Newspace parameters
| Level: | \( N \) | \(=\) | \( 378 = 2 \cdot 3^{3} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 378.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(118.081539633\) |
| 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: |
\( x^{4} - 17119x^{2} - 57106x + 65954672 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{6} \) |
| 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
\( x^{4} - 17119x^{2} - 57106x + 65954672 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 9\nu^{2} - 39\nu - 77038 ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{3} - 219\nu^{2} - 27120\nu + 1746042 ) / 50 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 5\beta_{2} + 13\beta _1 + 77038 ) / 9 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 150\beta_{3} + 365\beta_{2} + 28069\beta _1 + 385648 ) / 9 \)
|
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 |
|
−8.00000 | 0 | 64.0000 | −461.060 | 0 | 343.000 | −512.000 | 0 | 3688.48 | ||||||||||||||||||||||||||||||
| 1.2 | −8.00000 | 0 | 64.0000 | −344.918 | 0 | 343.000 | −512.000 | 0 | 2759.35 | |||||||||||||||||||||||||||||||
| 1.3 | −8.00000 | 0 | 64.0000 | 121.357 | 0 | 343.000 | −512.000 | 0 | −970.853 | |||||||||||||||||||||||||||||||
| 1.4 | −8.00000 | 0 | 64.0000 | 168.622 | 0 | 343.000 | −512.000 | 0 | −1348.98 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
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 | 378.8.a.m | ✓ | 4 |
| 3.b | odd | 2 | 1 | 378.8.a.t | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 378.8.a.m | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 378.8.a.t | yes | 4 | 3.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 516T_{5}^{3} - 54225T_{5}^{2} - 29621700T_{5} + 3254256000 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(378))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 8)^{4} \)
|
| $3$ |
\( T^{4} \)
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| $5$ |
\( T^{4} + \cdots + 3254256000 \)
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| $7$ |
\( (T - 343)^{4} \)
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| $11$ |
\( T^{4} + \cdots + 107308140097536 \)
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| $13$ |
\( T^{4} + \cdots + 11\!\cdots\!80 \)
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| $17$ |
\( T^{4} + \cdots + 25\!\cdots\!13 \)
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| $19$ |
\( T^{4} + \cdots + 13\!\cdots\!92 \)
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| $23$ |
\( T^{4} + \cdots + 60\!\cdots\!60 \)
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| $29$ |
\( T^{4} + \cdots + 11\!\cdots\!84 \)
|
| $31$ |
\( T^{4} + \cdots - 60\!\cdots\!40 \)
|
| $37$ |
\( T^{4} + \cdots + 21\!\cdots\!16 \)
|
| $41$ |
\( T^{4} + \cdots + 52\!\cdots\!08 \)
|
| $43$ |
\( T^{4} + \cdots + 85\!\cdots\!93 \)
|
| $47$ |
\( T^{4} + \cdots + 29\!\cdots\!12 \)
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| $53$ |
\( T^{4} + \cdots + 16\!\cdots\!68 \)
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| $59$ |
\( T^{4} + \cdots + 66\!\cdots\!45 \)
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| $61$ |
\( T^{4} + \cdots + 43\!\cdots\!64 \)
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| $67$ |
\( T^{4} + \cdots + 33\!\cdots\!00 \)
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| $71$ |
\( T^{4} + \cdots - 23\!\cdots\!84 \)
|
| $73$ |
\( T^{4} + \cdots + 37\!\cdots\!32 \)
|
| $79$ |
\( T^{4} + \cdots - 28\!\cdots\!40 \)
|
| $83$ |
\( T^{4} + \cdots - 92\!\cdots\!96 \)
|
| $89$ |
\( T^{4} + \cdots + 69\!\cdots\!88 \)
|
| $97$ |
\( T^{4} + \cdots + 16\!\cdots\!04 \)
|
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