Newspace parameters
| Level: | \( N \) | \(=\) | \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 570.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(91.4187772934\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 787x - 4530 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - 787x - 4530 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 9\nu - 526 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 8\beta_{2} + 9\beta _1 + 1052 ) / 2 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
4.00000 | −9.00000 | 16.0000 | −25.0000 | −36.0000 | −152.799 | 64.0000 | 81.0000 | −100.000 | |||||||||||||||||||||||||||
| 1.2 | 4.00000 | −9.00000 | 16.0000 | −25.0000 | −36.0000 | 26.8435 | 64.0000 | 81.0000 | −100.000 | ||||||||||||||||||||||||||||
| 1.3 | 4.00000 | −9.00000 | 16.0000 | −25.0000 | −36.0000 | 135.955 | 64.0000 | 81.0000 | −100.000 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(5\) | \(1\) |
| \(19\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 570.6.a.e | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 570.6.a.e | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{3} - 10T_{7}^{2} - 21226T_{7} + 557644 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(570))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{3} \)
|
| $3$ |
\( (T + 9)^{3} \)
|
| $5$ |
\( (T + 25)^{3} \)
|
| $7$ |
\( T^{3} - 10 T^{2} + \cdots + 557644 \)
|
| $11$ |
\( T^{3} + 572 T^{2} + \cdots - 1214424 \)
|
| $13$ |
\( T^{3} - 376 T^{2} + \cdots + 12753168 \)
|
| $17$ |
\( T^{3} - 758 T^{2} + \cdots + 124016112 \)
|
| $19$ |
\( (T - 361)^{3} \)
|
| $23$ |
\( T^{3} + \cdots + 3211873176 \)
|
| $29$ |
\( T^{3} + \cdots - 8883547452 \)
|
| $31$ |
\( T^{3} + \cdots - 1387442304 \)
|
| $37$ |
\( T^{3} + \cdots + 234425887168 \)
|
| $41$ |
\( T^{3} + \cdots - 347579414100 \)
|
| $43$ |
\( T^{3} + \cdots - 3039503311260 \)
|
| $47$ |
\( T^{3} + \cdots - 114730359336 \)
|
| $53$ |
\( T^{3} + \cdots + 8115365685024 \)
|
| $59$ |
\( T^{3} + \cdots - 8690848839072 \)
|
| $61$ |
\( T^{3} + \cdots - 489230088288 \)
|
| $67$ |
\( T^{3} + \cdots - 39210324278400 \)
|
| $71$ |
\( T^{3} + \cdots - 41793891053760 \)
|
| $73$ |
\( T^{3} + \cdots - 60088853187256 \)
|
| $79$ |
\( T^{3} + \cdots - 73073301583616 \)
|
| $83$ |
\( T^{3} + \cdots + 171562448805000 \)
|
| $89$ |
\( T^{3} + \cdots + 53506479041100 \)
|
| $97$ |
\( T^{3} + \cdots + 109640080369016 \)
|
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