Properties

Label 500.1.h.a
Level $500$
Weight $1$
Character orbit 500.h
Analytic conductor $0.250$
Analytic rank $0$
Dimension $8$
Projective image $D_{5}$
CM discriminant -4
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 500 = 2^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 500.h (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.249532506317\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
Defining polynomial: \(x^{8} - x^{6} + x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 100)
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.6250000.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{20} q^{2} + \zeta_{20}^{2} q^{4} -\zeta_{20}^{3} q^{8} -\zeta_{20}^{4} q^{9} +O(q^{10})\) \( q -\zeta_{20} q^{2} + \zeta_{20}^{2} q^{4} -\zeta_{20}^{3} q^{8} -\zeta_{20}^{4} q^{9} + ( \zeta_{20} - \zeta_{20}^{7} ) q^{13} + \zeta_{20}^{4} q^{16} + ( \zeta_{20}^{7} - \zeta_{20}^{9} ) q^{17} + \zeta_{20}^{5} q^{18} + ( -\zeta_{20}^{2} + \zeta_{20}^{8} ) q^{26} + ( \zeta_{20}^{6} - \zeta_{20}^{8} ) q^{29} -\zeta_{20}^{5} q^{32} + ( -1 - \zeta_{20}^{8} ) q^{34} -\zeta_{20}^{6} q^{36} + ( \zeta_{20}^{3} - \zeta_{20}^{5} ) q^{37} + ( -\zeta_{20}^{2} - \zeta_{20}^{6} ) q^{41} - q^{49} + ( \zeta_{20}^{3} - \zeta_{20}^{9} ) q^{52} + ( \zeta_{20}^{5} + \zeta_{20}^{9} ) q^{53} + ( -\zeta_{20}^{7} + \zeta_{20}^{9} ) q^{58} + ( \zeta_{20}^{4} + \zeta_{20}^{8} ) q^{61} + \zeta_{20}^{6} q^{64} + ( \zeta_{20} + \zeta_{20}^{9} ) q^{68} + \zeta_{20}^{7} q^{72} + ( -\zeta_{20}^{3} + \zeta_{20}^{9} ) q^{73} + ( -\zeta_{20}^{4} + \zeta_{20}^{6} ) q^{74} + \zeta_{20}^{8} q^{81} + ( \zeta_{20}^{3} + \zeta_{20}^{7} ) q^{82} + ( -1 + \zeta_{20}^{2} ) q^{89} + ( -\zeta_{20} + \zeta_{20}^{3} ) q^{97} + \zeta_{20} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{4} + 2q^{9} + O(q^{10}) \) \( 8q + 2q^{4} + 2q^{9} - 2q^{16} - 4q^{26} + 4q^{29} - 6q^{34} - 2q^{36} - 4q^{41} - 8q^{49} - 4q^{61} + 2q^{64} + 4q^{74} - 2q^{81} - 6q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/500\mathbb{Z}\right)^\times\).

