Newspace parameters
| Level: | \( N \) | \(=\) | \( 4304 = 2^{4} \cdot 269 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4304.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(34.3676130300\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.4913.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 6x^{2} + x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 538) |
| 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
\( x^{4} - x^{3} - 6x^{2} + x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 2\nu^{2} + 4\nu - 3 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 6\nu - 3 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 6\beta _1 + 3 \)
|
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 | −3.04948 | 0 | 1.48793 | 0 | −0.344151 | 0 | 6.29934 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.34415 | 0 | 3.90570 | 0 | 0.487928 | 0 | −1.19326 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.512072 | 0 | −1.04948 | 0 | 2.90570 | 0 | −2.73778 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.90570 | 0 | 0.655849 | 0 | −2.04948 | 0 | 0.631706 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(269\) | \( -1 \) |
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 | 4304.2.a.e | 4 | |
| 4.b | odd | 2 | 1 | 538.2.a.c | ✓ | 4 | |
| 12.b | even | 2 | 1 | 4842.2.a.j | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 538.2.a.c | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 4304.2.a.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 4842.2.a.j | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 3T_{3}^{3} - 3T_{3}^{2} - 10T_{3} - 4 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4304))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 3 T^{3} + \cdots - 4 \)
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| $5$ |
\( T^{4} - 5 T^{3} + \cdots - 4 \)
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| $7$ |
\( T^{4} - T^{3} - 6 T^{2} + \cdots + 1 \)
|
| $11$ |
\( T^{4} + 7 T^{3} + \cdots - 13 \)
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| $13$ |
\( T^{4} - 4 T^{3} + \cdots + 52 \)
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| $17$ |
\( T^{4} - 4 T^{3} + \cdots - 16 \)
|
| $19$ |
\( T^{4} + 2 T^{3} + \cdots + 16 \)
|
| $23$ |
\( T^{4} + 16 T^{3} + \cdots + 52 \)
|
| $29$ |
\( T^{4} - 51 T^{2} + \cdots + 17 \)
|
| $31$ |
\( T^{4} - T^{3} + \cdots - 67 \)
|
| $37$ |
\( T^{4} + 2 T^{3} + \cdots - 188 \)
|
| $41$ |
\( T^{4} + T^{3} + \cdots + 52 \)
|
| $43$ |
\( T^{4} + 9 T^{3} + \cdots + 1937 \)
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| $47$ |
\( T^{4} + 13 T^{3} + \cdots - 1172 \)
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| $53$ |
\( (T^{2} - 14 T + 32)^{2} \)
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| $59$ |
\( T^{4} + 19 T^{3} + \cdots - 188 \)
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| $61$ |
\( T^{4} + 20 T^{3} + \cdots + 268 \)
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| $67$ |
\( T^{4} + 15 T^{3} + \cdots + 101 \)
|
| $71$ |
\( T^{4} + 15 T^{3} + \cdots - 2228 \)
|
| $73$ |
\( T^{4} + 7 T^{3} + \cdots + 6481 \)
|
| $79$ |
\( T^{4} - 17 T^{3} + \cdots + 68 \)
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| $83$ |
\( T^{4} + 6 T^{3} + \cdots + 1024 \)
|
| $89$ |
\( T^{4} - 20 T^{3} + \cdots - 191 \)
|
| $97$ |
\( T^{4} - 3 T^{3} + \cdots + 27196 \)
|
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