Newspace parameters
| Level: | \( N \) | \(=\) | \( 308 = 2^{2} \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 308.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(18.1725882818\) |
| 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: |
\( x^{4} - x^{3} - 43x^{2} - 11x + 222 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 43x^{2} - 11x + 222 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 21 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu^{2} - 30\nu + 32 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 21 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_{2} + 36\beta _1 + 31 \)
|
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 | −6.25471 | 0 | 17.1213 | 0 | −7.00000 | 0 | 12.1213 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −3.79239 | 0 | −7.61779 | 0 | −7.00000 | 0 | −12.6178 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.21425 | 0 | −20.5256 | 0 | −7.00000 | 0 | −25.5256 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 5.83284 | 0 | 12.0220 | 0 | −7.00000 | 0 | 7.02203 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
| \(11\) | \(-1\) |
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 | 308.4.a.d | ✓ | 4 |
| 4.b | odd | 2 | 1 | 1232.4.a.v | 4 | ||
| 7.b | odd | 2 | 1 | 2156.4.a.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 308.4.a.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1232.4.a.v | 4 | 4.b | odd | 2 | 1 | ||
| 2156.4.a.f | 4 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 3T_{3}^{3} - 40T_{3}^{2} - 96T_{3} + 168 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(308))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 3 T^{3} + \cdots + 168 \)
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| $5$ |
\( T^{4} - T^{3} + \cdots + 32184 \)
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| $7$ |
\( (T + 7)^{4} \)
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| $11$ |
\( (T - 11)^{4} \)
|
| $13$ |
\( T^{4} - 98 T^{3} + \cdots - 5715264 \)
|
| $17$ |
\( T^{4} - 64 T^{3} + \cdots + 80892704 \)
|
| $19$ |
\( T^{4} - 114 T^{3} + \cdots + 186624 \)
|
| $23$ |
\( T^{4} - 231 T^{3} + \cdots + 6309184 \)
|
| $29$ |
\( T^{4} - 268 T^{3} + \cdots - 62229648 \)
|
| $31$ |
\( T^{4} - 33 T^{3} + \cdots + 668400408 \)
|
| $37$ |
\( T^{4} + \cdots + 2194340904 \)
|
| $41$ |
\( T^{4} + \cdots - 6396705696 \)
|
| $43$ |
\( T^{4} - 44 T^{3} + \cdots - 221937664 \)
|
| $47$ |
\( T^{4} + \cdots - 3932145984 \)
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| $53$ |
\( T^{4} + \cdots - 1987381584 \)
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| $59$ |
\( T^{4} + \cdots + 4667157144 \)
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| $61$ |
\( T^{4} + \cdots + 1375017408 \)
|
| $67$ |
\( T^{4} + \cdots + 2114400384 \)
|
| $71$ |
\( T^{4} + \cdots - 156603934656 \)
|
| $73$ |
\( T^{4} + \cdots - 8143492032 \)
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| $79$ |
\( T^{4} + \cdots - 14872658688 \)
|
| $83$ |
\( T^{4} + \cdots + 163701302784 \)
|
| $89$ |
\( T^{4} + \cdots - 58144933944 \)
|
| $97$ |
\( T^{4} + \cdots + 5739946776 \)
|
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