Properties

Label 500.1.d.b
Level $500$
Weight $1$
Character orbit 500.d
Self dual yes
Analytic conductor $0.250$
Analytic rank $0$
Dimension $2$
Projective image $D_{5}$
CM discriminant -20
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(0.249532506317\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \( x^{2} - x - 1 \) 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
Artin image: $D_5$
Artin field: Galois closure of 5.1.250000.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
499.1
−0.618034
1.61803
1.00000 −1.61803 1.00000 0 −1.61803 0.618034 1.00000 1.61803 0
499.2 1.00000 0.618034 1.00000 0 0.618034 −1.61803 1.00000 −0.618034 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 \(\Q(\sqrt{-5}) \)

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} - 1 \) acting on \(S_{1}^{\mathrm{new}}(500, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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