Newspace parameters
| Level: | \( N \) | \(=\) | \( 6897 = 3 \cdot 11^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6897.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(55.0728222741\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.65858461.1 |
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| Defining polynomial: |
\( x^{6} - 3x^{5} - 5x^{4} + 12x^{3} + 6x^{2} - 11x + 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{6} - 3x^{5} - 5x^{4} + 12x^{3} + 6x^{2} - 11x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 3\nu^{3} - 3\nu^{2} + 6\nu + 1 \)
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| \(\beta_{3}\) | \(=\) |
\( -\nu^{4} + 4\nu^{3} - 9\nu + 4 \)
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| \(\beta_{4}\) | \(=\) |
\( -\nu^{5} + 3\nu^{4} + 4\nu^{3} - 8\nu^{2} - 5\nu + 2 \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 3\nu^{4} - 4\nu^{3} + 9\nu^{2} + 3\nu - 4 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 2\beta _1 + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( 3\beta_{5} + 3\beta_{4} + \beta_{3} + \beta_{2} + 9\beta _1 + 1 \)
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| \(\nu^{4}\) | \(=\) |
\( 12\beta_{5} + 12\beta_{4} + 3\beta_{3} + 4\beta_{2} + 27\beta _1 + 8 \)
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| \(\nu^{5}\) | \(=\) |
\( 40\beta_{5} + 39\beta_{4} + 13\beta_{3} + 16\beta_{2} + 96\beta _1 + 14 \)
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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 |
|
−2.34821 | 1.00000 | 3.51410 | −1.66891 | −2.34821 | 3.95376 | −3.55543 | 1.00000 | 3.91895 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −0.625866 | 1.00000 | −1.60829 | −0.429053 | −0.625866 | 1.51680 | 2.25831 | 1.00000 | 0.268530 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0.123362 | 1.00000 | −1.98478 | −1.78556 | 0.123362 | −2.58236 | −0.491571 | 1.00000 | −0.220271 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.903006 | 1.00000 | −1.18458 | 3.13062 | 0.903006 | 3.44368 | −2.87570 | 1.00000 | 2.82697 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.36513 | 1.00000 | 3.59384 | 2.63713 | 2.36513 | 0.900686 | 3.76964 | 1.00000 | 6.23715 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.58258 | 1.00000 | 4.66971 | −3.88423 | 2.58258 | 4.76743 | 6.89475 | 1.00000 | −10.0313 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(11\) | \( -1 \) |
| \(19\) | \( -1 \) |
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 | 6897.2.a.y | yes | 6 |
| 11.b | odd | 2 | 1 | 6897.2.a.t | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6897.2.a.t | ✓ | 6 | 11.b | odd | 2 | 1 | |
| 6897.2.a.y | yes | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6897))\):
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\( T_{2}^{6} - 3T_{2}^{5} - 5T_{2}^{4} + 18T_{2}^{3} - 3T_{2}^{2} - 8T_{2} + 1 \)
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\( T_{5}^{6} + 2T_{5}^{5} - 17T_{5}^{4} - 30T_{5}^{3} + 59T_{5}^{2} + 125T_{5} + 41 \)
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\( T_{7}^{6} - 12T_{7}^{5} + 42T_{7}^{4} + 6T_{7}^{3} - 292T_{7}^{2} + 489T_{7} - 229 \)
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\( T_{13}^{6} - 7T_{13}^{5} - 45T_{13}^{4} + 358T_{13}^{3} - 27T_{13}^{2} - 1664T_{13} - 1237 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 3 T^{5} + \cdots + 1 \)
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| $3$ |
\( (T - 1)^{6} \)
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| $5$ |
\( T^{6} + 2 T^{5} + \cdots + 41 \)
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| $7$ |
\( T^{6} - 12 T^{5} + \cdots - 229 \)
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| $11$ |
\( T^{6} \)
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| $13$ |
\( T^{6} - 7 T^{5} + \cdots - 1237 \)
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| $17$ |
\( T^{6} + 5 T^{5} + \cdots + 55 \)
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| $19$ |
\( (T - 1)^{6} \)
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| $23$ |
\( T^{6} + 11 T^{5} + \cdots + 1 \)
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| $29$ |
\( T^{6} - 7 T^{5} + \cdots - 715 \)
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| $31$ |
\( T^{6} + 2 T^{5} + \cdots - 379 \)
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| $37$ |
\( T^{6} - 4 T^{5} + \cdots + 17917 \)
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| $41$ |
\( T^{6} + 22 T^{5} + \cdots - 10327 \)
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| $43$ |
\( T^{6} - 6 T^{5} + \cdots + 7447 \)
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| $47$ |
\( T^{6} + 21 T^{5} + \cdots + 817 \)
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| $53$ |
\( T^{6} - 5 T^{5} + \cdots - 828043 \)
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| $59$ |
\( T^{6} - 6 T^{5} + \cdots - 11405 \)
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| $61$ |
\( T^{6} - 17 T^{5} + \cdots + 6571 \)
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| $67$ |
\( T^{6} - 12 T^{5} + \cdots + 719 \)
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| $71$ |
\( T^{6} - 2 T^{5} + \cdots + 298471 \)
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| $73$ |
\( T^{6} - 16 T^{5} + \cdots + 829 \)
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| $79$ |
\( T^{6} - 36 T^{5} + \cdots - 59491 \)
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| $83$ |
\( T^{6} - 11 T^{5} + \cdots + 7421 \)
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| $89$ |
\( T^{6} + 3 T^{5} + \cdots - 523639 \)
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| $97$ |
\( T^{6} - 8 T^{5} + \cdots - 1324219 \)
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