Properties

Label 8.21
Level 8
Weight 21
Dimension 19
Nonzero newspaces 1
Newforms 2
Sturm bound 84
Trace bound 0

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Defining parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 21 \)
Nonzero newspaces: \( 1 \)
Newforms: \( 2 \)
Sturm bound: \(84\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{21}(\Gamma_1(8))\).

Total New Old
Modular forms 43 21 22
Cusp forms 37 19 18
Eisenstein series 6 2 4

Trace form

\(19q \) \(\mathstrut +\mathstrut 626q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 1773556q^{4} \) \(\mathstrut +\mathstrut 25323676q^{6} \) \(\mathstrut -\mathstrut 2031549544q^{8} \) \(\mathstrut +\mathstrut 19758444937q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(19q \) \(\mathstrut +\mathstrut 626q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut +\mathstrut 1773556q^{4} \) \(\mathstrut +\mathstrut 25323676q^{6} \) \(\mathstrut -\mathstrut 2031549544q^{8} \) \(\mathstrut +\mathstrut 19758444937q^{9} \) \(\mathstrut +\mathstrut 742754160q^{10} \) \(\mathstrut -\mathstrut 14520628706q^{11} \) \(\mathstrut -\mathstrut 92518091912q^{12} \) \(\mathstrut -\mathstrut 431248136928q^{14} \) \(\mathstrut +\mathstrut 140976757264q^{16} \) \(\mathstrut +\mathstrut 990389375398q^{17} \) \(\mathstrut +\mathstrut 5272759222774q^{18} \) \(\mathstrut -\mathstrut 1375336086082q^{19} \) \(\mathstrut -\mathstrut 32520172742880q^{20} \) \(\mathstrut -\mathstrut 109675699217572q^{22} \) \(\mathstrut +\mathstrut 144938556063376q^{24} \) \(\mathstrut -\mathstrut 300360045368285q^{25} \) \(\mathstrut +\mathstrut 154289718058128q^{26} \) \(\mathstrut -\mathstrut 514588541331140q^{27} \) \(\mathstrut -\mathstrut 394619539621440q^{28} \) \(\mathstrut +\mathstrut 223250517248160q^{30} \) \(\mathstrut -\mathstrut 1707914682142624q^{32} \) \(\mathstrut -\mathstrut 379273727638900q^{33} \) \(\mathstrut -\mathstrut 4373483391631324q^{34} \) \(\mathstrut +\mathstrut 3615863153468160q^{35} \) \(\mathstrut +\mathstrut 11585741011529500q^{36} \) \(\mathstrut +\mathstrut 11866377694661788q^{38} \) \(\mathstrut +\mathstrut 11813880412973760q^{40} \) \(\mathstrut -\mathstrut 8427241614192506q^{41} \) \(\mathstrut +\mathstrut 38127588900300480q^{42} \) \(\mathstrut -\mathstrut 24717352370364898q^{43} \) \(\mathstrut -\mathstrut 11262977142561032q^{44} \) \(\mathstrut -\mathstrut 21739818881100192q^{46} \) \(\mathstrut +\mathstrut 153956946591726688q^{48} \) \(\mathstrut -\mathstrut 173173121721250637q^{49} \) \(\mathstrut +\mathstrut 139361869584456530q^{50} \) \(\mathstrut -\mathstrut 417243550710750724q^{51} \) \(\mathstrut -\mathstrut 44583718369992480q^{52} \) \(\mathstrut +\mathstrut 131060624290419256q^{54} \) \(\mathstrut -\mathstrut 505574909383001472q^{56} \) \(\mathstrut -\mathstrut 638353406270838580q^{57} \) \(\mathstrut -\mathstrut 677523738697093680q^{58} \) \(\mathstrut -\mathstrut 80800002484130978q^{59} \) \(\mathstrut +\mathstrut 1631640690429240000q^{60} \) \(\mathstrut +\mathstrut 1780090172849178240q^{62} \) \(\mathstrut +\mathstrut 1328363317618417216q^{64} \) \(\mathstrut +\mathstrut 1575343920200472960q^{65} \) \(\mathstrut +\mathstrut 3043614561170466056q^{66} \) \(\mathstrut -\mathstrut 1808783156240800642q^{67} \) \(\mathstrut -\mathstrut 2362666796221870232q^{68} \) \(\mathstrut -\mathstrut 6151558949299572480q^{70} \) \(\mathstrut +\mathstrut 4959295908955144264q^{72} \) \(\mathstrut +\mathstrut 3866104143546483398q^{73} \) \(\mathstrut -\mathstrut 5642095430673385488q^{74} \) \(\mathstrut +\mathstrut 24308156015351409310q^{75} \) \(\mathstrut -\mathstrut 9480723535297927816q^{76} \) \(\mathstrut -\mathstrut 14599907290310144160q^{78} \) \(\mathstrut -\mathstrut 22104885212702947200q^{80} \) \(\mathstrut +\mathstrut 13599596957313525631q^{81} \) \(\mathstrut -\mathstrut 15707154812006670172q^{82} \) \(\mathstrut -\mathstrut 77448386448570160322q^{83} \) \(\mathstrut +\mathstrut 17095266896298568320q^{84} \) \(\mathstrut +\mathstrut 36403346004507897820q^{86} \) \(\mathstrut -\mathstrut 13164074256796170352q^{88} \) \(\mathstrut +\mathstrut 33127276190960144518q^{89} \) \(\mathstrut -\mathstrut 33204348719592139440q^{90} \) \(\mathstrut +\mathstrut 161045136122144660736q^{91} \) \(\mathstrut -\mathstrut 932896292396925120q^{92} \) \(\mathstrut -\mathstrut 107259275077774974528q^{94} \) \(\mathstrut +\mathstrut 197187886676221266496q^{96} \) \(\mathstrut +\mathstrut 77267514659308382822q^{97} \) \(\mathstrut +\mathstrut 236544665851892453426q^{98} \) \(\mathstrut -\mathstrut 342412845288286744966q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{21}^{\mathrm{new}}(\Gamma_1(8))\)

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
8.21.c \(\chi_{8}(7, \cdot)\) None 0 1
8.21.d \(\chi_{8}(3, \cdot)\) 8.21.d.a 1 1
8.21.d.b 18

Decomposition of \(S_{21}^{\mathrm{old}}(\Gamma_1(8))\) into lower level spaces

\( S_{21}^{\mathrm{old}}(\Gamma_1(8)) \cong \) \(S_{21}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 2}\)