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: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.3101016.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 312x + 1740 \)
|
| 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
\( x^{3} - x^{2} - 312x + 1740 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 7\nu - 211 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 7\beta _1 + 211 \)
|
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 | −16.5332 | 0 | 25.0000 | 0 | 0 | 0 | 30.3467 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 9.22631 | 0 | 25.0000 | 0 | 0 | 0 | −157.875 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 17.3069 | 0 | 25.0000 | 0 | 0 | 0 | 56.5286 | 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.i | 3 | |
| 7.b | odd | 2 | 1 | 140.6.a.c | ✓ | 3 | |
| 28.d | even | 2 | 1 | 560.6.a.u | 3 | ||
| 35.c | odd | 2 | 1 | 700.6.a.j | 3 | ||
| 35.f | even | 4 | 2 | 700.6.e.h | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 140.6.a.c | ✓ | 3 | 7.b | odd | 2 | 1 | |
| 560.6.a.u | 3 | 28.d | even | 2 | 1 | ||
| 700.6.a.j | 3 | 35.c | odd | 2 | 1 | ||
| 700.6.e.h | 6 | 35.f | even | 4 | 2 | ||
| 980.6.a.i | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 10T_{3}^{2} - 279T_{3} + 2640 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(980))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} - 10 T^{2} + \cdots + 2640 \)
|
| $5$ |
\( (T - 25)^{3} \)
|
| $7$ |
\( T^{3} \)
|
| $11$ |
\( T^{3} - 58 T^{2} + \cdots - 14472500 \)
|
| $13$ |
\( T^{3} + 148 T^{2} + \cdots - 266840838 \)
|
| $17$ |
\( T^{3} + \cdots - 4796110110 \)
|
| $19$ |
\( T^{3} + \cdots - 8250023232 \)
|
| $23$ |
\( T^{3} + \cdots + 11010917376 \)
|
| $29$ |
\( T^{3} + \cdots + 4810314762 \)
|
| $31$ |
\( T^{3} + \cdots - 19217358464 \)
|
| $37$ |
\( T^{3} + \cdots - 16558995800 \)
|
| $41$ |
\( T^{3} + \cdots - 285705250976 \)
|
| $43$ |
\( T^{3} + \cdots - 230990747600 \)
|
| $47$ |
\( T^{3} + \cdots - 3541720333592 \)
|
| $53$ |
\( T^{3} + \cdots + 4215757774880 \)
|
| $59$ |
\( T^{3} + \cdots + 11147794362880 \)
|
| $61$ |
\( T^{3} + \cdots - 27664467900480 \)
|
| $67$ |
\( T^{3} + \cdots + 78576282671040 \)
|
| $71$ |
\( T^{3} + \cdots - 17256845706240 \)
|
| $73$ |
\( T^{3} + \cdots + 157999195437960 \)
|
| $79$ |
\( T^{3} + \cdots + 5100296528456 \)
|
| $83$ |
\( T^{3} + \cdots + 249960926161280 \)
|
| $89$ |
\( T^{3} + \cdots - 102863482208800 \)
|
| $97$ |
\( T^{3} + \cdots - 12\!\cdots\!18 \)
|
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