Newspace parameters
| Level: | \( N \) | \(=\) | \( 348 = 2^{2} \cdot 3 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 348.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(20.5326646820\) |
| 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: |
\( x^{4} - x^{3} - 85x^{2} + 130x + 1224 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{4} - x^{3} - 85x^{2} + 130x + 1224 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 3\nu^{2} - 49\nu - 84 ) / 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 6\nu^{2} - 46\nu - 213 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{3} - 4\beta_{2} - \beta _1 + 86 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -3\beta_{3} + 12\beta_{2} + 26\beta _1 - 45 \)
|
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 | −3.00000 | 0 | −20.3750 | 0 | −25.1587 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −3.00000 | 0 | −4.25736 | 0 | −4.02057 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | −3.00000 | 0 | 6.89206 | 0 | 33.0799 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | −3.00000 | 0 | 16.7402 | 0 | 13.0994 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(29\) | \(-1\) |
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 | 348.4.a.d | ✓ | 4 |
| 3.b | odd | 2 | 1 | 1044.4.a.h | 4 | ||
| 4.b | odd | 2 | 1 | 1392.4.a.q | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 348.4.a.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1044.4.a.h | 4 | 3.b | odd | 2 | 1 | ||
| 1392.4.a.q | 4 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + T_{5}^{3} - 380T_{5}^{2} + 792T_{5} + 10008 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(348))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T + 3)^{4} \)
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| $5$ |
\( T^{4} + T^{3} + \cdots + 10008 \)
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| $7$ |
\( T^{4} - 17 T^{3} + \cdots + 43832 \)
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| $11$ |
\( T^{4} + 14 T^{3} + \cdots - 1052712 \)
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| $13$ |
\( T^{4} + 20 T^{3} + \cdots - 901872 \)
|
| $17$ |
\( T^{4} + 143 T^{3} + \cdots + 5815242 \)
|
| $19$ |
\( T^{4} - 17 T^{3} + \cdots + 27099408 \)
|
| $23$ |
\( T^{4} - 170 T^{3} + \cdots - 17292672 \)
|
| $29$ |
\( (T - 29)^{4} \)
|
| $31$ |
\( T^{4} - 378 T^{3} + \cdots - 26692992 \)
|
| $37$ |
\( T^{4} + \cdots - 2044995000 \)
|
| $41$ |
\( T^{4} + \cdots + 4577778792 \)
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| $43$ |
\( T^{4} + \cdots + 4593494784 \)
|
| $47$ |
\( T^{4} - 31 T^{3} + \cdots - 392190792 \)
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| $53$ |
\( T^{4} - 778 T^{3} + \cdots + 354684624 \)
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| $59$ |
\( T^{4} + \cdots - 86888501712 \)
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| $61$ |
\( T^{4} + \cdots - 1477900752 \)
|
| $67$ |
\( T^{4} + \cdots - 34290992592 \)
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| $71$ |
\( T^{4} + \cdots + 226831497216 \)
|
| $73$ |
\( T^{4} + \cdots + 3141989424 \)
|
| $79$ |
\( T^{4} + \cdots - 22521738816 \)
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| $83$ |
\( T^{4} + \cdots + 98624653056 \)
|
| $89$ |
\( T^{4} + \cdots - 60528532068 \)
|
| $97$ |
\( T^{4} + \cdots + 709606794816 \)
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