Newspace parameters
| Level: | \( N \) | \(=\) | \( 5041 = 71^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5041.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(40.2525876589\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | 7.7.294755098673.1 |
|
|
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| Defining polynomial: |
\( x^{7} - 14x^{5} + 56x^{3} - 56x - 21 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $N(\mathrm{U}(1))$ |
$q$-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
\( x^{7} - 14x^{5} + 56x^{3} - 56x - 21 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 8\nu^{2} + 6\nu + 8 \)
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| \(\beta_{4}\) | \(=\) |
\( -\nu^{4} + 2\nu^{3} + 8\nu^{2} - 12\nu - 8 \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 10\nu^{3} - 3\nu^{2} + 20\nu + 12 \)
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| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - 12\nu^{4} + 36\nu^{2} - 5\nu - 16 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 6\beta_1 \)
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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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| \(\nu^{6}\) | \(=\) |
\( \beta_{6} + 12\beta_{4} + 24\beta_{3} + 60\beta_{2} + 5\beta _1 + 160 \)
|
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 |
|
−2.82423 | −2.22920 | 5.97625 | 1.03268 | 6.29575 | 0 | −11.2298 | 1.96932 | −2.91653 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.88136 | 3.15890 | 1.53952 | 4.01238 | −5.94303 | 0 | 0.866336 | 6.97864 | −7.54874 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.64039 | 0.857975 | 0.690885 | −4.47197 | −1.40741 | 0 | 2.14746 | −2.26388 | 7.33578 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.478208 | −3.46294 | −1.77132 | −2.81836 | −1.65601 | 0 | −1.80347 | 8.99196 | −1.34776 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.778691 | 0.683179 | −1.39364 | 0.957531 | 0.531985 | 0 | −2.64260 | −2.53327 | 0.745620 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.47768 | 3.08111 | 4.13888 | −2.75809 | 7.63398 | 0 | 5.29945 | 6.49322 | −6.83366 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.61140 | −2.08902 | 4.81943 | 4.04583 | −5.45528 | 0 | 7.36266 | 1.36401 | 10.5653 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(71\) | \( -1 \) |
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 | 5041.2.a.f | ✓ | 7 |
| 71.b | odd | 2 | 1 | CM | 5041.2.a.f | ✓ | 7 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5041.2.a.f | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 5041.2.a.f | ✓ | 7 | 71.b | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5041))\):
|
\( T_{2}^{7} - 14T_{2}^{5} + 56T_{2}^{3} - 56T_{2} + 21 \)
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\( T_{7} \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 14 T^{5} + \cdots + 21 \)
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| $3$ |
\( T^{7} - 21 T^{5} + \cdots + 92 \)
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| $5$ |
\( T^{7} - 35 T^{5} + \cdots + 558 \)
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| $7$ |
\( T^{7} \)
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| $11$ |
\( T^{7} \)
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| $13$ |
\( T^{7} \)
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| $17$ |
\( T^{7} \)
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| $19$ |
\( T^{7} - 133 T^{5} + \cdots + 10316 \)
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| $23$ |
\( T^{7} \)
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| $29$ |
\( T^{7} - 203 T^{5} + \cdots + 37194 \)
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| $31$ |
\( T^{7} \)
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| $37$ |
\( T^{7} - 259 T^{5} + \cdots - 419186 \)
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| $41$ |
\( T^{7} \)
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| $43$ |
\( T^{7} - 301 T^{5} + \cdots - 488348 \)
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| $47$ |
\( T^{7} \)
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| $53$ |
\( T^{7} \)
|
| $59$ |
\( T^{7} \)
|
| $61$ |
\( T^{7} \)
|
| $67$ |
\( T^{7} \)
|
| $71$ |
\( T^{7} \)
|
| $73$ |
\( T^{7} - 511 T^{5} + \cdots + 2294102 \)
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| $79$ |
\( T^{7} - 553 T^{5} + \cdots - 8763256 \)
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| $83$ |
\( T^{7} - 581 T^{5} + \cdots - 8683452 \)
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| $89$ |
\( T^{7} - 623 T^{5} + \cdots + 8955930 \)
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| $97$ |
\( T^{7} \)
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