# Properties

 Label 300.1 Level 300 Weight 1 Dimension 8 Nonzero newspaces 3 Newform subspaces 4 Sturm bound 4800 Trace bound 9

## Defining parameters

 Level: $$N$$ = $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$3$$ Newform subspaces: $$4$$ Sturm bound: $$4800$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(300))$$.

Total New Old
Modular forms 296 48 248
Cusp forms 16 8 8
Eisenstein series 280 40 240

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

 $$8q - 4q^{6} + O(q^{10})$$ $$8q - 4q^{6} - 4q^{16} - 4q^{21} - 4q^{31} + 4q^{36} + 8q^{46} - 12q^{61} + 4q^{91} + 4q^{96} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(300))$$

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
300.1.b $$\chi_{300}(149, \cdot)$$ 300.1.b.a 2 1
300.1.c $$\chi_{300}(151, \cdot)$$ None 0 1
300.1.f $$\chi_{300}(199, \cdot)$$ None 0 1
300.1.g $$\chi_{300}(101, \cdot)$$ 300.1.g.a 1 1
300.1.g.b 1
300.1.k $$\chi_{300}(157, \cdot)$$ None 0 2
300.1.l $$\chi_{300}(107, \cdot)$$ 300.1.l.a 4 2
300.1.p $$\chi_{300}(31, \cdot)$$ None 0 4
300.1.q $$\chi_{300}(29, \cdot)$$ None 0 4
300.1.s $$\chi_{300}(41, \cdot)$$ None 0 4
300.1.t $$\chi_{300}(19, \cdot)$$ None 0 4
300.1.u $$\chi_{300}(23, \cdot)$$ None 0 8
300.1.v $$\chi_{300}(13, \cdot)$$ None 0 8

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(300))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(300)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 2}$$