Properties

Label 1000.1
Level 1000
Weight 1
Dimension 8
Nonzero newspaces 2
Newform subspaces 3
Sturm bound 60000
Trace bound 4

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Defining parameters

Level: \( N \) = \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) = \( 1 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 3 \)
Sturm bound: \(60000\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(1000))\).

Total New Old
Modular forms 1192 264 928
Cusp forms 112 8 104
Eisenstein series 1080 256 824

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 8 0 0 0

Trace form

\( 8q + O(q^{10}) \) \( 8q - 4q^{11} + 8q^{16} - 4q^{26} + 8q^{36} - 4q^{41} - 4q^{46} - 4q^{56} - 4q^{76} + 8q^{81} - 8q^{91} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(1000))\)

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
1000.1.b \(\chi_{1000}(751, \cdot)\) None 0 1
1000.1.e \(\chi_{1000}(499, \cdot)\) 1000.1.e.a 2 1
1000.1.e.b 2
1000.1.g \(\chi_{1000}(251, \cdot)\) 1000.1.g.a 4 1
1000.1.h \(\chi_{1000}(999, \cdot)\) None 0 1
1000.1.i \(\chi_{1000}(557, \cdot)\) None 0 2
1000.1.l \(\chi_{1000}(57, \cdot)\) None 0 2
1000.1.n \(\chi_{1000}(51, \cdot)\) None 0 4
1000.1.p \(\chi_{1000}(199, \cdot)\) None 0 4
1000.1.r \(\chi_{1000}(151, \cdot)\) None 0 4
1000.1.s \(\chi_{1000}(99, \cdot)\) None 0 4
1000.1.u \(\chi_{1000}(257, \cdot)\) None 0 8
1000.1.x \(\chi_{1000}(93, \cdot)\) None 0 8
1000.1.z \(\chi_{1000}(39, \cdot)\) None 0 20
1000.1.ba \(\chi_{1000}(19, \cdot)\) None 0 20
1000.1.bc \(\chi_{1000}(11, \cdot)\) None 0 20
1000.1.bf \(\chi_{1000}(31, \cdot)\) None 0 20
1000.1.bg \(\chi_{1000}(17, \cdot)\) None 0 40
1000.1.bj \(\chi_{1000}(13, \cdot)\) None 0 40

Decomposition of \(S_{1}^{\mathrm{old}}(\Gamma_1(1000))\) into lower level spaces

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

Hecke characteristic polynomials

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