## Defining parameters

 Level: $$N$$ = $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$24$$ Sturm bound: $$884736$$ Trace bound: $$49$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(2304))$$.

Total New Old
Modular forms 297728 131796 165932
Cusp forms 292096 130860 161236
Eisenstein series 5632 936 4696

## Trace form

 $$130860q - 96q^{2} - 96q^{3} - 96q^{4} - 96q^{5} - 128q^{6} - 72q^{7} - 96q^{8} - 160q^{9} + O(q^{10})$$ $$130860q - 96q^{2} - 96q^{3} - 96q^{4} - 96q^{5} - 128q^{6} - 72q^{7} - 96q^{8} - 160q^{9} - 288q^{10} - 72q^{11} - 128q^{12} - 96q^{13} - 96q^{14} - 96q^{15} - 96q^{16} - 144q^{17} - 128q^{18} - 216q^{19} - 96q^{20} - 128q^{21} - 96q^{22} - 72q^{23} - 128q^{24} - 120q^{25} - 96q^{26} - 96q^{27} - 288q^{28} - 96q^{29} - 128q^{30} - 64q^{31} - 96q^{32} - 224q^{33} - 96q^{34} - 72q^{35} - 128q^{36} - 288q^{37} - 96q^{38} - 96q^{39} - 96q^{40} - 120q^{41} - 128q^{42} - 72q^{43} - 96q^{44} - 128q^{45} - 288q^{46} - 72q^{47} - 128q^{48} + 52q^{49} - 96q^{50} - 96q^{51} - 96q^{52} + 224q^{53} - 128q^{54} + 296q^{55} - 96q^{56} - 160q^{57} - 96q^{58} + 184q^{59} - 128q^{60} + 32q^{61} - 96q^{62} - 96q^{63} - 288q^{64} - 344q^{65} - 128q^{66} - 392q^{67} - 96q^{68} - 128q^{69} - 96q^{70} - 584q^{71} - 128q^{72} - 1000q^{73} - 96q^{74} - 96q^{75} - 96q^{76} - 544q^{77} - 128q^{78} - 584q^{79} - 96q^{80} - 192q^{81} - 288q^{82} - 72q^{83} - 128q^{84} - 296q^{85} - 96q^{86} - 96q^{87} - 96q^{88} - 120q^{89} - 128q^{90} - 216q^{91} - 96q^{92} - 128q^{93} - 96q^{94} - 96q^{95} - 128q^{96} - 168q^{97} - 96q^{98} - 96q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(2304))$$

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
2304.3.b $$\chi_{2304}(127, \cdot)$$ 2304.3.b.a 2 1
2304.3.b.b 2
2304.3.b.c 2
2304.3.b.d 2
2304.3.b.e 2
2304.3.b.f 2
2304.3.b.g 2
2304.3.b.h 2
2304.3.b.i 2
2304.3.b.j 4
2304.3.b.k 4
2304.3.b.l 4
2304.3.b.m 4
2304.3.b.n 4
2304.3.b.o 4
2304.3.b.p 4
2304.3.b.q 8
2304.3.b.r 8
2304.3.b.s 8
2304.3.b.t 8
2304.3.e $$\chi_{2304}(1025, \cdot)$$ 2304.3.e.a 2 1
2304.3.e.b 2
2304.3.e.c 2
2304.3.e.d 2
2304.3.e.e 4
2304.3.e.f 4
2304.3.e.g 4
2304.3.e.h 4
2304.3.e.i 4
2304.3.e.j 4
2304.3.e.k 4
2304.3.e.l 4
2304.3.e.m 8
2304.3.e.n 8
2304.3.e.o 8
2304.3.g $$\chi_{2304}(1279, \cdot)$$ 2304.3.g.a 1 1
2304.3.g.b 1
2304.3.g.c 1
2304.3.g.d 1
2304.3.g.e 1
2304.3.g.f 1
2304.3.g.g 2
2304.3.g.h 2
2304.3.g.i 2
2304.3.g.j 2
2304.3.g.k 2
2304.3.g.l 2
2304.3.g.m 2
2304.3.g.n 2
2304.3.g.o 4
2304.3.g.p 4
2304.3.g.q 4
2304.3.g.r 4
2304.3.g.s 4
2304.3.g.t 4
2304.3.g.u 4
2304.3.g.v 4
2304.3.g.w 4
2304.3.g.x 4
2304.3.g.y 8
2304.3.g.z 8
2304.3.h $$\chi_{2304}(2177, \cdot)$$ 2304.3.h.a 4 1
2304.3.h.b 4
2304.3.h.c 4
2304.3.h.d 4
2304.3.h.e 4
2304.3.h.f 4
2304.3.h.g 4
2304.3.h.h 4
2304.3.h.i 8
2304.3.h.j 8
2304.3.h.k 8
2304.3.h.l 8
2304.3.j $$\chi_{2304}(449, \cdot)$$ n/a 128 2
2304.3.m $$\chi_{2304}(703, \cdot)$$ n/a 160 2
2304.3.n $$\chi_{2304}(641, \cdot)$$ n/a 376 2
2304.3.o $$\chi_{2304}(511, \cdot)$$ n/a 376 2
2304.3.q $$\chi_{2304}(257, \cdot)$$ n/a 376 2
2304.3.t $$\chi_{2304}(895, \cdot)$$ n/a 376 2
2304.3.u $$\chi_{2304}(415, \cdot)$$ n/a 312 4
2304.3.x $$\chi_{2304}(161, \cdot)$$ n/a 256 4
2304.3.z $$\chi_{2304}(319, \cdot)$$ n/a 768 4
2304.3.ba $$\chi_{2304}(65, \cdot)$$ n/a 768 4
2304.3.bc $$\chi_{2304}(17, \cdot)$$ n/a 512 8
2304.3.bf $$\chi_{2304}(271, \cdot)$$ n/a 632 8
2304.3.bh $$\chi_{2304}(31, \cdot)$$ n/a 1504 8
2304.3.bi $$\chi_{2304}(353, \cdot)$$ n/a 1504 8
2304.3.bk $$\chi_{2304}(55, \cdot)$$ None 0 16
2304.3.bn $$\chi_{2304}(89, \cdot)$$ None 0 16
2304.3.bo $$\chi_{2304}(79, \cdot)$$ n/a 3040 16
2304.3.br $$\chi_{2304}(113, \cdot)$$ n/a 3040 16
2304.3.bs $$\chi_{2304}(53, \cdot)$$ n/a 8192 32
2304.3.bv $$\chi_{2304}(19, \cdot)$$ n/a 10208 32
2304.3.bx $$\chi_{2304}(41, \cdot)$$ None 0 32
2304.3.by $$\chi_{2304}(7, \cdot)$$ None 0 32
2304.3.cb $$\chi_{2304}(5, \cdot)$$ n/a 49024 64
2304.3.cc $$\chi_{2304}(43, \cdot)$$ n/a 49024 64

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

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(2304))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(2304)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 18}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 9}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 14}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 15}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 7}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 10}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 9}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(96))$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(128))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(144))$$$$^{\oplus 5}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(192))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(256))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(288))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(384))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(576))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(768))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(1152))$$$$^{\oplus 2}$$