## Defining parameters

 Level: $$N$$ = $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$4$$ Newforms: $$7$$ Sturm bound: $$19200$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(600))$$.

Total New Old
Modular forms 746 112 634
Cusp forms 74 30 44
Eisenstein series 672 82 590

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 30 0 0 0

## Trace form

 $$30q + 2q^{4} - 2q^{6} - 4q^{7} + 2q^{9} + O(q^{10})$$ $$30q + 2q^{4} - 2q^{6} - 4q^{7} + 2q^{9} - 4q^{10} - 4q^{15} - 14q^{16} - 4q^{22} + 14q^{24} + 16q^{28} - 4q^{31} - 4q^{33} - 2q^{36} - 4q^{42} - 10q^{49} - 12q^{51} - 6q^{54} - 4q^{55} - 4q^{58} - 4q^{63} + 2q^{64} + 12q^{66} - 4q^{70} - 4q^{73} - 12q^{79} - 2q^{81} - 4q^{87} - 4q^{88} - 2q^{96} + 16q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(600))$$

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
600.1.c $$\chi_{600}(449, \cdot)$$ None 0 1
600.1.e $$\chi_{600}(151, \cdot)$$ None 0 1
600.1.g $$\chi_{600}(451, \cdot)$$ None 0 1
600.1.i $$\chi_{600}(149, \cdot)$$ None 0 1
600.1.j $$\chi_{600}(199, \cdot)$$ None 0 1
600.1.l $$\chi_{600}(401, \cdot)$$ None 0 1
600.1.n $$\chi_{600}(101, \cdot)$$ 600.1.n.a 1 1
600.1.n.b 1
600.1.p $$\chi_{600}(499, \cdot)$$ None 0 1
600.1.q $$\chi_{600}(107, \cdot)$$ 600.1.q.a 4 2
600.1.q.b 8
600.1.t $$\chi_{600}(157, \cdot)$$ None 0 2
600.1.u $$\chi_{600}(193, \cdot)$$ None 0 2
600.1.x $$\chi_{600}(143, \cdot)$$ None 0 2
600.1.z $$\chi_{600}(29, \cdot)$$ 600.1.z.a 8 4
600.1.bb $$\chi_{600}(91, \cdot)$$ None 0 4
600.1.bd $$\chi_{600}(31, \cdot)$$ None 0 4
600.1.bf $$\chi_{600}(89, \cdot)$$ None 0 4
600.1.bh $$\chi_{600}(19, \cdot)$$ None 0 4
600.1.bj $$\chi_{600}(221, \cdot)$$ 600.1.bj.a 4 4
600.1.bj.b 4
600.1.bl $$\chi_{600}(41, \cdot)$$ None 0 4
600.1.bn $$\chi_{600}(79, \cdot)$$ None 0 4
600.1.bo $$\chi_{600}(23, \cdot)$$ None 0 8
600.1.br $$\chi_{600}(73, \cdot)$$ None 0 8
600.1.bs $$\chi_{600}(13, \cdot)$$ None 0 8
600.1.bv $$\chi_{600}(83, \cdot)$$ None 0 8

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(600))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(600)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(120))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(300))$$$$^{\oplus 2}$$