Properties

Label 400.1
Level 400
Weight 1
Dimension 5
Nonzero newspaces 2
Newforms 2
Sturm bound 9600
Trace bound 1

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Defining parameters

Level: \( N \) = \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) = \( 1 \)
Nonzero newspaces: \( 2 \)
Newforms: \( 2 \)
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

\(5q \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(5q \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut -\mathstrut q^{25} \) \(\mathstrut -\mathstrut 4q^{29} \) \(\mathstrut -\mathstrut 5q^{37} \) \(\mathstrut -\mathstrut q^{45} \) \(\mathstrut -\mathstrut 3q^{49} \) \(\mathstrut -\mathstrut 5q^{53} \) \(\mathstrut -\mathstrut 5q^{65} \) \(\mathstrut +\mathstrut 5q^{85} \) \(\mathstrut +\mathstrut q^{89} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 the 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

\( S_{1}^{\mathrm{old}}(\Gamma_1(400)) \cong \) \(S_{1}^{\mathrm{new}}(\Gamma_1(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(100))\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(200))\)\(^{\oplus 2}\)