Newspace parameters
| Level: | \( N \) | \(=\) | \( 822 = 2 \cdot 3 \cdot 137 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 822.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(6.56370304615\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.961.1 |
|
|
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| Defining polynomial: |
\( x^{3} - x^{2} - 10x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{3} - x^{2} - 10x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} + \nu - 8 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} - \beta _1 + 8 \)
|
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 |
|
−1.00000 | −1.00000 | 1.00000 | −4.29707 | 1.00000 | 3.08387 | −1.00000 | 1.00000 | 4.29707 | |||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.00000 | 1.00000 | −1.78680 | 1.00000 | −3.29707 | −1.00000 | 1.00000 | 1.78680 | ||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.00000 | 1.00000 | 2.08387 | 1.00000 | −0.786802 | −1.00000 | 1.00000 | −2.08387 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( +1 \) |
| \(137\) | \( +1 \) |
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 | 822.2.a.i | ✓ | 3 |
| 3.b | odd | 2 | 1 | 2466.2.a.s | 3 | ||
| 4.b | odd | 2 | 1 | 6576.2.a.w | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 822.2.a.i | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 2466.2.a.s | 3 | 3.b | odd | 2 | 1 | ||
| 6576.2.a.w | 3 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(822))\):
|
\( T_{5}^{3} + 4T_{5}^{2} - 5T_{5} - 16 \)
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\( T_{7}^{3} + T_{7}^{2} - 10T_{7} - 8 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{3} \)
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| $3$ |
\( (T + 1)^{3} \)
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| $5$ |
\( T^{3} + 4 T^{2} + \cdots - 16 \)
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| $7$ |
\( T^{3} + T^{2} - 10T - 8 \)
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| $11$ |
\( T^{3} - 5 T^{2} + \cdots + 8 \)
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| $13$ |
\( T^{3} + 2 T^{2} + \cdots - 64 \)
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| $17$ |
\( T^{3} + 3 T^{2} + \cdots - 92 \)
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| $19$ |
\( T^{3} - 8 T^{2} + \cdots + 4 \)
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| $23$ |
\( T^{3} + 17 T^{2} + \cdots + 128 \)
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| $29$ |
\( T^{3} + 6 T^{2} + \cdots + 8 \)
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| $31$ |
\( T^{3} - T^{2} - 10T + 8 \)
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| $37$ |
\( T^{3} + 17 T^{2} + \cdots + 128 \)
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| $41$ |
\( T^{3} + 16 T^{2} + \cdots - 32 \)
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| $43$ |
\( T^{3} - 6 T^{2} + \cdots + 736 \)
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| $47$ |
\( T^{3} + 4 T^{2} + \cdots - 326 \)
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| $53$ |
\( T^{3} + 31 T^{2} + \cdots + 992 \)
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| $59$ |
\( T^{3} + 7 T^{2} + \cdots - 1504 \)
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| $61$ |
\( T^{3} - 7 T^{2} + \cdots + 16 \)
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| $67$ |
\( T^{3} - 2 T^{2} + \cdots + 64 \)
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| $71$ |
\( T^{3} + 11 T^{2} + \cdots + 16 \)
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| $73$ |
\( T^{3} + 2 T^{2} + \cdots - 2 \)
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| $79$ |
\( T^{3} + 11 T^{2} + \cdots - 356 \)
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| $83$ |
\( T^{3} + 15 T^{2} + \cdots - 464 \)
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| $89$ |
\( T^{3} - 13 T^{2} + \cdots + 712 \)
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| $97$ |
\( T^{3} + 10 T^{2} + \cdots - 2048 \)
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