( T^{12} - \)\(70\!\cdots\!08\)\( T^{14} + \)\(15\!\cdots\!37\)\( T^{16} - \)\(22\!\cdots\!98\)\( T^{18} + \)\(17\!\cdots\!01\)\( T^{20} \)
$59$ \( 1 - \)\(10\!\cdots\!90\)\( T^{2} + \)\(57\!\cdots\!85\)\( T^{4} - \)\(19\!\cdots\!60\)\( T^{6} + \)\(48\!\cdots\!90\)\( T^{8} - \)\(90\!\cdots\!52\)\( T^{10} + \)\(12\!\cdots\!90\)\( T^{12} - \)\(13\!\cdots\!60\)\( T^{14} + \)\(10\!\cdots\!85\)\( T^{16} - \)\(51\!\cdots\!90\)\( T^{18} + \)\(12\!\cdots\!01\)\( T^{20} \)
$61$ \( ( 1 - \)\(18\!\cdots\!10\)\( T + \)\(22\!\cdots\!45\)\( T^{2} - \)\(35\!\cdots\!20\)\( T^{3} + \)\(21\!\cdots\!10\)\( T^{4} - \)\(25\!\cdots\!52\)\( T^{5} + \)\(10\!\cdots\!10\)\( T^{6} - \)\(88\!\cdots\!20\)\( T^{7} + \)\(28\!\cdots\!45\)\( T^{8} - \)\(11\!\cdots\!10\)\( T^{9} + \)\(32\!\cdots\!01\)\( T^{10} )^{2} \)
$67$ \( ( 1 - \)\(32\!\cdots\!74\)\( T + \)\(90\!\cdots\!49\)\( T^{2} - \)\(15\!\cdots\!60\)\( T^{3} + \)\(25\!\cdots\!78\)\( T^{4} - \)\(28\!\cdots\!12\)\( T^{5} + \)\(30\!\cdots\!62\)\( T^{6} - \)\(23\!\cdots\!60\)\( T^{7} + \)\(16\!\cdots\!61\)\( T^{8} - \)\(71\!\cdots\!94\)\( T^{9} + \)\(27\!\cdots\!49\)\( T^{10} )^{2} \)
$71$ \( 1 - \)\(72\!\cdots\!30\)\( T^{2} + \)\(24\!\cdots\!05\)\( T^{4} - \)\(51\!\cdots\!80\)\( T^{6} + \)\(73\!\cdots\!30\)\( T^{8} - \)\(75\!\cdots\!52\)\( T^{10} + \)\(56\!\cdots\!30\)\( T^{12} - \)\(30\!\cdots\!80\)\( T^{14} + \)\(11\!\cdots\!05\)\( T^{16} - \)\(25\!\cdots\!30\)\( T^{18} + \)\(26\!\cdots\!01\)\( T^{20} \)
$73$ \( ( 1 - \)\(10\!\cdots\!46\)\( T + \)\(97\!\cdots\!69\)\( T^{2} - \)\(60\!\cdots\!20\)\( T^{3} + \)\(38\!\cdots\!18\)\( T^{4} - \)\(19\!\cdots\!08\)\( T^{5} + \)\(87\!\cdots\!62\)\( T^{6} - \)\(30\!\cdots\!20\)\( T^{7} + \)\(11\!\cdots\!01\)\( T^{8} - \)\(28\!\cdots\!06\)\( T^{9} + \)\(58\!\cdots\!49\)\( T^{10} )^{2} \)
$79$ \( ( 1 - \)\(64\!\cdots\!38\)\( T + \)\(29\!\cdots\!97\)\( T^{2} - \)\(93\!\cdots\!08\)\( T^{3} + \)\(23\!\cdots\!82\)\( T^{4} - \)\(47\!\cdots\!68\)\( T^{5} + \)\(78\!\cdots\!42\)\( T^{6} - \)\(10\!\cdots\!88\)\( T^{7} + \)\(10\!\cdots\!77\)\( T^{8} - \)\(76\!\cdots\!98\)\( T^{9} + \)\(39\!\cdots\!01\)\( T^{10} )^{2} \)
$83$ \( 1 - \)\(99\!\cdots\!98\)\( T^{2} + \)\(47\!\cdots\!37\)\( T^{4} - \)\(15\!\cdots\!08\)\( T^{6} + \)\(38\!\cdots\!02\)\( T^{8} - \)\(77\!\cdots\!68\)\( T^{10} + \)\(12\!\cdots\!82\)\( T^{12} - \)\(15\!\cdots\!48\)\( T^{14} + \)\(14\!\cdots\!77\)\( T^{16} - \)\(96\!\cdots\!78\)\( T^{18} + \)\(30\!\cdots\!01\)\( T^{20} \)
$89$ \( 1 - \)\(85\!\cdots\!90\)\( T^{2} + \)\(29\!\cdots\!85\)\( T^{4} - \)\(36\!\cdots\!60\)\( T^{6} - \)\(51\!\cdots\!10\)\( T^{8} + \)\(23\!\cdots\!48\)\( T^{10} - \)\(18\!\cdots\!10\)\( T^{12} - \)\(47\!\cdots\!60\)\( T^{14} + \)\(13\!\cdots\!85\)\( T^{16} - \)\(14\!\cdots\!90\)\( T^{18} + \)\(62\!\cdots\!01\)\( T^{20} \)
$97$ \( ( 1 + \)\(59\!\cdots\!46\)\( T + \)\(15\!\cdots\!29\)\( T^{2} + \)\(59\!\cdots\!40\)\( T^{3} + \)\(93\!\cdots\!78\)\( T^{4} + \)\(27\!\cdots\!08\)\( T^{5} + \)\(33\!\cdots\!82\)\( T^{6} + \)\(75\!\cdots\!40\)\( T^{7} + \)\(67\!\cdots\!61\)\( T^{8} + \)\(95\!\cdots\!66\)\( T^{9} + \)\(56\!\cdots\!49\)\( T^{10} )^{2} \)
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