\(n\) \(251\) \(377\)
\(\chi(n)\) \(-1\) \(\zeta_{20}^{2}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
0.951057 0.309017i
−0.951057 + 0.309017i
0.587785 + 0.809017i
−0.587785 0.809017i
0.587785 0.809017i
−0.587785 + 0.809017i
0.951057 + 0.309017i
−0.951057 0.309017i
−0.951057 + 0.309017i 0 0.809017 0.587785i 0 0 0 −0.587785 + 0.809017i −0.309017 + 0.951057i 0
99.2 0.951057 0.309017i 0 0.809017 0.587785i 0 0 0 0.587785 0.809017i −0.309017 + 0.951057i 0
199.1 −0.587785 0.809017i 0 −0.309017 + 0.951057i 0 0 0 0.951057 0.309017i 0.809017 + 0.587785i 0
199.2 0.587785 + 0.809017i 0 −0.309017 + 0.951057i 0 0 0 −0.951057 + 0.309017i 0.809017 + 0.587785i 0
299.1 −0.587785 + 0.809017i 0 −0.309017 0.951057i 0 0 0 0.951057 + 0.309017i 0.809017 0.587785i 0
299.2 0.587785 0.809017i 0 −0.309017 0.951057i 0 0 0 −0.951057 0.309017i 0.809017 0.587785i 0
399.1 −0.951057 0.309017i 0 0.809017 + 0.587785i 0 0 0 −0.587785 0.809017i −0.309017 0.951057i 0
399.2 0.951057 + 0.309017i 0 0.809017 + 0.587785i 0 0 0 0.587785 + 0.809017i −0.309017 0.951057i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 399.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
5.b even 2 1 inner
20.d odd 2 1 inner
25.d even 5 1 inner
25.e even 10 1 inner
100.h odd 10 1 inner
100.j odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 500.1.h.a 8
4.b odd 2 1 CM 500.1.h.a 8
5.b even 2 1 inner 500.1.h.a 8
5.c odd 4 1 100.1.j.a 4
5.c odd 4 1 500.1.j.a 4
15.e even 4 1 900.1.x.a 4
20.d odd 2 1 inner 500.1.h.a 8
20.e even 4 1 100.1.j.a 4
20.e even 4 1 500.1.j.a 4
25.d even 5 1 inner 500.1.h.a 8
25.d even 5 1 2500.1.d.a 4
25.d even 5 2 2500.1.h.e 8
25.e even 10 1 inner 500.1.h.a 8
25.e even 10 1 2500.1.d.a 4
25.e even 10 2 2500.1.h.e 8
25.f odd 20 1 100.1.j.a 4
25.f odd 20 1 500.1.j.a 4
25.f odd 20 1 2500.1.b.a 2
25.f odd 20 1 2500.1.b.b 2
25.f odd 20 2 2500.1.j.a 4
25.f odd 20 2 2500.1.j.b 4
40.i odd 4 1 1600.1.bh.a 4
40.k even 4 1 1600.1.bh.a 4
60.l odd 4 1 900.1.x.a 4
75.l even 20 1 900.1.x.a 4
100.h odd 10 1 inner 500.1.h.a 8
100.h odd 10 1 2500.1.d.a 4
100.h odd 10 2 2500.1.h.e 8
100.j odd 10 1 inner 500.1.h.a 8
100.j odd 10 1 2500.1.d.a 4
100.j odd 10 2 2500.1.h.e 8
100.l even 20 1 100.1.j.a 4
100.l even 20 1 500.1.j.a 4
100.l even 20 1 2500.1.b.a 2
100.l even 20 1 2500.1.b.b 2
100.l even 20 2 2500.1.j.a 4
100.l even 20 2 2500.1.j.b 4
200.v even 20 1 1600.1.bh.a 4
200.x odd 20 1 1600.1.bh.a 4
300.u odd 20 1 900.1.x.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.1.j.a 4 5.c odd 4 1
100.1.j.a 4 20.e even 4 1
100.1.j.a 4 25.f odd 20 1
100.1.j.a 4 100.l even 20 1
500.1.h.a 8 1.a even 1 1 trivial
500.1.h.a 8 4.b odd 2 1 CM
500.1.h.a 8 5.b even 2 1 inner
500.1.h.a 8 20.d odd 2 1 inner
500.1.h.a 8 25.d even 5 1 inner
500.1.h.a 8 25.e even 10 1 inner
500.1.h.a 8 100.h odd 10 1 inner
500.1.h.a 8 100.j odd 10 1 inner
500.1.j.a 4 5.c odd 4 1
500.1.j.a 4 20.e even 4 1
500.1.j.a 4 25.f odd 20 1
500.1.j.a 4 100.l even 20 1
900.1.x.a 4 15.e even 4 1
900.1.x.a 4 60.l odd 4 1
900.1.x.a 4 75.l even 20 1
900.1.x.a 4 300.u odd 20 1
1600.1.bh.a 4 40.i odd 4 1
1600.1.bh.a 4 40.k even 4 1
1600.1.bh.a 4 200.v even 20 1
1600.1.bh.a 4 200.x odd 20 1
2500.1.b.a 2 25.f odd 20 1
2500.1.b.a 2 100.l even 20 1
2500.1.b.b 2 25.f odd 20 1
2500.1.b.b 2 100.l even 20 1
2500.1.d.a 4 25.d even 5 1
2500.1.d.a 4 25.e even 10 1
2500.1.d.a 4 100.h odd 10 1
2500.1.d.a 4 100.j odd 10 1
2500.1.h.e 8 25.d even 5 2
2500.1.h.e 8 25.e even 10 2
2500.1.h.e 8 100.h odd 10 2
2500.1.h.e 8 100.j odd 10 2
2500.1.j.a 4 25.f odd 20 2
2500.1.j.a 4 100.l even 20 2
2500.1.j.b 4 25.f odd 20 2
2500.1.j.b 4 100.l even 20 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(500, [\chi])\).

Hecke characteristic polynomials

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