Properties

Label 3.40
Level 3
Weight 40
Dimension 6
Nonzero newspaces 1
Newform subspaces 2
Sturm bound 26
Trace bound 0

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Defining parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 40 \)
Nonzero newspaces: \( 1 \)
Newform subspaces: \( 2 \)
Sturm bound: \(26\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{40}(\Gamma_1(3))\).

Total New Old
Modular forms 14 6 8
Cusp forms 12 6 6
Eisenstein series 2 0 2

Trace form

\( 6q - 573426q^{2} + 1963342879332q^{4} - 44024924515140q^{5} - 1906775943962058q^{6} + 12264571701237576q^{7} - 1079218644984778536q^{8} + 8105110306037952534q^{9} + O(q^{10}) \) \( 6q - 573426q^{2} + 1963342879332q^{4} - 44024924515140q^{5} - 1906775943962058q^{6} + 12264571701237576q^{7} - 1079218644984778536q^{8} + 8105110306037952534q^{9} - 115898380405622295660q^{10} - 571211034516869157936q^{11} + 56320368566559610356q^{12} - 8460174433960988093004q^{13} + 95502602618274444522048q^{14} + 72919149344487096447240q^{15} + 1110814602363343838184720q^{16} + 554019200736986579220540q^{17} - 774613497058353161626914q^{18} - 17641039943993072444609712q^{19} + 63911451821095103202739800q^{20} + 17921138587575253620741144q^{21} - 283854013938352428867146520q^{22} + 159269819766648838812718992q^{23} - 2294182465350240325797938376q^{24} - 3418286073287737864059729750q^{25} + 15675657337740897001941438132q^{26} + 32515480615590040941793982208q^{28} + 26674098393747166485724914876q^{29} - 42028325321349092285108072700q^{30} - 286585477093183719358957705944q^{31} - 739781900001635776455453596064q^{32} + 570798669897838781495058834648q^{33} + 1707041741814522936546659815644q^{34} - 37792972135968456268721598480q^{35} + 2652185100926670238828917604548q^{36} - 2332468211575438555589825991564q^{37} + 3339380699150000394430883930328q^{38} - 22553951747432040455157351602928q^{39} + 5349597399964545985924367423760q^{40} + 33839684955553531563387451258956q^{41} + 38102954817093220350371699725248q^{42} - 256113246515210539209406739772816q^{43} + 170276129693854237111626230690736q^{44} - 59471144901700687412891380727460q^{45} + 542242270259368168393765843724016q^{46} - 1099638359767160247788038598820048q^{47} + 1263153813646161391910378489633616q^{48} + 791974071269664423178924272643830q^{49} + 1911141163510892326097466612696450q^{50} - 1038388582453945486198168360546992q^{51} - 