Newspace parameters
| Level: | \( N \) | \(=\) | \( 280 = 2^{3} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 280.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(44.9074695476\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 1151x^{3} - 5642x^{2} + 193596x + 1258056 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 1151x^{3} - 5642x^{2} + 193596x + 1258056 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 7\nu - 460 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 6\nu^{3} - 1157\nu^{2} - 7838\nu + 137328 ) / 210 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{4} + 15\nu^{3} + 989\nu^{2} - 8122\nu - 131091 ) / 105 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 7\beta _1 + 460 \)
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| \(\nu^{3}\) | \(=\) |
\( 5\beta_{4} + 10\beta_{3} + 8\beta_{2} + 816\beta _1 + 3383 \)
|
| \(\nu^{4}\) | \(=\) |
\( -30\beta_{4} + 150\beta_{3} + 1109\beta_{2} + 11041\beta _1 + 374594 \)
|
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 | −30.3496 | 0 | 25.0000 | 0 | 49.0000 | 0 | 678.095 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −11.5231 | 0 | 25.0000 | 0 | 49.0000 | 0 | −110.219 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 10.0384 | 0 | 25.0000 | 0 | 49.0000 | 0 | −142.230 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 16.5022 | 0 | 25.0000 | 0 | 49.0000 | 0 | 29.3221 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 30.3320 | 0 | 25.0000 | 0 | 49.0000 | 0 | 677.031 | 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 | 280.6.a.j | ✓ | 5 |
| 4.b | odd | 2 | 1 | 560.6.a.y | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 280.6.a.j | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 560.6.a.y | 5 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} - 15T_{3}^{4} - 1061T_{3}^{3} + 15731T_{3}^{2} + 129072T_{3} - 1757232 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(280))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} - 15 T^{4} + \cdots - 1757232 \)
|
| $5$ |
\( (T - 25)^{5} \)
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| $7$ |
\( (T - 49)^{5} \)
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| $11$ |
\( T^{5} + \cdots - 20384754585232 \)
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| $13$ |
\( T^{5} + \cdots - 9709872570412 \)
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| $17$ |
\( T^{5} + \cdots - 871869567890580 \)
|
| $19$ |
\( T^{5} + \cdots + 11\!\cdots\!84 \)
|
| $23$ |
\( T^{5} + \cdots + 23\!\cdots\!84 \)
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| $29$ |
\( T^{5} + \cdots - 44\!\cdots\!04 \)
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| $31$ |
\( T^{5} + \cdots + 23\!\cdots\!60 \)
|
| $37$ |
\( T^{5} + \cdots + 72\!\cdots\!52 \)
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| $41$ |
\( T^{5} + \cdots + 12\!\cdots\!96 \)
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| $43$ |
\( T^{5} + \cdots + 91\!\cdots\!72 \)
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| $47$ |
\( T^{5} + \cdots + 62\!\cdots\!32 \)
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| $53$ |
\( T^{5} + \cdots - 20\!\cdots\!48 \)
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| $59$ |
\( T^{5} + \cdots - 20\!\cdots\!44 \)
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| $61$ |
\( T^{5} + \cdots + 15\!\cdots\!80 \)
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| $67$ |
\( T^{5} + \cdots - 83\!\cdots\!84 \)
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| $71$ |
\( T^{5} + \cdots + 34\!\cdots\!00 \)
|
| $73$ |
\( T^{5} + \cdots - 23\!\cdots\!32 \)
|
| $79$ |
\( T^{5} + \cdots + 78\!\cdots\!16 \)
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| $83$ |
\( T^{5} + \cdots - 16\!\cdots\!04 \)
|
| $89$ |
\( T^{5} + \cdots + 18\!\cdots\!44 \)
|
| $97$ |
\( T^{5} + \cdots - 96\!\cdots\!88 \)
|
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