# Properties

 Label 21.33 Level 21 Weight 33 Dimension 358 Nonzero newspaces 4 Newform subspaces 6 Sturm bound 1056 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$21 = 3 \cdot 7$$ Weight: $$k$$ = $$33$$ Nonzero newspaces: $$4$$ Newform subspaces: $$6$$ Sturm bound: $$1056$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{33}(\Gamma_1(21))$$.

Total New Old
Modular forms 524 366 158
Cusp forms 500 358 142
Eisenstein series 24 8 16

## Trace form

 $$358q + 85821018q^{3} + 34404200538q^{4} + 337551521994q^{5} + 4849061577690q^{6} + 25814421548190q^{7} + 905858340105042q^{8} - 2664335140274988q^{9} + O(q^{10})$$ $$358q + 85821018q^{3} + 34404200538q^{4} + 337551521994q^{5} + 4849061577690q^{6} + 25814421548190q^{7} + 905858340105042q^{8} - 2664335140274988q^{9} + 3184075770068076q^{10} - 167157257243943114q^{11} - 618375879284046522q^{12} + 394828728859629130q^{13} + 12851778428248442616q^{14} + 28172109602802744480q^{15} - 155557396865448844286q^{16} + 77398333383202350336q^{17} + 235642817195312315604q^{18} - 975563611460804432846q^{19} - 6060926699667790548552q^{21} + 1902619733575359333636q^{22} + 31461360671699680688640q^{23} - 67837377824624911570854q^{24} + 103807014015513333345712q^{25} - 91622349693517678124166q^{26} - 226493464409346372136656q^{27} - 18404826410857804020086q^{28} + 503537673408455045833092q^{29} - 499127766494624768031384q^{30} - 391122469333800831620834q^{31} - 10624026479121911877184890q^{32} + 6363984227956104202484910q^{33} - 12092009778313335537767616q^{34} + 20867482951466450759491626q^{35} - 235260251605647495436014186q^{36} + 31785236974221879005611728q^{37} + 93778187170932179905821390q^{38} + 150334546805654548345765542q^{39} - 152601784588802315740129536q^{40} + 259708826358721956305249424q^{42} - 206189082817161268595015124q^{43} - 1467514495507448246278484232q^{44} + 801332411706508095089772318q^{45} - 1945642980241556619280412172q^{46} + 2711859951888026489715988986q^{47} - 15449848535613249747586966890q^{48} + 16242527230296474380171905852q^{49} + 2142606657460004701529554854q^{50} - 5411953769842519850240385792q^{51} + 19101969362231853440726653720q^{52} - 11833748472372660713728363716q^{53} + 9666621355747236548967733812q^{54} + 58245284167757238785268625284q^{55} + 40721099301498495875514099858q^{56} - 65059931949306168985432535124q^{57} + 333701399439065424093340997088q^{58} - 192748765444386490738617028524q^{59} - 184647969570236588592287226924q^{60} + 44803823009386720408383417238q^{61} - 62030619157497460583124044214q^{63} + 94722199451301125681336660646q^{64} - 134886077030051592943040841942q^{65} - 708404187217382032581545648112q^{66} - 1100966365601109118036444855522q^{67} + 1932264158131046514460420864116q^{68} - 2464726377300221407514345915976q^{69} + 14267754387562113097888185156q^{70} - 877984033190160797962039716120q^{71} - 4220134977108320958961202974314q^{72} + 4119385500294121648958652329620q^{73} + 340766942290361028855529938450q^{74} - 3727126323535981732856153994576q^{75} + 1465296704050196431751716433260q^{76} - 2124037394941955355626139650664q^{77} - 18776859039125103235811773616040q^{78} + 8892556467863104888940294304074q^{79} - 16812269777054808036686580546960q^{80} - 1354691258368240797592495346520q^{81} - 3552648828739689546708676037892q^{82} + 23909287281765268427635378963410q^{84} - 50543626727024480800067153263656q^{85} + 121680094502886575532766519988610q^{86} + 31366952309476866039643222774032q^{87} - 74577713437195531176888625241112q^{88} + 102038877956780628559173843197220q^{89} - 94074444368717700624809371372716q^{90} - 119929274422865779302930885543116q^{91} + 336829786160011782417047495688300q^{92} + 181488847735047018816582854696868q^{93} - 836804892188627301950730732496500q^{94} - 198067568672425350745847146292766q^{95} + 1039725226398089907281184181250358q^{96} - 522203710196673046262981729105144q^{97} - 1070711111106940764855879001779462q^{98} + 314797934059105133715619731910200q^{99} + O(q^{100})$$

## Decomposition of $$S_{33}^{\mathrm{new}}(\Gamma_1(21))$$

We only show spaces with odd 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
21.33.b $$\chi_{21}(8, \cdot)$$ 21.33.b.a 64 1
21.33.d $$\chi_{21}(13, \cdot)$$ 21.33.d.a 42 1
21.33.f $$\chi_{21}(10, \cdot)$$ 21.33.f.a 42 2
21.33.f.b 44
21.33.h $$\chi_{21}(2, \cdot)$$ 21.33.h.a 2 2
21.33.h.b 164

## Decomposition of $$S_{33}^{\mathrm{old}}(\Gamma_1(21))$$ into lower level spaces

$$S_{33}^{\mathrm{old}}(\Gamma_1(21)) \cong$$ $$S_{33}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 2}$$$$\oplus$$$$S_{33}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 2}$$