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: |
\( x^{7} - x^{6} - 68982 x^{5} - 1651764 x^{4} + 1100124770 x^{3} - 1398985356 x^{2} + \cdots + 198660733545904 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{21}\cdot 3^{3}\cdot 5^{2} \) |
| 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
\( x^{7} - x^{6} - 68982 x^{5} - 1651764 x^{4} + 1100124770 x^{3} - 1398985356 x^{2} + \cdots + 198660733545904 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 11\!\cdots\!39 \nu^{6} + \cdots - 18\!\cdots\!24 ) / 18\!\cdots\!20 \)
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| \(\beta_{2}\) | \(=\) |
\( ( - 23\!\cdots\!41 \nu^{6} + \cdots - 72\!\cdots\!24 ) / 18\!\cdots\!20 \)
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| \(\beta_{3}\) | \(=\) |
\( ( - 66\!\cdots\!07 \nu^{6} + \cdots + 44\!\cdots\!04 ) / 36\!\cdots\!44 \)
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| \(\beta_{4}\) | \(=\) |
\( ( - 16\!\cdots\!49 \nu^{6} + \cdots + 72\!\cdots\!24 ) / 22\!\cdots\!40 \)
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| \(\beta_{5}\) | \(=\) |
\( ( 73\!\cdots\!31 \nu^{6} + \cdots - 17\!\cdots\!76 ) / 90\!\cdots\!60 \)
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| \(\beta_{6}\) | \(=\) |
\( ( - 77\!\cdots\!65 \nu^{6} + \cdots + 25\!\cdots\!24 ) / 90\!\cdots\!36 \)
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| \(\nu\) | \(=\) |
\( ( -2\beta_{6} - \beta_{5} + 11\beta_{4} - 15\beta_{3} + 15\beta_{2} + 676\beta _1 + 114 ) / 2880 \)
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| \(\nu^{2}\) | \(=\) |
\( ( 50\beta_{6} + 1141\beta_{5} + 577\beta_{4} - 45\beta_{3} + 1875\beta_{2} - 68722\beta _1 + 28410126 ) / 1440 \)
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| \(\nu^{3}\) | \(=\) |
\( ( - 68188 \beta_{6} + 94777 \beta_{5} + 258061 \beta_{4} - 161085 \beta_{3} + 545700 \beta_{2} + \cdots + 1059852882 ) / 1440 \)
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| \(\nu^{4}\) | \(=\) |
\( ( - 782606 \beta_{6} + 21437741 \beta_{5} + 16505941 \beta_{4} - 6602165 \beta_{3} + \cdots + 351752913226 ) / 480 \)
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| \(\nu^{5}\) | \(=\) |
\( ( - 598031304 \beta_{6} + 1357142823 \beta_{5} + 2320408277 \beta_{4} - 1069167695 \beta_{3} + \cdots + 20402211467128 ) / 240 \)
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| \(\nu^{6}\) | \(=\) |
\( ( - 32018883554 \beta_{6} + 341162223704 \beta_{5} + 316707680186 \beta_{4} - 145314320880 \beta_{3} + \cdots + 50\!\cdots\!42 ) / 144 \)
|
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 | −260.185 | 0 | 625.000 | 0 | 2401.00 | 0 | 48013.3 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −214.959 | 0 | 625.000 | 0 | 2401.00 | 0 | 26524.4 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −78.6397 | 0 | 625.000 | 0 | 2401.00 | 0 | −13498.8 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −43.1367 | 0 | 625.000 | 0 | 2401.00 | 0 | −17822.2 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 113.026 | 0 | 625.000 | 0 | 2401.00 | 0 | −6908.15 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 127.408 | 0 | 625.000 | 0 | 2401.00 | 0 | −3450.12 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 205.486 | 0 | 625.000 | 0 | 2401.00 | 0 | 22541.6 | 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.e | ✓ | 7 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 280.10.a.e | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{7} + 151 T_{3}^{6} - 85190 T_{3}^{5} - 8703282 T_{3}^{4} + 2144551221 T_{3}^{3} + \cdots - 561416576137200 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(280))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
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| $3$ |
\( T^{7} + \cdots - 561416576137200 \)
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| $5$ |
\( (T - 625)^{7} \)
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| $7$ |
\( (T - 2401)^{7} \)
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| $11$ |
\( T^{7} + \cdots + 71\!\cdots\!00 \)
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| $13$ |
\( T^{7} + \cdots - 37\!\cdots\!00 \)
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| $17$ |
\( T^{7} + \cdots - 33\!\cdots\!76 \)
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| $19$ |
\( T^{7} + \cdots + 74\!\cdots\!52 \)
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| $23$ |
\( T^{7} + \cdots + 10\!\cdots\!00 \)
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| $29$ |
\( T^{7} + \cdots - 31\!\cdots\!04 \)
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| $31$ |
\( T^{7} + \cdots - 33\!\cdots\!04 \)
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| $37$ |
\( T^{7} + \cdots - 44\!\cdots\!28 \)
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| $41$ |
\( T^{7} + \cdots - 79\!\cdots\!20 \)
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| $43$ |
\( T^{7} + \cdots + 27\!\cdots\!40 \)
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| $47$ |
\( T^{7} + \cdots - 29\!\cdots\!32 \)
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| $53$ |
\( T^{7} + \cdots + 14\!\cdots\!00 \)
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| $59$ |
\( T^{7} + \cdots - 13\!\cdots\!00 \)
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| $61$ |
\( T^{7} + \cdots - 57\!\cdots\!56 \)
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| $67$ |
\( T^{7} + \cdots - 44\!\cdots\!40 \)
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| $71$ |
\( T^{7} + \cdots + 11\!\cdots\!00 \)
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| $73$ |
\( T^{7} + \cdots + 11\!\cdots\!00 \)
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| $79$ |
\( T^{7} + \cdots - 30\!\cdots\!28 \)
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| $83$ |
\( T^{7} + \cdots - 74\!\cdots\!88 \)
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| $89$ |
\( T^{7} + \cdots + 48\!\cdots\!00 \)
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| $97$ |
\( T^{7} + \cdots - 12\!\cdots\!24 \)
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