Properties

Label 4000.1.bf
Level 4000
Weight 1
Character orbit bf
Rep. character \(\chi_{4000}(799,\cdot)\)
Character field \(\Q(\zeta_{10})\)
Dimension 16
Newform subspaces 2
Sturm bound 600
Trace bound 3

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Defining parameters

Level: \( N \) = \( 4000 = 2^{5} \cdot 5^{3} \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 4000.bf (of order \(10\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 100 \)
Character field: \(\Q(\zeta_{10})\)
Newform subspaces: \( 2 \)
Sturm bound: \(600\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(4000, [\chi])\).

Total New Old
Modular forms 232 16 216
Cusp forms 72 16 56
Eisenstein series 160 0 160

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 0 0 0 16

Trace form

\( 16q + O(q^{10}) \) \( 16q + 8q^{21} + 12q^{29} + 8q^{49} - 4q^{61} - 4q^{69} + 4q^{81} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(4000, [\chi])\) into newform subspaces

Label Dim. \(A\) Field Image CM RM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
4000.1.bf.a \(8\) \(1.996\) \(\Q(\zeta_{20})\) \(A_{5}\) None None \(0\) \(-2\) \(0\) \(4\) \(q-\zeta_{20}^{2}q^{3}+(-\zeta_{20}^{4}+\zeta_{20}^{6})q^{7}+\cdots\)
4000.1.bf.b \(8\) \(1.996\) \(\Q(\zeta_{20})\) \(A_{5}\) None None \(0\) \(2\) \(0\) \(-4\) \(q+\zeta_{20}^{2}q^{3}+(\zeta_{20}^{4}-\zeta_{20}^{6})q^{7}+(-\zeta_{20}^{3}+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{1}^{\mathrm{old}}(4000, [\chi])\) into lower level spaces

\( S_{1}^{\mathrm{old}}(4000, [\chi]) \cong \) \(S_{1}^{\mathrm{new}}(400, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(500, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(2000, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \))(\( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \))
$5$ 1
$7$ (\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \))
$11$ (\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))
$13$ (\( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \))(\( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \))
$17$ (\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))
$19$ (\( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \))(\( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \))
$23$ (\( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \))(\( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \))
$29$ (\( ( 1 - T )^{8}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 - T )^{8}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))
$31$ (\( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \))(\( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \))
$37$ (\( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \))(\( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \))
$41$ (\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))
$43$ (\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \))
$47$ (\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \))
$53$ (\( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \))(\( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \))
$59$ (\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))
$61$ (\( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \))(\( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \))
$67$ (\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))
$71$ (\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))
$73$ (\( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \))(\( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \))
$79$ (\( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \))(\( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \))
$83$ (\( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \))(\( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \))
$89$ (\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))
$97$ (\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))
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