10504983187080960548300430699080040q^{52} - 4748029987074233351934795532216308q^{53} - 2575771559118686957935558350159162q^{54} + 5730931660897976001502479307836480q^{55} + 42446685016142089005851784665740800q^{56} + 4330701126027555176493779476400712q^{57} + 4414009909237468880095804564249956q^{58} + 28075115963403615488942702335747392q^{59} + 3141812294591909997719737950123960q^{60} - 98072442237581081104384450621333500q^{61} - 431729203205837443249638574037796528q^{62} + 16567617749140350294424152157536264q^{63} + 566046408677741660865862203628803648q^{64} + 188896325623857507679470258627558600q^{65} - 95029780989839448100567186495005624q^{66} - 719565492822486165959824430036410560q^{67} + 2386007139747900053821942682324075592q^{68} - 782916874011467083179540377393385312q^{69} - 49677242895639345578166823722929280q^{70} - 2291395561353761673647760223133886768q^{71} - 1457864360322407134596401792145001704q^{72} - 3188188566745146458586434775765463908q^{73} + 6376799948286966204813738390479600388q^{74} - 1195798922274124252482775880117493600q^{75} + 7343466463406418604363467671734069968q^{76} - 3211133558017914689459449427402455008q^{77} - 3293659951234924145990159548860293340q^{78} - 17345327817555538671747155514653772600q^{79} + 26923539487891958920201521619835549280q^{80} + 10948802178840438764154311867139503526q^{81} - 65631228389550365548344068064794318388q^{82} + 128963596434416394623192348983261811088q^{83} - 110990254797967712638412098138832218368q^{84} + 24410279317296195756156922864792870680q^{85} + 168319335493655257016317091149676329704q^{86} - 138564437138487256163732885756373887016q^{87} - 446439663712583439539945964084749262816q^{88} + 318125047136214332290809004755356916828q^{89} - 156561526246452727686689064321199033740q^{90} - 26575612687701136739930280164334402384q^{91} + 830205694689296494460595899957403326496q^{92} - 324180830878217909112421819525046783544q^{93} + 334875738498277147397433632365216002432q^{94} + 1304491927260843040638612796719285594240q^{95} - 1740012976543512186029633319566394225696q^{96} - 2329461700517669779575516407239574299860q^{97} + 3673032743283526674450830836720516013694q^{98} - 771621407130879474841548656451019568304q^{99} + O(q^{100}) \)

Decomposition of \(S_{40}^{\mathrm{new}}(\Gamma_1(3))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
3.40.a \(\chi_{3}(1, \cdot)\) 3.40.a.a 3 1
3.40.a.b 3

