Properties

Label 9.6
Level 9
Weight 6
Dimension 9
Nonzero newspaces 2
Newforms 2
Sturm bound 36
Trace bound 1

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 6 \)
Nonzero newspaces: \( 2 \)
Newforms: \( 2 \)
Sturm bound: \(36\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_1(9))\).

Total New Old
Modular forms 19 14 5
Cusp forms 11 9 2
Eisenstein series 8 5 3

Trace form

\(9q \) \(\mathstrut +\mathstrut 9q^{2} \) \(\mathstrut -\mathstrut 12q^{3} \) \(\mathstrut -\mathstrut 45q^{4} \) \(\mathstrut +\mathstrut 72q^{5} \) \(\mathstrut +\mathstrut 171q^{6} \) \(\mathstrut -\mathstrut 12q^{7} \) \(\mathstrut -\mathstrut 918q^{8} \) \(\mathstrut -\mathstrut 414q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(9q \) \(\mathstrut +\mathstrut 9q^{2} \) \(\mathstrut -\mathstrut 12q^{3} \) \(\mathstrut -\mathstrut 45q^{4} \) \(\mathstrut +\mathstrut 72q^{5} \) \(\mathstrut +\mathstrut 171q^{6} \) \(\mathstrut -\mathstrut 12q^{7} \) \(\mathstrut -\mathstrut 918q^{8} \) \(\mathstrut -\mathstrut 414q^{9} \) \(\mathstrut +\mathstrut 24q^{10} \) \(\mathstrut +\mathstrut 1008q^{11} \) \(\mathstrut +\mathstrut 2724q^{12} \) \(\mathstrut +\mathstrut 456q^{13} \) \(\mathstrut +\mathstrut 1152q^{14} \) \(\mathstrut -\mathstrut 2052q^{15} \) \(\mathstrut -\mathstrut 1425q^{16} \) \(\mathstrut -\mathstrut 5238q^{17} \) \(\mathstrut -\mathstrut 8100q^{18} \) \(\mathstrut +\mathstrut 396q^{19} \) \(\mathstrut +\mathstrut 6660q^{20} \) \(\mathstrut +\mathstrut 8670q^{21} \) \(\mathstrut +\mathstrut 4395q^{22} \) \(\mathstrut +\mathstrut 9684q^{23} \) \(\mathstrut +\mathstrut 549q^{24} \) \(\mathstrut -\mathstrut 4743q^{25} \) \(\mathstrut -\mathstrut 21060q^{26} \) \(\mathstrut -\mathstrut 10152q^{27} \) \(\mathstrut -\mathstrut 1764q^{28} \) \(\mathstrut +\mathstrut 7380q^{29} \) \(\mathstrut +\mathstrut 22104q^{30} \) \(\mathstrut +\mathstrut 5532q^{31} \) \(\mathstrut +\mathstrut 7263q^{32} \) \(\mathstrut -\mathstrut 8820q^{33} \) \(\mathstrut +\mathstrut 4833q^{34} \) \(\mathstrut -\mathstrut 15984q^{35} \) \(\mathstrut -\mathstrut 14589q^{36} \) \(\mathstrut -\mathstrut 17586q^{37} \) \(\mathstrut -\mathstrut 14481q^{38} \) \(\mathstrut +\mathstrut 8220q^{39} \) \(\mathstrut -\mathstrut 7728q^{40} \) \(\mathstrut +\mathstrut 8118q^{41} \) \(\mathstrut +\mathstrut 10098q^{42} \) \(\mathstrut +\mathstrut 3552q^{43} \) \(\mathstrut +\mathstrut 51786q^{44} \) \(\mathstrut +\mathstrut 43038q^{45} \) \(\mathstrut +\mathstrut 51000q^{46} \) \(\mathstrut +\mathstrut 18612q^{47} \) \(\mathstrut -\mathstrut 57405q^{48} \) \(\mathstrut -\mathstrut 6117q^{49} \) \(\mathstrut -\mathstrut 75591q^{50} \) \(\mathstrut -\mathstrut 9396q^{51} \) \(\mathstrut -\mathstrut 29958q^{52} \) \(\mathstrut -\mathstrut 12798q^{53} \) \(\mathstrut +\mathstrut 8181q^{54} \) \(\mathstrut -\mathstrut 39504q^{55} \) \(\mathstrut -\mathstrut 54450q^{56} \) \(\mathstrut -\mathstrut 104298q^{57} \) \(\mathstrut -\mathstrut 19500q^{58} \) \(\mathstrut +\mathstrut 20160q^{59} \) \(\mathstrut +\mathstrut 88308q^{60} \) \(\mathstrut +\mathstrut 87900q^{61} \) \(\mathstrut +\mathstrut 268164q^{62} \) \(\mathstrut +\mathstrut 133524q^{63} \) \(\mathstrut +\mathstrut 6786q^{64} \) \(\mathstrut -\mathstrut 16974q^{65} \) \(\mathstrut -\mathstrut 100998q^{66} \) \(\mathstrut -\mathstrut 30216q^{67} \) \(\mathstrut -\mathstrut 126657q^{68} \) \(\mathstrut -\mathstrut 125982q^{69} \) \(\mathstrut +\mathstrut 786q^{70} \) \(\mathstrut -\mathstrut 67608q^{71} \) \(\mathstrut +\mathstrut 35451q^{72} \) \(\mathstrut +\mathstrut 81342q^{73} \) \(\mathstrut -\mathstrut 174780q^{74} \) \(\mathstrut -\mathstrut 18732q^{75} \) \(\mathstrut -\mathstrut 51795q^{76} \) \(\mathstrut +\mathstrut 16974q^{77} \) \(\mathstrut +\mathstrut 181422q^{78} \) \(\mathstrut -\mathstrut 37596q^{79} \) \(\mathstrut +\mathstrut 131328q^{80} \) \(\mathstrut +\mathstrut 194562q^{81} \) \(\mathstrut -\mathstrut 192378q^{82} \) \(\mathstrut +\mathstrut 35244q^{83} \) \(\mathstrut +\mathstrut 108354q^{84} \) \(\mathstrut +\mathstrut 34128q^{85} \) \(\mathstrut +\mathstrut 41949q^{86} \) \(\mathstrut -\mathstrut 142956q^{87} \) \(\mathstrut +\mathstrut 9771q^{88} \) \(\mathstrut -\mathstrut 357642q^{89} \) \(\mathstrut -\mathstrut 539892q^{90} \) \(\mathstrut +\mathstrut 85872q^{91} \) \(\mathstrut +\mathstrut 137394q^{92} \) \(\mathstrut +\mathstrut 113898q^{93} \) \(\mathstrut +\mathstrut 281496q^{94} \) \(\mathstrut +\mathstrut 298224q^{95} \) \(\mathstrut +\mathstrut 42768q^{96} \) \(\mathstrut +\mathstrut 83418q^{97} \) \(\mathstrut -\mathstrut 33588q^{98} \) \(\mathstrut +\mathstrut 33696q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_1(9))\)

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
9.6.a \(\chi_{9}(1, \cdot)\) 9.6.a.a 1 1
9.6.c \(\chi_{9}(4, \cdot)\) 9.6.c.a 8 2

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_1(9))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_1(9)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 2}\)