Newspace parameters
| Level: | \( N \) | \(=\) | \( 280 = 2^{3} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 280.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(87.4678071356\) |
| 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: |
\( x^{4} - x^{3} - 2283x^{2} - 2749x + 794610 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 2283x^{2} - 2749x + 794610 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 541\nu^{2} - 4153\nu - 622472 ) / 8197 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -17\nu^{3} + 171\nu^{2} + 28445\nu - 138481 ) / 1171 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 4 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 119\beta_{2} + 9\beta _1 + 9164 ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -541\beta_{3} + 1197\beta_{2} + 3437\beta _1 + 30358 ) / 8 \)
|
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 | −63.2885 | 0 | 125.000 | 0 | 343.000 | 0 | 1818.43 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −57.6233 | 0 | 125.000 | 0 | 343.000 | 0 | 1133.44 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 20.4471 | 0 | 125.000 | 0 | 343.000 | 0 | −1768.92 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 45.4647 | 0 | 125.000 | 0 | 343.000 | 0 | −119.962 | 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 | 280.8.a.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | 560.8.a.q | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 280.8.a.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 560.8.a.q | 4 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 55T_{3}^{3} - 3393T_{3}^{2} - 127971T_{3} + 3390228 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(280))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 55 T^{3} + \cdots + 3390228 \)
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| $5$ |
\( (T - 125)^{4} \)
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| $7$ |
\( (T - 343)^{4} \)
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| $11$ |
\( T^{4} + \cdots + 22849882602660 \)
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| $13$ |
\( T^{4} + \cdots + 183932394746506 \)
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| $17$ |
\( T^{4} + \cdots + 37\!\cdots\!10 \)
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| $19$ |
\( T^{4} + \cdots + 27\!\cdots\!12 \)
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| $23$ |
\( T^{4} + \cdots + 12\!\cdots\!96 \)
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| $29$ |
\( T^{4} + \cdots + 37\!\cdots\!22 \)
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| $31$ |
\( T^{4} + \cdots - 78\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots + 22\!\cdots\!24 \)
|
| $41$ |
\( T^{4} + \cdots - 32\!\cdots\!84 \)
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| $43$ |
\( T^{4} + \cdots + 58\!\cdots\!24 \)
|
| $47$ |
\( T^{4} + \cdots + 26\!\cdots\!44 \)
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| $53$ |
\( T^{4} + \cdots - 17\!\cdots\!64 \)
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| $59$ |
\( T^{4} + \cdots + 14\!\cdots\!72 \)
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| $61$ |
\( T^{4} + \cdots + 10\!\cdots\!80 \)
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| $67$ |
\( T^{4} + \cdots + 11\!\cdots\!48 \)
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| $71$ |
\( T^{4} + \cdots + 58\!\cdots\!88 \)
|
| $73$ |
\( T^{4} + \cdots - 17\!\cdots\!80 \)
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| $79$ |
\( T^{4} + \cdots - 56\!\cdots\!76 \)
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| $83$ |
\( T^{4} + \cdots - 13\!\cdots\!12 \)
|
| $89$ |
\( T^{4} + \cdots + 11\!\cdots\!36 \)
|
| $97$ |
\( T^{4} + \cdots + 29\!\cdots\!22 \)
|
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