Decomposition of \(S_{40}^{\mathrm{old}}(\Gamma_1(3))\) into lower level spaces

\( S_{40}^{\mathrm{old}}(\Gamma_1(3)) \cong \) \(S_{40}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 1107000 T + 934408108032 T^{2} + 888191526050463744 T^{3} + \)\(51\!\cdots\!16\)\( T^{4} + \)\(33\!\cdots\!00\)\( T^{5} + \)\(16\!\cdots\!72\)\( T^{6} \))(\( 1 - 533574 T + 488263000704 T^{2} - 261222362820182016 T^{3} + \)\(26\!\cdots\!52\)\( T^{4} - \)\(16\!\cdots\!56\)\( T^{5} + \)\(16\!\cdots\!72\)\( T^{6} \))
$3$ (\( ( 1 - 1162261467 T )^{3} \))(\( ( 1 + 1162261467 T )^{3} \))
$5$ (\( 1 - 9357049429290 T + \)\(38\!\cdots\!75\)\( T^{2} - \)\(43\!\cdots\!00\)\( T^{3} + \)\(70\!\cdots\!75\)\( T^{4} - \)\(30\!\cdots\!50\)\( T^{5} + \)\(60\!\cdots\!25\)\( T^{6} \))(\( 1 + 53381973944430 T + \)\(47\!\cdots\!75\)\( T^{2} + \)\(18\!\cdots\!00\)\( T^{3} + \)\(86\!\cdots\!75\)\( T^{4} + \)\(17\!\cdots\!50\)\( T^{5} + \)\(60\!\cdots\!25\)\( T^{6} \))
$7$ (\( 1 - 13841884377460104 T + \)\(15\!\cdots\!29\)\( T^{2} - \)\(35\!\cdots\!44\)\( T^{3} + \)\(13\!\cdots\!47\)\( T^{4} - \)\(11\!\cdots\!96\)\( T^{5} + \)\(75\!\cdots\!07\)\( T^{6} \))(\( 1 + 1577312676222528 T + \)\(90\!\cdots\!85\)\( T^{2} + \)\(12\!\cdots\!48\)\( T^{3} + \)\(82\!\cdots\!55\)\( T^{4} + \)\(13\!\cdots\!72\)\( T^{5} + \)\(75\!\cdots\!07\)\( T^{6} \))
$11$ (\( 1 + 40050327611147243796 T + \)\(80\!\cdots\!73\)\( T^{2} + \)\(36\!\cdots\!00\)\( T^{3} + \)\(33\!\cdots\!43\)\( T^{4} + \)\(67\!\cdots\!76\)\( T^{5} + \)\(69\!\cdots\!71\)\( T^{6} \))(\( 1 + \)\(53\!\cdots\!40\)\( T + \)\(21\!\cdots\!21\)\( T^{2} + \)\(49\!\cdots\!16\)\( T^{3} + \)\(89\!\cdots\!11\)\( T^{4} + \)\(89\!\cdots\!40\)\( T^{5} + \)\(69\!\cdots\!71\)\( T^{6} \))
$13$ (\( 1 + \)\(13\!\cdots\!94\)\( T + \)\(78\!\cdots\!31\)\( T^{2} + \)\(32\!\cdots\!08\)\( T^{3} + \)\(21\!\cdots\!87\)\( T^{4} + \)\(10\!\cdots\!26\)\( T^{5} + \)\(21\!\cdots\!33\)\( T^{6} \))(\( 1 - \)\(54\!\cdots\!90\)\( T + \)\(77\!\cdots\!39\)\( T^{2} - \)\(30\!\cdots\!24\)\( T^{3} + \)\(21\!\cdots\!03\)\( T^{4} - \)\(42\!\cdots\!10\)\( T^{5} + \)\(21\!\cdots\!33\)\( T^{6} \))
$17$ (\( 1 + \)\(16\!\cdots\!18\)\( T + \)\(19\!\cdots\!67\)\( T^{2} + \)\(51\!\cdots\!24\)\( T^{3} + \)\(18\!\cdots\!51\)\( T^{4} + \)\(16\!\cdots\!62\)\( T^{5} + \)\(91\!\cdots\!77\)\( T^{6} \))(\( 1 - \)\(72\!\cdots\!58\)\( T + \)\(50\!\cdots\!47\)\( T^{2} - \)\(63\!\cdots\!04\)\( T^{3} + \)\(49\!\cdots\!91\)\( T^{4} - \)\(68\!\cdots\!22\)\( T^{5} + \)\(91\!\cdots\!77\)\( T^{6} \))
