Newspace parameters
| Level: | \( N \) | \(=\) | \( 399 = 3 \cdot 7 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 399.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(3.18603104065\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.404.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 5x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{3} - x^{2} - 5x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
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 |
|
−2.74483 | 1.00000 | 5.53407 | −2.53407 | −2.74483 | −1.00000 | −9.70041 | 1.00000 | 6.95558 | |||||||||||||||||||||||||||
| 1.2 | 1.39593 | 1.00000 | −0.0513742 | 3.05137 | 1.39593 | −1.00000 | −2.86358 | 1.00000 | 4.25951 | ||||||||||||||||||||||||||||
| 1.3 | 2.34889 | 1.00000 | 3.51730 | −0.517304 | 2.34889 | −1.00000 | 3.56399 | 1.00000 | −1.21509 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(7\) | \( +1 \) |
| \(19\) | \( +1 \) |
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 | 399.2.a.e | ✓ | 3 |
| 3.b | odd | 2 | 1 | 1197.2.a.m | 3 | ||
| 4.b | odd | 2 | 1 | 6384.2.a.bu | 3 | ||
| 5.b | even | 2 | 1 | 9975.2.a.x | 3 | ||
| 7.b | odd | 2 | 1 | 2793.2.a.w | 3 | ||
| 19.b | odd | 2 | 1 | 7581.2.a.l | 3 | ||
| 21.c | even | 2 | 1 | 8379.2.a.bq | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 399.2.a.e | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 1197.2.a.m | 3 | 3.b | odd | 2 | 1 | ||
| 2793.2.a.w | 3 | 7.b | odd | 2 | 1 | ||
| 6384.2.a.bu | 3 | 4.b | odd | 2 | 1 | ||
| 7581.2.a.l | 3 | 19.b | odd | 2 | 1 | ||
| 8379.2.a.bq | 3 | 21.c | even | 2 | 1 | ||
| 9975.2.a.x | 3 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(399))\):
|
\( T_{2}^{3} - T_{2}^{2} - 7T_{2} + 9 \)
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\( T_{5}^{3} - 8T_{5} - 4 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - T^{2} - 7T + 9 \)
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| $3$ |
\( (T - 1)^{3} \)
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| $5$ |
\( T^{3} - 8T - 4 \)
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| $7$ |
\( (T + 1)^{3} \)
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| $11$ |
\( T^{3} - 4 T^{2} + \cdots + 48 \)
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| $13$ |
\( T^{3} - 2 T^{2} + \cdots - 8 \)
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| $17$ |
\( T^{3} - 12 T^{2} + \cdots - 28 \)
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| $19$ |
\( (T + 1)^{3} \)
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| $23$ |
\( T^{3} + 8T^{2} - 16 \)
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| $29$ |
\( T^{3} + 8 T^{2} + \cdots - 292 \)
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| $31$ |
\( T^{3} + 8 T^{2} + \cdots - 112 \)
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| $37$ |
\( T^{3} + 2 T^{2} + \cdots - 232 \)
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| $41$ |
\( T^{3} - 22 T^{2} + \cdots - 216 \)
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| $43$ |
\( T^{3} - 16 T^{2} + \cdots - 48 \)
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| $47$ |
\( T^{3} + 2 T^{2} + \cdots - 196 \)
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| $53$ |
\( T^{3} - 156T - 412 \)
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| $59$ |
\( T^{3} - 12 T^{2} + \cdots + 704 \)
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| $61$ |
\( T^{3} - 14 T^{2} + \cdots + 472 \)
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| $67$ |
\( T^{3} + 8 T^{2} + \cdots - 288 \)
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| $71$ |
\( T^{3} + 26 T^{2} + \cdots - 172 \)
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| $73$ |
\( T^{3} - 6 T^{2} + \cdots + 88 \)
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| $79$ |
\( T^{3} - 12 T^{2} + \cdots + 704 \)
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| $83$ |
\( T^{3} - 2 T^{2} + \cdots + 84 \)
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| $89$ |
\( T^{3} + 2 T^{2} + \cdots - 1176 \)
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| $97$ |
\( T^{3} - 22 T^{2} + \cdots + 888 \)
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