Properties

Label 3.41.b
Level 3
Weight 41
Character orbit b
Rep. character \(\chi_{3}(2,\cdot)\)
Character field \(\Q\)
Dimension 12
Newform subspaces 1
Sturm bound 13
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 41 \)
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: \(13\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{41}(3, [\chi])\).

Total New Old
Modular forms 14 14 0
Cusp forms 12 12 0
Eisenstein series 2 2 0

Trace form

\( 12q - 372082572q^{3} - 4781373657072q^{4} + 8471862485327952q^{6} - 94341749124766584q^{7} - 23605502262041939220q^{9} + O(q^{10}) \) \( 12q - 372082572q^{3} - 4781373657072q^{4} + 8471862485327952q^{6} - 94341749124766584q^{7} - 23605502262041939220q^{9} - 129384136354706290080q^{10} - 4708039830028985428272q^{12} + 704368845864154901976q^{13} + 320935800901459838631840q^{15} + 6054431850962872672227456q^{16} + 10227710367381761273509920q^{18} + 37616619688417835612151720q^{19} - 179163719456723608834234728q^{21} + 846315150249292772829325920q^{22} - 18213480358094369517645250944q^{24} + 8842265325510151276196474700q^{25} - 70717074113738779393083858252q^{27} + 469404222007499612173589889696q^{28} - 207809636542026559976510503200q^{30} - 914600368399753689553849145016q^{31} + 157344642221562801276041757600q^{33} - 14575908480758621487300381842304q^{34} + 41100171486790863962401658870544q^{36} + 5673106137824212681873457220696q^{37} - 9206461430638842876438079090008q^{39} + 50495897877443363581635475618560q^{40} - 939238658319640212873301303645920q^{42} + 1012654832326802922602871229857576q^{43} + 196109968512215856818385626472000q^{45} - 3082500781506234684371181468390336q^{46} + 6532128822166367998693520969705088q^{48} + 8498166150175390075155260408801892q^{49} + 27565453699222767413826960590504832q^{51} - 57328509042562386918101767400001504q^{52} - 102847607224782098537542351912875792q^{54} - 9637639495259752880102706884884800q^{55} - 40307213604482059940636368892395464q^{57} + 772129436848561910287203884295052320q^{58} - 114385571579949295944121547557720320q^{60} - 494443282701607071691140418439862696q^{61} + 1379291076525018665154441723571240776q^{63} - 6231783699524672226208093043936541696q^{64} + 2652084094607847915810184973553481440q^{66} + 5401062423529881431761320436868275176q^{67} + 7110274695178254111533494284632123328q^{69} + 516103243900107216646679160954379200q^{70} - 27544975046740790141119542770107057920q^{72} + 30000154498324943042431295514883418136q^{73} + 43820294796704764666539599709819928500q^{75} - 215478934792040263381331019050660960352q^{76} + 234946633516471169601955850003339878560q^{78} - 128704557118405295534397879672035334840q^{79} + 473157359902659920298963194247360443532q^{81} - 