Properties

Label 3.36.a
Level 3
Weight 36
Character orbit a
Rep. character \(\chi_{3}(1,\cdot)\)
Character field \(\Q\)
Dimension 5
Newforms 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: \(\Q\)
Newforms: \( 2 \)
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.

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

Trace form

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

Decomposition of \(S_{36}^{\mathrm{new}}(\Gamma_0(3))\) into irreducible Hecke orbits

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

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