# Properties

 Label 384.10 Level 384 Weight 10 Dimension 15504 Nonzero newspaces 10 Sturm bound 81920 Trace bound 25

## Defining parameters

 Level: $$N$$ = $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ = $$10$$ Nonzero newspaces: $$10$$ Sturm bound: $$81920$$ Trace bound: $$25$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{10}(\Gamma_1(384))$$.

Total New Old
Modular forms 37184 15600 21584
Cusp forms 36544 15504 21040
Eisenstein series 640 96 544

## Trace form

 $$15504 q - 12 q^{3} - 32 q^{4} - 16 q^{6} - 24 q^{7} - 20 q^{9} + O(q^{10})$$ $$15504 q - 12 q^{3} - 32 q^{4} - 16 q^{6} - 24 q^{7} - 20 q^{9} - 32 q^{10} - 16 q^{12} - 32 q^{13} - 8 q^{15} - 32 q^{16} - 16 q^{18} - 24 q^{19} + 78716 q^{21} - 32 q^{22} - 6931952 q^{23} - 16 q^{24} + 13774072 q^{25} - 12650100 q^{27} - 32 q^{28} - 2533600 q^{29} - 16 q^{30} + 44328976 q^{31} - 151296 q^{33} - 32 q^{34} - 76481232 q^{35} - 16 q^{36} - 4859136 q^{37} + 72274188 q^{39} - 32 q^{40} - 30247648 q^{41} - 16 q^{42} - 76124936 q^{43} + 7812484 q^{45} - 32 q^{46} - 16 q^{48} - 322828904 q^{49} + 731033184 q^{50} + 180107376 q^{51} - 100910048 q^{52} - 599262784 q^{53} - 323116144 q^{54} - 280217688 q^{55} + 812652064 q^{56} + 632898988 q^{57} + 3420613120 q^{58} + 576150464 q^{59} + 461878832 q^{60} - 720968864 q^{61} - 2390454336 q^{62} - 252047408 q^{63} - 5459239328 q^{64} - 1004897408 q^{65} + 372019376 q^{66} + 1007094936 q^{67} + 4017311808 q^{68} + 1529361956 q^{69} + 7476368224 q^{70} + 476202944 q^{71} - 16 q^{72} - 3549598888 q^{73} - 8728733216 q^{74} - 1162395584 q^{75} - 3554843936 q^{76} + 1648353728 q^{77} + 6439686608 q^{78} + 2269128656 q^{79} - 3649420896 q^{80} + 362395336 q^{81} - 32 q^{82} - 16 q^{84} - 31250032 q^{85} + 2350951612 q^{87} - 32 q^{88} - 16 q^{90} - 24 q^{91} - 2143933120 q^{93} - 32 q^{94} - 16 q^{96} - 64 q^{97} + 1556825012 q^{99} + O(q^{100})$$

## Decomposition of $$S_{10}^{\mathrm{new}}(\Gamma_1(384))$$

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
384.10.a $$\chi_{384}(1, \cdot)$$ 384.10.a.a 4 1
384.10.a.b 4
384.10.a.c 4
384.10.a.d 4
384.10.a.e 4
384.10.a.f 4
384.10.a.g 4
384.10.a.h 4
384.10.a.i 5
384.10.a.j 5
384.10.a.k 5
384.10.a.l 5
384.10.a.m 5
384.10.a.n 5
384.10.a.o 5
384.10.a.p 5
384.10.c $$\chi_{384}(383, \cdot)$$ n/a 144 1
384.10.d $$\chi_{384}(193, \cdot)$$ 384.10.d.a 8 1
384.10.d.b 8
384.10.d.c 10
384.10.d.d 10
384.10.d.e 16
384.10.d.f 20
384.10.f $$\chi_{384}(191, \cdot)$$ n/a 144 1
384.10.j $$\chi_{384}(97, \cdot)$$ n/a 144 2
384.10.k $$\chi_{384}(95, \cdot)$$ n/a 280 2
384.10.n $$\chi_{384}(49, \cdot)$$ n/a 288 4
384.10.o $$\chi_{384}(47, \cdot)$$ n/a 568 4
384.10.r $$\chi_{384}(25, \cdot)$$ None 0 8
384.10.s $$\chi_{384}(23, \cdot)$$ None 0 8
384.10.v $$\chi_{384}(13, \cdot)$$ n/a 4608 16
384.10.w $$\chi_{384}(11, \cdot)$$ n/a 9184 16

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

## Decomposition of $$S_{10}^{\mathrm{old}}(\Gamma_1(384))$$ into lower level spaces

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