Properties

Label 3.36.a
Level 3
Weight 36
Character orbit a
Rep. character \(\chi_{3}(1,\cdot)\)
Character field \(\Q\)
Dimension 5
Newform subspaces 2
Sturm bound 12
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 36 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)
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 \(M_{36}(\Gamma_0(3))\).

Total New Old
Modular forms 13 5 8
Cusp forms 11 5 6
Eisenstein series 2 0 2

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)Dim.
\(+\)\(3\)
\(-\)\(2\)

Trace form

\( 5q - 148242q^{2} - 129140163q^{3} + 89631692804q^{4} + 1434896738670q^{5} + 3411624826134q^{6} - 713274934414880q^{7} - 25884103158515976q^{8} + 83385908498332845q^{9} + O(q^{10}) \) \( 5q - 148242q^{2} - 129140163q^{3} + 89631692804q^{4} + 1434896738670q^{5} + 3411624826134q^{6} - 713274934414880q^{7} - 25884103158515976q^{8} + 83385908498332845q^{9} - 1237325895294970860q^{10} + 1951912322902904172q^{11} + 3183845728329041652q^{12} + 80092932089464218334q^{13} - 163661807843817841248q^{14} - 529791801950135340450q^{15} - 1624934440130383112176q^{16} + 8915090677778688661674q^{17} - 2472258769521971521698q^{18} + 12752449246032371443828q^{19} + 59516639483894300092440q^{20} - 218214671980101706154304q^{21} - 519321466626604832071512q^{22} + 2300799335026855019769480q^{23} + 1482811913733911374506264q^{24} + 3849405855791926758755075q^{25} - 8499712129693383091578540q^{26} - 2153693963075557766310747q^{27} + 53180296064407219444765120q^{28} + 30909672135818816060166q^{29} + 131494821896832682795598340q^{30} - 122958169184937165760781096q^{31} + 355435458612944113091904480q^{32} - 632890114361808194928946068q^{33} - 582179574327733019775453732q^{34} - 379897165326179675561125440q^{35} + 1494804026941004501836669476q^{36} - 3436963830781587858776868170q^{37} + 4326394742024006333308026648q^{38} - 2593457939840324435573210514q^{39} - 11985118729325414773626803760q^{40} - 53189554457094285710502903246q^{41} + 35138898632596499814877823520q^{42} + 56844975969227231301683699692q^{43} + 179114156554011793095382044720q^{44} + 23930033631058567284528523230q^{45} - 341571880622250261376296674256q^{46} + 240328948723045336711504850496q^{47} - 366596375750148891605445878832q^{48} + 802789826455702254915243383757q^{49} - 356186228865319312804012540350q^{50} + 446007778207236743219441959770q^{51} - 441714863191873022725669400936q^{52} - 420542982748345546138778219250q^{53} + 56896287116530085070397314246q^{54} - 4719096094374741745614624676920q^{55} - 5090932230417875092882709930880q^{56} - 5790135177913208297984503593228q^{57} + 