Properties

Label 500.1.b.a
Level $500$
Weight $1$
Character orbit 500.b
Analytic conductor $0.250$
Analytic rank $0$
Dimension $4$
Projective image $D_{5}$
CM discriminant -20
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 500 = 2^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 500.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.249532506317\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.250000.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - \beta_1 q^{3} - q^{4} + \beta_{2} q^{6} + ( - \beta_{3} - \beta_1) q^{7} - \beta_{3} q^{8} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - \beta_1 q^{3} - q^{4} + \beta_{2} q^{6} + ( - \beta_{3} - \beta_1) q^{7} - \beta_{3} q^{8} + \beta_{2} q^{9} + \beta_1 q^{12} + (\beta_{2} + 1) q^{14} + q^{16} + \beta_1 q^{18} - q^{21} + (\beta_{3} + \beta_1) q^{23} - \beta_{2} q^{24} - \beta_{3} q^{27} + (\beta_{3} + \beta_1) q^{28} - \beta_{2} q^{29} + \beta_{3} q^{32} - \beta_{2} q^{36} + \beta_{2} q^{41} - \beta_{3} q^{42} - \beta_1 q^{43} + ( - \beta_{2} - 1) q^{46} + \beta_1 q^{47} - \beta_1 q^{48} + ( - \beta_{2} - 1) q^{49} + q^{54} + ( - \beta_{2} - 1) q^{56} - \beta_1 q^{58} + ( - \beta_{2} - 1) q^{61} - \beta_{3} q^{63} - q^{64} + 2 \beta_{3} q^{67} + q^{69} - \beta_1 q^{72} + \beta_1 q^{82} + (\beta_{3} + \beta_1) q^{83} + q^{84} + \beta_{2} q^{86} + (\beta_{3} - \beta_1) q^{87} + (\beta_{2} + 1) q^{89} + ( - \beta_{3} - \beta_1) q^{92} - \beta_{2} q^{94} + \beta_{2} q^{96} + ( - \beta_{3} - \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 2 q^{6} - 2 q^{9} + 2 q^{14} + 4 q^{16} - 4 q^{21} + 2 q^{24} + 2 q^{29} + 2 q^{36} - 2 q^{41} - 2 q^{46} - 2 q^{49} + 4 q^{54} - 2 q^{56} - 2 q^{61} - 4 q^{64} + 4 q^{69} + 4 q^{84} - 2 q^{86} + 2 q^{89} + 2 q^{94} - 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 2\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(251\) \(377\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
1.61803i
0.618034i
0.618034i
1.61803i
1.00000i 1.61803i −1.00000 0 −1.61803 0.618034i 1.00000i −1.61803 0
251.2 1.00000i 0.618034i −1.00000 0 0.618034 1.61803i 1.00000i 0.618034 0
251.3 1.00000i 0.618034i −1.00000 0 0.618034 1.61803i 1.00000i 0.618034 0
251.4 1.00000i 1.61803i −1.00000 0 −1.61803 0.618034i 1.00000i −1.61803 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 500.1.b.a 4
4.b odd 2 1 inner 500.1.b.a 4
5.b even 2 1 inner 500.1.b.a 4
5.c odd 4 1 500.1.d.a 2
5.c odd 4 1 500.1.d.b 2
20.d odd 2 1 CM 500.1.b.a 4
20.e even 4 1 500.1.d.a 2
20.e even 4 1 500.1.d.b 2
25.d even 5 2 2500.1.j.c 8
25.d even 5 2 2500.1.j.d 8
25.e even 10 2 2500.1.j.c 8
25.e even 10 2 2500.1.j.d 8
25.f odd 20 2 2500.1.h.a 4
25.f odd 20 2 2500.1.h.b 4
25.f odd 20 2 2500.1.h.c 4
25.f odd 20 2 2500.1.h.d 4
100.h odd 10 2 2500.1.j.c 8
100.h odd 10 2 2500.1.j.d 8
100.j odd 10 2 2500.1.j.c 8
100.j odd 10 2 2500.1.j.d 8
100.l even 20 2 2500.1.h.a 4
100.l even 20 2 2500.1.h.b 4
100.l even 20 2 2500.1.h.c 4
100.l even 20 2 2500.1.h.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
500.1.b.a 4 1.a even 1 1 trivial
500.1.b.a 4 4.b odd 2 1 inner
500.1.b.a 4 5.b even 2 1 inner
500.1.b.a 4 20.d odd 2 1 CM
500.1.d.a 2 5.c odd 4 1
500.1.d.a 2 20.e even 4 1
500.1.d.b 2 5.c odd 4 1
500.1.d.b 2 20.e even 4 1
2500.1.h.a 4 25.f odd 20 2
2500.1.h.a 4 100.l even 20 2
2500.1.h.b 4 25.f odd 20 2
2500.1.h.b 4 100.l even 20 2
2500.1.h.c 4 25.f odd 20 2
2500.1.h.c 4 100.l even 20 2
2500.1.h.d 4 25.f odd 20 2
2500.1.h.d 4 100.l even 20 2
2500.1.j.c 8 25.d even 5 2
2500.1.j.c 8 25.e even 10 2
2500.1.j.c 8 100.h odd 10 2
2500.1.j.c 8 100.j odd 10 2
2500.1.j.d 8 25.d even 5 2
2500.1.j.d 8 25.e even 10 2
2500.1.j.d 8 100.h odd 10 2
2500.1.j.d 8 100.j odd 10 2

Hecke kernels

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

Hecke characteristic polynomials

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