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Properties

Label 324.8
Level 324
Weight 8
Dimension 9352
Nonzero newspaces 8
Sturm bound 46656
Trace bound 1

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Defining parameters

Level: \( N \) = \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 8 \)
Sturm bound: \(46656\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(324))\).

Total New Old
Modular forms 20682 9464 11218
Cusp forms 20142 9352 10790
Eisenstein series 540 112 428

Trace form

\( 9352 q - 12 q^{2} - 20 q^{4} + 189 q^{5} - 18 q^{6} + 249 q^{7} - 9 q^{8} - 36 q^{9} + O(q^{10}) \) \( 9352 q - 12 q^{2} - 20 q^{4} + 189 q^{5} - 18 q^{6} + 249 q^{7} - 9 q^{8} - 36 q^{9} + 227 q^{10} - 8319 q^{11} - 18 q^{12} + 14399 q^{13} + 43473 q^{14} - 31688 q^{16} - 58974 q^{17} - 18 q^{18} - 131958 q^{19} + 130797 q^{20} + 185967 q^{21} + 105648 q^{22} + 348141 q^{23} - 18 q^{24} - 248655 q^{25} - 27 q^{26} - 416961 q^{27} + 152103 q^{28} + 738435 q^{29} - 18 q^{30} + 856575 q^{31} - 308922 q^{32} + 154215 q^{33} + 584750 q^{34} - 2827824 q^{35} - 18 q^{36} - 449710 q^{37} - 252918 q^{38} - 1746145 q^{40} - 1148775 q^{41} + 4341267 q^{42} + 3417849 q^{43} + 8102439 q^{44} - 1459062 q^{45} - 4609881 q^{46} - 10838223 q^{47} - 9962685 q^{48} - 2329319 q^{49} + 1622208 q^{50} + 4677309 q^{51} + 7819721 q^{52} + 13126686 q^{53} + 17669844 q^{54} + 6249438 q^{55} + 3456189 q^{56} - 3123774 q^{57} - 6184369 q^{58} - 24381663 q^{59} - 28050651 q^{60} - 6908497 q^{61} - 25222635 q^{62} + 1006938 q^{63} + 9299113 q^{64} + 23522985 q^{65} + 33755958 q^{66} - 4308453 q^{67} - 4337502 q^{68} - 9814626 q^{69} + 5694489 q^{70} - 17658060 q^{71} - 18 q^{72} + 5372567 q^{73} - 1590663 q^{74} - 18281250 q^{75} - 18467166 q^{76} + 8574435 q^{77} + 19665 q^{78} - 18150243 q^{79} + 35458452 q^{81} + 18332246 q^{82} + 31933089 q^{83} - 19701 q^{84} + 20360164 q^{85} + 16265676 q^{86} - 92159568 q^{87} - 9815484 q^{88} - 177696609 q^{89} - 8220321 q^{90} + 2745957 q^{91} - 207789753 q^{92} + 157006170 q^{93} - 21667083 q^{94} + 82418124 q^{95} + 64345419 q^{96} - 58005493 q^{97} + 356672817 q^{98} - 156150594 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(324))\)

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 available newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
324.8.a \(\chi_{324}(1, \cdot)\) 324.8.a.a 3 1
324.8.a.b 3
324.8.a.c 7
324.8.a.d 7
324.8.a.e 8
324.8.b \(\chi_{324}(323, \cdot)\) n/a 164 1
324.8.e \(\chi_{324}(109, \cdot)\) 324.8.e.a 2 2
324.8.e.b 2
324.8.e.c 2
324.8.e.d 2
324.8.e.e 2
324.8.e.f 2
324.8.e.g 4
324.8.e.h 4
324.8.e.i 4
324.8.e.j 4
324.8.e.k 6
324.8.e.l 6
324.8.e.m 16
324.8.h \(\chi_{324}(107, \cdot)\) n/a 332 2
324.8.i \(\chi_{324}(37, \cdot)\) n/a 126 6
324.8.l \(\chi_{324}(35, \cdot)\) n/a 744 6
324.8.m \(\chi_{324}(13, \cdot)\) n/a 1134 18
324.8.p \(\chi_{324}(11, \cdot)\) n/a 6768 18

"n/a" means that newforms for that character have not been added to the database yet

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(324))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(324)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 10}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 9}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(36))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(54))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(81))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(108))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(162))\)\(^{\oplus 2}\)