Newspace parameters
| Level: | \( N \) | \(=\) | \( 4160 = 2^{6} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4160.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(33.2177672409\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{16})^+\) |
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| Defining polynomial: |
\( x^{4} - 4x^{2} + 2 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 2080) |
| 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 \(\nu = \zeta_{16} + \zeta_{16}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} - 2\nu \)
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| \(\beta_{2}\) | \(=\) |
\( -\nu^{3} + 4\nu \)
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| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} - 4 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 4 ) / 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{2} + 2\beta_1 \)
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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 | −2.61313 | 0 | 1.00000 | 0 | −3.69552 | 0 | 3.82843 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.08239 | 0 | 1.00000 | 0 | 1.53073 | 0 | −1.82843 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.08239 | 0 | 1.00000 | 0 | −1.53073 | 0 | −1.82843 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.61313 | 0 | 1.00000 | 0 | 3.69552 | 0 | 3.82843 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(13\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 4160.2.a.bu | 4 | |
| 4.b | odd | 2 | 1 | inner | 4160.2.a.bu | 4 | |
| 8.b | even | 2 | 1 | 2080.2.a.q | ✓ | 4 | |
| 8.d | odd | 2 | 1 | 2080.2.a.q | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2080.2.a.q | ✓ | 4 | 8.b | even | 2 | 1 | |
| 2080.2.a.q | ✓ | 4 | 8.d | odd | 2 | 1 | |
| 4160.2.a.bu | 4 | 1.a | even | 1 | 1 | trivial | |
| 4160.2.a.bu | 4 | 4.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4160))\):
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\( T_{3}^{4} - 8T_{3}^{2} + 8 \)
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\( T_{7}^{4} - 16T_{7}^{2} + 32 \)
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\( T_{11}^{4} - 8T_{11}^{2} + 8 \)
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\( T_{17}^{2} + 4T_{17} - 28 \)
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\( T_{19}^{4} - 8T_{19}^{2} + 8 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( T^{4} - 8T^{2} + 8 \)
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| $5$ |
\( (T - 1)^{4} \)
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| $7$ |
\( T^{4} - 16T^{2} + 32 \)
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| $11$ |
\( T^{4} - 8T^{2} + 8 \)
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| $13$ |
\( (T - 1)^{4} \)
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| $17$ |
\( (T^{2} + 4 T - 28)^{2} \)
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| $19$ |
\( T^{4} - 8T^{2} + 8 \)
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| $23$ |
\( T^{4} - 104T^{2} + 2312 \)
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| $29$ |
\( (T^{2} - 12 T + 28)^{2} \)
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| $31$ |
\( T^{4} - 104T^{2} + 2312 \)
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| $37$ |
\( (T^{2} + 4 T - 68)^{2} \)
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| $41$ |
\( (T + 2)^{4} \)
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| $43$ |
\( T^{4} - 136T^{2} + 4232 \)
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| $47$ |
\( T^{4} - 144T^{2} + 2592 \)
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| $53$ |
\( (T^{2} - 4 T - 124)^{2} \)
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| $59$ |
\( T^{4} - 136T^{2} + 4232 \)
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| $61$ |
\( (T^{2} - 12 T + 28)^{2} \)
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| $67$ |
\( T^{4} - 272 T^{2} + 16928 \)
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| $71$ |
\( T^{4} - 136T^{2} + 392 \)
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| $73$ |
\( (T^{2} + 4 T - 4)^{2} \)
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| $79$ |
\( T^{4} - 160T^{2} + 6272 \)
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| $83$ |
\( T^{4} - 16T^{2} + 32 \)
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| $89$ |
\( (T^{2} + 12 T + 4)^{2} \)
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| $97$ |
\( (T^{2} - 12 T - 92)^{2} \)
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