Newspace parameters
| Level: | \( N \) | \(=\) | \( 264 = 2^{3} \cdot 3 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 264.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(42.3413284306\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 2117x^{2} + 1518x + 1092672 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 2117x^{2} + 1518x + 1092672 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + \nu^{2} - 1094\nu - 2544 ) / 15 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 59\nu^{2} + 1004\nu - 60996 ) / 45 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{3} + \beta_{2} + 3\beta _1 + 4239 ) / 4 \)
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| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{3} + 59\beta_{2} + 2185\beta _1 + 8125 ) / 4 \)
|
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 | 9.00000 | 0 | −78.9037 | 0 | 207.206 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 9.00000 | 0 | −72.8555 | 0 | −184.422 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 9.00000 | 0 | 47.0872 | 0 | −114.978 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 9.00000 | 0 | 60.6720 | 0 | 130.194 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( -1 \) |
| \(11\) | \( -1 \) |
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 | 264.6.a.i | ✓ | 4 |
| 3.b | odd | 2 | 1 | 792.6.a.m | 4 | ||
| 4.b | odd | 2 | 1 | 528.6.a.z | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 264.6.a.i | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 528.6.a.z | 4 | 4.b | odd | 2 | 1 | ||
| 792.6.a.m | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 44T_{5}^{3} - 7748T_{5}^{2} - 185904T_{5} + 16422912 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(264))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T - 9)^{4} \)
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| $5$ |
\( T^{4} + 44 T^{3} + \cdots + 16422912 \)
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| $7$ |
\( T^{4} - 38 T^{3} + \cdots + 572028800 \)
|
| $11$ |
\( (T - 121)^{4} \)
|
| $13$ |
\( T^{4} + \cdots - 23772849408 \)
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| $17$ |
\( T^{4} + \cdots + 1442025007776 \)
|
| $19$ |
\( T^{4} + \cdots - 55216000000 \)
|
| $23$ |
\( T^{4} + \cdots - 1331666503680 \)
|
| $29$ |
\( T^{4} + \cdots - 521650037147328 \)
|
| $31$ |
\( T^{4} + \cdots - 15017840640000 \)
|
| $37$ |
\( T^{4} + \cdots - 77\!\cdots\!80 \)
|
| $41$ |
\( T^{4} + \cdots - 18\!\cdots\!12 \)
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| $43$ |
\( T^{4} + \cdots + 34\!\cdots\!88 \)
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| $47$ |
\( T^{4} + \cdots - 92\!\cdots\!40 \)
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| $53$ |
\( T^{4} + \cdots + 32\!\cdots\!68 \)
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| $59$ |
\( T^{4} + \cdots + 10\!\cdots\!92 \)
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| $61$ |
\( T^{4} + \cdots + 89\!\cdots\!00 \)
|
| $67$ |
\( T^{4} + \cdots - 17\!\cdots\!88 \)
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| $71$ |
\( T^{4} + \cdots + 16\!\cdots\!80 \)
|
| $73$ |
\( T^{4} + \cdots - 33\!\cdots\!60 \)
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| $79$ |
\( T^{4} + \cdots + 29\!\cdots\!04 \)
|
| $83$ |
\( T^{4} + \cdots - 25\!\cdots\!00 \)
|
| $89$ |
\( T^{4} + \cdots + 19\!\cdots\!40 \)
|
| $97$ |
\( T^{4} + \cdots + 41\!\cdots\!40 \)
|
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