16528504324592073402015579466596q^{58} + 19526992971932091149542270994412q^{59} - 15166049261937686536170748128840q^{60} - 27738567850968050765013234760322q^{61} + 86071886939891705608614484052400q^{62} - 11895415683054708969039112146720q^{63} - 85342070906504957985351659515840q^{64} - 3442292245616854221978242984940q^{65} - 35363475315490252092136720304760q^{66} + 185371889425048703381547073947124q^{67} + 419308640906343280137197033130504q^{68} - 169582772745083819463841300198968q^{69} - 985792143822122845734777417407040q^{70} + 413207616969898733271065798331000q^{71} - 431673891507484271704592059606344q^{72} - 722545434631882110139874803796030q^{73} + 1950543872560668471816255070861092q^{74} - 1723798507030118942206480733467125q^{75} - 820039376334225227691266295365552q^{76} + 6129758858469887216803084728453120q^{77} - 4316404119544128353848272405826236q^{78} - 3729572519079692046656359737107480q^{79} + 8799280713314725844148275888738400q^{80} + 1390641947218467556286428881158805q^{81} - 4208079896200069759137693606108660q^{82} + 1077932265473174157200621650549236q^{83} - 16011501370423823600965253673997632q^{84} - 1335321643323156859792325991910500q^{85} + 50076000751340258784704784063203112q^{86} - 20381543372820767573178418053677706q^{87} - 27916091705317544050926332640599136q^{88} + 35450883863324503243217633720327058q^{89} - 20635108777536841317831164571179340q^{90} - 61416375185455464480097922316938176q^{91} + 17938960440142164513430957923038880q^{92} - 15466362475906764331309710113949000q^{93} - 86800543980059079782841270078748224q^{94} + 155074682399177631629171912512592760q^{95} - 8888826680820510399896890489820064q^{96} + 212629410612226095930593684860939594q^{97} - 144633623466817135851886200566855394q^{98} + 32552396470869976256394214159025868q^{99} + O(q^{100}) \)

Decomposition of \(S_{36}^{\mathrm{new}}(\Gamma_0(3))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3
3.36.a.a \(2\) \(23.279\) \(\Q(\sqrt{2196841}) \) None \(-60912\) \(258280326\) \(-1\!\cdots\!40\) \(-1\!\cdots\!44\) \(-\) \(q+(-30456-\beta )q^{2}+3^{17}q^{3}+(28571469952+\cdots)q^{4}+\cdots\)
3.36.a.b \(3\) \(23.279\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-87330\) \(-387420489\) \(27\!\cdots\!10\) \(48\!\cdots\!64\) \(+\) \(q+(-29110+\beta _{1})q^{2}-3^{17}q^{3}+(10829584300+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{36}^{\mathrm{old}}(\Gamma_0(3))\) into lower level spaces