639159997421964256448917103585883414720q^{82} + 1283288903941350744781821238287455403552q^{84} - 1077142486191382871466217243736209678080q^{85} + 1076668306862385320797179121273520148960q^{87} - 2918662713596496795370587920448329114880q^{88} + 6571793457940570799819318674019816438880q^{90} - 9859184204686878150356379414618151910256q^{91} + 11506410044121210986218054652959643683416q^{93} - 16857264118330945289191940748512432348544q^{94} + 53068562519213731340201280418423192237056q^{96} - 41841010071967618660163746722085070705064q^{97} + 54341076855402233475300783474292013590080q^{99} + O(q^{100}) \)

Decomposition of \(S_{41}^{\mathrm{new}}(3, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
3.41.b.a \(12\) \(30.403\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(-372082572\) \(0\) \(-9\!\cdots\!84\) \(q+\beta _{1}q^{2}+(-31006881-471\beta _{1}+\cdots)q^{3}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4206382938120 T^{2} + \)\(86\!\cdots\!04\)\( T^{4} - \)\(11\!\cdots\!40\)\( T^{6} + \)\(10\!\cdots\!00\)\( T^{8} - \)\(69\!\cdots\!40\)\( T^{10} + \)\(45\!\cdots\!40\)\( T^{12} - \)\(83\!\cdots\!40\)\( T^{14} + \)\(15\!\cdots\!00\)\( T^{16} - \)\(20\!\cdots\!40\)\( T^{18} + \)\(18\!\cdots\!04\)\( T^{20} - \)\(10\!\cdots\!20\)\( T^{22} + \)\(31\!\cdots\!76\)\( T^{24} \)
$3$ \( 1 + 372082572 T + 11871973851214037202 T^{2} + \)\(27\!\cdots\!12\)\( T^{3} - \)\(39\!\cdots\!21\)\( T^{4} + \)\(22\!\cdots\!76\)\( T^{5} - \)\(16\!\cdots\!44\)\( T^{6} + \)\(27\!\cdots\!76\)\( T^{7} - \)\(57\!\cdots\!21\)\( T^{8} + \)\(50\!\cdots\!12\)\( T^{9} + \)\(25\!\cdots\!02\)\( T^{10} + \)\(98\!\cdots\!72\)\( T^{11} + \)\(32\!\cdots\!01\)\( T^{12} \)
$5$ \( 1 - \)\(58\!\cdots\!00\)\( T^{2} + \)\(16\!\cdots\!50\)\( T^{4} - \)\(31\!\cdots\!00\)\( T^{6} + \)\(42\!\cdots\!75\)\( T^{8} - \)\(47\!\cdots\!00\)\( T^{10} + \)\(45\!\cdots\!00\)\( T^{12} - \)\(39\!\cdots\!00\)\( T^{14} + \)\(29\!\cdots\!75\)\( T^{16} - \)\(17\!\cdots\!00\)\( T^{18} + \)\(78\!\cdots\!50\)\( T^{20} - \)\(22\!\cdots\!00\)\( T^{22} + \)\(32\!\cdots\!25\)\( T^{24} \)
$7$ \( ( 1 + 47170874562383292 T + \)\(18\!\cdots\!62\)\( T^{2} + \)\(77\!\cdots\!12\)\( T^{3} + \)\(17\!\cdots\!39\)\( T^{4} + \)\(54\!\cdots\!76\)\( T^{5} + \)\(12\!\cdots\!56\)\( T^{6} + \)\(34\!\cdots\!76\)\( T^{7} + \)\(70\!\cdots\!39\)\( T^{8} + \)\(19\!\cdots\!12\)\( T^{9} + \)\(29\!\cdots\!62\)\( T^{10} + \)\(49\!\cdots\!92\)\( T^{11} + \)\(66\!\cdots\!01\)\( T^{12} )^{2} \)
$11$ \( 1 - \)\(27\!\cdots\!32\)\( T^{2} + \)\(37\!\cdots\!66\)\( T^{4} - \)\(35\!\cdots\!20\)\( T^{6} + \)\(25\!\cdots\!95\)\( T^{8} - \)\(15\!\cdots\!92\)\( T^{10} + \)\(75\!\cdots\!64\)\( T^{12} - \)\(31\!\cdots\!92\)\( T^{14} + \)\(10\!\cdots\!95\)\( T^{16} - \)\(30\!\cdots\!20\)\( T^{18} + \)\(66\!\cdots\!66\)\( T^{20} - \)\(99\!\cdots\!32\)\( T^{22} + \)\(73\!\cdots\!01\)\( T^{24} \)
