Properties

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

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.249532506317\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

Character values

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

\(n\) \(251\) \(377\)
\(\chi(n)\) \(-1\) \(-\zeta_{10}^{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
0.809017 + 0.587785i
−0.309017 + 0.951057i
−0.309017 0.951057i
0.809017 0.587785i
0.809017 0.587785i 0 0.309017 0.951057i 0 0 0 −0.309017 0.951057i −0.809017 0.587785i 0
151.1 −0.309017 0.951057i 0 −0.809017 + 0.587785i 0 0 0 0.809017 + 0.587785i 0.309017 0.951057i 0
351.1 −0.309017 + 0.951057i 0 −0.809017 0.587785i 0 0 0 0.809017 0.587785i 0.309017 + 0.951057i 0
451.1 0.809017 + 0.587785i 0 0.309017 + 0.951057i 0 0 0 −0.309017 + 0.951057i −0.809017 + 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
25.d even 5 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.j.a 4
4.b odd 2 1 CM 500.1.j.a 4
5.b even 2 1 100.1.j.a 4
5.c odd 4 2 500.1.h.a 8
15.d odd 2 1 900.1.x.a 4
20.d odd 2 1 100.1.j.a 4
20.e even 4 2 500.1.h.a 8
25.d even 5 1 inner 500.1.j.a 4
25.d even 5 1 2500.1.b.a 2
25.d even 5 2 2500.1.j.b 4
25.e even 10 1 100.1.j.a 4
25.e even 10 1 2500.1.b.b 2
25.e even 10 2 2500.1.j.a 4
25.f odd 20 2 500.1.h.a 8
25.f odd 20 2 2500.1.d.a 4
25.f odd 20 4 2500.1.h.e 8
40.e odd 2 1 1600.1.bh.a 4
40.f even 2 1 1600.1.bh.a 4
60.h even 2 1 900.1.x.a 4
75.h odd 10 1 900.1.x.a 4
100.h odd 10 1 100.1.j.a 4
100.h odd 10 1 2500.1.b.b 2
100.h odd 10 2 2500.1.j.a 4
100.j odd 10 1 inner 500.1.j.a 4
100.j odd 10 1 2500.1.b.a 2
100.j odd 10 2 2500.1.j.b 4
100.l even 20 2 500.1.h.a 8
100.l even 20 2 2500.1.d.a 4
100.l even 20 4 2500.1.h.e 8
200.o even 10 1 1600.1.bh.a 4
200.s odd 10 1 1600.1.bh.a 4
300.r even 10 1 900.1.x.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.1.j.a 4 5.b even 2 1
100.1.j.a 4 20.d odd 2 1
100.1.j.a 4 25.e even 10 1
100.1.j.a 4 100.h odd 10 1
500.1.h.a 8 5.c odd 4 2
500.1.h.a 8 20.e even 4 2
500.1.h.a 8 25.f odd 20 2
500.1.h.a 8 100.l even 20 2
500.1.j.a 4 1.a even 1 1 trivial
500.1.j.a 4 4.b odd 2 1 CM
500.1.j.a 4 25.d even 5 1 inner
500.1.j.a 4 100.j odd 10 1 inner
900.1.x.a 4 15.d odd 2 1
900.1.x.a 4 60.h even 2 1
900.1.x.a 4 75.h odd 10 1
900.1.x.a 4 300.r even 10 1
1600.1.bh.a 4 40.e odd 2 1
1600.1.bh.a 4 40.f even 2 1
1600.1.bh.a 4 200.o even 10 1
1600.1.bh.a 4 200.s odd 10 1
2500.1.b.a 2 25.d even 5 1
2500.1.b.a 2 100.j odd 10 1
2500.1.b.b 2 25.e even 10 1
2500.1.b.b 2 100.h odd 10 1
2500.1.d.a 4 25.f odd 20 2
2500.1.d.a 4 100.l even 20 2
2500.1.h.e 8 25.f odd 20 4
2500.1.h.e 8 100.l even 20 4
2500.1.j.a 4 25.e even 10 2
2500.1.j.a 4 100.h odd 10 2
2500.1.j.b 4 25.d even 5 2
2500.1.j.b 4 100.j odd 10 2

Hecke kernels

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

Hecke characteristic polynomials

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