## Defining parameters

 Level: $$N$$ = $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$24$$ Sturm bound: $$1179648$$ Trace bound: $$49$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(2304))$$.

Total New Old
Modular forms 445184 197460 247724
Cusp forms 439552 196524 243028
Eisenstein series 5632 936 4696

## Trace form

 $$196524 q - 96 q^{2} - 96 q^{3} - 96 q^{4} - 96 q^{5} - 128 q^{6} - 72 q^{7} - 96 q^{8} - 160 q^{9} + O(q^{10})$$ $$196524 q - 96 q^{2} - 96 q^{3} - 96 q^{4} - 96 q^{5} - 128 q^{6} - 72 q^{7} - 96 q^{8} - 160 q^{9} - 288 q^{10} - 72 q^{11} - 128 q^{12} - 96 q^{13} - 96 q^{14} - 96 q^{15} - 96 q^{16} - 144 q^{17} - 128 q^{18} - 216 q^{19} - 96 q^{20} - 128 q^{21} - 96 q^{22} - 72 q^{23} - 128 q^{24} - 120 q^{25} - 96 q^{26} - 96 q^{27} - 288 q^{28} - 96 q^{29} - 128 q^{30} - 80 q^{31} - 96 q^{32} - 224 q^{33} - 96 q^{34} - 72 q^{35} - 128 q^{36} - 288 q^{37} - 96 q^{38} - 96 q^{39} - 96 q^{40} - 120 q^{41} - 128 q^{42} - 72 q^{43} - 96 q^{44} - 128 q^{45} - 288 q^{46} - 72 q^{47} - 128 q^{48} - 1516 q^{49} - 96 q^{50} - 96 q^{51} - 96 q^{52} - 1600 q^{53} - 128 q^{54} - 792 q^{55} - 96 q^{56} - 160 q^{57} - 96 q^{58} + 2680 q^{59} - 128 q^{60} + 3552 q^{61} - 96 q^{62} - 96 q^{63} - 288 q^{64} + 3688 q^{65} - 128 q^{66} + 4008 q^{67} - 96 q^{68} - 128 q^{69} - 96 q^{70} + 376 q^{71} - 128 q^{72} - 2088 q^{73} - 96 q^{74} - 96 q^{75} - 96 q^{76} - 3904 q^{77} - 128 q^{78} - 5736 q^{79} - 96 q^{80} - 192 q^{81} - 288 q^{82} - 72 q^{83} - 128 q^{84} - 1096 q^{85} - 96 q^{86} - 96 q^{87} - 96 q^{88} - 120 q^{89} - 128 q^{90} - 216 q^{91} - 96 q^{92} - 128 q^{93} - 96 q^{94} - 48 q^{95} - 128 q^{96} - 168 q^{97} - 96 q^{98} - 96 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(2304))$$

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
2304.4.a $$\chi_{2304}(1, \cdot)$$ 2304.4.a.a 1 1
2304.4.a.b 1
2304.4.a.c 1
2304.4.a.d 1
2304.4.a.e 1
2304.4.a.f 1
2304.4.a.g 1
2304.4.a.h 1
2304.4.a.i 1
2304.4.a.j 1
2304.4.a.k 1
2304.4.a.l 1
2304.4.a.m 1
2304.4.a.n 1
2304.4.a.o 1
2304.4.a.p 1
2304.4.a.q 2
2304.4.a.r 2
2304.4.a.s 2
2304.4.a.t 2
2304.4.a.u 2
2304.4.a.v 2
2304.4.a.w 2
2304.4.a.x 2
2304.4.a.y 2
2304.4.a.z 2
2304.4.a.ba 2
2304.4.a.bb 2
2304.4.a.bc 2
2304.4.a.bd 2
2304.4.a.be 2
2304.4.a.bf 2
2304.4.a.bg 2
2304.4.a.bh 2
2304.4.a.bi 2
2304.4.a.bj 2
2304.4.a.bk 2
2304.4.a.bl 2
2304.4.a.bm 2
2304.4.a.bn 2
2304.4.a.bo 2
2304.4.a.bp 2
2304.4.a.bq 2
2304.4.a.br 2
2304.4.a.bs 2
2304.4.a.bt 3
2304.4.a.bu 3
2304.4.a.bv 3
2304.4.a.bw 3
2304.4.a.bx 4
2304.4.a.by 4
2304.4.a.bz 4
2304.4.a.ca 4
2304.4.a.cb 4
2304.4.a.cc 4
2304.4.a.cd 8
2304.4.c $$\chi_{2304}(2303, \cdot)$$ 2304.4.c.a 2 1
2304.4.c.b 2
2304.4.c.c 2
2304.4.c.d 2
2304.4.c.e 4
2304.4.c.f 4
2304.4.c.g 6
2304.4.c.h 6
2304.4.c.i 6
2304.4.c.j 6
2304.4.c.k 8
2304.4.c.l 8
2304.4.c.m 16
2304.4.c.n 24
2304.4.d $$\chi_{2304}(1153, \cdot)$$ n/a 118 1
2304.4.f $$\chi_{2304}(1151, \cdot)$$ 2304.4.f.a 4 1
2304.4.f.b 4
2304.4.f.c 4
2304.4.f.d 4
2304.4.f.e 8
2304.4.f.f 8
2304.4.f.g 8
2304.4.f.h 8
2304.4.f.i 12
2304.4.f.j 12
2304.4.f.k 12
2304.4.f.l 12
2304.4.i $$\chi_{2304}(769, \cdot)$$ n/a 568 2
2304.4.k $$\chi_{2304}(577, \cdot)$$ n/a 240 2
2304.4.l $$\chi_{2304}(575, \cdot)$$ n/a 192 2
2304.4.p $$\chi_{2304}(383, \cdot)$$ n/a 568 2
2304.4.r $$\chi_{2304}(385, \cdot)$$ n/a 568 2
2304.4.s $$\chi_{2304}(767, \cdot)$$ n/a 568 2
2304.4.v $$\chi_{2304}(289, \cdot)$$ n/a 472 4
2304.4.w $$\chi_{2304}(287, \cdot)$$ n/a 384 4
2304.4.y $$\chi_{2304}(191, \cdot)$$ n/a 1152 4
2304.4.bb $$\chi_{2304}(193, \cdot)$$ n/a 1152 4
2304.4.bd $$\chi_{2304}(145, \cdot)$$ n/a 952 8
2304.4.be $$\chi_{2304}(143, \cdot)$$ n/a 768 8
2304.4.bg $$\chi_{2304}(97, \cdot)$$ n/a 2272 8
2304.4.bj $$\chi_{2304}(95, \cdot)$$ n/a 2272 8
2304.4.bl $$\chi_{2304}(73, \cdot)$$ None 0 16
2304.4.bm $$\chi_{2304}(71, \cdot)$$ None 0 16
2304.4.bp $$\chi_{2304}(47, \cdot)$$ n/a 4576 16
2304.4.bq $$\chi_{2304}(49, \cdot)$$ n/a 4576 16
2304.4.bt $$\chi_{2304}(37, \cdot)$$ n/a 15328 32
2304.4.bu $$\chi_{2304}(35, \cdot)$$ n/a 12288 32
2304.4.bw $$\chi_{2304}(23, \cdot)$$ None 0 32
2304.4.bz $$\chi_{2304}(25, \cdot)$$ None 0 32
2304.4.ca $$\chi_{2304}(13, \cdot)$$ n/a 73600 64
2304.4.cd $$\chi_{2304}(11, \cdot)$$ n/a 73600 64

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(2304))$$ into lower level spaces

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