$19$ (\( 1 + \)\(69\!\cdots\!88\)\( T + \)\(18\!\cdots\!53\)\( T^{2} + \)\(90\!\cdots\!24\)\( T^{3} + \)\(13\!\cdots\!87\)\( T^{4} + \)\(38\!\cdots\!08\)\( T^{5} + \)\(41\!\cdots\!39\)\( T^{6} \))(\( 1 + \)\(10\!\cdots\!24\)\( T + \)\(14\!\cdots\!97\)\( T^{2} + \)\(13\!\cdots\!92\)\( T^{3} + \)\(10\!\cdots\!63\)\( T^{4} + \)\(59\!\cdots\!84\)\( T^{5} + \)\(41\!\cdots\!39\)\( T^{6} \))
$23$ (\( 1 + \)\(25\!\cdots\!72\)\( T + \)\(25\!\cdots\!61\)\( T^{2} + \)\(38\!\cdots\!28\)\( T^{3} + \)\(32\!\cdots\!07\)\( T^{4} + \)\(42\!\cdots\!68\)\( T^{5} + \)\(20\!\cdots\!03\)\( T^{6} \))(\( 1 - \)\(41\!\cdots\!64\)\( T + \)\(57\!\cdots\!65\)\( T^{2} + \)\(34\!\cdots\!04\)\( T^{3} + \)\(74\!\cdots\!55\)\( T^{4} - \)\(68\!\cdots\!16\)\( T^{5} + \)\(20\!\cdots\!03\)\( T^{6} \))
$29$ (\( 1 + \)\(46\!\cdots\!86\)\( T + \)\(35\!\cdots\!11\)\( T^{2} + \)\(95\!\cdots\!08\)\( T^{3} + \)\(38\!\cdots\!59\)\( T^{4} + \)\(53\!\cdots\!46\)\( T^{5} + \)\(12\!\cdots\!09\)\( T^{6} \))(\( 1 - \)\(72\!\cdots\!62\)\( T + \)\(49\!\cdots\!87\)\( T^{2} - \)\(16\!\cdots\!56\)\( T^{3} + \)\(53\!\cdots\!03\)\( T^{4} - \)\(85\!\cdots\!82\)\( T^{5} + \)\(12\!\cdots\!09\)\( T^{6} \))
$31$ (\( 1 + \)\(28\!\cdots\!88\)\( T + \)\(49\!\cdots\!13\)\( T^{2} + \)\(65\!\cdots\!96\)\( T^{3} + \)\(71\!\cdots\!23\)\( T^{4} + \)\(59\!\cdots\!08\)\( T^{5} + \)\(30\!\cdots\!11\)\( T^{6} \))(\( 1 + \)\(38\!\cdots\!56\)\( T + \)\(20\!\cdots\!97\)\( T^{2} + \)\(76\!\cdots\!52\)\( T^{3} + \)\(29\!\cdots\!87\)\( T^{4} + \)\(81\!\cdots\!96\)\( T^{5} + \)\(30\!\cdots\!11\)\( T^{6} \))
$37$ (\( 1 + \)\(52\!\cdots\!86\)\( T + \)\(36\!\cdots\!59\)\( T^{2} + \)\(12\!\cdots\!56\)\( T^{3} + \)\(52\!\cdots\!07\)\( T^{4} + \)\(10\!\cdots\!94\)\( T^{5} + \)\(30\!\cdots\!17\)\( T^{6} \))(\( 1 - \)\(29\!\cdots\!22\)\( T + \)\(24\!\cdots\!35\)\( T^{2} - \)\(30\!\cdots\!32\)\( T^{3} + \)\(35\!\cdots\!55\)\( T^{4} - \)\(60\!\cdots\!38\)\( T^{5} + \)\(30\!\cdots\!17\)\( T^{6} \))
$41$ (\( 1 - \)\(47\!\cdots\!82\)\( T + \)\(26\!\cdots\!23\)\( T^{2} - \)\(68\!\cdots\!04\)\( T^{3} + \)\(21\!\cdots\!03\)\( T^{4} - \)\(29\!\cdots\!22\)\( T^{5} + \)\(49\!\cdots\!81\)\( T^{6} \))(\( 1 + \)\(13\!\cdots\!26\)\( T + \)\(11\!\cdots\!27\)\( T^{2} + \)\(14\!\cdots\!72\)\( T^{3} + \)\(91\!\cdots\!47\)\( T^{4} + \)\(86\!\cdots\!46\)\( T^{5} + \)\(49\!\cdots\!81\)\( T^{6} \))