$13$ \( ( 1 - \)\(35\!\cdots\!88\)\( T + \)\(15\!\cdots\!82\)\( T^{2} + \)\(34\!\cdots\!52\)\( T^{3} + \)\(11\!\cdots\!79\)\( T^{4} + \)\(31\!\cdots\!96\)\( T^{5} + \)\(53\!\cdots\!96\)\( T^{6} + \)\(11\!\cdots\!96\)\( T^{7} + \)\(15\!\cdots\!79\)\( T^{8} + \)\(16\!\cdots\!52\)\( T^{9} + \)\(26\!\cdots\!82\)\( T^{10} - \)\(21\!\cdots\!88\)\( T^{11} + \)\(22\!\cdots\!01\)\( T^{12} )^{2} \)
$17$ \( 1 - \)\(81\!\cdots\!80\)\( T^{2} + \)\(33\!\cdots\!34\)\( T^{4} - \)\(93\!\cdots\!60\)\( T^{6} + \)\(21\!\cdots\!55\)\( T^{8} - \)\(41\!\cdots\!60\)\( T^{10} + \)\(72\!\cdots\!20\)\( T^{12} - \)\(11\!\cdots\!60\)\( T^{14} + \)\(15\!\cdots\!55\)\( T^{16} - \)\(19\!\cdots\!60\)\( T^{18} + \)\(18\!\cdots\!34\)\( T^{20} - \)\(12\!\cdots\!80\)\( T^{22} + \)\(41\!\cdots\!01\)\( T^{24} \)
$19$ \( ( 1 - \)\(18\!\cdots\!60\)\( T + \)\(39\!\cdots\!74\)\( T^{2} + \)\(16\!\cdots\!80\)\( T^{3} + \)\(81\!\cdots\!95\)\( T^{4} + \)\(26\!\cdots\!80\)\( T^{5} + \)\(14\!\cdots\!60\)\( T^{6} + \)\(36\!\cdots\!80\)\( T^{7} + \)\(16\!\cdots\!95\)\( T^{8} + \)\(47\!\cdots\!80\)\( T^{9} + \)\(15\!\cdots\!74\)\( T^{10} - \)\(10\!\cdots\!60\)\( T^{11} + \)\(79\!\cdots\!01\)\( T^{12} )^{2} \)
$23$ \( 1 - \)\(19\!\cdots\!20\)\( T^{2} + \)\(17\!\cdots\!54\)\( T^{4} - \)\(10\!\cdots\!40\)\( T^{6} + \)\(49\!\cdots\!75\)\( T^{8} - \)\(18\!\cdots\!40\)\( T^{10} + \)\(57\!\cdots\!40\)\( T^{12} - \)\(15\!\cdots\!40\)\( T^{14} + \)\(37\!\cdots\!75\)\( T^{16} - \)\(70\!\cdots\!40\)\( T^{18} + \)\(10\!\cdots\!54\)\( T^{20} - \)\(94\!\cdots\!20\)\( T^{22} + \)\(42\!\cdots\!01\)\( T^{24} \)
$29$ \( 1 - \)\(11\!\cdots\!92\)\( T^{2} + \)\(67\!\cdots\!66\)\( T^{4} - \)\(27\!\cdots\!20\)\( T^{6} + \)\(97\!\cdots\!95\)\( T^{8} - \)\(31\!\cdots\!92\)\( T^{10} + \)\(98\!\cdots\!84\)\( T^{12} - \)\(31\!\cdots\!92\)\( T^{14} + \)\(94\!\cdots\!95\)\( T^{16} - \)\(25\!\cdots\!20\)\( T^{18} + \)\(62\!\cdots\!66\)\( T^{20} - \)\(10\!\cdots\!92\)\( T^{22} + \)\(89\!\cdots\!01\)\( T^{24} \)
$31$ \( ( 1 + \)\(45\!\cdots\!08\)\( T + \)\(19\!\cdots\!66\)\( T^{2} + \)\(58\!\cdots\!80\)\( T^{3} + \)\(17\!\cdots\!95\)\( T^{4} + \)\(38\!\cdots\!08\)\( T^{5} + \)\(94\!\cdots\!84\)\( T^{6} + \)\(17\!\cdots\!08\)\( T^{7} + \)\(34\!\cdots\!95\)\( T^{8} + \)\(53\!\cdots\!80\)\( T^{9} + \)\(80\!\cdots\!66\)\( T^{10} + \)\(85\!\cdots\!08\)\( T^{11} + \)\(84\!\cdots\!01\)\( T^{12} )^{2} \)