\( S_{36}^{\mathrm{old}}(\Gamma_0(3)) \cong \) \(S_{36}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 60912 T + 7643404288 T^{2} + 2092920383471616 T^{3} + \)\(11\!\cdots\!24\)\( T^{4} \))(\( 1 + 87330 T + 39108495552 T^{2} + 8420840235761664 T^{3} + \)\(13\!\cdots\!36\)\( T^{4} + \)\(10\!\cdots\!20\)\( T^{5} + \)\(40\!\cdots\!32\)\( T^{6} \))
$3$ (\( ( 1 - 129140163 T )^{2} \))(\( ( 1 + 129140163 T )^{3} \))
$5$ (\( 1 + 1333779496740 T + \)\(61\!\cdots\!50\)\( T^{2} + \)\(38\!\cdots\!00\)\( T^{3} + \)\(84\!\cdots\!25\)\( T^{4} \))(\( 1 - 2768676235410 T + \)\(38\!\cdots\!75\)\( T^{2} - \)\(23\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!75\)\( T^{4} - \)\(23\!\cdots\!50\)\( T^{5} + \)\(24\!\cdots\!25\)\( T^{6} \))
$7$ (\( 1 + 1201512782952944 T + \)\(11\!\cdots\!86\)\( T^{2} + \)\(45\!\cdots\!92\)\( T^{3} + \)\(14\!\cdots\!49\)\( T^{4} \))(\( 1 - 488237848538064 T + \)\(26\!\cdots\!09\)\( T^{2} + \)\(84\!\cdots\!96\)\( T^{3} + \)\(10\!\cdots\!87\)\( T^{4} - \)\(70\!\cdots\!36\)\( T^{5} + \)\(54\!\cdots\!07\)\( T^{6} \))
$11$ (\( 1 + 1474443852221320632 T + \)\(53\!\cdots\!34\)\( T^{2} + \)\(41\!\cdots\!32\)\( T^{3} + \)\(78\!\cdots\!01\)\( T^{4} \))(\( 1 - 3426356175124224804 T + \)\(53\!\cdots\!33\)\( T^{2} - \)\(70\!\cdots\!60\)\( T^{3} + \)\(14\!\cdots\!83\)\( T^{4} - \)\(27\!\cdots\!04\)\( T^{5} + \)\(22\!\cdots\!51\)\( T^{6} \))
$13$ (\( 1 - 30005213658205678828 T + \)\(55\!\cdots\!06\)\( T^{2} - \)\(29\!\cdots\!96\)\( T^{3} + \)\(94\!\cdots\!49\)\( T^{4} \))(\( 1 - 50087718431258539506 T + \)\(27\!\cdots\!11\)\( T^{2} - \)\(77\!\cdots\!12\)\( T^{3} + \)\(27\!\cdots\!27\)\( T^{4} - \)\(47\!\cdots\!94\)\( T^{5} + \)\(92\!\cdots\!93\)\( T^{6} \))
$17$ (\( 1 - \)\(61\!\cdots\!32\)\( T + \)\(31\!\cdots\!42\)\( T^{2} - \)\(71\!\cdots\!76\)\( T^{3} + \)\(13\!\cdots\!49\)\( T^{4} \))(\( 1 - \)\(27\!\cdots\!42\)\( T + \)\(30\!\cdots\!67\)\( T^{2} - \)\(59\!\cdots\!56\)\( T^{3} + \)\(35\!\cdots\!31\)\( T^{4} - \)\(36\!\cdots\!58\)\( T^{5} + \)\(15\!\cdots\!57\)\( T^{6} \))
$19$ (\( 1 + \)\(16\!\cdots\!64\)\( T + \)\(56\!\cdots\!78\)\( T^{2} + \)\(91\!\cdots\!36\)\( T^{3} + \)\(32\!\cdots\!01\)\( T^{4} \))(\( 1 - \)\(28\!\cdots\!92\)\( T + \)\(13\!\cdots\!93\)\( T^{2} - \)\(22\!\cdots\!96\)\( T^{3} + \)\(78\!\cdots\!07\)\( T^{4} - \)\(93\!\cdots\!92\)\( T^{5} + \)\(18\!\cdots\!99\)\( T^{6} \))
$23$ (\( 1 - \)\(49\!\cdots\!72\)\( T + \)\(39\!\cdots\!14\)\( T^{2} - \)\(22\!\cdots\!04\)\( T^{3} + \)\(20\!\cdots\!49\)\( T^{4} \))(\( 1 - \)\(18\!\cdots\!08\)\( T + \)\(22\!\cdots\!61\)\( T^{2} - \)\(17\!\cdots\!12\)\( T^{3} + \)\(10\!\cdots\!27\)\( T^{4} - \)\(37\!\cdots\!92\)\( T^{5} + \)\(95\!\cdots\!43\)\( T^{6} \))
