# Properties

 Label 768.1 Level 768 Weight 1 Dimension 12 Nonzero newspaces 2 Newform subspaces 5 Sturm bound 32768 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$2$$ Newform subspaces: $$5$$ Sturm bound: $$32768$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(768))$$.

Total New Old
Modular forms 732 116 616
Cusp forms 28 12 16
Eisenstein series 704 104 600

The following table gives the dimensions of subspaces with specified projective image type.

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 12 0 0 0

## Trace form

 $$12 q + O(q^{10})$$ $$12 q + 4 q^{25} - 4 q^{33} - 12 q^{49} - 4 q^{57} - 4 q^{81} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(768))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
768.1.b $$\chi_{768}(127, \cdot)$$ None 0 1
768.1.e $$\chi_{768}(257, \cdot)$$ 768.1.e.a 1 1
768.1.e.b 1
768.1.e.c 2
768.1.g $$\chi_{768}(511, \cdot)$$ None 0 1
768.1.h $$\chi_{768}(641, \cdot)$$ None 0 1
768.1.i $$\chi_{768}(65, \cdot)$$ 768.1.i.a 4 2
768.1.i.b 4
768.1.l $$\chi_{768}(319, \cdot)$$ None 0 2
768.1.m $$\chi_{768}(31, \cdot)$$ None 0 4
768.1.p $$\chi_{768}(161, \cdot)$$ None 0 4
768.1.q $$\chi_{768}(17, \cdot)$$ None 0 8
768.1.t $$\chi_{768}(79, \cdot)$$ None 0 8
768.1.u $$\chi_{768}(7, \cdot)$$ None 0 16
768.1.x $$\chi_{768}(41, \cdot)$$ None 0 16
768.1.y $$\chi_{768}(5, \cdot)$$ None 0 32
768.1.bb $$\chi_{768}(19, \cdot)$$ None 0 32

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(768))$$ into lower level spaces

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