Properties

 Label 768.2 Level 768 Weight 2 Dimension 6808 Nonzero newspaces 12 Newform subspaces 59 Sturm bound 65536 Trace bound 49

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

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

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 available 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

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