$37$ \( ( 1 - \)\(28\!\cdots\!48\)\( T + \)\(13\!\cdots\!62\)\( T^{2} - \)\(24\!\cdots\!08\)\( T^{3} + \)\(12\!\cdots\!79\)\( T^{4} - \)\(72\!\cdots\!84\)\( T^{5} + \)\(72\!\cdots\!36\)\( T^{6} - \)\(38\!\cdots\!84\)\( T^{7} + \)\(34\!\cdots\!79\)\( T^{8} - \)\(36\!\cdots\!08\)\( T^{9} + \)\(10\!\cdots\!62\)\( T^{10} - \)\(12\!\cdots\!48\)\( T^{11} + \)\(23\!\cdots\!01\)\( T^{12} )^{2} \)
$41$ \( 1 - \)\(29\!\cdots\!32\)\( T^{2} + \)\(40\!\cdots\!66\)\( T^{4} - \)\(35\!\cdots\!20\)\( T^{6} + \)\(22\!\cdots\!95\)\( T^{8} - \)\(10\!\cdots\!92\)\( T^{10} + \)\(38\!\cdots\!64\)\( T^{12} - \)\(10\!\cdots\!92\)\( T^{14} + \)\(24\!\cdots\!95\)\( T^{16} - \)\(41\!\cdots\!20\)\( T^{18} + \)\(49\!\cdots\!66\)\( T^{20} - \)\(37\!\cdots\!32\)\( T^{22} + \)\(13\!\cdots\!01\)\( T^{24} \)
$43$ \( ( 1 - \)\(50\!\cdots\!88\)\( T + \)\(73\!\cdots\!42\)\( T^{2} - \)\(23\!\cdots\!08\)\( T^{3} + \)\(17\!\cdots\!59\)\( T^{4} - \)\(34\!\cdots\!84\)\( T^{5} + \)\(25\!\cdots\!36\)\( T^{6} - \)\(76\!\cdots\!84\)\( T^{7} + \)\(82\!\cdots\!59\)\( T^{8} - \)\(24\!\cdots\!08\)\( T^{9} + \)\(16\!\cdots\!42\)\( T^{10} - \)\(25\!\cdots\!88\)\( T^{11} + \)\(10\!\cdots\!01\)\( T^{12} )^{2} \)
$47$ \( 1 - \)\(40\!\cdots\!80\)\( T^{2} + \)\(86\!\cdots\!34\)\( T^{4} - \)\(13\!\cdots\!60\)\( T^{6} + \)\(15\!\cdots\!55\)\( T^{8} - \)\(15\!\cdots\!60\)\( T^{10} + \)\(12\!\cdots\!20\)\( T^{12} - \)\(90\!\cdots\!60\)\( T^{14} + \)\(53\!\cdots\!55\)\( T^{16} - \)\(26\!\cdots\!60\)\( T^{18} + \)\(10\!\cdots\!34\)\( T^{20} - \)\(27\!\cdots\!80\)\( T^{22} + \)\(40\!\cdots\!01\)\( T^{24} \)
$53$ \( 1 - \)\(61\!\cdots\!80\)\( T^{2} + \)\(16\!\cdots\!34\)\( T^{4} - \)\(27\!\cdots\!60\)\( T^{6} + \)\(29\!\cdots\!55\)\( T^{8} - \)\(21\!\cdots\!60\)\( T^{10} + \)\(15\!\cdots\!20\)\( T^{12} - \)\(18\!\cdots\!60\)\( T^{14} + \)\(22\!\cdots\!55\)\( T^{16} - \)\(18\!\cdots\!60\)\( T^{18} + \)\(99\!\cdots\!34\)\( T^{20} - \)\(31\!\cdots\!80\)\( T^{22} + \)\(44\!\cdots\!01\)\( T^{24} \)
$59$ \( 1 - \)\(46\!\cdots\!32\)\( T^{2} + \)\(98\!\cdots\!66\)\( T^{4} - \)\(13\!\cdots\!20\)\( T^{6} + \)\(13\!\cdots\!95\)\( T^{8} - \)\(11\!\cdots\!92\)\( T^{10} + \)\(79\!\cdots\!64\)\( T^{12} - \)\(51\!\cdots\!92\)\( T^{14} + \)\(28\!\cdots\!95\)\( T^{16} - \)\(13\!\cdots\!20\)\( T^{18} + \)\(46\!\cdots\!66\)\( T^{20} - \)\(10\!\cdots\!32\)\( T^{22} + \)\(10\!\cdots\!01\)\( T^{24} \)
$61$ \( ( 1 + \)\(24\!\cdots\!48\)\( T + \)\(13\!\cdots\!66\)\( T^{2} + \)\(29\!\cdots\!80\)\( T^{3} + \)\(76\!\cdots\!95\)\( T^{4} + \)\(14\!\cdots\!08\)\( T^{5} + \)\(25\!\cdots\!04\)\( T^{6} + \)\(38\!\cdots\!08\)\( T^{7} + \)\(51\!\cdots\!95\)\( T^{8} + \)\(51\!\cdots\!80\)\( T^{9} + \)\(58\!\cdots\!66\)\( T^{10} + \)\(28\!\cdots\!48\)\( T^{11} + \)\(30\!\cdots\!01\)\( T^{12} )^{2} \)
