Properties

Label 4000.1.bh
Level 4000
Weight 1
Character orbit bh
Rep. character \(\chi_{4000}(351,\cdot)\)
Character field \(\Q(\zeta_{10})\)
Dimension 8
Newform subspaces 1
Sturm bound 600
Trace bound 0

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Defining parameters

Level: \( N \) = \( 4000 = 2^{5} \cdot 5^{3} \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 4000.bh (of order \(10\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 100 \)
Character field: \(\Q(\zeta_{10})\)
Newform subspaces: \( 1 \)
Sturm bound: \(600\)
Trace bound: \(0\)

Dimensions

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

Total New Old
Modular forms 248 8 240
Cusp forms 88 8 80
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 8

Trace form

\( 8q + O(q^{10}) \) \( 8q - 6q^{13} + 4q^{21} - 6q^{29} - 2q^{37} - 4q^{49} - 2q^{53} + 4q^{57} - 2q^{61} + 2q^{69} + 2q^{73} + 2q^{81} + 8q^{93} - 4q^{97} + O(q^{100}) \)

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.bh.a \(8\) \(1.996\) \(\Q(\zeta_{20})\) \(A_{5}\) None None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{20}q^{3}+(\zeta_{20}^{3}+\zeta_{20}^{7})q^{7}+(-1+\cdots)q^{13}+\cdots\)

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

\( S_{1}^{\mathrm{old}}(4000, [\chi]) \cong \) \(S_{1}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(500, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(800, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(2000, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

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