Newspace parameters
| Level: | \( N \) | \(=\) | \( 639 = 3^{2} \cdot 71 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 639.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(5.10244068916\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.257.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 4x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 71) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 4x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
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 |
|
−1.91223 | 0 | 1.65662 | −3.25561 | 0 | −3.82446 | 0.656620 | 0 | 6.22547 | |||||||||||||||||||||||||||
| 1.2 | 0.713538 | 0 | −1.49086 | −3.77733 | 0 | 1.42708 | −2.49086 | 0 | −2.69527 | ||||||||||||||||||||||||||||
| 1.3 | 2.19869 | 0 | 2.83424 | 2.03293 | 0 | 4.39738 | 1.83424 | 0 | 4.46980 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(71\) | \( +1 \) |
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 | 639.2.a.h | 3 | |
| 3.b | odd | 2 | 1 | 71.2.a.a | ✓ | 3 | |
| 12.b | even | 2 | 1 | 1136.2.a.h | 3 | ||
| 15.d | odd | 2 | 1 | 1775.2.a.f | 3 | ||
| 21.c | even | 2 | 1 | 3479.2.a.k | 3 | ||
| 24.f | even | 2 | 1 | 4544.2.a.u | 3 | ||
| 24.h | odd | 2 | 1 | 4544.2.a.r | 3 | ||
| 33.d | even | 2 | 1 | 8591.2.a.g | 3 | ||
| 213.b | even | 2 | 1 | 5041.2.a.a | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 71.2.a.a | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 639.2.a.h | 3 | 1.a | even | 1 | 1 | trivial | |
| 1136.2.a.h | 3 | 12.b | even | 2 | 1 | ||
| 1775.2.a.f | 3 | 15.d | odd | 2 | 1 | ||
| 3479.2.a.k | 3 | 21.c | even | 2 | 1 | ||
| 4544.2.a.r | 3 | 24.h | odd | 2 | 1 | ||
| 4544.2.a.u | 3 | 24.f | even | 2 | 1 | ||
| 5041.2.a.a | 3 | 213.b | even | 2 | 1 | ||
| 8591.2.a.g | 3 | 33.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(639))\):
|
\( T_{2}^{3} - T_{2}^{2} - 4T_{2} + 3 \)
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\( T_{5}^{3} + 5T_{5}^{2} - 2T_{5} - 25 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - T^{2} - 4T + 3 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + 5 T^{2} + \cdots - 25 \)
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| $7$ |
\( T^{3} - 2 T^{2} + \cdots + 24 \)
|
| $11$ |
\( T^{3} - 20T - 24 \)
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| $13$ |
\( T^{3} + 6 T^{2} + \cdots - 56 \)
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| $17$ |
\( T^{3} - 2 T^{2} + \cdots + 24 \)
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| $19$ |
\( T^{3} - T^{2} + \cdots - 25 \)
|
| $23$ |
\( (T - 4)^{3} \)
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| $29$ |
\( T^{3} + 11 T^{2} + \cdots - 71 \)
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| $31$ |
\( (T - 4)^{3} \)
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| $37$ |
\( T^{3} + 15 T^{2} + \cdots + 97 \)
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| $41$ |
\( T^{3} - 2 T^{2} + \cdots - 56 \)
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| $43$ |
\( T^{3} - 13 T^{2} + \cdots - 45 \)
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| $47$ |
\( T^{3} + 4 T^{2} + \cdots - 40 \)
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| $53$ |
\( T^{3} - 18 T^{2} + \cdots + 456 \)
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| $59$ |
\( T^{3} - 4 T^{2} + \cdots + 152 \)
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| $61$ |
\( T^{3} - 16 T^{2} + \cdots + 320 \)
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| $67$ |
\( T^{3} + 12 T^{2} + \cdots - 40 \)
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| $71$ |
\( (T + 1)^{3} \)
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| $73$ |
\( T^{3} - 27 T^{2} + \cdots - 461 \)
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| $79$ |
\( T^{3} + 3 T^{2} + \cdots + 15 \)
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| $83$ |
\( T^{3} - 19 T^{2} + \cdots - 63 \)
|
| $89$ |
\( T^{3} + T^{2} + \cdots + 27 \)
|
| $97$ |
\( T^{3} - 22 T^{2} + \cdots - 280 \)
|
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