Properties

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

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Defining parameters

Level: \( N \) = \( 2000\( 2000 = 2^{4} \cdot 5^{3} \) \)
Weight: \( k \) = \( 1 \)
Nonzero newspaces: \( 5 \)
Newform subspaces: \( 5 \)
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

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

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( 1 - T^{2} + T^{4} - T^{6} + T^{8} \))(\( 1 - T^{2} + T^{4} - T^{6} + T^{8} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( 1 - T^{10} + T^{20} - T^{30} + T^{40} \))
$5$ (\( 1 - T^{5} + T^{10} - T^{15} + T^{20} \))
$7$ (\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( 1 - T^{2} + T^{4} - T^{6} + T^{8} \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 - T )^{8}( 1 + T )^{8} \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{5} \))
$11$ (\( ( 1 - T )^{4}( 1 + T )^{4} \))(\( ( 1 - T )^{4}( 1 + T )^{4} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \))
$13$ (\( ( 1 + T^{2} )^{4} \))(\( ( 1 - T )^{4}( 1 + T )^{4} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))(\( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \))
$17$ (\( ( 1 + T^{2} )^{4} \))(\( ( 1 - T )^{4}( 1 + T )^{4} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))(\( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \))
$19$ (\( ( 1 - T )^{4}( 1 + T )^{4} \))(\( ( 1 - T )^{4}( 1 + T )^{4} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \))
$23$ (\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( 1 - T^{2} + T^{4} - T^{6} + T^{8} \))(\( 1 - T^{2} + T^{4} - T^{6} + T^{8} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( 1 - T^{10} + T^{20} - T^{30} + T^{40} \))
$29$ (\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \))(\( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \))
$31$ (\( ( 1 - T )^{4}( 1 + T )^{4} \))(\( ( 1 - T )^{4}( 1 + T )^{4} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \))
$37$ (\( ( 1 + T^{2} )^{4} \))(\( ( 1 - T )^{4}( 1 + T )^{4} \))(\( ( 1 - T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \))(\( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \))
$41$ (\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \))(\( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2} \))
$43$ (\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( 1 - T^{2} + T^{4} - T^{6} + T^{8} \))(\( ( 1 + T^{2} )^{4} \))(\( ( 1 - T )^{8}( 1 + T )^{8} \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{5} \))
$47$ (\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( 1 - T^{2} + T^{4} - T^{6} + T^{8} \))(\( 1 - T^{2} + T^{4} - T^{6} + T^{8} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( 1 - T^{10} + T^{20} - T^{30} + T^{40} \))
$53$ (\( ( 1 + T^{2} )^{4} \))(\( ( 1 - T )^{4}( 1 + T )^{4} \))(\( ( 1 - T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \))(\( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \))
$59$ (\( ( 1 - T )^{4}( 1 + T )^{4} \))(\( ( 1 - T )^{4}( 1 + T )^{4} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \))
$61$ (\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \))(\( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{2} \))
$67$ (\( ( 1 - T )^{4}( 1 + T )^{4} \))(\( ( 1 + T^{2} )^{4} \))(\( 1 - T^{2} + T^{4} - T^{6} + T^{8} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( 1 - T^{10} + T^{20} - T^{30} + T^{40} \))
$71$ (\( ( 1 - T )^{4}( 1 + T )^{4} \))(\( ( 1 - T )^{4}( 1 + T )^{4} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \))
$73$ (\( ( 1 + T^{2} )^{4} \))(\( ( 1 - T )^{4}( 1 + T )^{4} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))(\( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \))
$79$ (\( ( 1 - T )^{4}( 1 + T )^{4} \))(\( ( 1 - T )^{4}( 1 + T )^{4} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \))
$83$ (\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( 1 - T^{2} + T^{4} - T^{6} + T^{8} \))(\( 1 - T^{2} + T^{4} - T^{6} + T^{8} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( 1 - T^{10} + T^{20} - T^{30} + T^{40} \))
$89$ (\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 + T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \))
$97$ (\( ( 1 + T^{2} )^{4} \))(\( ( 1 - T )^{4}( 1 + T )^{4} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))(\( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \))
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