$29$ (\( 1 + \)\(78\!\cdots\!48\)\( T + \)\(45\!\cdots\!38\)\( T^{2} + \)\(12\!\cdots\!52\)\( T^{3} + \)\(23\!\cdots\!01\)\( T^{4} \))(\( 1 - \)\(78\!\cdots\!14\)\( T + \)\(56\!\cdots\!11\)\( T^{2} - \)\(22\!\cdots\!52\)\( T^{3} + \)\(85\!\cdots\!39\)\( T^{4} - \)\(18\!\cdots\!14\)\( T^{5} + \)\(35\!\cdots\!49\)\( T^{6} \))
$31$ (\( 1 + \)\(12\!\cdots\!48\)\( T + \)\(33\!\cdots\!02\)\( T^{2} + \)\(19\!\cdots\!48\)\( T^{3} + \)\(24\!\cdots\!01\)\( T^{4} \))(\( 1 + \)\(15\!\cdots\!48\)\( T + \)\(33\!\cdots\!53\)\( T^{2} - \)\(11\!\cdots\!04\)\( T^{3} + \)\(52\!\cdots\!03\)\( T^{4} + \)\(39\!\cdots\!48\)\( T^{5} + \)\(39\!\cdots\!51\)\( T^{6} \))
$37$ (\( 1 - \)\(16\!\cdots\!16\)\( T + \)\(20\!\cdots\!06\)\( T^{2} - \)\(12\!\cdots\!88\)\( T^{3} + \)\(59\!\cdots\!49\)\( T^{4} \))(\( 1 + \)\(51\!\cdots\!86\)\( T + \)\(29\!\cdots\!19\)\( T^{2} + \)\(78\!\cdots\!96\)\( T^{3} + \)\(22\!\cdots\!67\)\( T^{4} + \)\(30\!\cdots\!14\)\( T^{5} + \)\(45\!\cdots\!57\)\( T^{6} \))
$41$ (\( 1 + \)\(32\!\cdots\!88\)\( T + \)\(68\!\cdots\!22\)\( T^{2} + \)\(92\!\cdots\!88\)\( T^{3} + \)\(78\!\cdots\!01\)\( T^{4} \))(\( 1 + \)\(20\!\cdots\!58\)\( T + \)\(97\!\cdots\!83\)\( T^{2} + \)\(11\!\cdots\!16\)\( T^{3} + \)\(27\!\cdots\!83\)\( T^{4} + \)\(15\!\cdots\!58\)\( T^{5} + \)\(21\!\cdots\!01\)\( T^{6} \))
$43$ (\( 1 - \)\(10\!\cdots\!20\)\( T - \)\(10\!\cdots\!10\)\( T^{2} - \)\(15\!\cdots\!40\)\( T^{3} + \)\(22\!\cdots\!49\)\( T^{4} \))(\( 1 - \)\(46\!\cdots\!72\)\( T + \)\(36\!\cdots\!57\)\( T^{2} - \)\(11\!\cdots\!00\)\( T^{3} + \)\(53\!\cdots\!99\)\( T^{4} - \)\(10\!\cdots\!28\)\( T^{5} + \)\(32\!\cdots\!43\)\( T^{6} \))
$47$ (\( 1 + \)\(52\!\cdots\!20\)\( T + \)\(66\!\cdots\!90\)\( T^{2} + \)\(17\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!49\)\( T^{4} \))(\( 1 - \)\(29\!\cdots\!16\)\( T + \)\(84\!\cdots\!53\)\( T^{2} - \)\(18\!\cdots\!60\)\( T^{3} + \)\(28\!\cdots\!79\)\( T^{4} - \)\(32\!\cdots\!84\)\( T^{5} + \)\(37\!\cdots\!07\)\( T^{6} \))
$53$ (\( 1 + \)\(31\!\cdots\!68\)\( T - \)\(16\!\cdots\!26\)\( T^{2} + \)\(69\!\cdots\!76\)\( T^{3} + \)\(50\!\cdots\!49\)\( T^{4} \))(\( 1 + \)\(10\!\cdots\!82\)\( T + \)\(58\!\cdots\!31\)\( T^{2} + \)\(17\!\cdots\!48\)\( T^{3} + \)\(13\!\cdots\!67\)\( T^{4} + \)\(54\!\cdots\!18\)\( T^{5} + \)\(11\!\cdots\!93\)\( T^{6} \))
$59$ (\( 1 - \)\(82\!\cdots\!64\)\( T + \)\(20\!\cdots\!58\)\( T^{2} - \)\(78\!\cdots\!36\)\( T^{3} + \)\(91\!\cdots\!01\)\( T^{4} \))(\( 1 - \)\(11\!\cdots\!48\)\( T + \)\(26\!\cdots\!33\)\( T^{2} - \)\(20\!\cdots\!44\)\( T^{3} + \)\(25\!\cdots\!67\)\( T^{4} - \)\(10\!\cdots\!48\)\( T^{5} + \)\(86\!\cdots\!99\)\( T^{6} \))
