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: |
\( x^{2} - x - 1 \)
|
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.
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 |
|
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 | |||||||||||||||||||||||||
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 \(S_{1}^{\mathrm{new}}(500, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T - 1)^{2} \)
$3$
\( T^{2} + T - 1 \)
$5$
\( T^{2} \)
$7$
\( T^{2} + T - 1 \)
$11$
\( T^{2} \)
$13$
\( T^{2} \)
$17$
\( T^{2} \)
$19$
\( T^{2} \)
$23$
\( T^{2} + T - 1 \)
$29$
\( T^{2} + T - 1 \)
$31$
\( T^{2} \)
$37$
\( T^{2} \)
$41$
\( T^{2} + T - 1 \)
$43$
\( T^{2} + T - 1 \)
$47$
\( T^{2} + T - 1 \)
$53$
\( T^{2} \)
$59$
\( T^{2} \)
$61$
\( T^{2} + T - 1 \)
$67$
\( (T - 2)^{2} \)
$71$
\( T^{2} \)
$73$
\( T^{2} \)
$79$
\( T^{2} \)
$83$
\( T^{2} + T - 1 \)
$89$
\( T^{2} + T - 1 \)
$97$
\( T^{2} \)
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