Newspace parameters
| Level: | \( N \) | \(=\) | \( 490 = 2 \cdot 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 490.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(78.5880717084\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{3} - x^{2} - 438x - 1536 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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
\( x^{3} - x^{2} - 438x - 1536 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 6\nu - 290 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 6\beta _1 + 290 \)
|
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 | −19.3265 | 16.0000 | 25.0000 | −77.3060 | 0 | 64.0000 | 130.513 | 100.000 | |||||||||||||||||||||||||||
| 1.2 | 4.00000 | −4.64808 | 16.0000 | 25.0000 | −18.5923 | 0 | 64.0000 | −221.395 | 100.000 | ||||||||||||||||||||||||||||
| 1.3 | 4.00000 | 21.9746 | 16.0000 | 25.0000 | 87.8983 | 0 | 64.0000 | 239.882 | 100.000 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
| \(7\) | \(-1\) |
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 | 490.6.a.x | ✓ | 3 |
| 7.b | odd | 2 | 1 | 490.6.a.y | yes | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 490.6.a.x | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 490.6.a.y | yes | 3 | 7.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 2T_{3}^{2} - 437T_{3} - 1974 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(490))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{3} \)
|
| $3$ |
\( T^{3} + 2 T^{2} + \cdots - 1974 \)
|
| $5$ |
\( (T - 25)^{3} \)
|
| $7$ |
\( T^{3} \)
|
| $11$ |
\( T^{3} + 144 T^{2} + \cdots + 36714100 \)
|
| $13$ |
\( T^{3} - 288 T^{2} + \cdots - 27112106 \)
|
| $17$ |
\( T^{3} + \cdots + 1725187024 \)
|
| $19$ |
\( T^{3} - 1134 T^{2} + \cdots + 549437112 \)
|
| $23$ |
\( T^{3} - 1872 T^{2} + \cdots - 511879424 \)
|
| $29$ |
\( T^{3} + \cdots - 298814968674 \)
|
| $31$ |
\( T^{3} + \cdots - 131675935312 \)
|
| $37$ |
\( T^{3} + \cdots + 843065971840 \)
|
| $41$ |
\( T^{3} - 6204 T^{2} + \cdots - 329347312 \)
|
| $43$ |
\( T^{3} + \cdots + 734044145936 \)
|
| $47$ |
\( T^{3} + \cdots + 144830063108 \)
|
| $53$ |
\( T^{3} + \cdots + 3065842096760 \)
|
| $59$ |
\( T^{3} + \cdots - 18977297698072 \)
|
| $61$ |
\( T^{3} + \cdots + 232565751040 \)
|
| $67$ |
\( T^{3} + \cdots - 134174283407600 \)
|
| $71$ |
\( T^{3} + \cdots - 506529050112 \)
|
| $73$ |
\( T^{3} + \cdots - 133474914210928 \)
|
| $79$ |
\( T^{3} + \cdots - 40492333854100 \)
|
| $83$ |
\( T^{3} + \cdots + 265902053984 \)
|
| $89$ |
\( T^{3} + \cdots - 346860567680000 \)
|
| $97$ |
\( T^{3} + \cdots + 364270677687000 \)
|
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