## Defining parameters

 Level: $$N$$ = $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$24$$ Sturm bound: $$589824$$ Trace bound: $$49$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(2304))$$.

Total New Old
Modular forms 150272 66132 84140
Cusp forms 144641 65196 79445
Eisenstein series 5631 936 4695

## Trace form

 $$65196q - 96q^{2} - 96q^{3} - 96q^{4} - 96q^{5} - 128q^{6} - 72q^{7} - 96q^{8} - 160q^{9} + O(q^{10})$$ $$65196q - 96q^{2} - 96q^{3} - 96q^{4} - 96q^{5} - 128q^{6} - 72q^{7} - 96q^{8} - 160q^{9} - 288q^{10} - 72q^{11} - 128q^{12} - 96q^{13} - 96q^{14} - 96q^{15} - 96q^{16} - 144q^{17} - 128q^{18} - 216q^{19} - 96q^{20} - 128q^{21} - 96q^{22} - 72q^{23} - 128q^{24} - 120q^{25} - 96q^{26} - 96q^{27} - 288q^{28} - 96q^{29} - 128q^{30} - 80q^{31} - 96q^{32} - 224q^{33} - 96q^{34} - 72q^{35} - 128q^{36} - 288q^{37} - 96q^{38} - 96q^{39} - 96q^{40} - 120q^{41} - 128q^{42} - 72q^{43} - 96q^{44} - 128q^{45} - 288q^{46} - 72q^{47} - 128q^{48} - 172q^{49} - 96q^{50} - 96q^{51} - 96q^{52} - 128q^{53} - 128q^{54} - 280q^{55} - 96q^{56} - 160q^{57} - 96q^{58} - 136q^{59} - 128q^{60} - 160q^{61} - 96q^{62} - 96q^{63} - 288q^{64} - 280q^{65} - 128q^{66} - 152q^{67} - 96q^{68} - 128q^{69} - 96q^{70} - 136q^{71} - 128q^{72} - 424q^{73} - 96q^{74} - 96q^{75} - 96q^{76} - 128q^{77} - 128q^{78} - 104q^{79} - 96q^{80} - 192q^{81} - 288q^{82} - 72q^{83} - 128q^{84} - 136q^{85} - 96q^{86} - 96q^{87} - 96q^{88} - 120q^{89} - 128q^{90} - 216q^{91} - 96q^{92} - 128q^{93} - 96q^{94} - 48q^{95} - 128q^{96} - 168q^{97} - 96q^{98} - 96q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(2304))$$

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 the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
2304.2.a $$\chi_{2304}(1, \cdot)$$ 2304.2.a.a 1 1
2304.2.a.b 1
2304.2.a.c 1
2304.2.a.d 1
2304.2.a.e 1
2304.2.a.f 1
2304.2.a.g 1
2304.2.a.h 1
2304.2.a.i 1
2304.2.a.j 1
2304.2.a.k 1
2304.2.a.l 1
2304.2.a.m 1
2304.2.a.n 1
2304.2.a.o 1
2304.2.a.p 1
2304.2.a.q 2
2304.2.a.r 2
2304.2.a.s 2
2304.2.a.t 2
2304.2.a.u 2
2304.2.a.v 2
2304.2.a.w 2
2304.2.a.x 2
2304.2.a.y 2
2304.2.a.z 4
2304.2.c $$\chi_{2304}(2303, \cdot)$$ 2304.2.c.a 2 1
2304.2.c.b 2
2304.2.c.c 2
2304.2.c.d 2
2304.2.c.e 2
2304.2.c.f 2
2304.2.c.g 2
2304.2.c.h 2
2304.2.c.i 8
2304.2.c.j 8
2304.2.d $$\chi_{2304}(1153, \cdot)$$ 2304.2.d.a 2 1
2304.2.d.b 2
2304.2.d.c 2
2304.2.d.d 2
2304.2.d.e 2
2304.2.d.f 2
2304.2.d.g 2
2304.2.d.h 2
2304.2.d.i 2
2304.2.d.j 2
2304.2.d.k 2
2304.2.d.l 2
2304.2.d.m 2
2304.2.d.n 2
2304.2.d.o 2
2304.2.d.p 2
2304.2.d.q 2
2304.2.d.r 2
2304.2.d.s 2
2304.2.f $$\chi_{2304}(1151, \cdot)$$ 2304.2.f.a 4 1
2304.2.f.b 4
2304.2.f.c 4
2304.2.f.d 4
2304.2.f.e 4
2304.2.f.f 4
2304.2.f.g 4
2304.2.f.h 4
2304.2.i $$\chi_{2304}(769, \cdot)$$ n/a 184 2
2304.2.k $$\chi_{2304}(577, \cdot)$$ 2304.2.k.a 4 2
2304.2.k.b 4
2304.2.k.c 4
2304.2.k.d 4
2304.2.k.e 8
2304.2.k.f 8
2304.2.k.g 8
2304.2.k.h 8
2304.2.k.i 8
2304.2.k.j 8
2304.2.k.k 8
2304.2.k.l 8
2304.2.l $$\chi_{2304}(575, \cdot)$$ 2304.2.l.a 8 2
2304.2.l.b 8
2304.2.l.c 8
2304.2.l.d 8
2304.2.l.e 8
2304.2.l.f 8
2304.2.l.g 8
2304.2.l.h 8
2304.2.p $$\chi_{2304}(383, \cdot)$$ n/a 184 2
2304.2.r $$\chi_{2304}(385, \cdot)$$ n/a 184 2
2304.2.s $$\chi_{2304}(767, \cdot)$$ n/a 184 2
2304.2.v $$\chi_{2304}(289, \cdot)$$ n/a 152 4
2304.2.w $$\chi_{2304}(287, \cdot)$$ n/a 128 4
2304.2.y $$\chi_{2304}(191, \cdot)$$ n/a 384 4
2304.2.bb $$\chi_{2304}(193, \cdot)$$ n/a 384 4
2304.2.bd $$\chi_{2304}(145, \cdot)$$ n/a 312 8
2304.2.be $$\chi_{2304}(143, \cdot)$$ n/a 256 8
2304.2.bg $$\chi_{2304}(97, \cdot)$$ n/a 736 8
2304.2.bj $$\chi_{2304}(95, \cdot)$$ n/a 736 8
2304.2.bl $$\chi_{2304}(73, \cdot)$$ None 0 16
2304.2.bm $$\chi_{2304}(71, \cdot)$$ None 0 16
2304.2.bp $$\chi_{2304}(47, \cdot)$$ n/a 1504 16
2304.2.bq $$\chi_{2304}(49, \cdot)$$ n/a 1504 16
2304.2.bt $$\chi_{2304}(37, \cdot)$$ n/a 5088 32
2304.2.bu $$\chi_{2304}(35, \cdot)$$ n/a 4096 32
2304.2.bw $$\chi_{2304}(23, \cdot)$$ None 0 32
2304.2.bz $$\chi_{2304}(25, \cdot)$$ None 0 32
2304.2.ca $$\chi_{2304}(13, \cdot)$$ n/a 24448 64
2304.2.cd $$\chi_{2304}(11, \cdot)$$ n/a 24448 64

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(2304))$$ into lower level spaces

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