Newspace parameters
| Level: | \( N \) | \(=\) | \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 980.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(157.176143417\) |
| 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: |
\( x^{3} - 499x - 210 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 140) |
| 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
\( x^{3} - 499x - 210 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 333 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 333 \)
|
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 |
|
0 | −24.5458 | 0 | −25.0000 | 0 | 0 | 0 | 359.498 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −1.57901 | 0 | −25.0000 | 0 | 0 | 0 | −240.507 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 20.1248 | 0 | −25.0000 | 0 | 0 | 0 | 162.009 | 0 | ||||||||||||||||||||||||||||
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 | 980.6.a.h | 3 | |
| 7.b | odd | 2 | 1 | 140.6.a.d | ✓ | 3 | |
| 28.d | even | 2 | 1 | 560.6.a.t | 3 | ||
| 35.c | odd | 2 | 1 | 700.6.a.i | 3 | ||
| 35.f | even | 4 | 2 | 700.6.e.g | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 140.6.a.d | ✓ | 3 | 7.b | odd | 2 | 1 | |
| 560.6.a.t | 3 | 28.d | even | 2 | 1 | ||
| 700.6.a.i | 3 | 35.c | odd | 2 | 1 | ||
| 700.6.e.g | 6 | 35.f | even | 4 | 2 | ||
| 980.6.a.h | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 6T_{3}^{2} - 487T_{3} - 780 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(980))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + 6 T^{2} + \cdots - 780 \)
|
| $5$ |
\( (T + 25)^{3} \)
|
| $7$ |
\( T^{3} \)
|
| $11$ |
\( T^{3} + 14 T^{2} + \cdots + 12436740 \)
|
| $13$ |
\( T^{3} - 4 T^{2} + \cdots + 6140734 \)
|
| $17$ |
\( T^{3} + 44 T^{2} + \cdots - 279119070 \)
|
| $19$ |
\( T^{3} + \cdots - 11429521264 \)
|
| $23$ |
\( T^{3} + \cdots + 2083660704 \)
|
| $29$ |
\( T^{3} + \cdots + 17580868722 \)
|
| $31$ |
\( T^{3} + \cdots - 211802104832 \)
|
| $37$ |
\( T^{3} + \cdots - 47286923800 \)
|
| $41$ |
\( T^{3} + \cdots - 478579953600 \)
|
| $43$ |
\( T^{3} + \cdots + 750561676176 \)
|
| $47$ |
\( T^{3} + \cdots + 173657376144 \)
|
| $53$ |
\( T^{3} + \cdots + 743911257600 \)
|
| $59$ |
\( T^{3} + \cdots - 41176040028480 \)
|
| $61$ |
\( T^{3} + \cdots + 169955356480 \)
|
| $67$ |
\( T^{3} + \cdots - 15170685707520 \)
|
| $71$ |
\( T^{3} + \cdots + 113425504819200 \)
|
| $73$ |
\( T^{3} + \cdots + 11043630664360 \)
|
| $79$ |
\( T^{3} + \cdots - 274092525845520 \)
|
| $83$ |
\( T^{3} + \cdots + 40289422939200 \)
|
| $89$ |
\( T^{3} + \cdots + 305457269205600 \)
|
| $97$ |
\( T^{3} + \cdots + 41652594334882 \)
|
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