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Properties

Label 288.8
Level 288
Weight 8
Dimension 7137
Nonzero newspaces 12
Sturm bound 36864
Trace bound 13

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Defining parameters

Level: \( N \) = \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 12 \)
Sturm bound: \(36864\)
Trace bound: \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(288))\).

Total New Old
Modular forms 16384 7227 9157
Cusp forms 15872 7137 8735
Eisenstein series 512 90 422

Trace form

\( 7137 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 290 q^{5} - 16 q^{6} + 678 q^{7} - 12 q^{8} - 24 q^{9} + O(q^{10}) \) \( 7137 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 290 q^{5} - 16 q^{6} + 678 q^{7} - 12 q^{8} - 24 q^{9} - 13036 q^{10} - 6 q^{11} - 16 q^{12} + 13606 q^{13} - 26204 q^{14} + 4362 q^{15} - 52792 q^{16} - 66106 q^{17} - 16 q^{18} + 121140 q^{19} + 81988 q^{20} - 4704 q^{21} - 374248 q^{22} - 434142 q^{23} - 16 q^{24} + 32767 q^{25} - 363992 q^{26} + 238032 q^{27} - 195496 q^{28} + 323638 q^{29} - 1040 q^{30} - 1161766 q^{31} + 535648 q^{32} - 269084 q^{33} - 201080 q^{34} + 816492 q^{35} - 730120 q^{36} + 2108966 q^{37} + 5109436 q^{38} + 146334 q^{39} - 2866336 q^{40} - 3794206 q^{41} - 5990976 q^{42} - 1139522 q^{43} - 1984756 q^{44} + 2210912 q^{45} + 8147388 q^{46} + 2586462 q^{47} + 10088712 q^{48} + 3865393 q^{49} + 326556 q^{50} - 1125980 q^{51} - 6236772 q^{52} - 13736130 q^{53} - 14891288 q^{54} - 4799288 q^{55} - 10509568 q^{56} - 720360 q^{57} + 16629912 q^{58} + 10005574 q^{59} + 17197080 q^{60} - 5728826 q^{61} - 18150816 q^{62} - 11485218 q^{63} + 16915512 q^{64} - 14576536 q^{65} - 16 q^{66} - 1164418 q^{67} - 9157688 q^{68} + 14060848 q^{69} - 8149488 q^{70} + 5269900 q^{71} - 16 q^{72} + 677598 q^{73} + 10931428 q^{74} - 2254012 q^{75} + 14443508 q^{76} - 39457084 q^{77} + 39347576 q^{78} - 5942886 q^{79} - 104949200 q^{80} - 9215800 q^{81} - 82802116 q^{82} + 43046454 q^{83} + 17056800 q^{84} + 6941676 q^{85} + 124258088 q^{86} - 83645306 q^{87} + 110752120 q^{88} - 25358550 q^{89} + 31733984 q^{90} + 16922112 q^{91} - 81318896 q^{92} - 16 q^{93} - 163612192 q^{94} + 171106312 q^{95} - 112869000 q^{96} + 7063342 q^{97} - 67942016 q^{98} + 33890246 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(288))\)

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
288.8.a \(\chi_{288}(1, \cdot)\) 288.8.a.a 1 1
288.8.a.b 1
288.8.a.c 1
288.8.a.d 1
288.8.a.e 1
288.8.a.f 2
288.8.a.g 2
288.8.a.h 2
288.8.a.i 2
288.8.a.j 2
288.8.a.k 2
288.8.a.l 2
288.8.a.m 2
288.8.a.n 2
288.8.a.o 2
288.8.a.p 2
288.8.a.q 4
288.8.a.r 4
288.8.c \(\chi_{288}(287, \cdot)\) 288.8.c.a 12 1
288.8.c.b 16
288.8.d \(\chi_{288}(145, \cdot)\) 288.8.d.a 2 1
288.8.d.b 6
288.8.d.c 12
288.8.d.d 14
288.8.f \(\chi_{288}(143, \cdot)\) 288.8.f.a 28 1
288.8.i \(\chi_{288}(97, \cdot)\) n/a 168 2
288.8.k \(\chi_{288}(73, \cdot)\) None 0 2
288.8.l \(\chi_{288}(71, \cdot)\) None 0 2
288.8.p \(\chi_{288}(47, \cdot)\) n/a 164 2
288.8.r \(\chi_{288}(49, \cdot)\) n/a 164 2
288.8.s \(\chi_{288}(95, \cdot)\) n/a 168 2
288.8.v \(\chi_{288}(37, \cdot)\) n/a 556 4
288.8.w \(\chi_{288}(35, \cdot)\) n/a 448 4
288.8.y \(\chi_{288}(23, \cdot)\) None 0 4
288.8.bb \(\chi_{288}(25, \cdot)\) None 0 4
288.8.bc \(\chi_{288}(13, \cdot)\) n/a 2672 8
288.8.bf \(\chi_{288}(11, \cdot)\) n/a 2672 8

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

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(288))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(288)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 15}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 10}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 9}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(32))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(72))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(96))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(144))\)\(^{\oplus 2}\)