# Properties

 Label 384.8 Level 384 Weight 8 Dimension 12048 Nonzero newspaces 10 Sturm bound 65536 Trace bound 25

## Defining parameters

 Level: $$N$$ = $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ = $$8$$ Nonzero newspaces: $$10$$ Sturm bound: $$65536$$ Trace bound: $$25$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{8}(\Gamma_1(384))$$.

Total New Old
Modular forms 28992 12144 16848
Cusp forms 28352 12048 16304
Eisenstein series 640 96 544

## Trace form

 $$12048 q - 12 q^{3} - 32 q^{4} - 16 q^{6} - 24 q^{7} - 20 q^{9} + O(q^{10})$$ $$12048 q - 12 q^{3} - 32 q^{4} - 16 q^{6} - 24 q^{7} - 20 q^{9} - 32 q^{10} - 16 q^{12} - 32 q^{13} - 8 q^{15} - 32 q^{16} - 16 q^{18} - 24 q^{19} + 8732 q^{21} - 32 q^{22} - 286832 q^{23} - 16 q^{24} + 128952 q^{25} + 476076 q^{27} - 32 q^{28} - 206752 q^{29} - 16 q^{30} - 1430000 q^{31} - 396096 q^{33} - 32 q^{34} + 1633008 q^{35} - 16 q^{36} + 1662272 q^{37} - 283956 q^{39} - 32 q^{40} - 3530272 q^{41} - 16 q^{42} + 732344 q^{43} + 312484 q^{45} - 32 q^{46} - 16 q^{48} - 6588392 q^{49} - 9269664 q^{50} + 3002064 q^{51} + 40819744 q^{52} + 7262528 q^{53} - 629872 q^{54} - 8382040 q^{55} - 42949088 q^{56} - 12408788 q^{57} - 41432576 q^{58} - 3671872 q^{59} + 8672816 q^{60} + 18239072 q^{61} + 41390016 q^{62} + 4000720 q^{63} + 91348576 q^{64} + 22830208 q^{65} + 15028400 q^{66} - 3881384 q^{67} - 35000256 q^{68} - 19170556 q^{69} - 143961248 q^{70} - 24697408 q^{71} - 16 q^{72} - 20273064 q^{73} + 16857568 q^{74} + 10948000 q^{75} + 155771616 q^{76} + 47657792 q^{77} + 58970576 q^{78} + 523728 q^{79} - 126217824 q^{80} + 27028744 q^{81} - 32 q^{82} - 16 q^{84} - 1250032 q^{85} - 40789124 q^{87} - 32 q^{88} - 16 q^{90} - 24 q^{91} + 28999520 q^{93} - 32 q^{94} - 16 q^{96} - 64 q^{97} + 9729716 q^{99} + O(q^{100})$$

## Decomposition of $$S_{8}^{\mathrm{new}}(\Gamma_1(384))$$

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
384.8.a $$\chi_{384}(1, \cdot)$$ 384.8.a.a 1 1
384.8.a.b 1
384.8.a.c 1
384.8.a.d 1
384.8.a.e 2
384.8.a.f 2
384.8.a.g 2
384.8.a.h 2
384.8.a.i 3
384.8.a.j 3
384.8.a.k 3
384.8.a.l 3
384.8.a.m 4
384.8.a.n 4
384.8.a.o 4
384.8.a.p 4
384.8.a.q 4
384.8.a.r 4
384.8.a.s 4
384.8.a.t 4
384.8.c $$\chi_{384}(383, \cdot)$$ n/a 112 1
384.8.d $$\chi_{384}(193, \cdot)$$ 384.8.d.a 6 1
384.8.d.b 6
384.8.d.c 8
384.8.d.d 8
384.8.d.e 12
384.8.d.f 16
384.8.f $$\chi_{384}(191, \cdot)$$ n/a 112 1
384.8.j $$\chi_{384}(97, \cdot)$$ n/a 112 2
384.8.k $$\chi_{384}(95, \cdot)$$ n/a 216 2
384.8.n $$\chi_{384}(49, \cdot)$$ n/a 224 4
384.8.o $$\chi_{384}(47, \cdot)$$ n/a 440 4
384.8.r $$\chi_{384}(25, \cdot)$$ None 0 8
384.8.s $$\chi_{384}(23, \cdot)$$ None 0 8
384.8.v $$\chi_{384}(13, \cdot)$$ n/a 3584 16
384.8.w $$\chi_{384}(11, \cdot)$$ n/a 7136 16

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

## Decomposition of $$S_{8}^{\mathrm{old}}(\Gamma_1(384))$$ into lower level spaces

$$S_{8}^{\mathrm{old}}(\Gamma_1(384)) \cong$$ $$S_{8}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 14}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 8}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 7}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 10}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 6}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 8}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 5}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 6}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 4}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 4}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(96))$$$$^{\oplus 3}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(128))$$$$^{\oplus 2}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(192))$$$$^{\oplus 2}$$