# Properties

 Label 384.9 Level 384 Weight 9 Dimension 13776 Nonzero newspaces 10 Sturm bound 73728 Trace bound 25

## Defining parameters

 Level: $$N$$ = $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ = $$9$$ Nonzero newspaces: $$10$$ Sturm bound: $$73728$$ Trace bound: $$25$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{9}(\Gamma_1(384))$$.

Total New Old
Modular forms 33088 13872 19216
Cusp forms 32448 13776 18672
Eisenstein series 640 96 544

## Trace form

 $$13776 q - 12 q^{3} - 32 q^{4} - 16 q^{6} - 24 q^{7} - 20 q^{9} + O(q^{10})$$ $$13776 q - 12 q^{3} - 32 q^{4} - 16 q^{6} - 24 q^{7} - 20 q^{9} - 32 q^{10} - 16 q^{12} - 32 q^{13} - 16 q^{15} - 32 q^{16} - 16 q^{18} - 24 q^{19} - 26260 q^{21} - 32 q^{22} + 1691136 q^{23} - 16 q^{24} - 2833192 q^{25} - 51084 q^{27} - 32 q^{28} + 4264704 q^{29} - 16 q^{30} - 16 q^{31} - 6610720 q^{33} - 32 q^{34} - 4831488 q^{35} - 16 q^{36} + 9440992 q^{37} + 7650036 q^{39} - 32 q^{40} - 17498880 q^{41} - 16 q^{42} - 3719448 q^{43} + 1562484 q^{45} - 32 q^{46} - 16 q^{48} + 46118360 q^{49} - 20580768 q^{50} - 27697800 q^{51} - 183792608 q^{52} + 21434880 q^{53} + 39680912 q^{54} + 92653544 q^{55} + 227499552 q^{56} + 12400108 q^{57} - 203505152 q^{58} - 89877504 q^{59} - 327416272 q^{60} - 195808288 q^{61} - 175726656 q^{62} + 8 q^{63} + 387071584 q^{64} + 239533056 q^{65} + 454631600 q^{66} + 186760936 q^{67} + 327458880 q^{68} + 34572404 q^{69} - 371559584 q^{70} - 159664128 q^{71} - 16 q^{72} - 502953000 q^{73} - 680115744 q^{74} - 24274568 q^{75} + 637215456 q^{76} + 379857408 q^{77} + 735959504 q^{78} + 144406496 q^{79} - 808638048 q^{80} - 182266136 q^{81} - 32 q^{82} - 16 q^{84} - 6250032 q^{85} + 149712628 q^{87} - 32 q^{88} - 16 q^{90} - 24 q^{91} - 276511888 q^{93} - 32 q^{94} - 16 q^{96} - 64 q^{97} + 630017012 q^{99} + O(q^{100})$$

## Decomposition of $$S_{9}^{\mathrm{new}}(\Gamma_1(384))$$

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
384.9.b $$\chi_{384}(319, \cdot)$$ 384.9.b.a 8 1
384.9.b.b 8
384.9.b.c 16
384.9.b.d 32
384.9.e $$\chi_{384}(257, \cdot)$$ n/a 128 1
384.9.g $$\chi_{384}(127, \cdot)$$ 384.9.g.a 32 1
384.9.g.b 32
384.9.h $$\chi_{384}(65, \cdot)$$ n/a 128 1
384.9.i $$\chi_{384}(161, \cdot)$$ n/a 248 2
384.9.l $$\chi_{384}(31, \cdot)$$ n/a 128 2
384.9.m $$\chi_{384}(79, \cdot)$$ n/a 256 4
384.9.p $$\chi_{384}(17, \cdot)$$ n/a 504 4
384.9.q $$\chi_{384}(41, \cdot)$$ None 0 8
384.9.t $$\chi_{384}(7, \cdot)$$ None 0 8
384.9.u $$\chi_{384}(19, \cdot)$$ n/a 4096 16
384.9.x $$\chi_{384}(5, \cdot)$$ n/a 8160 16

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

## Decomposition of $$S_{9}^{\mathrm{old}}(\Gamma_1(384))$$ into lower level spaces

$$S_{9}^{\mathrm{old}}(\Gamma_1(384)) \cong$$ $$S_{9}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 8}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 12}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 7}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 10}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 6}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 8}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 5}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 6}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 4}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 4}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(96))$$$$^{\oplus 3}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(128))$$$$^{\oplus 2}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(\Gamma_1(192))$$$$^{\oplus 2}$$