Newspace parameters
| Level: | \( N \) | \(=\) | \( 1197 = 3^{2} \cdot 7 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1197.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(9.55809312195\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.12730624.1 |
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| Defining polynomial: |
\( x^{6} - 8x^{4} + 13x^{2} - 4 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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
\( x^{6} - 8x^{4} + 13x^{2} - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{4} - 5\nu^{2} + 1 \)
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| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 6\nu^{3} + \nu ) / 2 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 8\nu^{3} + 11\nu ) / 2 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 7\nu^{2} + 6 \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 7\nu^{3} + 8\nu \)
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| \(\nu\) | \(=\) |
\( ( \beta_{5} - \beta_{3} - \beta_{2} ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{4} + \beta _1 + 5 ) / 2 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 5\beta_{5} - 7\beta_{3} - 3\beta_{2} ) / 2 \)
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| \(\nu^{4}\) | \(=\) |
\( ( -5\beta_{4} + 7\beta _1 + 23 ) / 2 \)
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| \(\nu^{5}\) | \(=\) |
\( ( 29\beta_{5} - 41\beta_{3} - 13\beta_{2} ) / 2 \)
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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 |
|
−2.51912 | 0 | 4.34596 | −2.12216 | 0 | −1.00000 | −5.90976 | 0 | 5.34596 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.61036 | 0 | 0.593272 | −0.989386 | 0 | −1.00000 | 2.26534 | 0 | 1.59327 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.246506 | 0 | −1.93923 | 3.81019 | 0 | −1.00000 | 0.971044 | 0 | −0.939235 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.246506 | 0 | −1.93923 | −3.81019 | 0 | −1.00000 | −0.971044 | 0 | −0.939235 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.61036 | 0 | 0.593272 | 0.989386 | 0 | −1.00000 | −2.26534 | 0 | 1.59327 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.51912 | 0 | 4.34596 | 2.12216 | 0 | −1.00000 | 5.90976 | 0 | 5.34596 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(7\) | \( +1 \) |
| \(19\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1197.2.a.q | ✓ | 6 |
| 3.b | odd | 2 | 1 | inner | 1197.2.a.q | ✓ | 6 |
| 7.b | odd | 2 | 1 | 8379.2.a.ch | 6 | ||
| 21.c | even | 2 | 1 | 8379.2.a.ch | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1197.2.a.q | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 1197.2.a.q | ✓ | 6 | 3.b | odd | 2 | 1 | inner |
| 8379.2.a.ch | 6 | 7.b | odd | 2 | 1 | ||
| 8379.2.a.ch | 6 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1197))\):
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\( T_{2}^{6} - 9T_{2}^{4} + 17T_{2}^{2} - 1 \)
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\( T_{5}^{6} - 20T_{5}^{4} + 84T_{5}^{2} - 64 \)
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\( T_{11}^{6} - 36T_{11}^{4} + 272T_{11}^{2} - 64 \)
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\( T_{13}^{3} + 2T_{13}^{2} - 32T_{13} - 80 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 9 T^{4} + \cdots - 1 \)
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| $3$ |
\( T^{6} \)
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| $5$ |
\( T^{6} - 20 T^{4} + \cdots - 64 \)
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| $7$ |
\( (T + 1)^{6} \)
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| $11$ |
\( T^{6} - 36 T^{4} + \cdots - 64 \)
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| $13$ |
\( (T^{3} + 2 T^{2} - 32 T - 80)^{2} \)
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| $17$ |
\( T^{6} - 20 T^{4} + \cdots - 64 \)
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| $19$ |
\( (T - 1)^{6} \)
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| $23$ |
\( T^{6} - 132 T^{4} + \cdots - 78400 \)
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| $29$ |
\( T^{6} - 32 T^{4} + \cdots - 784 \)
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| $31$ |
\( (T^{3} - 4 T^{2} + \cdots + 448)^{2} \)
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| $37$ |
\( (T^{3} - 6 T^{2} - 28 T + 40)^{2} \)
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| $41$ |
\( T^{6} - 36 T^{4} + \cdots - 64 \)
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| $43$ |
\( (T^{3} - 20 T^{2} + \cdots - 16)^{2} \)
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| $47$ |
\( T^{6} - 128 T^{4} + \cdots - 10000 \)
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| $53$ |
\( T^{6} - 336 T^{4} + \cdots - 868624 \)
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| $59$ |
\( T^{6} - 272 T^{4} + \cdots - 16384 \)
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| $61$ |
\( (T^{3} + 6 T^{2} - 28 T - 40)^{2} \)
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| $67$ |
\( (T^{3} - 12 T^{2} + \cdots + 832)^{2} \)
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| $71$ |
\( T^{6} - 164 T^{4} + \cdots - 141376 \)
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| $73$ |
\( (T^{3} - 10 T^{2} + \cdots + 664)^{2} \)
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| $79$ |
\( (T^{3} - 8 T^{2} - 104 T + 32)^{2} \)
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| $83$ |
\( T^{6} - 272 T^{4} + \cdots - 355216 \)
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| $89$ |
\( T^{6} - 36 T^{4} + \cdots - 64 \)
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| $97$ |
\( (T + 2)^{6} \)
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