Newspace parameters
| Level: | \( N \) | \(=\) | \( 1629 = 3^{2} \cdot 181 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1629.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(13.0076304893\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.170701.1 |
|
|
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| Defining polynomial: |
\( x^{5} - x^{4} - 6x^{3} + 4x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 543) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 6x^{3} + 4x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 6\nu^{2} + 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{4} + 2\nu^{3} + 4\nu^{2} - 4\nu - 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{4} + 2\nu^{3} + 5\nu^{2} - 5\nu - 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} + \beta _1 + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( 2\beta_{4} - \beta_{3} + \beta_{2} + 6\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 8\beta_{4} - 7\beta_{3} + 2\beta_{2} + 12\beta _1 + 11 \)
|
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 |
|
−2.55337 | 0 | 4.51968 | 3.80110 | 0 | 3.74959 | −6.43367 | 0 | −9.70561 | |||||||||||||||||||||||||||||||||
| 1.2 | −0.319710 | 0 | −1.89779 | −2.14004 | 0 | 5.03818 | 1.24616 | 0 | 0.684193 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.435090 | 0 | −1.81070 | −1.04050 | 0 | −4.97235 | −1.65800 | 0 | −0.452712 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.34554 | 0 | −0.189509 | −4.08349 | 0 | 0.669650 | −2.94608 | 0 | −5.49452 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.09244 | 0 | 2.37831 | 0.462928 | 0 | 4.51493 | 0.791590 | 0 | 0.968650 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(181\) | \( +1 \) |
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 | 1629.2.a.b | 5 | |
| 3.b | odd | 2 | 1 | 543.2.a.b | ✓ | 5 | |
| 12.b | even | 2 | 1 | 8688.2.a.z | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 543.2.a.b | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 1629.2.a.b | 5 | 1.a | even | 1 | 1 | trivial | |
| 8688.2.a.z | 5 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - T_{2}^{4} - 6T_{2}^{3} + 8T_{2}^{2} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1629))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - T^{4} - 6 T^{3} + \cdots - 1 \)
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| $3$ |
\( T^{5} \)
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| $5$ |
\( T^{5} + 3 T^{4} + \cdots + 16 \)
|
| $7$ |
\( T^{5} - 9 T^{4} + \cdots + 284 \)
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| $11$ |
\( T^{5} - 4 T^{4} + \cdots + 52 \)
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| $13$ |
\( T^{5} - 9 T^{4} + \cdots + 7 \)
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| $17$ |
\( T^{5} + 9 T^{4} + \cdots + 1492 \)
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| $19$ |
\( T^{5} - 6 T^{4} + \cdots - 368 \)
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| $23$ |
\( T^{5} - 5 T^{4} + \cdots + 197 \)
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| $29$ |
\( T^{5} - 20 T^{4} + \cdots - 628 \)
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| $31$ |
\( T^{5} - 13 T^{4} + \cdots + 52 \)
|
| $37$ |
\( T^{5} + 5 T^{4} + \cdots + 269 \)
|
| $41$ |
\( T^{5} - 2 T^{4} + \cdots - 7 \)
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| $43$ |
\( T^{5} - 4 T^{4} + \cdots + 1163 \)
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| $47$ |
\( T^{5} - 2 T^{4} + \cdots - 1472 \)
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| $53$ |
\( T^{5} + 19 T^{4} + \cdots - 2743 \)
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| $59$ |
\( T^{5} - 25 T^{4} + \cdots + 15856 \)
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| $61$ |
\( T^{5} - 17 T^{4} + \cdots + 212 \)
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| $67$ |
\( T^{5} + 9 T^{4} + \cdots + 24173 \)
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| $71$ |
\( T^{5} + 9 T^{4} + \cdots + 81989 \)
|
| $73$ |
\( T^{5} - 7 T^{4} + \cdots + 5357 \)
|
| $79$ |
\( T^{5} - 2 T^{4} + \cdots - 9103 \)
|
| $83$ |
\( T^{5} + 14 T^{4} + \cdots - 329 \)
|
| $89$ |
\( T^{5} - 12 T^{4} + \cdots - 5071 \)
|
| $97$ |
\( T^{5} + 8 T^{4} + \cdots - 16324 \)
|
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