$61$ (\( 1 + \)\(40\!\cdots\!40\)\( T + \)\(98\!\cdots\!18\)\( T^{2} + \)\(12\!\cdots\!40\)\( T^{3} + \)\(93\!\cdots\!01\)\( T^{4} \))(\( 1 - \)\(12\!\cdots\!18\)\( T + \)\(44\!\cdots\!99\)\( T^{2} - \)\(42\!\cdots\!64\)\( T^{3} + \)\(13\!\cdots\!99\)\( T^{4} - \)\(12\!\cdots\!18\)\( T^{5} + \)\(28\!\cdots\!01\)\( T^{6} \))
$67$ (\( 1 - \)\(96\!\cdots\!12\)\( T + \)\(11\!\cdots\!22\)\( T^{2} - \)\(78\!\cdots\!16\)\( T^{3} + \)\(66\!\cdots\!49\)\( T^{4} \))(\( 1 - \)\(17\!\cdots\!12\)\( T + \)\(26\!\cdots\!77\)\( T^{2} - \)\(23\!\cdots\!96\)\( T^{3} + \)\(21\!\cdots\!11\)\( T^{4} - \)\(11\!\cdots\!88\)\( T^{5} + \)\(54\!\cdots\!07\)\( T^{6} \))
$71$ (\( 1 - \)\(44\!\cdots\!24\)\( T + \)\(99\!\cdots\!46\)\( T^{2} - \)\(27\!\cdots\!24\)\( T^{3} + \)\(38\!\cdots\!01\)\( T^{4} \))(\( 1 + \)\(27\!\cdots\!24\)\( T + \)\(62\!\cdots\!45\)\( T^{2} - \)\(11\!\cdots\!40\)\( T^{3} + \)\(39\!\cdots\!95\)\( T^{4} + \)\(10\!\cdots\!24\)\( T^{5} + \)\(24\!\cdots\!51\)\( T^{6} \))
$73$ (\( 1 + \)\(13\!\cdots\!08\)\( T + \)\(29\!\cdots\!54\)\( T^{2} + \)\(22\!\cdots\!56\)\( T^{3} + \)\(27\!\cdots\!49\)\( T^{4} \))(\( 1 + \)\(58\!\cdots\!22\)\( T + \)\(54\!\cdots\!71\)\( T^{2} + \)\(18\!\cdots\!08\)\( T^{3} + \)\(89\!\cdots\!47\)\( T^{4} + \)\(15\!\cdots\!78\)\( T^{5} + \)\(44\!\cdots\!93\)\( T^{6} \))
$79$ (\( 1 + \)\(10\!\cdots\!80\)\( T + \)\(29\!\cdots\!98\)\( T^{2} + \)\(27\!\cdots\!20\)\( T^{3} + \)\(68\!\cdots\!01\)\( T^{4} \))(\( 1 + \)\(26\!\cdots\!00\)\( T + \)\(99\!\cdots\!97\)\( T^{2} + \)\(14\!\cdots\!00\)\( T^{3} + \)\(25\!\cdots\!03\)\( T^{4} + \)\(18\!\cdots\!00\)\( T^{5} + \)\(17\!\cdots\!99\)\( T^{6} \))
$83$ (\( 1 + \)\(55\!\cdots\!44\)\( T + \)\(36\!\cdots\!82\)\( T^{2} + \)\(80\!\cdots\!08\)\( T^{3} + \)\(21\!\cdots\!49\)\( T^{4} \))(\( 1 - \)\(65\!\cdots\!80\)\( T + \)\(11\!\cdots\!73\)\( T^{2} + \)\(21\!\cdots\!32\)\( T^{3} + \)\(17\!\cdots\!11\)\( T^{4} - \)\(14\!\cdots\!20\)\( T^{5} + \)\(31\!\cdots\!43\)\( T^{6} \))
$89$ (\( 1 - \)\(24\!\cdots\!36\)\( T + \)\(40\!\cdots\!58\)\( T^{2} - \)\(40\!\cdots\!64\)\( T^{3} + \)\(28\!\cdots\!01\)\( T^{4} \))(\( 1 - \)\(11\!\cdots\!22\)\( T + \)\(24\!\cdots\!03\)\( T^{2} - \)\(97\!\cdots\!16\)\( T^{3} + \)\(40\!\cdots\!47\)\( T^{4} - \)\(32\!\cdots\!22\)\( T^{5} + \)\(48\!\cdots\!49\)\( T^{6} \))
$97$ (\( 1 - \)\(83\!\cdots\!52\)\( T + \)\(66\!\cdots\!62\)\( T^{2} - \)\(28\!\cdots\!36\)\( T^{3} + \)\(11\!\cdots\!49\)\( T^{4} \))(\( 1 - \)\(21\!\cdots\!42\)\( T + \)\(24\!\cdots\!67\)\( T^{2} - \)\(17\!\cdots\!56\)\( T^{3} + \)\(83\!\cdots\!31\)\( T^{4} - \)\(25\!\cdots\!58\)\( T^{5} + \)\(40\!\cdots\!57\)\( T^{6} \))
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