Newspace parameters
| Level: | \( N \) | \(=\) | \( 71 \) |
| Weight: | \( k \) | \(=\) | \( 11 \) |
| Character orbit: | \([\chi]\) | \(=\) | 71.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(45.1103649398\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | 7.7.294755098673.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| 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
\( x^{7} - 14x^{5} + 56x^{3} - 56x - 21 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 8\nu^{2} + 6\nu + 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{4} + 2\nu^{3} + 8\nu^{2} - 12\nu - 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 10\nu^{3} - 3\nu^{2} + 20\nu + 12 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - 12\nu^{4} + 36\nu^{2} - 5\nu - 16 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + 8\beta_{2} + 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 10\beta_{4} + 10\beta_{3} + 3\beta_{2} + 40\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} + 12\beta_{4} + 24\beta_{3} + 60\beta_{2} + 5\beta _1 + 160 \)
|
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 |
|
−63.7034 | 390.171 | 3034.12 | 6226.80 | −24855.2 | 0 | −128052. | 93184.2 | −396668. | |||||||||||||||||||||||||||||||||||||||
| 70.2 | −44.5303 | −477.255 | 958.945 | −1910.09 | 21252.3 | 0 | 2896.92 | 168724. | 85056.9 | ||||||||||||||||||||||||||||||||||||||||
| 70.3 | −34.9065 | −225.808 | 194.467 | −861.095 | 7882.18 | 0 | 28956.1 | −8059.70 | 30057.9 | ||||||||||||||||||||||||||||||||||||||||
| 70.4 | 8.17505 | 469.814 | −957.169 | −5376.73 | 3840.75 | 0 | −16196.1 | 161676. | −43955.0 | ||||||||||||||||||||||||||||||||||||||||
| 70.5 | 20.1756 | 16.7214 | −616.944 | −5843.58 | 337.364 | 0 | −33107.1 | −58769.4 | −117898. | ||||||||||||||||||||||||||||||||||||||||
| 70.6 | 54.7244 | −369.320 | 1970.76 | 4302.96 | −20210.8 | 0 | 51810.8 | 77347.9 | 235477. | ||||||||||||||||||||||||||||||||||||||||
| 70.7 | 60.0651 | 195.677 | 2583.82 | 3461.73 | 11753.4 | 0 | 93690.9 | −20759.4 | 207929. | ||||||||||||||||||||||||||||||||||||||||
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 | 71.11.b.a | ✓ | 7 |
| 71.b | odd | 2 | 1 | CM | 71.11.b.a | ✓ | 7 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 71.11.b.a | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 71.11.b.a | ✓ | 7 | 71.b | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - 7168T_{2}^{5} + 14680064T_{2}^{3} - 7516192768T_{2} + 53684057575 \)
acting on \(S_{11}^{\mathrm{new}}(71, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{7} - 7168 T^{5} + \cdots + 53684057575 \)
$3$
\( T^{7} - 413343 T^{5} + \cdots + 23\!\cdots\!50 \)
$5$
\( T^{7} - 68359375 T^{5} + \cdots - 47\!\cdots\!74 \)
$7$
\( T^{7} \)
$11$
\( T^{7} \)
$13$
\( T^{7} \)
$17$
\( T^{7} \)
$19$
\( T^{7} - 42917463804607 T^{5} + \cdots - 18\!\cdots\!98 \)
$23$
\( T^{7} \)
$29$
\( T^{7} + \cdots + 45\!\cdots\!02 \)
$31$
\( T^{7} \)
$37$
\( T^{7} + \cdots + 56\!\cdots\!50 \)
$41$
\( T^{7} \)
$43$
\( T^{7} + \cdots + 47\!\cdots\!50 \)
$47$
\( T^{7} \)
$53$
\( T^{7} \)
$59$
\( T^{7} \)
$61$
\( T^{7} \)
$67$
\( T^{7} \)
$71$
\( (T + 1804229351)^{7} \)
$73$
\( T^{7} + \cdots - 30\!\cdots\!50 \)
$79$
\( T^{7} + \cdots - 51\!\cdots\!98 \)
$83$
\( T^{7} + \cdots - 26\!\cdots\!50 \)
$89$
\( T^{7} + \cdots + 15\!\cdots\!98 \)
$97$
\( T^{7} \)
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