Properties

Label 2000.1
Level 2000
Weight 1
Dimension 40
Nonzero newspaces 5
Newforms 5
Sturm bound 240000
Trace bound 9

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Defining parameters

Level: \( N \) = \( 2000 = 2^{4} \cdot 5^{3} \)
Weight: \( k \) = \( 1 \)
Nonzero newspaces: \( 5 \)
Newforms: \( 5 \)
Sturm bound: \(240000\)
Trace bound: \(9\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(2000))\).

Total New Old
Modular forms 2749 616 2133
Cusp forms 229 40 189
Eisenstein series 2520 576 1944

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 40 0 0 0

Trace form

\(40q \) \(\mathstrut -\mathstrut q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(40q \) \(\mathstrut -\mathstrut q^{9} \) \(\mathstrut +\mathstrut 2q^{29} \) \(\mathstrut +\mathstrut 10q^{37} \) \(\mathstrut +\mathstrut 10q^{41} \) \(\mathstrut +\mathstrut 9q^{49} \) \(\mathstrut +\mathstrut 10q^{53} \) \(\mathstrut +\mathstrut 10q^{61} \) \(\mathstrut +\mathstrut 5q^{65} \) \(\mathstrut +\mathstrut 5q^{81} \) \(\mathstrut -\mathstrut 5q^{85} \) \(\mathstrut -\mathstrut 8q^{89} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(2000))\)

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
2000.1.b \(\chi_{2000}(751, \cdot)\) 2000.1.b.a 4 1
2000.1.e \(\chi_{2000}(999, \cdot)\) None 0 1
2000.1.g \(\chi_{2000}(1751, \cdot)\) None 0 1
2000.1.h \(\chi_{2000}(1999, \cdot)\) 2000.1.h.a 4 1
2000.1.i \(\chi_{2000}(1557, \cdot)\) None 0 2
2000.1.k \(\chi_{2000}(499, \cdot)\) None 0 2
2000.1.m \(\chi_{2000}(57, \cdot)\) None 0 2
2000.1.p \(\chi_{2000}(193, \cdot)\) None 0 2
2000.1.r \(\chi_{2000}(251, \cdot)\) None 0 2
2000.1.t \(\chi_{2000}(557, \cdot)\) None 0 2
2000.1.v \(\chi_{2000}(151, \cdot)\) None 0 4
2000.1.x \(\chi_{2000}(399, \cdot)\) 2000.1.x.a 4 4
2000.1.z \(\chi_{2000}(351, \cdot)\) 2000.1.z.a 8 4
2000.1.ba \(\chi_{2000}(199, \cdot)\) None 0 4
2000.1.bc \(\chi_{2000}(157, \cdot)\) None 0 8
2000.1.bf \(\chi_{2000}(99, \cdot)\) None 0 8
2000.1.bg \(\chi_{2000}(257, \cdot)\) None 0 8
2000.1.bj \(\chi_{2000}(393, \cdot)\) None 0 8
2000.1.bk \(\chi_{2000}(51, \cdot)\) None 0 8
2000.1.bn \(\chi_{2000}(93, \cdot)\) None 0 8
2000.1.bp \(\chi_{2000}(79, \cdot)\) 2000.1.bp.a 20 20
2000.1.bq \(\chi_{2000}(39, \cdot)\) None 0 20
2000.1.bs \(\chi_{2000}(71, \cdot)\) None 0 20
2000.1.bv \(\chi_{2000}(31, \cdot)\) None 0 20
2000.1.bw \(\chi_{2000}(11, \cdot)\) None 0 40
2000.1.bz \(\chi_{2000}(13, \cdot)\) None 0 40
2000.1.ca \(\chi_{2000}(17, \cdot)\) None 0 40
2000.1.cd \(\chi_{2000}(73, \cdot)\) None 0 40
2000.1.ce \(\chi_{2000}(53, \cdot)\) None 0 40
2000.1.ch \(\chi_{2000}(19, \cdot)\) None 0 40

Decomposition of \(S_{1}^{\mathrm{old}}(\Gamma_1(2000))\) into lower level spaces

\( S_{1}^{\mathrm{old}}(\Gamma_1(2000)) \cong \) \(S_{1}^{\mathrm{new}}(\Gamma_1(80))\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(100))\)\(^{\oplus 6}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(200))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(400))\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(500))\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(1000))\)\(^{\oplus 2}\)