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

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

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

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