Properties

Label 29.3.c
Level 29
Weight 3
Character orbit c
Rep. character \(\chi_{29}(12,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 8
Newform subspaces 1
Sturm bound 7
Trace bound 0

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Defining parameters

Level: \( N \) = \( 29 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 29.c (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 29 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 1 \)
Sturm bound: \(7\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(29, [\chi])\).

Total New Old
Modular forms 12 12 0
Cusp forms 8 8 0
Eisenstein series 4 4 0

Trace form

\( 8q + 2q^{2} - 2q^{3} - 4q^{7} - 42q^{8} + O(q^{10}) \) \( 8q + 2q^{2} - 2q^{3} - 4q^{7} - 42q^{8} + 6q^{10} - 6q^{11} + 54q^{12} - 40q^{14} - 10q^{15} - 32q^{16} + 12q^{17} + 20q^{18} - 16q^{19} + 108q^{20} - 36q^{21} + 168q^{24} + 104q^{25} - 54q^{26} - 98q^{27} + 128q^{29} - 220q^{30} - 10q^{31} - 106q^{32} - 252q^{36} - 84q^{37} - 90q^{39} + 226q^{40} + 20q^{41} - 190q^{43} + 42q^{44} + 292q^{45} + 12q^{46} + 58q^{47} + 354q^{48} - 72q^{49} - 60q^{50} - 144q^{52} + 252q^{53} + 400q^{54} - 74q^{55} - 192q^{56} + 326q^{58} - 40q^{59} - 258q^{60} - 208q^{61} + 36q^{65} - 414q^{66} - 296q^{68} + 120q^{69} + 44q^{70} - 636q^{72} - 188q^{73} - 64q^{74} - 12q^{75} + 592q^{76} + 180q^{77} + 600q^{78} - 382q^{79} - 124q^{81} + 228q^{82} + 280q^{83} - 124q^{84} + 32q^{85} + 34q^{87} + 20q^{88} - 64q^{89} + 128q^{90} - 460q^{94} - 380q^{95} - 44q^{97} - 66q^{98} + 552q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(29, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
29.3.c.a \(8\) \(0.790\) \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(2\) \(-2\) \(0\) \(-4\) \(q-\beta _{6}q^{2}+(\beta _{2}+\beta _{6})q^{3}+(\beta _{3}+\beta _{4}-\beta _{5}+\cdots)q^{4}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 2 T^{2} + 10 T^{3} - 26 T^{4} + 18 T^{5} + 66 T^{6} - 90 T^{7} + 321 T^{8} - 360 T^{9} + 1056 T^{10} + 1152 T^{11} - 6656 T^{12} + 10240 T^{13} + 8192 T^{14} - 32768 T^{15} + 65536 T^{16} \)
$3$ \( 1 + 2 T + 2 T^{2} + 40 T^{3} + 10 T^{4} - 30 T^{5} + 720 T^{6} + 1242 T^{7} + 3483 T^{8} + 11178 T^{9} + 58320 T^{10} - 21870 T^{11} + 65610 T^{12} + 2361960 T^{13} + 1062882 T^{14} + 9565938 T^{15} + 43046721 T^{16} \)
$5$ \( 1 - 152 T^{2} + 11042 T^{4} - 495504 T^{6} + 14942291 T^{8} - 309690000 T^{10} + 4313281250 T^{12} - 37109375000 T^{14} + 152587890625 T^{16} \)
$7$ \( ( 1 + 2 T + 118 T^{2} + 262 T^{3} + 6782 T^{4} + 12838 T^{5} + 283318 T^{6} + 235298 T^{7} + 5764801 T^{8} )^{2} \)
$11$ \( 1 + 6 T + 18 T^{2} - 984 T^{3} + 5458 T^{4} + 25326 T^{5} + 537840 T^{6} - 21947202 T^{7} - 268164717 T^{8} - 2655611442 T^{9} + 7874515440 T^{10} + 44866553886 T^{11} + 1169970772498 T^{12} - 25522425807384 T^{13} + 56491710780978 T^{14} + 2278499001499446 T^{15} + 45949729863572161 T^{16} \)
