Properties

Label 2008.1
Level 2008
Weight 1
Dimension 130
Nonzero newspaces 4
Newform subspaces 5
Sturm bound 252000
Trace bound 1

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Defining parameters

Level: \( N \) = \( 2008\( 2008 = 2^{3} \cdot 251 \) \)
Weight: \( k \) = \( 1 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 5 \)
Sturm bound: \(252000\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(2008))\).

Total New Old
Modular forms 1656 628 1028
Cusp forms 156 130 26
Eisenstein series 1500 498 1002

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 130 0 0 0

Trace form

\( 130q - q^{2} - 2q^{3} + 5q^{4} - 2q^{6} - 2q^{7} - q^{8} + 3q^{9} + O(q^{10}) \) \( 130q - q^{2} - 2q^{3} + 5q^{4} - 2q^{6} - 2q^{7} - q^{8} + 3q^{9} - 2q^{11} - 2q^{12} + 5q^{16} - 4q^{17} - 3q^{18} - 2q^{19} - 4q^{22} - 2q^{23} - 2q^{24} + 5q^{25} - 4q^{27} - 2q^{28} - 2q^{31} - q^{32} - 4q^{33} - 2q^{34} + 3q^{36} - 4q^{38} - 4q^{41} - 2q^{43} - 2q^{44} - 2q^{48} + 3q^{49} - q^{50} - 4q^{51} - 4q^{54} - 4q^{57} - 2q^{58} - 2q^{59} - 2q^{63} + 5q^{64} - 4q^{66} - 2q^{67} - 4q^{68} - 3q^{72} - 4q^{73} - 2q^{74} - 2q^{75} - 2q^{76} - 2q^{79} + q^{81} - 2q^{82} - 2q^{83} - 4q^{86} - 4q^{88} - 4q^{89} - 2q^{92} - 2q^{96} - 2q^{97} - q^{98} - 6q^{99} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(2008))\)

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
2008.1.c \(\chi_{2008}(501, \cdot)\) 2008.1.c.a 3 1
2008.1.c.b 3
2008.1.d \(\chi_{2008}(503, \cdot)\) None 0 1
2008.1.g \(\chi_{2008}(1507, \cdot)\) None 0 1
2008.1.h \(\chi_{2008}(1505, \cdot)\) None 0 1
2008.1.j \(\chi_{2008}(219, \cdot)\) 2008.1.j.a 4 4
2008.1.l \(\chi_{2008}(353, \cdot)\) None 0 4
2008.1.m \(\chi_{2008}(389, \cdot)\) None 0 4
2008.1.p \(\chi_{2008}(271, \cdot)\) None 0 4
2008.1.s \(\chi_{2008}(377, \cdot)\) None 0 20
2008.1.t \(\chi_{2008}(63, \cdot)\) None 0 20
2008.1.w \(\chi_{2008}(51, \cdot)\) 2008.1.w.a 20 20
2008.1.x \(\chi_{2008}(157, \cdot)\) None 0 20
2008.1.z \(\chi_{2008}(33, \cdot)\) None 0 100
2008.1.bb \(\chi_{2008}(7, \cdot)\) None 0 100
2008.1.bd \(\chi_{2008}(3, \cdot)\) 2008.1.bd.a 100 100
2008.1.bf \(\chi_{2008}(29, \cdot)\) None 0 100

Decomposition of \(S_{1}^{\mathrm{old}}(\Gamma_1(2008))\) into lower level spaces

