Space of modular forms of level 400 and weight 1

• Label: 400.1
• Level: 400
• Weight: 1
• Dimension: 5
• Nonzero newspaces: 2
• Newform subspaces: 2
• Sturm bound: 9600
• Trace bound: 1

Defining parameters

Level:	$N$	=	$400 = 2^{4} \cdot 5^{2}$
Weight:	$k$	=	$1$
Nonzero newspaces:			$2$
Newform subspaces:			$2$
Sturm bound:			$9600$
Trace bound:			$1$

Dimensions

The following table gives the dimensions of various subspaces of $M_{1}(\Gamma_1(400))$.

	Total	New	Old
Modular forms	419	97	322
Cusp forms	27	5	22
Eisenstein series	392	92	300

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

	$D_n$	$A_4$	$S_4$	$A_5$
Dimension	5	0	0	0

Trace form

$5 q + q^{5} + 2 q^{9} - q^{25} - 4 q^{29} - 5 q^{37} - q^{45} - 3 q^{49} - 5 q^{53} - 5 q^{65} + 5 q^{85} + q^{89}+O(q^{100})$

Decomposition of $S_{1}^{\mathrm{new}}(\Gamma_1(400))$

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 available newforms together with their dimension.

Label	$\chi$	Newforms	Dimension	$\chi$ degree
400.1.b	$\chi_{400}(351, \cdot)$	400.1.b.a	1	1
400.1.e	$\chi_{400}(199, \cdot)$	None	0	1
400.1.g	$\chi_{400}(151, \cdot)$	None	0	1
400.1.h	$\chi_{400}(399, \cdot)$	None	0	1
400.1.i	$\chi_{400}(93, \cdot)$	None	0	2
400.1.k	$\chi_{400}(99, \cdot)$	None	0	2
400.1.m	$\chi_{400}(57, \cdot)$	None	0	2
400.1.p	$\chi_{400}(193, \cdot)$	None	0	2
400.1.r	$\chi_{400}(51, \cdot)$	None	0	2
400.1.t	$\chi_{400}(157, \cdot)$	None	0	2
400.1.v	$\chi_{400}(71, \cdot)$	None	0	4
400.1.x	$\chi_{400}(79, \cdot)$	400.1.x.a	4	4
400.1.z	$\chi_{400}(31, \cdot)$	None	0	4
400.1.ba	$\chi_{400}(39, \cdot)$	None	0	4
400.1.bc	$\chi_{400}(53, \cdot)$	None	0	8
400.1.bf	$\chi_{400}(19, \cdot)$	None	0	8
400.1.bg	$\chi_{400}(17, \cdot)$	None	0	8
400.1.bj	$\chi_{400}(73, \cdot)$	None	0	8
400.1.bk	$\chi_{400}(11, \cdot)$	None	0	8
400.1.bn	$\chi_{400}(13, \cdot)$	None	0	8

Decomposition of $S_{1}^{\mathrm{old}}(\Gamma_1(400))$ into lower level spaces

$S_{1}^{\mathrm{old}}(\Gamma_1(400)) \cong$ $S_{1}^{\mathrm{new}}(\Gamma_1(1))$$^{\oplus 15}$$\oplus$$S_{1}^{\mathrm{new}}(\Gamma_1(2))$$^{\oplus 12}$$\oplus$$S_{1}^{\mathrm{new}}(\Gamma_1(4))$$^{\oplus 9}$$\oplus$$S_{1}^{\mathrm{new}}(\Gamma_1(5))$$^{\oplus 10}$$\oplus$$S_{1}^{\mathrm{new}}(\Gamma_1(8))$$^{\oplus 6}$$\oplus$$S_{1}^{\mathrm{new}}(\Gamma_1(10))$$^{\oplus 8}$$\oplus$$S_{1}^{\mathrm{new}}(\Gamma_1(16))$$^{\oplus 3}$$\oplus$$S_{1}^{\mathrm{new}}(\Gamma_1(20))$$^{\oplus 6}$$\oplus$$S_{1}^{\mathrm{new}}(\Gamma_1(25))$$^{\oplus 5}$$\oplus$$S_{1}^{\mathrm{new}}(\Gamma_1(40))$$^{\oplus 4}$$\oplus$$S_{1}^{\mathrm{new}}(\Gamma_1(50))$$^{\oplus 4}$$\oplus$$S_{1}^{\mathrm{new}}(\Gamma_1(80))$$^{\oplus 2}$$\oplus$$S_{1}^{\mathrm{new}}(\Gamma_1(100))$$^{\oplus 3}$$\oplus$$S_{1}^{\mathrm{new}}(\Gamma_1(200))$$^{\oplus 2}$