$13$ \( 1 - 900 T^{2} + 416090 T^{4} - 121830768 T^{6} + 24605646387 T^{8} - 3479608564848 T^{10} + 339417395700890 T^{12} - 20968276610232900 T^{14} + 665416609183179841 T^{16} \)
$17$ \( 1 - 12 T + 72 T^{2} - 156 T^{3} - 11400 T^{4} - 788916 T^{5} + 10299960 T^{6} - 276428772 T^{7} + 7918453918 T^{8} - 79887915108 T^{9} + 860262959160 T^{10} - 19042514385204 T^{11} - 79523634827400 T^{12} - 314495048470044 T^{13} + 41948801080542792 T^{14} - 2020533918712811148 T^{15} + 48661191875666868481 T^{16} \)
$19$ \( 1 + 16 T + 128 T^{2} - 9968 T^{3} - 125288 T^{4} + 124848 T^{5} + 67714944 T^{6} - 106114512 T^{7} + 873564318 T^{8} - 38307338832 T^{9} + 8824679217024 T^{10} + 5873584151088 T^{11} - 2127836646280808 T^{12} - 61114468457760368 T^{13} + 283304309640468608 T^{14} + 12784106972526145936 T^{15} + \)\(28\!\cdots\!81\)\( T^{16} \)
$23$ \( ( 1 + 1798 T^{2} - 204 T^{3} + 1362542 T^{4} - 107916 T^{5} + 503154118 T^{6} + 78310985281 T^{8} )^{2} \)
$29$ \( 1 - 128 T + 7400 T^{2} - 269120 T^{3} + 8004638 T^{4} - 226329920 T^{5} + 5233879400 T^{6} - 76137385088 T^{7} + 500246412961 T^{8} \)
$31$ \( 1 + 10 T + 50 T^{2} - 18752 T^{3} - 59126 T^{4} + 5169114 T^{5} + 230466192 T^{6} - 14955517326 T^{7} - 982612297341 T^{8} - 14372252150286 T^{9} + 212840368102032 T^{10} + 4587607702508634 T^{11} - 50428035479736566 T^{12} - 15369669637463980352 T^{13} + 39383139189427488050 T^{14} + \)\(75\!\cdots\!10\)\( T^{15} + \)\(72\!\cdots\!81\)\( T^{16} \)
$37$ \( 1 + 84 T + 3528 T^{2} + 71652 T^{3} - 2089980 T^{4} - 123205908 T^{5} - 408842280 T^{6} + 218820380604 T^{7} + 13702947378118 T^{8} + 299565101046876 T^{9} - 766236256327080 T^{10} - 316112651900424372 T^{11} - 7341011809105811580 T^{12} + \)\(34\!\cdots\!48\)\( T^{13} + \)\(23\!\cdots\!68\)\( T^{14} + \)\(75\!\cdots\!76\)\( T^{15} + \)\(12\!\cdots\!41\)\( T^{16} \)
$41$ \( 1 - 20 T + 200 T^{2} - 65948 T^{3} + 2077432 T^{4} + 115380012 T^{5} - 548517288 T^{6} + 112249030692 T^{7} - 9153258882402 T^{8} + 188690620593252 T^{9} - 1549978760256168 T^{10} + 548067084327830892 T^{11} + 16588139188583297272 T^{12} - \)\(88\!\cdots\!48\)\( T^{13} + \)\(45\!\cdots\!00\)\( T^{14} - \)\(75\!\cdots\!20\)\( T^{15} + \)\(63\!\cdots\!41\)\( T^{16} \)
$43$ \( 1 + 190 T + 18050 T^{2} + 1310896 T^{3} + 89172690 T^{4} + 5513705486 T^{5} + 297261149248 T^{6} + 14457675945470 T^{7} + 648656884794643 T^{8} + 26732242823174030 T^{9} + 1016276714310211648 T^{10} + 34854134122268986814 T^{11} + \)\(10\!\cdots\!90\)\( T^{12} + \)\(28\!\cdots\!04\)\( T^{13} + \)\(72\!\cdots\!50\)\( T^{14} + \)\(14\!\cdots\!10\)\( T^{15} + \)\(13\!\cdots\!01\)\( T^{16} \)
$47$ \( 1 - 58 T + 1682 T^{2} - 75904 T^{3} + 8447626 T^{4} - 469112154 T^{5} + 15880306608 T^{6} - 750498418338 T^{7} + 30625787988195 T^{8} - 1657851006108642 T^{9} + 77490830429232048 T^{10} - 5056660921417008666 T^{11} + \)\(20\!\cdots\!86\)\( T^{12} - \)\(39\!\cdots\!96\)\( T^{13} + \)\(19\!\cdots\!62\)\( T^{14} - \)\(14\!\cdots\!02\)\( T^{15} + \)\(56\!\cdots\!21\)\( T^{16} \)