$43$ (\( 1 + \)\(14\!\cdots\!88\)\( T + \)\(93\!\cdots\!97\)\( T^{2} + \)\(27\!\cdots\!20\)\( T^{3} + \)\(47\!\cdots\!79\)\( T^{4} + \)\(38\!\cdots\!12\)\( T^{5} + \)\(13\!\cdots\!43\)\( T^{6} \))(\( 1 + \)\(24\!\cdots\!28\)\( T + \)\(33\!\cdots\!77\)\( T^{2} + \)\(28\!\cdots\!60\)\( T^{3} + \)\(16\!\cdots\!39\)\( T^{4} + \)\(62\!\cdots\!72\)\( T^{5} + \)\(13\!\cdots\!43\)\( T^{6} \))
$47$ (\( 1 + \)\(13\!\cdots\!44\)\( T + \)\(28\!\cdots\!93\)\( T^{2} - \)\(30\!\cdots\!60\)\( T^{3} + \)\(45\!\cdots\!19\)\( T^{4} + \)\(34\!\cdots\!16\)\( T^{5} + \)\(43\!\cdots\!87\)\( T^{6} \))(\( 1 + \)\(10\!\cdots\!04\)\( T + \)\(86\!\cdots\!53\)\( T^{2} + \)\(39\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!99\)\( T^{4} + \)\(28\!\cdots\!56\)\( T^{5} + \)\(43\!\cdots\!87\)\( T^{6} \))
$53$ (\( 1 + \)\(11\!\cdots\!42\)\( T + \)\(91\!\cdots\!71\)\( T^{2} + \)\(43\!\cdots\!28\)\( T^{3} + \)\(16\!\cdots\!07\)\( T^{4} + \)\(34\!\cdots\!38\)\( T^{5} + \)\(54\!\cdots\!13\)\( T^{6} \))(\( 1 - \)\(62\!\cdots\!34\)\( T + \)\(47\!\cdots\!55\)\( T^{2} - \)\(22\!\cdots\!36\)\( T^{3} + \)\(84\!\cdots\!35\)\( T^{4} - \)\(19\!\cdots\!26\)\( T^{5} + \)\(54\!\cdots\!13\)\( T^{6} \))
$59$ (\( 1 - \)\(35\!\cdots\!48\)\( T + \)\(19\!\cdots\!33\)\( T^{2} - \)\(81\!\cdots\!04\)\( T^{3} + \)\(22\!\cdots\!87\)\( T^{4} - \)\(47\!\cdots\!08\)\( T^{5} + \)\(15\!\cdots\!19\)\( T^{6} \))(\( 1 + \)\(75\!\cdots\!56\)\( T + \)\(29\!\cdots\!37\)\( T^{2} + \)\(16\!\cdots\!68\)\( T^{3} + \)\(34\!\cdots\!43\)\( T^{4} + \)\(10\!\cdots\!76\)\( T^{5} + \)\(15\!\cdots\!19\)\( T^{6} \))
$61$ (\( 1 + \)\(26\!\cdots\!02\)\( T + \)\(89\!\cdots\!79\)\( T^{2} + \)\(22\!\cdots\!76\)\( T^{3} + \)\(37\!\cdots\!39\)\( T^{4} + \)\(47\!\cdots\!62\)\( T^{5} + \)\(76\!\cdots\!21\)\( T^{6} \))(\( 1 + \)\(71\!\cdots\!98\)\( T + \)\(73\!\cdots\!99\)\( T^{2} + \)\(33\!\cdots\!84\)\( T^{3} + \)\(31\!\cdots\!59\)\( T^{4} + \)\(12\!\cdots\!38\)\( T^{5} + \)\(76\!\cdots\!21\)\( T^{6} \))
$67$ (\( 1 + \)\(20\!\cdots\!88\)\( T + \)\(10\!\cdots\!57\)\( T^{2} + \)\(28\!\cdots\!64\)\( T^{3} + \)\(16\!\cdots\!71\)\( T^{4} + \)\(54\!\cdots\!92\)\( T^{5} + \)\(44\!\cdots\!27\)\( T^{6} \))(\( 1 + \)\(51\!\cdots\!72\)\( T + \)\(32\!\cdots\!37\)\( T^{2} + \)\(82\!\cdots\!56\)\( T^{3} + \)\(53\!\cdots\!11\)\( T^{4} + \)\(14\!\cdots\!48\)\( T^{5} + \)\(44\!\cdots\!27\)\( T^{6} \))
