Newspace parameters
| Level: | \( N \) | \(=\) | \( 580 = 2^{2} \cdot 5 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 580.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(4.63132331723\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{11})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 12x^{2} + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} + 12x^{2} + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 7\nu ) / 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 17\nu ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 6 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -7\beta_{2} + 17\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/580\mathbb{Z}\right)^\times\).
| \(n\) | \(117\) | \(291\) | \(321\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 521.1 |
|
0 | − | 1.41421i | 0 | 1.00000 | 0 | −4.31662 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||
| 521.2 | 0 | − | 1.41421i | 0 | 1.00000 | 0 | 2.31662 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||
| 521.3 | 0 | 1.41421i | 0 | 1.00000 | 0 | −4.31662 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
| 521.4 | 0 | 1.41421i | 0 | 1.00000 | 0 | 2.31662 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 29.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 580.2.d.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | 5220.2.l.d | 4 | ||
| 4.b | odd | 2 | 1 | 2320.2.g.h | 4 | ||
| 5.b | even | 2 | 1 | 2900.2.d.d | 4 | ||
| 5.c | odd | 4 | 2 | 2900.2.f.c | 8 | ||
| 29.b | even | 2 | 1 | inner | 580.2.d.c | ✓ | 4 |
| 87.d | odd | 2 | 1 | 5220.2.l.d | 4 | ||
| 116.d | odd | 2 | 1 | 2320.2.g.h | 4 | ||
| 145.d | even | 2 | 1 | 2900.2.d.d | 4 | ||
| 145.h | odd | 4 | 2 | 2900.2.f.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 580.2.d.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 580.2.d.c | ✓ | 4 | 29.b | even | 2 | 1 | inner |
| 2320.2.g.h | 4 | 4.b | odd | 2 | 1 | ||
| 2320.2.g.h | 4 | 116.d | odd | 2 | 1 | ||
| 2900.2.d.d | 4 | 5.b | even | 2 | 1 | ||
| 2900.2.d.d | 4 | 145.d | even | 2 | 1 | ||
| 2900.2.f.c | 8 | 5.c | odd | 4 | 2 | ||
| 2900.2.f.c | 8 | 145.h | odd | 4 | 2 | ||
| 5220.2.l.d | 4 | 3.b | odd | 2 | 1 | ||
| 5220.2.l.d | 4 | 87.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 2 \)
acting on \(S_{2}^{\mathrm{new}}(580, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} + 2)^{2} \)
|
| $5$ |
\( (T - 1)^{4} \)
|
| $7$ |
\( (T^{2} + 2 T - 10)^{2} \)
|
| $11$ |
\( (T^{2} + 2)^{2} \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} + 60T^{2} + 196 \)
|
| $19$ |
\( T^{4} + 60T^{2} + 196 \)
|
| $23$ |
\( (T^{2} + 6 T - 2)^{2} \)
|
| $29$ |
\( T^{4} + 14T^{2} + 841 \)
|
| $31$ |
\( (T^{2} + 22)^{2} \)
|
| $37$ |
\( T^{4} + 60T^{2} + 196 \)
|
| $41$ |
\( T^{4} + 48T^{2} + 400 \)
|
| $43$ |
\( T^{4} + 180T^{2} + 7396 \)
|
| $47$ |
\( (T^{2} + 18)^{2} \)
|
| $53$ |
\( (T^{2} + 4 T - 40)^{2} \)
|
| $59$ |
\( (T^{2} - 8 T - 28)^{2} \)
|
| $61$ |
\( T^{4} + 48T^{2} + 400 \)
|
| $67$ |
\( (T^{2} + 2 T - 98)^{2} \)
|
| $71$ |
\( (T^{2} - 44)^{2} \)
|
| $73$ |
\( T^{4} + 212T^{2} + 4900 \)
|
| $79$ |
\( T^{4} + 212T^{2} + 4900 \)
|
| $83$ |
\( (T^{2} - 2 T - 10)^{2} \)
|
| $89$ |
\( (T^{2} + 128)^{2} \)
|
| $97$ |
\( (T^{2} + 18)^{2} \)
|
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