Newspace parameters
| Level: | \( N \) | \(=\) | \( 280 = 2^{3} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 280.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(144.210034126\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
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| Defining polynomial: |
\( x^{7} - 3 x^{6} - 75886 x^{5} + 1263838 x^{4} + 1492027269 x^{3} - 43705600687 x^{2} + \cdots + 112929763661700 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{21}\cdot 3\cdot 5 \) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 3 x^{6} - 75886 x^{5} + 1263838 x^{4} + 1492027269 x^{3} - 43705600687 x^{2} + \cdots + 112929763661700 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 21\nu - 21692 \)
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| \(\beta_{3}\) | \(=\) |
\( ( 1154557 \nu^{6} + 187093904 \nu^{5} - 80308488126 \nu^{4} - 8732670272012 \nu^{3} + \cdots - 38\!\cdots\!00 ) / 66778542929520 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 9058969 \nu^{6} - 684377024 \nu^{5} + 618724990470 \nu^{4} + 47207270330012 \nu^{3} + \cdots + 26\!\cdots\!20 ) / 467449800506640 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 4393385 \nu^{6} - 154194511 \nu^{5} + 313455738273 \nu^{4} + 6464438882638 \nu^{3} + \cdots + 97\!\cdots\!95 ) / 29215612531665 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 5465461 \nu^{6} + 212175876 \nu^{5} - 405866763290 \nu^{4} - 10236188932728 \nu^{3} + \cdots - 14\!\cdots\!40 ) / 25969433361480 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 21\beta _1 + 21692 \)
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| \(\nu^{3}\) | \(=\) |
\( -2\beta_{6} - 5\beta_{5} + 26\beta_{4} + 10\beta_{3} - 5\beta_{2} + 36483\beta _1 - 459719 \)
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| \(\nu^{4}\) | \(=\) |
\( -1452\beta_{6} - 1801\beta_{5} - 1953\beta_{4} - 179\beta_{3} + 42840\beta_{2} - 1318747\beta _1 + 791539811 \)
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| \(\nu^{5}\) | \(=\) |
\( - 75512 \beta_{6} - 162178 \beta_{5} + 1275239 \beta_{4} + 938013 \beta_{3} - 1062124 \beta_{2} + \cdots - 28765390622 \)
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| \(\nu^{6}\) | \(=\) |
\( - 103888704 \beta_{6} - 136811282 \beta_{5} - 145841646 \beta_{4} - 30978538 \beta_{3} + \cdots + 31087009835678 \)
|
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 | −198.752 | 0 | −625.000 | 0 | −2401.00 | 0 | 19819.4 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −162.642 | 0 | −625.000 | 0 | −2401.00 | 0 | 6769.45 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −42.5903 | 0 | −625.000 | 0 | −2401.00 | 0 | −17869.1 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −36.5497 | 0 | −625.000 | 0 | −2401.00 | 0 | −18347.1 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 53.7160 | 0 | −625.000 | 0 | −2401.00 | 0 | −16797.6 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 180.506 | 0 | −625.000 | 0 | −2401.00 | 0 | 12899.5 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 217.312 | 0 | −625.000 | 0 | −2401.00 | 0 | 27541.4 | 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 | 280.10.a.f | ✓ | 7 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 280.10.a.f | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{7} - 11 T_{3}^{6} - 75838 T_{3}^{5} - 505078 T_{3}^{4} + 1499102613 T_{3}^{3} + \cdots - 106027430616576 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(280))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
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| $3$ |
\( T^{7} + \cdots - 106027430616576 \)
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| $5$ |
\( (T + 625)^{7} \)
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| $7$ |
\( (T + 2401)^{7} \)
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| $11$ |
\( T^{7} + \cdots + 65\!\cdots\!60 \)
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| $13$ |
\( T^{7} + \cdots + 12\!\cdots\!92 \)
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| $17$ |
\( T^{7} + \cdots + 91\!\cdots\!00 \)
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| $19$ |
\( T^{7} + \cdots + 11\!\cdots\!36 \)
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| $23$ |
\( T^{7} + \cdots - 15\!\cdots\!52 \)
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| $29$ |
\( T^{7} + \cdots + 15\!\cdots\!72 \)
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| $31$ |
\( T^{7} + \cdots - 67\!\cdots\!00 \)
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| $37$ |
\( T^{7} + \cdots + 27\!\cdots\!08 \)
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| $41$ |
\( T^{7} + \cdots + 27\!\cdots\!56 \)
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| $43$ |
\( T^{7} + \cdots - 43\!\cdots\!32 \)
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| $47$ |
\( T^{7} + \cdots + 47\!\cdots\!64 \)
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| $53$ |
\( T^{7} + \cdots - 96\!\cdots\!64 \)
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| $59$ |
\( T^{7} + \cdots - 16\!\cdots\!76 \)
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| $61$ |
\( T^{7} + \cdots - 27\!\cdots\!00 \)
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| $67$ |
\( T^{7} + \cdots + 56\!\cdots\!44 \)
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| $71$ |
\( T^{7} + \cdots + 56\!\cdots\!00 \)
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| $73$ |
\( T^{7} + \cdots + 10\!\cdots\!40 \)
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| $79$ |
\( T^{7} + \cdots - 10\!\cdots\!48 \)
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| $83$ |
\( T^{7} + \cdots - 16\!\cdots\!64 \)
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| $89$ |
\( T^{7} + \cdots + 15\!\cdots\!56 \)
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| $97$ |
\( T^{7} + \cdots - 14\!\cdots\!36 \)
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