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: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 637x^{2} - 2512x + 15636 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 637x^{2} - 2512x + 15636 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 6\nu^{2} - 559\nu - 660 ) / 30 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{2} - 6\nu - 319 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 5\beta_{3} + 6\beta _1 + 319 \)
|
| \(\nu^{3}\) | \(=\) |
\( 30\beta_{3} + 30\beta_{2} + 595\beta _1 + 2574 \)
|
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 | −19.6169 | 0 | −25.0000 | 0 | 0 | 0 | 141.822 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 4.63169 | 0 | −25.0000 | 0 | 0 | 0 | −221.547 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 16.0160 | 0 | −25.0000 | 0 | 0 | 0 | 13.5121 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 28.9692 | 0 | −25.0000 | 0 | 0 | 0 | 596.214 | 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.k | 4 | |
| 7.b | odd | 2 | 1 | 980.6.a.j | 4 | ||
| 7.d | odd | 6 | 2 | 140.6.i.c | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 140.6.i.c | ✓ | 8 | 7.d | odd | 6 | 2 | |
| 980.6.a.j | 4 | 7.b | odd | 2 | 1 | ||
| 980.6.a.k | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 30T_{3}^{3} - 301T_{3}^{2} + 11040T_{3} - 42156 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(980))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 30 T^{3} + \cdots - 42156 \)
|
| $5$ |
\( (T + 25)^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + 776 T^{3} + \cdots - 469438956 \)
|
| $13$ |
\( T^{4} - 590 T^{3} + \cdots + 617640956 \)
|
| $17$ |
\( T^{4} + \cdots + 108098753904 \)
|
| $19$ |
\( T^{4} + \cdots + 417087145576 \)
|
| $23$ |
\( T^{4} + \cdots - 2005803838971 \)
|
| $29$ |
\( T^{4} + \cdots + 10704301699716 \)
|
| $31$ |
\( T^{4} + \cdots - 225405687595264 \)
|
| $37$ |
\( T^{4} + \cdots - 98\!\cdots\!44 \)
|
| $41$ |
\( T^{4} + \cdots - 58859927331471 \)
|
| $43$ |
\( T^{4} + \cdots - 22\!\cdots\!84 \)
|
| $47$ |
\( T^{4} + \cdots + 18\!\cdots\!76 \)
|
| $53$ |
\( T^{4} + \cdots - 12\!\cdots\!76 \)
|
| $59$ |
\( T^{4} + \cdots - 58\!\cdots\!04 \)
|
| $61$ |
\( T^{4} + \cdots + 38\!\cdots\!16 \)
|
| $67$ |
\( T^{4} + \cdots - 14\!\cdots\!24 \)
|
| $71$ |
\( T^{4} + \cdots - 96\!\cdots\!16 \)
|
| $73$ |
\( T^{4} + \cdots - 32\!\cdots\!84 \)
|
| $79$ |
\( T^{4} + \cdots - 93\!\cdots\!96 \)
|
| $83$ |
\( T^{4} + \cdots - 18\!\cdots\!16 \)
|
| $89$ |
\( T^{4} + \cdots + 15\!\cdots\!76 \)
|
| $97$ |
\( T^{4} + \cdots - 56\!\cdots\!56 \)
|
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