Newspace parameters
| Level: | \( N \) | \(=\) | \( 6728 = 2^{3} \cdot 29^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6728.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(53.7233504799\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.50158625.1 |
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| Defining polynomial: |
\( x^{6} - x^{5} - 11x^{4} + 9x^{3} + 28x^{2} - 20x - 1 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 11x^{4} + 9x^{3} + 28x^{2} - 20x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 5\nu^{4} - 15\nu^{3} - 30\nu^{2} + 35\nu + 3 ) / 17 \)
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| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{5} + 2\nu^{4} + 28\nu^{3} - 12\nu^{2} - 54\nu + 8 ) / 17 \)
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| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} - 5\nu^{4} + 15\nu^{3} + 47\nu^{2} - 52\nu - 54 ) / 17 \)
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| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{5} + 2\nu^{4} + 45\nu^{3} - 29\nu^{2} - 156\nu + 76 ) / 17 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{2} + \beta _1 + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 7\beta _1 - 1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 7\beta_{4} + 10\beta_{2} + 10\beta _1 + 16 \)
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| \(\nu^{5}\) | \(=\) |
\( 10\beta_{5} + 10\beta_{4} - 15\beta_{3} + 12\beta_{2} + 50\beta _1 - 8 \)
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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 |
|
0 | −2.85072 | 0 | 3.61256 | 0 | 4.36475 | 0 | 5.12659 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.97463 | 0 | −2.22039 | 0 | 5.09418 | 0 | 0.899162 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.749617 | 0 | 0.212905 | 0 | −1.90136 | 0 | −2.43808 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.0469617 | 0 | −0.970976 | 0 | −2.07378 | 0 | −2.99779 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.98230 | 0 | −4.20743 | 0 | 3.15464 | 0 | 0.929516 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.54570 | 0 | 0.573330 | 0 | 0.361566 | 0 | 3.48060 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(29\) | \( -1 \) |
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 | 6728.2.a.v | ✓ | 6 |
| 29.b | even | 2 | 1 | 6728.2.a.w | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6728.2.a.v | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 6728.2.a.w | yes | 6 | 29.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6728))\):
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\( T_{3}^{6} + T_{3}^{5} - 11T_{3}^{4} - 9T_{3}^{3} + 28T_{3}^{2} + 20T_{3} - 1 \)
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\( T_{5}^{6} + 3T_{5}^{5} - 14T_{5}^{4} - 38T_{5}^{3} + 3T_{5}^{2} + 20T_{5} - 4 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
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| $3$ |
\( T^{6} + T^{5} - 11 T^{4} + \cdots - 1 \)
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| $5$ |
\( T^{6} + 3 T^{5} + \cdots - 4 \)
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| $7$ |
\( T^{6} - 9 T^{5} + \cdots + 100 \)
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| $11$ |
\( T^{6} + 2 T^{5} + \cdots - 121 \)
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| $13$ |
\( T^{6} - 16 T^{5} + \cdots - 580 \)
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| $17$ |
\( T^{6} - 99 T^{4} + \cdots + 181 \)
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| $19$ |
\( T^{6} + 20 T^{5} + \cdots - 7895 \)
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| $23$ |
\( T^{6} + 2 T^{5} + \cdots + 580 \)
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| $29$ |
\( T^{6} \)
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| $31$ |
\( T^{6} + 4 T^{5} + \cdots + 1216 \)
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| $37$ |
\( T^{6} - 8 T^{5} + \cdots + 164 \)
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| $41$ |
\( T^{6} - 8 T^{5} + \cdots - 704 \)
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| $43$ |
\( T^{6} - 17 T^{5} + \cdots + 3056 \)
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| $47$ |
\( T^{6} + 12 T^{5} + \cdots + 24244 \)
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| $53$ |
\( T^{6} - 9 T^{5} + \cdots + 2956 \)
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| $59$ |
\( T^{6} + 3 T^{5} + \cdots + 68381 \)
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| $61$ |
\( T^{6} - 32 T^{5} + \cdots - 484 \)
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| $67$ |
\( T^{6} - 14 T^{5} + \cdots + 841 \)
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| $71$ |
\( T^{6} + 11 T^{5} + \cdots - 6064 \)
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| $73$ |
\( T^{6} - 21 T^{5} + \cdots + 2801 \)
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| $79$ |
\( T^{6} - 4 T^{5} + \cdots - 1569164 \)
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| $83$ |
\( T^{6} + 21 T^{5} + \cdots - 22189 \)
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| $89$ |
\( T^{6} - 8 T^{5} + \cdots - 604189 \)
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| $97$ |
\( T^{6} - 3 T^{5} + \cdots - 2861 \)
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