Properties

Label 2.48.a
Level 2
Weight 48
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newforms 2
Sturm bound 12
Trace bound 1

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Defining parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 48 \)
Character orbit: \([\chi]\) = 2.a (trivial)
Character field: \(\Q\)
Newforms: \( 2 \)
Sturm bound: \(12\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{48}(\Gamma_0(2))\).

Total New Old
Modular forms 13 3 10
Cusp forms 11 3 8
Eisenstein series 2 0 2

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

\(2\)Dim.
\(+\)\(2\)
\(-\)\(1\)

Trace form

\(3q \) \(\mathstrut -\mathstrut 8388608q^{2} \) \(\mathstrut -\mathstrut 74344735548q^{3} \) \(\mathstrut +\mathstrut 211106232532992q^{4} \) \(\mathstrut +\mathstrut 38848460008174890q^{5} \) \(\mathstrut -\mathstrut 2675331984594567168q^{6} \) \(\mathstrut +\mathstrut 109845663619130304936q^{7} \) \(\mathstrut -\mathstrut 590295810358705651712q^{8} \) \(\mathstrut +\mathstrut 54423021867263863670511q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut -\mathstrut 8388608q^{2} \) \(\mathstrut -\mathstrut 74344735548q^{3} \) \(\mathstrut +\mathstrut 211106232532992q^{4} \) \(\mathstrut +\mathstrut 38848460008174890q^{5} \) \(\mathstrut -\mathstrut 2675331984594567168q^{6} \) \(\mathstrut +\mathstrut 109845663619130304936q^{7} \) \(\mathstrut -\mathstrut 590295810358705651712q^{8} \) \(\mathstrut +\mathstrut 54423021867263863670511q^{9} \) \(\mathstrut +\mathstrut 20899917608562594938880q^{10} \) \(\mathstrut +\mathstrut 6445844520254455949589036q^{11} \) \(\mathstrut -\mathstrut 5231545676733294808399872q^{12} \) \(\mathstrut -\mathstrut 216241597406825633160829038q^{13} \) \(\mathstrut -\mathstrut 1779990704520835159948263424q^{14} \) \(\mathstrut -\mathstrut 11069210953037625739698901320q^{15} \) \(\mathstrut +\mathstrut 14855280471424563298789490688q^{16} \) \(\mathstrut -\mathstrut 147671761715151713849969513514q^{17} \) \(\mathstrut -\mathstrut 253925967540802572186588020736q^{18} \) \(\mathstrut -\mathstrut 90049958047986224074569651180q^{19} \) \(\mathstrut +\mathstrut 2733717344011469540530543656960q^{20} \) \(\mathstrut +\mathstrut 31797866731443750893227190851296q^{21} \) \(\mathstrut +\mathstrut 34804469808406446312512651526144q^{22} \) \(\mathstrut -\mathstrut 169477041591559848893453118624648q^{23} \) \(\mathstrut -\mathstrut 188259752014257222546806173335552q^{24} \) \(\mathstrut -\mathstrut 790978048565991068045933134957275q^{25} \) \(\mathstrut -\mathstrut 284765363157310487150899366985728q^{26} \) \(\mathstrut +\mathstrut 12958638098603035500336753161222760q^{27} \) \(\mathstrut +\mathstrut 7729701402240313911912486880149504q^{28} \) \(\mathstrut +\mathstrut 14332625907484550456156170114348770q^{29} \) \(\mathstrut +\mathstrut 24665462643998001954509591999938560q^{30} \) \(\mathstrut +\mathstrut 96825893394040620657847712183926176q^{31} \) \(\mathstrut -\mathstrut 41538374868278621028243970633760768q^{32} \) \(\mathstrut +\mathstrut 435814384490191993172239318958622864q^{33} \) \(\mathstrut +\mathstrut 486637501562810690826354300756688896q^{34} \) \(\mathstrut -\mathstrut 690246997665433130952737277477527760q^{35} \) \(\mathstrut +\mathstrut 3829679703152904560106655983341666304q^{36} \) \(\mathstrut +\mathstrut 21658271658899710181509800192286325706q^{37} \) \(\mathstrut -\mathstrut 17889459547243398972801901305958236160q^{38} \) \(\mathstrut -\mathstrut 59857069485467476824804199925228464488q^{39} \) \(\mathstrut +\mathstrut 1470700955531196413237407233941176320q^{40} \) \(\mathstrut -\mathstrut 219313846503396117029867828221704166194q^{41} \) \(\mathstrut -\mathstrut 97921483154406343496952269602843262976q^{42} \) \(\mathstrut +\mathstrut 589877653037525515045352124391830893772q^{43} \) \(\mathstrut +\mathstrut 453585984054783146422395441318570491904q^{44} \) \(\mathstrut -\mathstrut 358024544607134041864258496512804812270q^{45} \) \(\mathstrut -\mathstrut 1568662456032431771010538737753881837568q^{46} \) \(\mathstrut +\mathstrut 578873128480237332871778371927474744176q^{47} \) \(\mathstrut -\mathstrut 368137299379809309759963731240322859008q^{48} \) \(\mathstrut +\mathstrut 1457791583895449138512085300971615625499q^{49} \) \(\mathstrut +\mathstrut 1882296673569186069043613508470780723200q^{50} \) \(\mathstrut +\mathstrut 41128728330665931775726227903070136204616q^{51} \) \(\mathstrut -\mathstrut 15216649648490323994218558692673802207232q^{52} \) \(\mathstrut +\mathstrut 6641352650970641048994808739590947060762q^{53} \) \(\mathstrut -\mathstrut 60828573211600186863720515650419110707200q^{54} \) \(\mathstrut -\mathstrut 9786896376947069384148151409323562868120q^{55} \) \(\mathstrut -\mathstrut 125255710525046560564588059985067728961536q^{56} \) \(\mathstrut +\mathstrut 1090675886322192583586028725640676080240q^{57} \) \(\mathstrut -\mathstrut 632508178830470849192526953243506586419200q^{58} \) \(\mathstrut +\mathstrut 619529183986549822297782427362015407328540q^{59} \) \(\mathstrut -\mathstrut 778926473802901002805159363486207684116480q^{60} \) \(\mathstrut +\mathstrut 2918499186202721946884896058367636565327746q^{61} \) \(\mathstrut -\mathstrut 2671178808652184235140222775854520919392256q^{62} \) \(\mathstrut +\mathstrut 4245350339535941942169308492774113138602312q^{63} \) \(\mathstrut +\mathstrut 1045347431181122959759486794030391945592832q^{64} \) \(\mathstrut +\mathstrut 3857288118116927373814849800498133541545980q^{65} \) \(\mathstrut -\mathstrut 21131997094294214741031556381858378826121216q^{66} \) \(\mathstrut +\mathstrut 7811326418220707137890139953431605518321156q^{67} \) \(\mathstrut -\mathstrut 10391476422398467746430473193712190064951296q^{68} \) \(\mathstrut +\mathstrut 47714557908452059369471633395141193288781472q^{69} \) \(\mathstrut -\mathstrut 11955746662009806357450830227097161480273920q^{70} \) \(\mathstrut +\mathstrut 20502294222510143366907230402092449839796456q^{71} \) \(\mathstrut -\mathstrut 17868451449944548853908680849596249700040704q^{72} \) \(\mathstrut -\mathstrut 124102644650709561672709195052522515329179538q^{73} \) \(\mathstrut -\mathstrut 77600797954860619473131110670534188396969984q^{74} \) \(\mathstrut -\mathstrut 122878767143293924741649096706603940800867300q^{75} \) \(\mathstrut -\mathstrut 6336702461088118064067453208039060427243520q^{76} \) \(\mathstrut +\mathstrut 11411626266451435249185883657111209648373152q^{77} \) \(\mathstrut +\mathstrut 914800651658672531465762061151649175026270208q^{78} \) \(\mathstrut +\mathstrut 547326521440352087747244737556746293617722640q^{79} \) \(\mathstrut +\mathstrut 192368256434786191367845175226260566510141440q^{80} \) \(\mathstrut -\mathstrut 1208755745524035424834499666208927534016001157q^{81} \) \(\mathstrut +\mathstrut 1241298429793695479218989762495630820459413504q^{82} \) \(\mathstrut -\mathstrut 2094841049563545648214697183506777580590670508q^{83} \) \(\mathstrut +\mathstrut 2237575949420418251980498085548305927628652544q^{84} \) \(\mathstrut -\mathstrut 5419768912185182211534230885918205540731715660q^{85} \) \(\mathstrut +\mathstrut 723879619556785631376512838461535289880870912q^{86} \) \(\mathstrut -\mathstrut 249861346808472892079697736409658232317733800q^{87} \) \(\mathstrut +\mathstrut 2449146832186983592555669649670684163876847616q^{88} \) \(\mathstrut -\mathstrut 3717296904499967445201901496448125141449778370q^{89} \) \(\mathstrut +\mathstrut 7191215453307250398868452367101678187614044160q^{90} \) \(\mathstrut -\mathstrut 11589141432976516906106193690226113912903623184q^{91} \) \(\mathstrut -\mathstrut 11925886583743796684784974774305942069441462272q^{92} \) \(\mathstrut +\mathstrut 62723041493778875442429184025157465626622267264q^{93} \) \(\mathstrut -\mathstrut 43416465978648308364159241788472676218551926784q^{94} \) \(\mathstrut +\mathstrut 12111303946348034174743315850165453325069317400q^{95} \) \(\mathstrut -\mathstrut 13247602328441731425408226815624856766175510528q^{96} \) \(\mathstrut -\mathstrut 8126597993497819634696338191273881278826460954q^{97} \) \(\mathstrut -\mathstrut 56263572844840869445593996530350616538538573824q^{98} \) \(\mathstrut +\mathstrut 260382064486578341778890903987058140146892976732q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{48}^{\mathrm{new}}(\Gamma_0(2))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.48.a.a \(1\) \(27.982\) \(\Q\) None \(8388608\) \(-196634580372\) \(20\!\cdots\!50\) \(-5\!\cdots\!96\) \(-\) \(q+2^{23}q^{2}-196634580372q^{3}+2^{46}q^{4}+\cdots\)
2.48.a.b \(2\) \(27.982\) \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(-16777216\) \(122289844824\) \(18\!\cdots\!40\) \(16\!\cdots\!32\) \(+\) \(q-2^{23}q^{2}+(61144922412-5\beta )q^{3}+\cdots\)

Decomposition of \(S_{48}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

\( S_{48}^{\mathrm{old}}(\Gamma_0(2)) \cong \) \(S_{48}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)