# Properties

 Label 384.4 Level 384 Weight 4 Dimension 5136 Nonzero newspaces 10 Sturm bound 32768 Trace bound 25

## Defining parameters

 Level: $$N$$ = $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$10$$ Sturm bound: $$32768$$ Trace bound: $$25$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(384))$$.

Total New Old
Modular forms 12608 5232 7376
Cusp forms 11968 5136 6832
Eisenstein series 640 96 544

## Trace form

 $$5136 q - 12 q^{3} - 32 q^{4} - 16 q^{6} - 24 q^{7} - 20 q^{9} + O(q^{10})$$ $$5136 q - 12 q^{3} - 32 q^{4} - 16 q^{6} - 24 q^{7} - 20 q^{9} - 32 q^{10} - 16 q^{12} - 32 q^{13} - 8 q^{15} - 32 q^{16} - 16 q^{18} - 24 q^{19} + 92 q^{21} - 32 q^{22} + 656 q^{23} - 16 q^{24} + 312 q^{25} - 276 q^{27} - 32 q^{28} - 800 q^{29} - 16 q^{30} - 1520 q^{31} - 960 q^{33} - 32 q^{34} - 912 q^{35} - 16 q^{36} - 64 q^{37} + 588 q^{39} - 32 q^{40} + 1888 q^{41} - 16 q^{42} + 1592 q^{43} + 484 q^{45} - 32 q^{46} - 16 q^{48} - 2792 q^{49} - 11424 q^{50} - 1392 q^{51} - 13280 q^{52} - 3008 q^{53} - 880 q^{54} - 600 q^{55} + 1568 q^{56} + 1324 q^{57} + 9472 q^{58} + 2752 q^{59} + 9776 q^{60} + 7264 q^{61} + 11712 q^{62} + 976 q^{63} + 24160 q^{64} + 7808 q^{65} + 10928 q^{66} + 4056 q^{67} + 8256 q^{68} + 1988 q^{69} + 8032 q^{70} + 448 q^{71} - 16 q^{72} - 3496 q^{73} - 10528 q^{74} - 2720 q^{75} - 23840 q^{76} - 7616 q^{77} - 14128 q^{78} - 5680 q^{79} - 20064 q^{80} - 3704 q^{81} - 32 q^{82} - 16 q^{84} - 2032 q^{85} + 1276 q^{87} - 32 q^{88} - 16 q^{90} - 24 q^{91} + 4256 q^{93} - 32 q^{94} - 16 q^{96} - 64 q^{97} + 5300 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(384))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
384.4.a $$\chi_{384}(1, \cdot)$$ 384.4.a.a 1 1
384.4.a.b 1
384.4.a.c 1
384.4.a.d 1
384.4.a.e 1
384.4.a.f 1
384.4.a.g 1
384.4.a.h 1
384.4.a.i 2
384.4.a.j 2
384.4.a.k 2
384.4.a.l 2
384.4.a.m 2
384.4.a.n 2
384.4.a.o 2
384.4.a.p 2
384.4.c $$\chi_{384}(383, \cdot)$$ 384.4.c.a 12 1
384.4.c.b 12
384.4.c.c 12
384.4.c.d 12
384.4.d $$\chi_{384}(193, \cdot)$$ 384.4.d.a 2 1
384.4.d.b 2
384.4.d.c 4
384.4.d.d 4
384.4.d.e 4
384.4.d.f 8
384.4.f $$\chi_{384}(191, \cdot)$$ 384.4.f.a 4 1
384.4.f.b 4
384.4.f.c 4
384.4.f.d 4
384.4.f.e 4
384.4.f.f 4
384.4.f.g 8
384.4.f.h 8
384.4.f.i 8
384.4.j $$\chi_{384}(97, \cdot)$$ 384.4.j.a 24 2
384.4.j.b 24
384.4.k $$\chi_{384}(95, \cdot)$$ 384.4.k.a 44 2
384.4.k.b 44
384.4.n $$\chi_{384}(49, \cdot)$$ 384.4.n.a 96 4
384.4.o $$\chi_{384}(47, \cdot)$$ n/a 184 4
384.4.r $$\chi_{384}(25, \cdot)$$ None 0 8
384.4.s $$\chi_{384}(23, \cdot)$$ None 0 8
384.4.v $$\chi_{384}(13, \cdot)$$ n/a 1536 16
384.4.w $$\chi_{384}(11, \cdot)$$ n/a 3040 16

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(384))$$ into lower level spaces

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