## Defining parameters

 Level: $$N$$ = $$700 = 2^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$4$$ Newform subspaces: $$5$$ Sturm bound: $$28800$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(700))$$.

Total New Old
Modular forms 898 230 668
Cusp forms 58 26 32
Eisenstein series 840 204 636

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 26 0 0 0

## Trace form

 $$26q + 2q^{4} - 4q^{6} + O(q^{10})$$ $$26q + 2q^{4} - 4q^{6} - 2q^{11} - 4q^{14} + 2q^{16} - 12q^{21} - 2q^{24} + 6q^{29} - 16q^{36} - 2q^{39} - 4q^{41} - 14q^{46} + 2q^{51} + 2q^{54} - 10q^{56} + 14q^{61} - 4q^{64} - 4q^{69} + 4q^{71} + 2q^{79} - 8q^{81} + 4q^{84} + 18q^{86} - 2q^{89} - 2q^{91} + 4q^{94} + 14q^{96} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(700))$$

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
700.1.b $$\chi_{700}(351, \cdot)$$ None 0 1
700.1.d $$\chi_{700}(601, \cdot)$$ 700.1.d.a 2 1
700.1.f $$\chi_{700}(99, \cdot)$$ None 0 1
700.1.h $$\chi_{700}(349, \cdot)$$ None 0 1
700.1.j $$\chi_{700}(307, \cdot)$$ 700.1.j.a 4 2
700.1.j.b 8
700.1.l $$\chi_{700}(57, \cdot)$$ None 0 2
700.1.o $$\chi_{700}(549, \cdot)$$ None 0 2
700.1.q $$\chi_{700}(499, \cdot)$$ None 0 2
700.1.s $$\chi_{700}(101, \cdot)$$ None 0 2
700.1.u $$\chi_{700}(51, \cdot)$$ 700.1.u.a 4 2
700.1.v $$\chi_{700}(69, \cdot)$$ None 0 4
700.1.x $$\chi_{700}(239, \cdot)$$ None 0 4
700.1.z $$\chi_{700}(41, \cdot)$$ None 0 4
700.1.bb $$\chi_{700}(71, \cdot)$$ None 0 4
700.1.bd $$\chi_{700}(93, \cdot)$$ None 0 4
700.1.bf $$\chi_{700}(143, \cdot)$$ 700.1.bf.a 8 4
700.1.bi $$\chi_{700}(113, \cdot)$$ None 0 8
700.1.bk $$\chi_{700}(27, \cdot)$$ None 0 8
700.1.bl $$\chi_{700}(11, \cdot)$$ None 0 8
700.1.bn $$\chi_{700}(61, \cdot)$$ None 0 8
700.1.bp $$\chi_{700}(39, \cdot)$$ None 0 8
700.1.br $$\chi_{700}(89, \cdot)$$ None 0 8
700.1.bs $$\chi_{700}(3, \cdot)$$ None 0 16
700.1.bu $$\chi_{700}(37, \cdot)$$ None 0 16

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(700))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(700)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(140))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(175))$$$$^{\oplus 3}$$