\( S_{1}^{\mathrm{old}}(\Gamma_1(2008)) \cong \) \(S_{1}^{\mathrm{new}}(\Gamma_1(251))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(1004))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + T )^{3} \))(\( ( 1 - T )^{3} \))(\( ( 1 - T )^{4} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \))(\( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{5} \))
$3$ (\( ( 1 - T )^{3}( 1 + T )^{3} \))(\( ( 1 - T )^{3}( 1 + T )^{3} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \))(\( ( 1 + T^{25} + T^{50} + T^{75} + T^{100} )^{2} \))
$5$ (\( ( 1 - T )^{3}( 1 + T )^{3} \))(\( ( 1 - T )^{3}( 1 + T )^{3} \))(\( ( 1 - T )^{4}( 1 + T )^{4} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \))(\( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{5} \))
$7$ (\( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \))(\( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \))(\( ( 1 - T^{25} + T^{50} - T^{75} + T^{100} )( 1 + T^{25} + T^{50} + T^{75} + T^{100} ) \))
$11$ (\( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \))(\( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \))(\( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \))(\( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{5}( 1 + T^{25} + T^{50} + T^{75} + T^{100} ) \))
$13$ (\( ( 1 - T )^{3}( 1 + T )^{3} \))(\( ( 1 - T )^{3}( 1 + T )^{3} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \))(\( ( 1 - T^{25} + T^{50} - T^{75} + T^{100} )( 1 + T^{25} + T^{50} + T^{75} + T^{100} ) \))
$17$ (\( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \))(\( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \))(\( ( 1 + T^{25} + T^{50} + T^{75} + T^{100} )^{2} \))
$19$ (\( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \))(\( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \))(\( ( 1 + T^{25} + T^{50} + T^{75} + T^{100} )^{2} \))
$23$ (\( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \))(\( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \))(\( ( 1 - T^{25} + T^{50} - T^{75} + T^{100} )( 1 + T^{25} + T^{50} + T^{75} + T^{100} ) \))
$29$ (\( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \))(\( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \))(\( ( 1 - T^{25} + T^{50} - T^{75} + T^{100} )( 1 + T^{25} + T^{50} + T^{75} + T^{100} ) \))
$31$ (\( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \))(\( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \))(\( ( 1 - T^{25} + T^{50} - T^{75} + T^{100} )( 1 + T^{25} + T^{50} + T^{75} + T^{100} ) \))
$37$ (\( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \))(\( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \))(\( ( 1 - T^{25} + T^{50} - T^{75} + T^{100} )( 1 + T^{25} + T^{50} + T^{75} + T^{100} ) \))
$41$ (\( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \))(\( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \))(\( ( 1 + T^{25} + T^{50} + T^{75} + T^{100} )^{2} \))
$43$ (\( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \))(\( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \))(\( ( 1 + T^{25} + T^{50} + T^{75} + T^{100} )^{2} \))
$47$ (\( ( 1 - T )^{3}( 1 + T )^{3} \))(\( ( 1 - T )^{3}( 1 + T )^{3} \))(\( ( 1 - T )^{4}( 1 + T )^{4} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \))(\( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{5} \))
$53$ (\( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \))(\( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \))(\( ( 1 - T^{25} + T^{50} - T^{75} + T^{100} )( 1 + T^{25} + T^{50} + T^{75} + T^{100} ) \))
$59$ (\( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \))(\( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \))(\( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \))(\( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{5}( 1 + T^{25} + T^{50} + T^{75} + T^{100} ) \))
$61$ (\( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \))(\( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \))(\( ( 1 - T^{25} + T^{50} - T^{75} + T^{100} )( 1 + T^{25} + T^{50} + T^{75} + T^{100} ) \))
$67$ (\( ( 1 - T )^{3}( 1 + T )^{3} \))(\( ( 1 - T )^{3}( 1 + T )^{3} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \))(\( ( 1 + T^{25} + T^{50} + T^{75} + T^{100} )^{2} \))
$71$ (\( ( 1 - T )^{3}( 1 + T )^{3} \))(\( ( 1 - T )^{3}( 1 + T )^{3} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \))(\( ( 1 - T^{25} + T^{50} - T^{75} + T^{100} )( 1 + T^{25} + T^{50} + T^{75} + T^{100} ) \))
$73$ (\( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \))(\( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \))(\( ( 1 + T^{25} + T^{50} + T^{75} + T^{100} )^{2} \))
$79$ (\( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \))(\( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T^{5} + T^{10} - T^{15} + T^{20} )( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \))(\( ( 1 - T^{25} + T^{50} - T^{75} + T^{100} )( 1 + T^{25} + T^{50} + T^{75} + T^{100} ) \))
$83$ (\( ( 1 - T )^{3}( 1 + T )^{3} \))(\( ( 1 - T )^{3}( 1 + T )^{3} \))(\( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{5}( 1 + T^{5} + T^{10} + T^{15} + T^{20} ) \))(\( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{5}( 1 + T^{25} + T^{50} + T^{75} + T^{100} ) \))
$89$ (\( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \))(\( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \))(\( ( 1 + T^{25} + T^{50} + T^{75} + T^{100} )^{2} \))
$97$ (\( ( 1 - T )^{3}( 1 + T )^{3} \))(\( ( 1 - T )^{3}( 1 + T )^{3} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 + T^{5} + T^{10} + T^{15} + T^{20} )^{2} \))(\( ( 1 + T^{25} + T^{50} + T^{75} + T^{100} )^{2} \))
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