# Properties

 Label 768.4 Level 768 Weight 4 Dimension 20632 Nonzero newspaces 12 Sturm bound 131072 Trace bound 49

## Defining parameters

 Level: $$N$$ = $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$12$$ Sturm bound: $$131072$$ Trace bound: $$49$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(768))$$.

Total New Old
Modular forms 49856 20840 29016
Cusp forms 48448 20632 27816
Eisenstein series 1408 208 1200

## Trace form

 $$20632 q - 24 q^{3} - 64 q^{4} - 32 q^{6} - 48 q^{7} - 40 q^{9} + O(q^{10})$$ $$20632 q - 24 q^{3} - 64 q^{4} - 32 q^{6} - 48 q^{7} - 40 q^{9} - 64 q^{10} - 32 q^{12} - 64 q^{13} - 24 q^{15} - 64 q^{16} - 32 q^{18} - 48 q^{19} - 32 q^{21} - 64 q^{22} - 32 q^{24} - 80 q^{25} - 24 q^{27} - 64 q^{28} - 32 q^{30} - 32 q^{31} - 56 q^{33} - 64 q^{34} - 32 q^{36} - 64 q^{37} - 24 q^{39} - 64 q^{40} - 32 q^{42} - 48 q^{43} - 32 q^{45} - 64 q^{46} - 32 q^{48} + 2648 q^{49} + 2952 q^{51} - 64 q^{52} + 3008 q^{53} - 32 q^{54} + 1104 q^{55} - 1384 q^{57} - 64 q^{58} - 5504 q^{59} - 32 q^{60} - 7360 q^{61} - 2048 q^{63} - 64 q^{64} - 7808 q^{65} - 32 q^{66} - 8208 q^{67} - 2144 q^{69} - 64 q^{70} - 896 q^{71} - 32 q^{72} + 3376 q^{73} + 4392 q^{75} - 64 q^{76} + 7616 q^{77} - 32 q^{78} + 11280 q^{79} - 48 q^{81} - 64 q^{82} - 32 q^{84} + 1936 q^{85} - 24 q^{87} - 64 q^{88} - 32 q^{90} - 48 q^{91} - 464 q^{93} - 64 q^{94} - 32 q^{96} - 112 q^{97} - 24 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(768))$$

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
768.4.a $$\chi_{768}(1, \cdot)$$ 768.4.a.a 1 1
768.4.a.b 1
768.4.a.c 1
768.4.a.d 1
768.4.a.e 2
768.4.a.f 2
768.4.a.g 2
768.4.a.h 2
768.4.a.i 2
768.4.a.j 2
768.4.a.k 2
768.4.a.l 2
768.4.a.m 2
768.4.a.n 2
768.4.a.o 2
768.4.a.p 2
768.4.a.q 3
768.4.a.r 3
768.4.a.s 3
768.4.a.t 3
768.4.a.u 4
768.4.a.v 4
768.4.c $$\chi_{768}(767, \cdot)$$ 768.4.c.a 2 1
768.4.c.b 2
768.4.c.c 2
768.4.c.d 2
768.4.c.e 2
768.4.c.f 2
768.4.c.g 2
768.4.c.h 2
768.4.c.i 2
768.4.c.j 2
768.4.c.k 4
768.4.c.l 4
768.4.c.m 4
768.4.c.n 4
768.4.c.o 4
768.4.c.p 4
768.4.c.q 4
768.4.c.r 4
768.4.c.s 8
768.4.c.t 8
768.4.c.u 8
768.4.c.v 16
768.4.d $$\chi_{768}(385, \cdot)$$ 768.4.d.a 2 1
768.4.d.b 2
768.4.d.c 2
768.4.d.d 2
768.4.d.e 2
768.4.d.f 2
768.4.d.g 2
768.4.d.h 2
768.4.d.i 2
768.4.d.j 2
768.4.d.k 2
768.4.d.l 2
768.4.d.m 2
768.4.d.n 2
768.4.d.o 2
768.4.d.p 2
768.4.d.q 4
768.4.d.r 4
768.4.d.s 4
768.4.d.t 4
768.4.f $$\chi_{768}(383, \cdot)$$ 768.4.f.a 4 1
768.4.f.b 8
768.4.f.c 8
768.4.f.d 12
768.4.f.e 12
768.4.f.f 12
768.4.f.g 12
768.4.f.h 12
768.4.f.i 12
768.4.j $$\chi_{768}(193, \cdot)$$ 768.4.j.a 4 2
768.4.j.b 4
768.4.j.c 12
768.4.j.d 12
768.4.j.e 16
768.4.j.f 16
768.4.j.g 16
768.4.j.h 16
768.4.k $$\chi_{768}(191, \cdot)$$ n/a 192 2
768.4.n $$\chi_{768}(97, \cdot)$$ n/a 192 4
768.4.o $$\chi_{768}(95, \cdot)$$ n/a 368 4
768.4.r $$\chi_{768}(49, \cdot)$$ n/a 384 8
768.4.s $$\chi_{768}(47, \cdot)$$ n/a 752 8
768.4.v $$\chi_{768}(25, \cdot)$$ None 0 16
768.4.w $$\chi_{768}(23, \cdot)$$ None 0 16
768.4.z $$\chi_{768}(13, \cdot)$$ n/a 6144 32
768.4.ba $$\chi_{768}(11, \cdot)$$ n/a 12224 32

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(768))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(768)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 18}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 16}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 9}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 14}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 7}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(96))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(128))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(192))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(256))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(384))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(768))$$$$^{\oplus 1}$$