Properties

Label 2004.1
Level 2004
Weight 1
Dimension 26
Nonzero newspaces 1
Newform subspaces 6
Sturm bound 223104
Trace bound 0

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Defining parameters

Level: \( N \) = \( 2004\( 2004 = 2^{2} \cdot 3 \cdot 167 \) \)
Weight: \( k \) = \( 1 \)
Nonzero newspaces: \( 1 \)
Newform subspaces: \( 6 \)
Sturm bound: \(223104\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(2004))\).

Total New Old
Modular forms 1716 358 1358
Cusp forms 56 26 30
Eisenstein series 1660 332 1328

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 26 0 0 0

Trace form

\( 26q + 4q^{4} - 4q^{6} + 4q^{9} + O(q^{10}) \) \( 26q + 4q^{4} - 4q^{6} + 4q^{9} + 4q^{16} - 4q^{24} - 18q^{25} + 4q^{36} - 11q^{42} - 11q^{48} - 18q^{49} + 7q^{54} + 4q^{64} + 11q^{72} + 4q^{81} + 11q^{84} - 8q^{85} - 4q^{96} - 8q^{97} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(2004))\)

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
2004.1.d \(\chi_{2004}(1337, \cdot)\) None 0 1
2004.1.e \(\chi_{2004}(1669, \cdot)\) None 0 1
2004.1.f \(\chi_{2004}(1003, \cdot)\) None 0 1
2004.1.g \(\chi_{2004}(2003, \cdot)\) 2004.1.g.a 1 1
2004.1.g.b 1
2004.1.g.c 2
2004.1.g.d 2
2004.1.g.e 10
2004.1.g.f 10
2004.1.k \(\chi_{2004}(23, \cdot)\) None 0 82
2004.1.l \(\chi_{2004}(7, \cdot)\) None 0 82
2004.1.m \(\chi_{2004}(13, \cdot)\) None 0 82
2004.1.n \(\chi_{2004}(29, \cdot)\) None 0 82

Decomposition of \(S_{1}^{\mathrm{old}}(\Gamma_1(2004))\) into lower level spaces

\( S_{1}^{\mathrm{old}}(\Gamma_1(2004)) \cong \) \(S_{1}^{\mathrm{new}}(\Gamma_1(167))\)\(^{\oplus 6}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \))(\( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \))
$3$ (\( 1 + T \))(\( 1 - T \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \))(\( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \))
$5$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( ( 1 + T^{2} )^{10} \))(\( ( 1 + T^{2} )^{10} \))
$7$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \))
$11$ (\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \))
$13$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{10}( 1 + T )^{10} \))(\( ( 1 - T )^{10}( 1 + T )^{10} \))
$17$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( ( 1 + T^{2} )^{10} \))(\( ( 1 + T^{2} )^{10} \))
$19$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \))
$23$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{10}( 1 + T )^{10} \))(\( ( 1 - T )^{10}( 1 + T )^{10} \))
$29$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \))
$31$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \))
$37$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{10}( 1 + T )^{10} \))(\( ( 1 - T )^{10}( 1 + T )^{10} \))
$41$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( ( 1 + T^{2} )^{10} \))(\( ( 1 + T^{2} )^{10} \))
$43$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( ( 1 + T^{2} )^{10} \))(\( ( 1 + T^{2} )^{10} \))
$47$ (\( ( 1 - T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \))
$53$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( ( 1 + T^{2} )^{10} \))(\( ( 1 + T^{2} )^{10} \))
$59$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{10}( 1 + T )^{10} \))(\( ( 1 - T )^{10}( 1 + T )^{10} \))
$61$ (\( ( 1 + T )^{2} \))(\( ( 1 + T )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \))
$67$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( ( 1 + T^{2} )^{10} \))(\( ( 1 + T^{2} )^{10} \))
$71$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{10}( 1 + T )^{10} \))(\( ( 1 - T )^{10}( 1 + T )^{10} \))
$73$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{10}( 1 + T )^{10} \))(\( ( 1 - T )^{10}( 1 + T )^{10} \))
$79$ (\( 1 + T^{2} \))(\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( 1 + T^{4} \))(\( ( 1 + T^{2} )^{10} \))(\( ( 1 + T^{2} )^{10} \))
$83$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{10}( 1 + T )^{10} \))(\( ( 1 - T )^{10}( 1 + T )^{10} \))
$89$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \))
$97$ (\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \))
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