# Properties

 Label 1000.1 Level 1000 Weight 1 Dimension 8 Nonzero newspaces 2 Newform subspaces 3 Sturm bound 60000 Trace bound 4

# Learn more about

## Defining parameters

 Level: $$N$$ = $$1000 = 2^{3} \cdot 5^{3}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$2$$ Newform subspaces: $$3$$ Sturm bound: $$60000$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(1000))$$.

Total New Old
Modular forms 1192 264 928
Cusp forms 112 8 104
Eisenstein series 1080 256 824

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 8 0 0 0

## Trace form

 $$8q + O(q^{10})$$ $$8q - 4q^{11} + 8q^{16} - 4q^{26} + 8q^{36} - 4q^{41} - 4q^{46} - 4q^{56} - 4q^{76} + 8q^{81} - 8q^{91} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(1000))$$

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 the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
1000.1.b $$\chi_{1000}(751, \cdot)$$ None 0 1
1000.1.e $$\chi_{1000}(499, \cdot)$$ 1000.1.e.a 2 1
1000.1.e.b 2
1000.1.g $$\chi_{1000}(251, \cdot)$$ 1000.1.g.a 4 1
1000.1.h $$\chi_{1000}(999, \cdot)$$ None 0 1
1000.1.i $$\chi_{1000}(557, \cdot)$$ None 0 2
1000.1.l $$\chi_{1000}(57, \cdot)$$ None 0 2
1000.1.n $$\chi_{1000}(51, \cdot)$$ None 0 4
1000.1.p $$\chi_{1000}(199, \cdot)$$ None 0 4
1000.1.r $$\chi_{1000}(151, \cdot)$$ None 0 4
1000.1.s $$\chi_{1000}(99, \cdot)$$ None 0 4
1000.1.u $$\chi_{1000}(257, \cdot)$$ None 0 8
1000.1.x $$\chi_{1000}(93, \cdot)$$ None 0 8
1000.1.z $$\chi_{1000}(39, \cdot)$$ None 0 20
1000.1.ba $$\chi_{1000}(19, \cdot)$$ None 0 20
1000.1.bc $$\chi_{1000}(11, \cdot)$$ None 0 20
1000.1.bf $$\chi_{1000}(31, \cdot)$$ None 0 20
1000.1.bg $$\chi_{1000}(17, \cdot)$$ None 0 40
1000.1.bj $$\chi_{1000}(13, \cdot)$$ None 0 40

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(1000))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(1000)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(500))$$$$^{\oplus 2}$$