Newspace parameters
| Level: | \( N \) | \(=\) | \( 71 \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 71.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(16.3338399370\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | 7.7.294755098673.1 |
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|
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| Defining polynomial: |
\( x^{7} - 14x^{5} + 56x^{3} - 56x - 21 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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
\( x^{7} - 14x^{5} + 56x^{3} - 56x - 21 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 8\nu^{2} + 6\nu + 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{4} + 2\nu^{3} + 8\nu^{2} - 12\nu - 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 10\nu^{3} - 3\nu^{2} + 20\nu + 12 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - 12\nu^{4} + 36\nu^{2} - 5\nu - 16 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 6\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + 8\beta_{2} + 24 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 10\beta_{4} + 10\beta_{3} + 3\beta_{2} + 40\beta_1 \)
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| \(\nu^{6}\) | \(=\) |
\( \beta_{6} + 12\beta_{4} + 24\beta_{3} + 60\beta_{2} + 5\beta _1 + 160 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/71\mathbb{Z}\right)^\times\).
| \(n\) | \(7\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 70.1 |
|
−15.8816 | −51.6906 | 188.225 | 165.767 | 820.930 | 0 | −1972.89 | 1942.92 | −2632.64 | |||||||||||||||||||||||||||||||||||||||
| 70.2 | −11.4210 | 39.7931 | 66.4398 | −219.335 | −454.478 | 0 | −27.8654 | 854.488 | 2505.03 | ||||||||||||||||||||||||||||||||||||||||
| 70.3 | −8.38300 | 53.3502 | 6.27476 | 145.561 | −447.235 | 0 | 483.911 | 2117.25 | −1220.24 | ||||||||||||||||||||||||||||||||||||||||
| 70.4 | 1.63981 | −20.0140 | −61.3110 | −68.1538 | −32.8192 | 0 | −205.487 | −328.440 | −111.760 | ||||||||||||||||||||||||||||||||||||||||
| 70.5 | 5.42816 | −44.4432 | −34.5350 | −230.548 | −241.245 | 0 | −534.864 | 1246.19 | −1251.45 | ||||||||||||||||||||||||||||||||||||||||
| 70.6 | 13.4658 | −3.72908 | 117.329 | 249.666 | −50.2152 | 0 | 718.117 | −715.094 | 3361.96 | ||||||||||||||||||||||||||||||||||||||||
| 70.7 | 15.1518 | 26.7336 | 165.577 | −42.9579 | 405.062 | 0 | 1539.08 | −14.3161 | −650.890 | ||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 71.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-71}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 71.7.b.a | ✓ | 7 |
| 71.b | odd | 2 | 1 | CM | 71.7.b.a | ✓ | 7 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 71.7.b.a | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 71.7.b.a | ✓ | 7 | 71.b | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - 448T_{2}^{5} + 57344T_{2}^{3} - 1835008T_{2} + 2761495 \)
acting on \(S_{7}^{\mathrm{new}}(71, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 448 T^{5} + \cdots + 2761495 \)
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| $3$ |
\( T^{7} + \cdots - 9730899370 \)
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| $5$ |
\( T^{7} + \cdots - 891875426652794 \)
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| $7$ |
\( T^{7} \)
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| $11$ |
\( T^{7} \)
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| $13$ |
\( T^{7} \)
|
| $17$ |
\( T^{7} \)
|
| $19$ |
\( T^{7} + \cdots + 72\!\cdots\!62 \)
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| $23$ |
\( T^{7} \)
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| $29$ |
\( T^{7} + \cdots + 67\!\cdots\!42 \)
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| $31$ |
\( T^{7} \)
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| $37$ |
\( T^{7} + \cdots - 37\!\cdots\!90 \)
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| $41$ |
\( T^{7} \)
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| $43$ |
\( T^{7} + \cdots - 39\!\cdots\!70 \)
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| $47$ |
\( T^{7} \)
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| $53$ |
\( T^{7} \)
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| $59$ |
\( T^{7} \)
|
| $61$ |
\( T^{7} \)
|
| $67$ |
\( T^{7} \)
|
| $71$ |
\( (T + 357911)^{7} \)
|
| $73$ |
\( T^{7} + \cdots - 13\!\cdots\!30 \)
|
| $79$ |
\( T^{7} + \cdots - 14\!\cdots\!58 \)
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| $83$ |
\( T^{7} + \cdots + 37\!\cdots\!50 \)
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| $89$ |
\( T^{7} + \cdots - 47\!\cdots\!22 \)
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| $97$ |
\( T^{7} \)
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