$53$ \( ( 1 - 126 T + 12120 T^{2} - 805920 T^{3} + 48326113 T^{4} - 2263829280 T^{5} + 95632629720 T^{6} - 2792709502254 T^{7} + 62259690411361 T^{8} )^{2} \)
$59$ \( ( 1 + 20 T + 4038 T^{2} - 142432 T^{3} + 879134 T^{4} - 495805792 T^{5} + 48929903718 T^{6} + 843610672820 T^{7} + 146830437604321 T^{8} )^{2} \)
$61$ \( 1 + 208 T + 21632 T^{2} + 2017096 T^{3} + 184647288 T^{4} + 13664724728 T^{5} + 882310746016 T^{6} + 58889137003472 T^{7} + 3818097286447198 T^{8} + 219126478789919312 T^{9} + 12216334301928919456 T^{10} + \)\(70\!\cdots\!08\)\( T^{11} + \)\(35\!\cdots\!28\)\( T^{12} + \)\(14\!\cdots\!96\)\( T^{13} + \)\(57\!\cdots\!72\)\( T^{14} + \)\(20\!\cdots\!28\)\( T^{15} + \)\(36\!\cdots\!61\)\( T^{16} \)
$67$ \( 1 - 11008 T^{2} + 27285916 T^{4} + 201250654976 T^{6} - 1694675724118778 T^{8} + 4055426299750628096 T^{10} + \)\(11\!\cdots\!56\)\( T^{12} - \)\(90\!\cdots\!88\)\( T^{14} + \)\(16\!\cdots\!81\)\( T^{16} \)
$71$ \( 1 - 21324 T^{2} + 250110380 T^{4} - 1967023408740 T^{6} + 11455271890903830 T^{8} - 49985371382433491940 T^{10} + \)\(16\!\cdots\!80\)\( T^{12} - \)\(34\!\cdots\!84\)\( T^{14} + \)\(41\!\cdots\!21\)\( T^{16} \)
$73$ \( 1 + 188 T + 17672 T^{2} + 1559180 T^{3} + 182719716 T^{4} + 18486242500 T^{5} + 1461911905048 T^{6} + 113727191207188 T^{7} + 8685700663672390 T^{8} + 606052201943104852 T^{9} + 41515726600322220568 T^{10} + \)\(27\!\cdots\!00\)\( T^{11} + \)\(14\!\cdots\!96\)\( T^{12} + \)\(67\!\cdots\!20\)\( T^{13} + \)\(40\!\cdots\!12\)\( T^{14} + \)\(22\!\cdots\!92\)\( T^{15} + \)\(65\!\cdots\!61\)\( T^{16} \)
$79$ \( 1 + 382 T + 72962 T^{2} + 10210992 T^{3} + 1201614634 T^{4} + 122408323406 T^{5} + 11219951427216 T^{6} + 967189248125926 T^{7} + 78871519191926211 T^{8} + 6036228097553904166 T^{9} + \)\(43\!\cdots\!96\)\( T^{10} + \)\(29\!\cdots\!26\)\( T^{11} + \)\(18\!\cdots\!74\)\( T^{12} + \)\(96\!\cdots\!92\)\( T^{13} + \)\(43\!\cdots\!42\)\( T^{14} + \)\(14\!\cdots\!42\)\( T^{15} + \)\(23\!\cdots\!21\)\( T^{16} \)
$83$ \( ( 1 - 140 T + 22718 T^{2} - 2108544 T^{3} + 207732422 T^{4} - 14525759616 T^{5} + 1078158136478 T^{6} - 45771652271660 T^{7} + 2252292232139041 T^{8} )^{2} \)
$89$ \( 1 + 64 T + 2048 T^{2} - 960272 T^{3} - 17872712 T^{4} - 244915632 T^{5} + 481989870720 T^{6} - 41121933793344 T^{7} - 1442843088822882 T^{8} - 325726837577077824 T^{9} + 30241124628273083520 T^{10} - \)\(12\!\cdots\!52\)\( T^{11} - \)\(70\!\cdots\!72\)\( T^{12} - \)\(29\!\cdots\!72\)\( T^{13} + \)\(50\!\cdots\!08\)\( T^{14} + \)\(12\!\cdots\!24\)\( T^{15} + \)\(15\!\cdots\!61\)\( T^{16} \)
$97$ \( 1 + 44 T + 968 T^{2} - 45508 T^{3} + 12434104 T^{4} + 5439303060 T^{5} + 228328611000 T^{6} + 46830606041796 T^{7} + 9294107220541086 T^{8} + 440629172247258564 T^{9} + 20213767763558691000 T^{10} + \)\(45\!\cdots\!40\)\( T^{11} + \)\(97\!\cdots\!44\)\( T^{12} - \)\(33\!\cdots\!92\)\( T^{13} + \)\(67\!\cdots\!88\)\( T^{14} + \)\(28\!\cdots\!36\)\( T^{15} + \)\(61\!\cdots\!21\)\( T^{16} \)
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