Properties

Label 500.1.h
Level 500
Weight 1
Character orbit h
Rep. character \(\chi_{500}(99,\cdot)\)
Character field \(\Q(\zeta_{10})\)
Dimension 8
Newform subspaces 1
Sturm bound 75
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 500 = 2^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 500.h (of order \(10\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 100 \)
Character field: \(\Q(\zeta_{10})\)
Newform subspaces: \( 1 \)
Sturm bound: \(75\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(500, [\chi])\).

Total New Old
Modular forms 48 32 16
Cusp forms 8 8 0
Eisenstein series 40 24 16

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 8 0 0 0

Trace form

\( 8q + 2q^{4} + 2q^{9} + O(q^{10}) \) \( 8q + 2q^{4} + 2q^{9} - 2q^{16} - 4q^{26} + 4q^{29} - 6q^{34} - 2q^{36} - 4q^{41} - 8q^{49} - 4q^{61} + 2q^{64} + 4q^{74} - 2q^{81} - 6q^{89} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(500, [\chi])\) into newform subspaces

Label Dim. \(A\) Field Image CM RM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
500.1.h.a \(8\) \(0.250\) \(\Q(\zeta_{20})\) \(D_{5}\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{20}q^{2}+\zeta_{20}^{2}q^{4}-\zeta_{20}^{3}q^{8}-\zeta_{20}^{4}q^{9}+\cdots\)

Hecke characteristic polynomials

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