## Defining parameters

 Level: $$N$$ = $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$6$$ Newform subspaces: $$10$$ Sturm bound: $$76800$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(1200))$$.

Total New Old
Modular forms 1750 213 1537
Cusp forms 182 28 154
Eisenstein series 1568 185 1383

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 28 0 0 0

## Trace form

 $$28q + O(q^{10})$$ $$28q + 4q^{16} + 4q^{19} - 4q^{21} - 4q^{24} + 4q^{31} + 4q^{34} - 12q^{36} - 12q^{46} + 4q^{49} - 4q^{51} - 4q^{54} - 8q^{61} - 4q^{69} - 4q^{76} - 8q^{79} - 4q^{81} - 4q^{91} - 4q^{94} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(1200))$$

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
1200.1.c $$\chi_{1200}(449, \cdot)$$ 1200.1.c.a 2 1
1200.1.e $$\chi_{1200}(751, \cdot)$$ None 0 1
1200.1.g $$\chi_{1200}(151, \cdot)$$ None 0 1
1200.1.i $$\chi_{1200}(1049, \cdot)$$ None 0 1
1200.1.j $$\chi_{1200}(799, \cdot)$$ None 0 1
1200.1.l $$\chi_{1200}(401, \cdot)$$ 1200.1.l.a 1 1
1200.1.l.b 1
1200.1.n $$\chi_{1200}(1001, \cdot)$$ None 0 1
1200.1.p $$\chi_{1200}(199, \cdot)$$ None 0 1
1200.1.q $$\chi_{1200}(499, \cdot)$$ None 0 2
1200.1.r $$\chi_{1200}(101, \cdot)$$ 1200.1.r.a 4 2
1200.1.u $$\chi_{1200}(407, \cdot)$$ None 0 2
1200.1.x $$\chi_{1200}(457, \cdot)$$ None 0 2
1200.1.z $$\chi_{1200}(107, \cdot)$$ 1200.1.z.a 2 2
1200.1.z.b 2
1200.1.ba $$\chi_{1200}(493, \cdot)$$ None 0 2
1200.1.bd $$\chi_{1200}(443, \cdot)$$ 1200.1.bd.a 2 2
1200.1.bd.b 2
1200.1.be $$\chi_{1200}(157, \cdot)$$ None 0 2
1200.1.bg $$\chi_{1200}(193, \cdot)$$ None 0 2
1200.1.bj $$\chi_{1200}(143, \cdot)$$ 1200.1.bj.a 4 2
1200.1.bj.b 8
1200.1.bm $$\chi_{1200}(149, \cdot)$$ None 0 2
1200.1.bn $$\chi_{1200}(451, \cdot)$$ None 0 2
1200.1.bp $$\chi_{1200}(89, \cdot)$$ None 0 4
1200.1.br $$\chi_{1200}(391, \cdot)$$ None 0 4
1200.1.bt $$\chi_{1200}(31, \cdot)$$ None 0 4
1200.1.bv $$\chi_{1200}(209, \cdot)$$ None 0 4
1200.1.bx $$\chi_{1200}(439, \cdot)$$ None 0 4
1200.1.bz $$\chi_{1200}(41, \cdot)$$ None 0 4
1200.1.cb $$\chi_{1200}(161, \cdot)$$ None 0 4
1200.1.cd $$\chi_{1200}(79, \cdot)$$ None 0 4
1200.1.cg $$\chi_{1200}(221, \cdot)$$ None 0 8
1200.1.ch $$\chi_{1200}(19, \cdot)$$ None 0 8
1200.1.ci $$\chi_{1200}(47, \cdot)$$ None 0 8
1200.1.cl $$\chi_{1200}(97, \cdot)$$ None 0 8
1200.1.cn $$\chi_{1200}(133, \cdot)$$ None 0 8
1200.1.co $$\chi_{1200}(203, \cdot)$$ None 0 8
1200.1.cr $$\chi_{1200}(13, \cdot)$$ None 0 8
1200.1.cs $$\chi_{1200}(83, \cdot)$$ None 0 8
1200.1.cu $$\chi_{1200}(73, \cdot)$$ None 0 8
1200.1.cx $$\chi_{1200}(23, \cdot)$$ None 0 8
1200.1.cy $$\chi_{1200}(91, \cdot)$$ None 0 8
1200.1.cz $$\chi_{1200}(29, \cdot)$$ None 0 8

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(1200))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(1200)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(120))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(240))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(300))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(400))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(600))$$$$^{\oplus 2}$$