Newspace parameters
| Level: | \( N \) | \(=\) | \( 343 = 7^{3} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 343.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(20.2376551320\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\Q(\zeta_{14})^+\) |
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|
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| Defining polynomial: |
\( x^{3} - x^{2} - 2x + 1 \)
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| 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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
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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 |
|
−1.04892 | 0 | −6.89977 | 0 | 0 | 0 | 15.6286 | −27.0000 | 0 | |||||||||||||||||||||||||||
| 1.2 | 3.35690 | 0 | 3.26875 | 0 | 0 | 0 | −15.8823 | −27.0000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 3.69202 | 0 | 5.63102 | 0 | 0 | 0 | −8.74632 | −27.0000 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-7}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 343.4.a.b | ✓ | 3 |
| 7.b | odd | 2 | 1 | CM | 343.4.a.b | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 343.4.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 343.4.a.b | ✓ | 3 | 7.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(343))\):
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\( T_{2}^{3} - 6T_{2}^{2} + 5T_{2} + 13 \)
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\( T_{3} \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 6 T^{2} + \cdots + 13 \)
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| $3$ |
\( T^{3} \)
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| $5$ |
\( T^{3} \)
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| $7$ |
\( T^{3} \)
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| $11$ |
\( T^{3} + T^{2} + \cdots - 48931 \)
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| $13$ |
\( T^{3} \)
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| $17$ |
\( T^{3} \)
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| $19$ |
\( T^{3} \)
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| $23$ |
\( T^{3} + 267 T^{2} + \cdots - 5131309 \)
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| $29$ |
\( T^{3} - 433 T^{2} + \cdots - 1059563 \)
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| $31$ |
\( T^{3} \)
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| $37$ |
\( T^{3} + 211 T^{2} + \cdots - 10637383 \)
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| $41$ |
\( T^{3} \)
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| $43$ |
\( T^{3} + 617 T^{2} + \cdots - 100468859 \)
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| $47$ |
\( T^{3} \)
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| $53$ |
\( T^{3} + 953 T^{2} + \cdots + 4090939 \)
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| $59$ |
\( T^{3} \)
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| $61$ |
\( T^{3} \)
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| $67$ |
\( T^{3} + 701 T^{2} + \cdots - 695869343 \)
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| $71$ |
\( T^{3} + 1639 T^{2} + \cdots + 72694271 \)
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| $73$ |
\( T^{3} \)
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| $79$ |
\( T^{3} - 377 T^{2} + \cdots + 107402903 \)
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| $83$ |
\( T^{3} \)
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| $89$ |
\( T^{3} \)
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| $97$ |
\( T^{3} \)
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