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: | \(5\) |
| Coefficient field: | 5.5.2179633.1 |
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| Defining polynomial: |
\( x^{5} - x^{4} - 7x^{3} + 4x^{2} + 9x - 4 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 627) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 7x^{3} + 4x^{2} + 9x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 2 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 6\nu^{2} + 2\nu + 5 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 7\beta_{2} + 2\beta _1 + 14 \)
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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 |
|
−2.66695 | −1.00000 | 5.11262 | 1.72144 | 2.66695 | 3.18859 | −8.30122 | 1.00000 | −4.59098 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.29410 | −1.00000 | −0.325316 | 2.82255 | 1.29410 | −2.68387 | 3.00918 | 1.00000 | −3.65264 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.426019 | −1.00000 | −1.81851 | −3.71870 | 0.426019 | 0.191751 | 1.62676 | 1.00000 | 1.58424 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.30941 | −1.00000 | −0.285435 | 3.72141 | −1.30941 | 3.27802 | −2.99258 | 1.00000 | 4.87286 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.07765 | −1.00000 | 2.31663 | −1.54669 | −2.07765 | −2.97449 | 0.657857 | 1.00000 | −3.21348 | ||||||||||||||||||||||||||||||||||
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.r | 5 | |
| 11.b | odd | 2 | 1 | 627.2.a.i | ✓ | 5 | |
| 33.d | even | 2 | 1 | 1881.2.a.l | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 627.2.a.i | ✓ | 5 | 11.b | odd | 2 | 1 | |
| 1881.2.a.l | 5 | 33.d | even | 2 | 1 | ||
| 6897.2.a.r | 5 | 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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\( T_{2}^{5} + T_{2}^{4} - 7T_{2}^{3} - 4T_{2}^{2} + 9T_{2} + 4 \)
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\( T_{5}^{5} - 3T_{5}^{4} - 16T_{5}^{3} + 49T_{5}^{2} + 30T_{5} - 104 \)
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\( T_{7}^{5} - T_{7}^{4} - 18T_{7}^{3} + 11T_{7}^{2} + 82T_{7} - 16 \)
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\( T_{13}^{5} + 16T_{13}^{4} + 95T_{13}^{3} + 257T_{13}^{2} + 309T_{13} + 130 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + T^{4} - 7 T^{3} + \cdots + 4 \)
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| $3$ |
\( (T + 1)^{5} \)
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| $5$ |
\( T^{5} - 3 T^{4} + \cdots - 104 \)
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| $7$ |
\( T^{5} - T^{4} + \cdots - 16 \)
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| $11$ |
\( T^{5} \)
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| $13$ |
\( T^{5} + 16 T^{4} + \cdots + 130 \)
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| $17$ |
\( T^{5} + T^{4} + \cdots - 74 \)
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| $19$ |
\( (T + 1)^{5} \)
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| $23$ |
\( T^{5} + 5 T^{4} + \cdots - 128 \)
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| $29$ |
\( T^{5} + 4 T^{4} + \cdots + 4696 \)
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| $31$ |
\( T^{5} - 4 T^{4} + \cdots + 128 \)
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| $37$ |
\( T^{5} - 7 T^{4} + \cdots - 22516 \)
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| $41$ |
\( T^{5} + 7 T^{4} + \cdots - 404 \)
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| $43$ |
\( T^{5} + 12 T^{4} + \cdots + 640 \)
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| $47$ |
\( T^{5} - 2 T^{4} + \cdots - 16 \)
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| $53$ |
\( T^{5} - 21 T^{4} + \cdots + 11542 \)
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| $59$ |
\( T^{5} - 14 T^{4} + \cdots - 43420 \)
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| $61$ |
\( T^{5} + 13 T^{4} + \cdots + 28124 \)
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| $67$ |
\( T^{5} - 13 T^{4} + \cdots - 28936 \)
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| $71$ |
\( T^{5} + 7 T^{4} + \cdots - 3904 \)
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| $73$ |
\( T^{5} + 16 T^{4} + \cdots + 1976 \)
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| $79$ |
\( T^{5} - 11 T^{4} + \cdots - 19456 \)
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| $83$ |
\( T^{5} - 20 T^{4} + \cdots - 9140 \)
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| $89$ |
\( T^{5} - 7 T^{4} + \cdots - 79582 \)
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| $97$ |
\( T^{5} - 12 T^{4} + \cdots - 11164 \)
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