$71$ (\( 1 + \)\(31\!\cdots\!44\)\( T + \)\(74\!\cdots\!05\)\( T^{2} + \)\(10\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!55\)\( T^{4} + \)\(78\!\cdots\!84\)\( T^{5} + \)\(39\!\cdots\!91\)\( T^{6} \))(\( 1 - \)\(84\!\cdots\!76\)\( T + \)\(40\!\cdots\!85\)\( T^{2} - \)\(20\!\cdots\!00\)\( T^{3} + \)\(63\!\cdots\!35\)\( T^{4} - \)\(21\!\cdots\!36\)\( T^{5} + \)\(39\!\cdots\!91\)\( T^{6} \))
$73$ (\( 1 + \)\(38\!\cdots\!22\)\( T + \)\(13\!\cdots\!91\)\( T^{2} + \)\(26\!\cdots\!28\)\( T^{3} + \)\(62\!\cdots\!67\)\( T^{4} + \)\(83\!\cdots\!18\)\( T^{5} + \)\(10\!\cdots\!53\)\( T^{6} \))(\( 1 - \)\(63\!\cdots\!14\)\( T + \)\(59\!\cdots\!15\)\( T^{2} - \)\(10\!\cdots\!96\)\( T^{3} + \)\(27\!\cdots\!55\)\( T^{4} - \)\(13\!\cdots\!66\)\( T^{5} + \)\(10\!\cdots\!53\)\( T^{6} \))
$79$ (\( 1 + \)\(10\!\cdots\!00\)\( T + \)\(24\!\cdots\!57\)\( T^{2} + \)\(27\!\cdots\!00\)\( T^{3} + \)\(24\!\cdots\!83\)\( T^{4} + \)\(10\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!59\)\( T^{6} \))(\( 1 + \)\(16\!\cdots\!00\)\( T + \)\(16\!\cdots\!57\)\( T^{2} + \)\(94\!\cdots\!00\)\( T^{3} + \)\(16\!\cdots\!83\)\( T^{4} + \)\(16\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!59\)\( T^{6} \))
$83$ (\( 1 - \)\(69\!\cdots\!40\)\( T + \)\(32\!\cdots\!33\)\( T^{2} - \)\(10\!\cdots\!88\)\( T^{3} + \)\(22\!\cdots\!51\)\( T^{4} - \)\(33\!\cdots\!60\)\( T^{5} + \)\(34\!\cdots\!23\)\( T^{6} \))(\( 1 - \)\(59\!\cdots\!48\)\( T + \)\(32\!\cdots\!21\)\( T^{2} - \)\(91\!\cdots\!88\)\( T^{3} + \)\(22\!\cdots\!87\)\( T^{4} - \)\(29\!\cdots\!32\)\( T^{5} + \)\(34\!\cdots\!23\)\( T^{6} \))
$89$ (\( 1 - \)\(13\!\cdots\!42\)\( T + \)\(37\!\cdots\!43\)\( T^{2} - \)\(29\!\cdots\!56\)\( T^{3} + \)\(39\!\cdots\!87\)\( T^{4} - \)\(15\!\cdots\!02\)\( T^{5} + \)\(11\!\cdots\!29\)\( T^{6} \))(\( 1 - \)\(18\!\cdots\!86\)\( T + \)\(36\!\cdots\!47\)\( T^{2} - \)\(36\!\cdots\!48\)\( T^{3} + \)\(38\!\cdots\!23\)\( T^{4} - \)\(20\!\cdots\!66\)\( T^{5} + \)\(11\!\cdots\!29\)\( T^{6} \))
$97$ (\( 1 + \)\(13\!\cdots\!18\)\( T + \)\(13\!\cdots\!07\)\( T^{2} + \)\(85\!\cdots\!04\)\( T^{3} + \)\(41\!\cdots\!31\)\( T^{4} + \)\(12\!\cdots\!02\)\( T^{5} + \)\(28\!\cdots\!37\)\( T^{6} \))(\( 1 + \)\(99\!\cdots\!42\)\( T + \)\(88\!\cdots\!87\)\( T^{2} + \)\(45\!\cdots\!16\)\( T^{3} + \)\(26\!\cdots\!71\)\( T^{4} + \)\(92\!\cdots\!38\)\( T^{5} + \)\(28\!\cdots\!37\)\( T^{6} \))
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