Newspace parameters
| Level: | \( N \) | \(=\) | \( 56 = 2^{3} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 56.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(28.8420068252\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 1266x - 4032 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 1266x - 4032 \)
:
| \(\beta_{1}\) | \(=\) |
\( 6\nu - 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4\nu^{2} - 16\nu - 3372 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 2 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 9\beta_{2} + 8\beta _1 + 10132 ) / 12 \)
|
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 | −195.372 | 0 | −426.015 | 0 | 2401.00 | 0 | 18487.2 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 49.3163 | 0 | 0.429496 | 0 | 2401.00 | 0 | −17250.9 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 230.056 | 0 | −2532.41 | 0 | 2401.00 | 0 | 33242.7 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(7\) | \( -1 \) |
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 | 56.10.a.b | ✓ | 3 |
| 4.b | odd | 2 | 1 | 112.10.a.g | 3 | ||
| 7.b | odd | 2 | 1 | 392.10.a.c | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 56.10.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 112.10.a.g | 3 | 4.b | odd | 2 | 1 | ||
| 392.10.a.c | 3 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 84T_{3}^{2} - 43236T_{3} + 2216592 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(56))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} - 84 T^{2} + \cdots + 2216592 \)
|
| $5$ |
\( T^{3} + 2958 T^{2} + \cdots - 463360 \)
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| $7$ |
\( (T - 2401)^{3} \)
|
| $11$ |
\( T^{3} + \cdots - 11400890351616 \)
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| $13$ |
\( T^{3} + \cdots + 664390599124608 \)
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| $17$ |
\( T^{3} + \cdots + 80\!\cdots\!00 \)
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| $19$ |
\( T^{3} + \cdots - 11\!\cdots\!52 \)
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| $23$ |
\( T^{3} + \cdots - 10\!\cdots\!28 \)
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| $29$ |
\( T^{3} + \cdots - 14\!\cdots\!04 \)
|
| $31$ |
\( T^{3} + \cdots + 16\!\cdots\!00 \)
|
| $37$ |
\( T^{3} + \cdots + 15\!\cdots\!08 \)
|
| $41$ |
\( T^{3} + \cdots + 32\!\cdots\!36 \)
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| $43$ |
\( T^{3} + \cdots + 39\!\cdots\!88 \)
|
| $47$ |
\( T^{3} + \cdots - 11\!\cdots\!76 \)
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| $53$ |
\( T^{3} + \cdots - 14\!\cdots\!16 \)
|
| $59$ |
\( T^{3} + \cdots - 13\!\cdots\!48 \)
|
| $61$ |
\( T^{3} + \cdots + 14\!\cdots\!40 \)
|
| $67$ |
\( T^{3} + \cdots + 97\!\cdots\!92 \)
|
| $71$ |
\( T^{3} + \cdots + 93\!\cdots\!00 \)
|
| $73$ |
\( T^{3} + \cdots - 39\!\cdots\!96 \)
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| $79$ |
\( T^{3} + \cdots + 39\!\cdots\!68 \)
|
| $83$ |
\( T^{3} + \cdots - 53\!\cdots\!12 \)
|
| $89$ |
\( T^{3} + \cdots + 95\!\cdots\!56 \)
|
| $97$ |
\( T^{3} + \cdots + 49\!\cdots\!28 \)
|
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