## Defining parameters

 Level: $$N$$ = $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$59$$ Sturm bound: $$65536$$ Trace bound: $$49$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(768))$$.

Total New Old
Modular forms 17088 7016 10072
Cusp forms 15681 6808 8873
Eisenstein series 1407 208 1199

## Trace form

 $$6808q - 24q^{3} - 64q^{4} - 32q^{6} - 48q^{7} - 40q^{9} + O(q^{10})$$ $$6808q - 24q^{3} - 64q^{4} - 32q^{6} - 48q^{7} - 40q^{9} - 64q^{10} - 32q^{12} - 64q^{13} - 24q^{15} - 64q^{16} - 32q^{18} - 48q^{19} - 32q^{21} - 64q^{22} - 32q^{24} - 80q^{25} - 24q^{27} - 64q^{28} - 32q^{30} - 32q^{31} - 56q^{33} - 64q^{34} - 32q^{36} - 64q^{37} - 24q^{39} - 64q^{40} - 32q^{42} - 48q^{43} - 32q^{45} - 64q^{46} - 32q^{48} - 40q^{49} + 8q^{51} - 64q^{52} + 64q^{53} - 32q^{54} + 80q^{55} + 24q^{57} - 64q^{58} + 128q^{59} - 32q^{60} + 64q^{61} - 64q^{64} + 128q^{65} - 32q^{66} + 112q^{67} + 32q^{69} - 64q^{70} + 128q^{71} - 32q^{72} + 48q^{73} + 40q^{75} - 64q^{76} + 64q^{77} - 32q^{78} + 16q^{79} - 48q^{81} - 64q^{82} - 32q^{84} + 16q^{85} - 24q^{87} - 64q^{88} - 32q^{90} - 48q^{91} - 80q^{93} - 64q^{94} - 32q^{96} - 112q^{97} - 24q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(768))$$

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 the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
768.2.a $$\chi_{768}(1, \cdot)$$ 768.2.a.a 1 1
768.2.a.b 1
768.2.a.c 1
768.2.a.d 1
768.2.a.e 1
768.2.a.f 1
768.2.a.g 1
768.2.a.h 1
768.2.a.i 2
768.2.a.j 2
768.2.a.k 2
768.2.a.l 2
768.2.c $$\chi_{768}(767, \cdot)$$ 768.2.c.a 2 1
768.2.c.b 2
768.2.c.c 2
768.2.c.d 2
768.2.c.e 2
768.2.c.f 2
768.2.c.g 4
768.2.c.h 4
768.2.c.i 4
768.2.c.j 4
768.2.d $$\chi_{768}(385, \cdot)$$ 768.2.d.a 2 1
768.2.d.b 2
768.2.d.c 2
768.2.d.d 2
768.2.d.e 2
768.2.d.f 2
768.2.d.g 2
768.2.d.h 2
768.2.f $$\chi_{768}(383, \cdot)$$ 768.2.f.a 4 1
768.2.f.b 4
768.2.f.c 4
768.2.f.d 4
768.2.f.e 4
768.2.f.f 4
768.2.f.g 4
768.2.j $$\chi_{768}(193, \cdot)$$ 768.2.j.a 4 2
768.2.j.b 4
768.2.j.c 4
768.2.j.d 4
768.2.j.e 8
768.2.j.f 8
768.2.k $$\chi_{768}(191, \cdot)$$ 768.2.k.a 4 2
768.2.k.b 4
768.2.k.c 4
768.2.k.d 4
768.2.k.e 8
768.2.k.f 8
768.2.k.g 16
768.2.k.h 16
768.2.n $$\chi_{768}(97, \cdot)$$ 768.2.n.a 32 4
768.2.n.b 32
768.2.o $$\chi_{768}(95, \cdot)$$ 768.2.o.a 56 4
768.2.o.b 56
768.2.r $$\chi_{768}(49, \cdot)$$ 768.2.r.a 128 8
768.2.s $$\chi_{768}(47, \cdot)$$ 768.2.s.a 240 8
768.2.v $$\chi_{768}(25, \cdot)$$ None 0 16
768.2.w $$\chi_{768}(23, \cdot)$$ None 0 16
768.2.z $$\chi_{768}(13, \cdot)$$ 768.2.z.a 2048 32
768.2.ba $$\chi_{768}(11, \cdot)$$ 768.2.ba.a 4032 32

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(768))$$ into lower level spaces

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