Newspace parameters
| Level: | \( N \) | \(=\) | \( 2 \) |
| Weight: | \( k \) | \(=\) | \( 56 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(38.3162935370\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{3} - x^{2} - 9638486409249116938x + 5032749353671885590018834040 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{22}\cdot 3^{6}\cdot 5^{3}\cdot 11 \) |
| 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
\( x^{3} - x^{2} - 9638486409249116938x + 5032749353671885590018834040 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 640\nu^{2} + 857514869760\nu - 4112420868232128183680 ) / 143886491 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -30142080\nu^{2} + 117149300824366080\nu + 193682685578616648184959360 ) / 143886491 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 47097\beta _1 + 364953600 ) / 1094860800 \)
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| \(\nu^{2}\) | \(=\) |
\( ( -55827791\beta_{2} + 7626907605753\beta _1 + 293133359467211543627212800 ) / 45619200 \)
|
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 |
|
−1.34218e8 | −2.19490e13 | 1.80144e16 | 1.96436e19 | 2.94595e21 | 3.30259e23 | −2.41785e24 | 3.07310e26 | −2.63652e27 | |||||||||||||||||||||||||||
| 1.2 | −1.34218e8 | −6.32966e10 | 1.80144e16 | −1.92471e19 | 8.49553e18 | −8.70955e22 | −2.41785e24 | −1.74445e26 | 2.58331e27 | ||||||||||||||||||||||||||||
| 1.3 | −1.34218e8 | 2.51193e13 | 1.80144e16 | 1.98512e19 | −3.37146e21 | 3.18308e22 | −2.41785e24 | 4.56531e26 | −2.66439e27 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
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 | 2.56.a.b | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2.56.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 3107008649316T_{3}^{2} - 551544670455362719563040848T_{3} - 34898202603313072418721305018586426048 \)
acting on \(S_{56}^{\mathrm{new}}(\Gamma_0(2))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 134217728)^{3} \)
|
| $3$ |
\( T^{3} + \cdots - 34\!\cdots\!48 \)
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| $5$ |
\( T^{3} + \cdots + 75\!\cdots\!00 \)
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| $7$ |
\( T^{3} + \cdots + 91\!\cdots\!04 \)
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| $11$ |
\( T^{3} + \cdots - 86\!\cdots\!48 \)
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| $13$ |
\( T^{3} + \cdots - 84\!\cdots\!08 \)
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| $17$ |
\( T^{3} + \cdots - 19\!\cdots\!76 \)
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| $19$ |
\( T^{3} + \cdots - 30\!\cdots\!00 \)
|
| $23$ |
\( T^{3} + \cdots + 70\!\cdots\!12 \)
|
| $29$ |
\( T^{3} + \cdots - 18\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots - 28\!\cdots\!88 \)
|
| $37$ |
\( T^{3} + \cdots + 57\!\cdots\!44 \)
|
| $41$ |
\( T^{3} + \cdots + 38\!\cdots\!72 \)
|
| $43$ |
\( T^{3} + \cdots - 59\!\cdots\!08 \)
|
| $47$ |
\( T^{3} + \cdots - 24\!\cdots\!56 \)
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| $53$ |
\( T^{3} + \cdots - 98\!\cdots\!88 \)
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| $59$ |
\( T^{3} + \cdots - 94\!\cdots\!00 \)
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| $61$ |
\( T^{3} + \cdots + 18\!\cdots\!32 \)
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| $67$ |
\( T^{3} + \cdots + 17\!\cdots\!04 \)
|
| $71$ |
\( T^{3} + \cdots + 37\!\cdots\!32 \)
|
| $73$ |
\( T^{3} + \cdots - 63\!\cdots\!28 \)
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| $79$ |
\( T^{3} + \cdots - 47\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots - 25\!\cdots\!68 \)
|
| $89$ |
\( T^{3} + \cdots - 72\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots - 30\!\cdots\!76 \)
|
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