Properties

Label 324.3
Level 324
Weight 3
Dimension 2632
Nonzero newspaces 8
Newform subspaces 38
Sturm bound 17496
Trace bound 1

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Defining parameters

Level: \( N \) = \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) = \( 3 \)
Nonzero newspaces: \( 8 \)
Newform subspaces: \( 38 \)
Sturm bound: \(17496\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(\Gamma_1(324))\).

Total New Old
Modular forms 6102 2744 3358
Cusp forms 5562 2632 2930
Eisenstein series 540 112 428

Trace form

\( 2632q - 12q^{2} - 20q^{4} - 15q^{5} - 18q^{6} + 3q^{7} - 15q^{8} - 36q^{9} + O(q^{10}) \) \( 2632q - 12q^{2} - 20q^{4} - 15q^{5} - 18q^{6} + 3q^{7} - 15q^{8} - 36q^{9} - 37q^{10} - 36q^{11} - 18q^{12} - 79q^{13} - 57q^{14} - 32q^{16} - 30q^{17} - 18q^{18} - 72q^{19} + 51q^{20} - 171q^{21} + 48q^{22} - 63q^{23} - 18q^{24} - 63q^{25} + 165q^{26} + 27q^{27} + 87q^{28} + 3q^{29} - 18q^{30} + 81q^{31} + 78q^{32} + 153q^{33} + 26q^{34} + 486q^{35} - 18q^{36} + 152q^{37} - 90q^{38} + 65q^{40} + 570q^{41} + 477q^{42} + 270q^{43} + 849q^{44} + 396q^{45} + 351q^{46} + 567q^{47} + 477q^{48} + 205q^{49} + 498q^{50} + 126q^{51} + 11q^{52} + 6q^{53} - 144q^{54} - 126q^{55} - 495q^{56} - 252q^{57} - 187q^{58} - 630q^{59} - 837q^{60} - 307q^{61} - 1203q^{62} - 540q^{63} - 695q^{64} - 735q^{65} - 954q^{66} - 612q^{67} - 1068q^{68} + 468q^{69} - 405q^{70} + 324q^{71} - 18q^{72} - 277q^{73} - 465q^{74} + 450q^{75} - 546q^{76} - 231q^{77} - 99q^{78} - 273q^{79} - 834q^{80} - 108q^{81} - 514q^{82} - 189q^{83} - 99q^{84} + 520q^{85} - 624q^{86} - 1008q^{87} - 708q^{88} + 321q^{89} - 1089q^{90} + 483q^{91} - 2091q^{92} + 792q^{93} - 651q^{94} + 612q^{95} - 1305q^{96} + 1160q^{97} - 2067q^{98} - 252q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(\Gamma_1(324))\)

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
324.3.c \(\chi_{324}(161, \cdot)\) 324.3.c.a 4 1
324.3.c.b 4
324.3.d \(\chi_{324}(163, \cdot)\) 324.3.d.a 2 1
324.3.d.b 2
324.3.d.c 2
324.3.d.d 2
324.3.d.e 6
324.3.d.f 6
324.3.d.g 8
324.3.d.h 8
324.3.d.i 8
324.3.f \(\chi_{324}(55, \cdot)\) 324.3.f.a 2 2
324.3.f.b 2
324.3.f.c 2
324.3.f.d 2
324.3.f.e 2
324.3.f.f 2
324.3.f.g 2
324.3.f.h 2
324.3.f.i 2
324.3.f.j 2
324.3.f.k 4
324.3.f.l 4
324.3.f.m 4
324.3.f.n 4
324.3.f.o 8
324.3.f.p 8
324.3.f.q 12
324.3.f.r 12
324.3.f.s 16
324.3.g \(\chi_{324}(53, \cdot)\) 324.3.g.a 2 2
324.3.g.b 2
324.3.g.c 4
324.3.g.d 8
324.3.j \(\chi_{324}(19, \cdot)\) 324.3.j.a 204 6
324.3.k \(\chi_{324}(17, \cdot)\) 324.3.k.a 36 6
324.3.n \(\chi_{324}(7, \cdot)\) 324.3.n.a 1908 18
324.3.o \(\chi_{324}(5, \cdot)\) 324.3.o.a 324 18

Decomposition of \(S_{3}^{\mathrm{old}}(\Gamma_1(324))\) into lower level spaces

\( S_{3}^{\mathrm{old}}(\Gamma_1(324)) \cong \) \(S_{3}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(36))\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(54))\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(81))\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(108))\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(162))\)\(^{\oplus 2}\)