Properties

Label 324.2
Level 324
Weight 2
Dimension 1288
Nonzero newspaces 8
Newform subspaces 21
Sturm bound 11664
Trace bound 1

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Defining parameters

Level: \( N \) = \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 8 \)
Newform subspaces: \( 21 \)
Sturm bound: \(11664\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(324))\).

Total New Old
Modular forms 3186 1400 1786
Cusp forms 2647 1288 1359
Eisenstein series 539 112 427

Trace form

\( 1288q - 12q^{2} - 20q^{4} - 27q^{5} - 18q^{6} - 3q^{7} - 9q^{8} - 36q^{9} + O(q^{10}) \) \( 1288q - 12q^{2} - 20q^{4} - 27q^{5} - 18q^{6} - 3q^{7} - 9q^{8} - 36q^{9} - 25q^{10} - 3q^{11} - 18q^{12} - 37q^{13} + 3q^{14} - 8q^{16} - 6q^{17} - 18q^{18} + 18q^{19} + 9q^{20} - 9q^{21} - 12q^{22} + 57q^{23} - 18q^{24} - 3q^{25} - 27q^{26} + 27q^{27} - 33q^{28} + 39q^{29} - 18q^{30} + 27q^{31} - 42q^{32} - 9q^{33} - 52q^{34} + 48q^{35} - 18q^{36} - 34q^{37} - 36q^{38} - 73q^{40} - 33q^{41} - 63q^{42} - 27q^{43} - 153q^{44} - 90q^{45} - 117q^{46} - 99q^{47} - 117q^{48} - 83q^{49} - 222q^{50} - 63q^{51} - 91q^{52} - 174q^{53} - 144q^{54} - 90q^{55} - 243q^{56} - 90q^{57} - 91q^{58} - 123q^{59} - 135q^{60} - 97q^{61} - 207q^{62} - 54q^{63} - 119q^{64} - 171q^{65} - 90q^{66} - 9q^{67} - 114q^{68} - 126q^{69} - 81q^{70} - 60q^{71} - 18q^{72} - 73q^{73} - 39q^{74} - 90q^{75} - 66q^{76} - 117q^{77} + 9q^{78} + 57q^{79} - 108q^{81} - 34q^{82} - 27q^{83} - 45q^{84} - 68q^{85} + 12q^{86} - 144q^{87} + 60q^{88} - 105q^{89} + 45q^{90} - 15q^{91} + 195q^{92} - 234q^{93} + 75q^{94} - 132q^{95} + 99q^{96} - 79q^{97} + 333q^{98} - 90q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(324))\)

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
324.2.a \(\chi_{324}(1, \cdot)\) 324.2.a.a 1 1
324.2.a.b 1
324.2.a.c 1
324.2.a.d 1
324.2.b \(\chi_{324}(323, \cdot)\) 324.2.b.a 4 1
324.2.b.b 8
324.2.b.c 8
324.2.e \(\chi_{324}(109, \cdot)\) 324.2.e.a 2 2
324.2.e.b 2
324.2.e.c 2
324.2.e.d 2
324.2.h \(\chi_{324}(107, \cdot)\) 324.2.h.a 4 2
324.2.h.b 4
324.2.h.c 4
324.2.h.d 8
324.2.h.e 8
324.2.h.f 16
324.2.i \(\chi_{324}(37, \cdot)\) 324.2.i.a 18 6
324.2.l \(\chi_{324}(35, \cdot)\) 324.2.l.a 96 6
324.2.m \(\chi_{324}(13, \cdot)\) 324.2.m.a 162 18
324.2.p \(\chi_{324}(11, \cdot)\) 324.2.p.a 936 18

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(324))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(324)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(36))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(54))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(81))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(108))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(162))\)\(^{\oplus 2}\)