Properties

Label 9.12
Level 9
Weight 12
Dimension 24
Nonzero newspaces 2
Newforms 4
Sturm bound 72
Trace bound 1

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Defining parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 12 \)
Nonzero newspaces: \( 2 \)
Newforms: \( 4 \)
Sturm bound: \(72\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_1(9))\).

Total New Old
Modular forms 37 29 8
Cusp forms 29 24 5
Eisenstein series 8 5 3

Trace form

\( 24q - 87q^{2} - 12q^{3} - 5709q^{4} - 6690q^{5} + 20583q^{6} + 80208q^{7} - 268662q^{8} + 135504q^{9} + O(q^{10}) \) \( 24q - 87q^{2} - 12q^{3} - 5709q^{4} - 6690q^{5} + 20583q^{6} + 80208q^{7} - 268662q^{8} + 135504q^{9} + 598272q^{10} - 1285224q^{11} + 1027860q^{12} + 1993482q^{13} - 2138160q^{14} - 6358608q^{15} - 12402033q^{16} + 31587372q^{17} + 8682876q^{18} - 22423992q^{19} - 5380428q^{20} + 55012206q^{21} + 23067111q^{22} - 103327248q^{23} - 211100355q^{24} + 64454136q^{25} + 408298740q^{26} - 83699352q^{27} - 93891588q^{28} - 164595138q^{29} - 23582592q^{30} + 223113816q^{31} + 295991343q^{32} + 31338342q^{33} - 738443115q^{34} - 246345144q^{35} - 19984653q^{36} + 329160288q^{37} + 2267563587q^{38} - 1999064976q^{39} - 2096470824q^{40} - 875080866q^{41} + 4171968882q^{42} + 1087474320q^{43} - 1508879958q^{44} + 1749349170q^{45} + 4673267304q^{46} - 2601257856q^{47} - 9335236125q^{48} - 3660542556q^{49} - 5696437551q^{50} + 1736777052q^{51} + 8295260514q^{52} + 12717900612q^{53} + 18127857753q^{54} + 525774624q^{55} - 21950029266q^{56} - 7855424196q^{57} - 17786993244q^{58} - 21871135776q^{59} - 10283356116q^{60} + 6712066530q^{61} + 93749877348q^{62} + 51565206888q^{63} - 2527339134q^{64} - 32140271670q^{65} - 93201828246q^{66} - 21859852680q^{67} - 99913140585q^{68} + 6907292550q^{69} + 78375656802q^{70} + 127093320720q^{71} + 161899013547q^{72} + 4062188496q^{73} - 165741990492q^{74} - 218383044348q^{75} - 89592389763q^{76} - 61958337330q^{77} + 150319870614q^{78} + 55453127760q^{79} + 414961963872q^{80} + 222307104312q^{81} - 7348822842q^{82} - 78440398548q^{83} - 711968015814q^{84} - 238810904964q^{85} - 140048966847q^{86} + 99715491216q^{87} + 115063627659q^{88} + 527063391108q^{89} + 620021743884q^{90} + 215235126648q^{91} - 496134212574q^{92} - 572981484354q^{93} - 317074813512q^{94} - 575223807408q^{95} + 118587589272q^{96} + 423150430338q^{97} + 1363519752264q^{98} + 621154334268q^{99} + O(q^{100}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_1(9))\)

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
9.12.a \(\chi_{9}(1, \cdot)\) 9.12.a.a 1 1
9.12.a.b 1
9.12.a.c 2
9.12.c \(\chi_{9}(4, \cdot)\) 9.12.c.a 20 2

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_1(9))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_1(9)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 2}\)