$67$ \( ( 1 - \)\(27\!\cdots\!88\)\( T + \)\(40\!\cdots\!62\)\( T^{2} - \)\(74\!\cdots\!28\)\( T^{3} + \)\(80\!\cdots\!19\)\( T^{4} - \)\(12\!\cdots\!44\)\( T^{5} + \)\(10\!\cdots\!16\)\( T^{6} - \)\(13\!\cdots\!44\)\( T^{7} + \)\(98\!\cdots\!19\)\( T^{8} - \)\(10\!\cdots\!28\)\( T^{9} + \)\(60\!\cdots\!62\)\( T^{10} - \)\(44\!\cdots\!88\)\( T^{11} + \)\(18\!\cdots\!01\)\( T^{12} )^{2} \)
$71$ \( 1 - \)\(64\!\cdots\!92\)\( T^{2} + \)\(20\!\cdots\!66\)\( T^{4} - \)\(44\!\cdots\!20\)\( T^{6} + \)\(70\!\cdots\!95\)\( T^{8} - \)\(92\!\cdots\!92\)\( T^{10} + \)\(10\!\cdots\!84\)\( T^{12} - \)\(11\!\cdots\!92\)\( T^{14} + \)\(11\!\cdots\!95\)\( T^{16} - \)\(88\!\cdots\!20\)\( T^{18} + \)\(52\!\cdots\!66\)\( T^{20} - \)\(20\!\cdots\!92\)\( T^{22} + \)\(40\!\cdots\!01\)\( T^{24} \)
$73$ \( ( 1 - \)\(15\!\cdots\!68\)\( T + \)\(13\!\cdots\!82\)\( T^{2} - \)\(27\!\cdots\!88\)\( T^{3} + \)\(84\!\cdots\!59\)\( T^{4} - \)\(19\!\cdots\!24\)\( T^{5} + \)\(33\!\cdots\!56\)\( T^{6} - \)\(64\!\cdots\!24\)\( T^{7} + \)\(98\!\cdots\!59\)\( T^{8} - \)\(10\!\cdots\!88\)\( T^{9} + \)\(18\!\cdots\!82\)\( T^{10} - \)\(69\!\cdots\!68\)\( T^{11} + \)\(15\!\cdots\!01\)\( T^{12} )^{2} \)
$79$ \( ( 1 + \)\(64\!\cdots\!20\)\( T + \)\(36\!\cdots\!34\)\( T^{2} + \)\(21\!\cdots\!40\)\( T^{3} + \)\(63\!\cdots\!55\)\( T^{4} + \)\(30\!\cdots\!40\)\( T^{5} + \)\(65\!\cdots\!20\)\( T^{6} + \)\(24\!\cdots\!40\)\( T^{7} + \)\(41\!\cdots\!55\)\( T^{8} + \)\(10\!\cdots\!40\)\( T^{9} + \)\(15\!\cdots\!34\)\( T^{10} + \)\(21\!\cdots\!20\)\( T^{11} + \)\(26\!\cdots\!01\)\( T^{12} )^{2} \)
$83$ \( 1 - \)\(48\!\cdots\!80\)\( T^{2} + \)\(11\!\cdots\!34\)\( T^{4} - \)\(17\!\cdots\!60\)\( T^{6} + \)\(19\!\cdots\!55\)\( T^{8} - \)\(16\!\cdots\!60\)\( T^{10} + \)\(10\!\cdots\!20\)\( T^{12} - \)\(55\!\cdots\!60\)\( T^{14} + \)\(22\!\cdots\!55\)\( T^{16} - \)\(67\!\cdots\!60\)\( T^{18} + \)\(14\!\cdots\!34\)\( T^{20} - \)\(20\!\cdots\!80\)\( T^{22} + \)\(14\!\cdots\!01\)\( T^{24} \)
$89$ \( 1 - \)\(54\!\cdots\!32\)\( T^{2} + \)\(14\!\cdots\!66\)\( T^{4} - \)\(27\!\cdots\!20\)\( T^{6} + \)\(40\!\cdots\!95\)\( T^{8} - \)\(49\!\cdots\!92\)\( T^{10} + \)\(50\!\cdots\!64\)\( T^{12} - \)\(44\!\cdots\!92\)\( T^{14} + \)\(32\!\cdots\!95\)\( T^{16} - \)\(19\!\cdots\!20\)\( T^{18} + \)\(93\!\cdots\!66\)\( T^{20} - \)\(30\!\cdots\!32\)\( T^{22} + \)\(50\!\cdots\!01\)\( T^{24} \)
$97$ \( ( 1 + \)\(20\!\cdots\!32\)\( T + \)\(30\!\cdots\!02\)\( T^{2} + \)\(30\!\cdots\!92\)\( T^{3} + \)\(26\!\cdots\!19\)\( T^{4} + \)\(17\!\cdots\!16\)\( T^{5} + \)\(10\!\cdots\!36\)\( T^{6} + \)\(53\!\cdots\!16\)\( T^{7} + \)\(22\!\cdots\!19\)\( T^{8} + \)\(79\!\cdots\!92\)\( T^{9} + \)\(23\!\cdots\!02\)\( T^{10} + \)\(47\!\cdots\!32\)\( T^{11} + \)\(66\!\cdots\!01\)\( T^{12} )^{2} \)
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