Properties

Label 4000.1.b
Level 4000
Weight 1
Character orbit b
Rep. character \(\chi_{4000}(2751,\cdot)\)
Character field \(\Q\)
Dimension 8
Newform subspaces 2
Sturm bound 600
Trace bound 13

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Defining parameters

Level: \( N \) \(=\) \( 4000 = 2^{5} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 4000.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 4 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(600\)
Trace bound: \(13\)

Dimensions

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

Total New Old
Modular forms 76 8 68
Cusp forms 36 8 28
Eisenstein series 40 0 40

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

\( 8q - 4q^{9} + O(q^{10}) \) \( 8q - 4q^{9} - 4q^{21} - 8q^{29} + 8q^{41} - 4q^{61} + O(q^{100}) \)

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.b.a \(4\) \(1.996\) \(\Q(i, \sqrt{5})\) \(A_{5}\) None None \(0\) \(0\) \(0\) \(0\) \(q+(-\beta _{1}-\beta _{3})q^{3}-\beta _{3}q^{7}+(-1-\beta _{2}+\cdots)q^{9}+\cdots\)
4000.1.b.b \(4\) \(1.996\) \(\Q(i, \sqrt{5})\) \(A_{5}\) None None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{1}q^{3}+\beta _{3}q^{7}+\beta _{2}q^{9}-\beta _{3}q^{11}+\cdots\)

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