Properties

Label 12.9
Level 12
Weight 9
Dimension 11
Nonzero newspaces 2
Newforms 3
Sturm bound 72
Trace bound 1

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Defining parameters

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

Dimensions

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

Total New Old
Modular forms 37 11 26
Cusp forms 27 11 16
Eisenstein series 10 0 10

Trace form

\(11q \) \(\mathstrut +\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 21q^{3} \) \(\mathstrut -\mathstrut 52q^{4} \) \(\mathstrut -\mathstrut 336q^{5} \) \(\mathstrut +\mathstrut 1134q^{6} \) \(\mathstrut -\mathstrut 2154q^{7} \) \(\mathstrut -\mathstrut 12960q^{8} \) \(\mathstrut -\mathstrut 13653q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(11q \) \(\mathstrut +\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 21q^{3} \) \(\mathstrut -\mathstrut 52q^{4} \) \(\mathstrut -\mathstrut 336q^{5} \) \(\mathstrut +\mathstrut 1134q^{6} \) \(\mathstrut -\mathstrut 2154q^{7} \) \(\mathstrut -\mathstrut 12960q^{8} \) \(\mathstrut -\mathstrut 13653q^{9} \) \(\mathstrut +\mathstrut 36628q^{10} \) \(\mathstrut -\mathstrut 11340q^{12} \) \(\mathstrut -\mathstrut 53258q^{13} \) \(\mathstrut +\mathstrut 52728q^{14} \) \(\mathstrut +\mathstrut 142560q^{15} \) \(\mathstrut +\mathstrut 99440q^{16} \) \(\mathstrut -\mathstrut 193200q^{17} \) \(\mathstrut -\mathstrut 13122q^{18} \) \(\mathstrut -\mathstrut 419178q^{19} \) \(\mathstrut +\mathstrut 335592q^{20} \) \(\mathstrut +\mathstrut 764166q^{21} \) \(\mathstrut -\mathstrut 556968q^{22} \) \(\mathstrut +\mathstrut 221616q^{24} \) \(\mathstrut -\mathstrut 1973253q^{25} \) \(\mathstrut +\mathstrut 21564q^{26} \) \(\mathstrut +\mathstrut 1477899q^{27} \) \(\mathstrut -\mathstrut 594672q^{28} \) \(\mathstrut +\mathstrut 2063472q^{29} \) \(\mathstrut +\mathstrut 46980q^{30} \) \(\mathstrut -\mathstrut 937578q^{31} \) \(\mathstrut -\mathstrut 3602784q^{32} \) \(\mathstrut -\mathstrut 777600q^{33} \) \(\mathstrut +\mathstrut 1568476q^{34} \) \(\mathstrut +\mathstrut 113724q^{36} \) \(\mathstrut +\mathstrut 10292470q^{37} \) \(\mathstrut +\mathstrut 3659400q^{38} \) \(\mathstrut -\mathstrut 2156298q^{39} \) \(\mathstrut +\mathstrut 1749184q^{40} \) \(\mathstrut -\mathstrut 8865456q^{41} \) \(\mathstrut -\mathstrut 5288328q^{42} \) \(\mathstrut +\mathstrut 5472726q^{43} \) \(\mathstrut +\mathstrut 2395920q^{44} \) \(\mathstrut -\mathstrut 13806288q^{45} \) \(\mathstrut -\mathstrut 13649856q^{46} \) \(\mathstrut +\mathstrut 10916208q^{48} \) \(\mathstrut -\mathstrut 799471q^{49} \) \(\mathstrut +\mathstrut 14581842q^{50} \) \(\mathstrut -\mathstrut 7413120q^{51} \) \(\mathstrut +\mathstrut 18592888q^{52} \) \(\mathstrut +\mathstrut 8706672q^{53} \) \(\mathstrut -\mathstrut 2480058q^{54} \) \(\mathstrut -\mathstrut 2566080q^{55} \) \(\mathstrut -\mathstrut 45565632q^{56} \) \(\mathstrut -\mathstrut 15072378q^{57} \) \(\mathstrut -\mathstrut 8816444q^{58} \) \(\mathstrut +\mathstrut 28348056q^{60} \) \(\mathstrut +\mathstrut 28369078q^{61} \) \(\mathstrut +\mathstrut 80783976q^{62} \) \(\mathstrut +\mathstrut 34876566q^{63} \) \(\mathstrut +\mathstrut 1268864q^{64} \) \(\mathstrut +\mathstrut 7293408q^{65} \) \(\mathstrut -\mathstrut 51205608q^{66} \) \(\mathstrut -\mathstrut 61469994q^{67} \) \(\mathstrut -\mathstrut 117288264q^{68} \) \(\mathstrut +\mathstrut 3623616q^{69} \) \(\mathstrut -\mathstrut 60373104q^{70} \) \(\mathstrut +\mathstrut 28343520q^{72} \) \(\mathstrut +\mathstrut 60447958q^{73} \) \(\mathstrut +\mathstrut 119548428q^{74} \) \(\mathstrut +\mathstrut 122666955q^{75} \) \(\mathstrut +\mathstrut 144621360q^{76} \) \(\mathstrut -\mathstrut 56971392q^{77} \) \(\mathstrut -\mathstrut 140630580q^{78} \) \(\mathstrut -\mathstrut 145689834q^{79} \) \(\mathstrut -\mathstrut 163857888q^{80} \) \(\mathstrut +\mathstrut 2604555q^{81} \) \(\mathstrut -\mathstrut 188383460q^{82} \) \(\mathstrut +\mathstrut 199712304q^{84} \) \(\mathstrut -\mathstrut 67764256q^{85} \) \(\mathstrut +\mathstrut 240327384q^{86} \) \(\mathstrut +\mathstrut 108773280q^{87} \) \(\mathstrut +\mathstrut 156323520q^{88} \) \(\mathstrut +\mathstrut 188992272q^{89} \) \(\mathstrut -\mathstrut 80105436q^{90} \) \(\mathstrut -\mathstrut 99306132q^{91} \) \(\mathstrut -\mathstrut 387657984q^{92} \) \(\mathstrut -\mathstrut 245827770q^{93} \) \(\mathstrut -\mathstrut 38749872q^{94} \) \(\mathstrut +\mathstrut 246092256q^{96} \) \(\mathstrut +\mathstrut 92717014q^{97} \) \(\mathstrut +\mathstrut 691081830q^{98} \) \(\mathstrut -\mathstrut 14541120q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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

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
12.9.c \(\chi_{12}(5, \cdot)\) 12.9.c.a 1 1
12.9.c.b 2
12.9.d \(\chi_{12}(7, \cdot)\) 12.9.d.a 8 1

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

\( S_{9}^{\mathrm{old}}(\Gamma_1(12)) \cong \) \(S_{9}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 2}\)