## Defining parameters

 Level: $$N$$ = $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$18$$ Sturm bound: $$124416$$ Trace bound: $$29$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(864))$$.

Total New Old
Modular forms 42432 18592 23840
Cusp forms 40512 18272 22240
Eisenstein series 1920 320 1600

## Trace form

 $$18272q - 32q^{2} - 36q^{3} - 56q^{4} - 32q^{5} - 48q^{6} - 42q^{7} - 32q^{8} - 72q^{9} + O(q^{10})$$ $$18272q - 32q^{2} - 36q^{3} - 56q^{4} - 32q^{5} - 48q^{6} - 42q^{7} - 32q^{8} - 72q^{9} - 56q^{10} - 22q^{11} - 48q^{12} - 56q^{13} - 32q^{14} - 36q^{15} - 56q^{16} + 32q^{17} - 48q^{18} - 10q^{19} - 32q^{20} - 48q^{21} - 56q^{22} - 26q^{23} - 48q^{24} - 116q^{25} - 32q^{26} - 36q^{27} - 128q^{28} - 128q^{29} - 48q^{30} - 106q^{31} - 32q^{32} - 120q^{33} - 56q^{34} - 134q^{35} - 48q^{36} - 216q^{37} - 536q^{38} - 36q^{39} - 504q^{40} - 240q^{41} - 48q^{42} - 106q^{43} - 296q^{44} - 48q^{45} - 120q^{46} - 34q^{47} - 48q^{48} + 36q^{49} + 280q^{50} - 90q^{51} + 264q^{52} + 256q^{53} - 48q^{54} - 68q^{55} + 688q^{56} - 408q^{57} + 648q^{58} - 598q^{59} - 48q^{60} - 472q^{61} + 360q^{62} - 516q^{63} - 56q^{64} - 940q^{65} - 48q^{66} - 298q^{67} + 96q^{68} - 240q^{69} - 56q^{70} - 14q^{71} - 48q^{72} + 12q^{73} - 32q^{74} + 300q^{75} - 56q^{76} + 928q^{77} - 48q^{78} - 170q^{79} + 928q^{80} + 936q^{81} + 912q^{82} - 22q^{83} - 48q^{84} + 840q^{85} + 904q^{86} + 540q^{87} + 504q^{88} + 480q^{89} - 48q^{90} - 326q^{91} + 424q^{92} - 48q^{93} + 104q^{94} + 282q^{95} - 48q^{96} - 76q^{97} - 408q^{98} - 36q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(864))$$

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
864.3.b $$\chi_{864}(271, \cdot)$$ 864.3.b.a 16 1
864.3.b.b 16
864.3.e $$\chi_{864}(161, \cdot)$$ 864.3.e.a 4 1
864.3.e.b 4
864.3.e.c 4
864.3.e.d 4
864.3.e.e 8
864.3.e.f 8
864.3.g $$\chi_{864}(703, \cdot)$$ 864.3.g.a 8 1
864.3.g.b 8
864.3.g.c 8
864.3.g.d 8
864.3.h $$\chi_{864}(593, \cdot)$$ 864.3.h.a 2 1
864.3.h.b 2
864.3.h.c 6
864.3.h.d 6
864.3.h.e 8
864.3.h.f 8
864.3.j $$\chi_{864}(377, \cdot)$$ None 0 2
864.3.m $$\chi_{864}(55, \cdot)$$ None 0 2
864.3.n $$\chi_{864}(17, \cdot)$$ 864.3.n.a 44 2
864.3.o $$\chi_{864}(127, \cdot)$$ 864.3.o.a 4 2
864.3.o.b 20
864.3.o.c 24
864.3.q $$\chi_{864}(449, \cdot)$$ 864.3.q.a 24 2
864.3.q.b 24
864.3.t $$\chi_{864}(559, \cdot)$$ 864.3.t.a 4 2
864.3.t.b 40
864.3.u $$\chi_{864}(163, \cdot)$$ n/a 512 4
864.3.x $$\chi_{864}(53, \cdot)$$ n/a 512 4
864.3.ba $$\chi_{864}(199, \cdot)$$ None 0 4
864.3.bb $$\chi_{864}(89, \cdot)$$ None 0 4
864.3.bd $$\chi_{864}(79, \cdot)$$ n/a 420 6
864.3.be $$\chi_{864}(31, \cdot)$$ n/a 432 6
864.3.bg $$\chi_{864}(65, \cdot)$$ n/a 432 6
864.3.bj $$\chi_{864}(113, \cdot)$$ n/a 420 6
864.3.bl $$\chi_{864}(19, \cdot)$$ n/a 752 8
864.3.bm $$\chi_{864}(125, \cdot)$$ n/a 752 8
864.3.bp $$\chi_{864}(7, \cdot)$$ None 0 12
864.3.bq $$\chi_{864}(41, \cdot)$$ None 0 12
864.3.bs $$\chi_{864}(5, \cdot)$$ n/a 6864 24
864.3.bv $$\chi_{864}(43, \cdot)$$ n/a 6864 24

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

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(864))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(864)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 10}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 9}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 5}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(96))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(108))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(144))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(216))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(288))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(432))$$$$^{\oplus 2}$$