Properties

Label 21.40
Level 21
Weight 40
Dimension 450
Nonzero newspaces 4
Newform subspaces 9
Sturm bound 1280
Trace bound 1

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Defining parameters

Level: \( N \) = \( 21 = 3 \cdot 7 \)
Weight: \( k \) = \( 40 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 9 \)
Sturm bound: \(1280\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{40}(\Gamma_1(21))\).

Total New Old
Modular forms 636 462 174
Cusp forms 612 450 162
Eisenstein series 24 12 12

Trace form

\( 450 q + 1146852 q^{2} + 2324522931 q^{3} - 3926685758670 q^{4} + 182940449356176 q^{5} + 157393447861140 q^{6} - 51742187713521534 q^{7} + 2319897806532023958 q^{8} - 5521717201486723845 q^{9} + O(q^{10}) \) \( 450 q + 1146852 q^{2} + 2324522931 q^{3} - 3926685758670 q^{4} + 182940449356176 q^{5} + 157393447861140 q^{6} - 51742187713521534 q^{7} + 2319897806532023958 q^{8} - 5521717201486723845 q^{9} + 45588323894145494448 q^{10} + 687643550296214786064 q^{11} - 1457734263512554922880 q^{12} - 1051252605904938898428 q^{13} - 166766934520668279300024 q^{14} - 53333991564655338018918 q^{15} - 746473729596232053721422 q^{16} - 3142749364918272595947588 q^{17} - 18262714540572552134298798 q^{18} + 34066597913375547604098270 q^{19} + 138497119120532437218021924 q^{20} + 228370819687205933303141097 q^{21} + 1028079295607748025265648544 q^{22} + 629481528790579917608611128 q^{23} - 830453776526575370434306176 q^{24} + 2055909887608476365102459154 q^{25} - 1391761490824833077342358246 q^{26} - 6280171596328326446562138252 q^{27} + 101185048901564282229374973162 q^{28} - 216410173750330013430887842080 q^{29} + 334575415411304249427287325750 q^{30} + 809595752435671070381257595970 q^{31} - 395397294478224270334623357810 q^{32} + 1392704646512727472539498855927 q^{33} - 12062374983332617412055757777404 q^{34} + 9410886555703809402306486609924 q^{35} + 38090987004997204470632146114410 q^{36} - 17796415323389286560128565002314 q^{37} - 68936921563712374334417045979114 q^{38} + 60537358469807863261901746523076 q^{39} - 113035675883279567982727364375064 q^{40} + 38834372987397754918688197362984 q^{41} - 462705642899935988307234703308738 q^{42} + 717461088498012206055882038953728 q^{43} - 1325475978672683923555097392871208 q^{44} + 193013609264963119336232221551351 q^{45} - 689738003965416497971289766638556 q^{46} + 1184952235907168266575275598544632 q^{47} - 4458303485116602667796381170382496 q^{48} + 6008978163403979791681913020796922 q^{49} - 10969546155154475025104146221022638 q^{50} + 1726537835905409273890110216584427 q^{51} + 29069479814609136859454809516770552 q^{52} - 6705670783184111616504402868926684 q^{53} + 17908629762632350115514710189767914 q^{54} - 59970192713910007319896941522227340 q^{55} + 165009233047700945518667235123350610 q^{56} - 47569818987460743360068273024677014 q^{57} - 221850511193230303552920459844610904 q^{58} + 129565752777363261600998162341505040 q^{59} - 151138574810919509375347748732254140 q^{60} + 177785295632270228906587171230910218 q^{61} - 179295793258854867621118358455711164 q^{62} - 302453808235816438003585748573830053 q^{63} - 4454469374465011383578752001859869946 q^{64} + 225626171562283799106335649259486584 q^{65} - 1199874004385358652032514013294341014 q^{66} + 889541122326262915566460675654195446 q^{67} - 1238064243811110472073598420092483232 q^{68} - 515510732805804154439066903302119672 q^{69} + 6750800805667894585190633780713376016 q^{70} + 1656788439144524216050593223392178068 q^{71} + 7798683240767595931679895325606227294 q^{72} + 7910952998692531105621590407180049570 q^{73} - 13197742371107932579564574760245802618 q^{74} + 26973904147423021880093510364416953296 q^{75} - 9663790967671570494559203411918631608 q^{76} - 23423486697380198112001499462335218036 q^{77} + 70071485868236245149509285224423500480 q^{78} + 29796939492687118147618868693624477106 q^{79} - 41705122598319270284683238572850627784 q^{80} - 62652598476349354971521493026393241057 q^{81} - 293200401876394454399776097581114739688 q^{82} + 50817919579096190144774991285246574152 q^{83} - 416945735727963727854598408428558292500 q^{84} + 385310803737072177974069232539962768908 q^{85} - 406275120662948270131047992977250835738 q^{86} + 78787172626134578452359964859566473072 q^{87} + 849299479636222957399402643531088504684 q^{88} - 416731062818414729004833799853053748752 q^{89} + 151020846372146404693378128642398067480 q^{90} + 168712561831494173454077652714526434708 q^{91} - 1152730058811202783346169862987206949056 q^{92} + 719090567979581316047111616202193060175 q^{93} + 424863883388229271359960400246644206760 q^{94} - 2463443571549255114437337733877513246016 q^{95} + 463581381224260982085216464386876412628 q^{96} + 457771081650059478607217222361649195524 q^{97} + 3170132259076775884193553095778947956134 q^{98} + 1106368574314906988720201492838294635970 q^{99} + O(q^{100}) \)

Decomposition of \(S_{40}^{\mathrm{new}}(\Gamma_1(21))\)

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
21.40.a \(\chi_{21}(1, \cdot)\) 21.40.a.a 9 1
21.40.a.b 10
21.40.a.c 10
21.40.a.d 11
21.40.c \(\chi_{21}(20, \cdot)\) 21.40.c.a 2 1
21.40.c.b 100
21.40.e \(\chi_{21}(4, \cdot)\) 21.40.e.a 50 2
21.40.e.b 54
21.40.g \(\chi_{21}(5, \cdot)\) 21.40.g.a 204 2

Decomposition of \(S_{40}^{\mathrm{old}}(\Gamma_1(21))\) into lower level spaces

\( S_{40}^{\mathrm{old}}(\Gamma_1(21)) \cong \) \(S_{40}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{40}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{40}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 2}\)