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: | 4.4.4752.1 |
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| Defining polynomial: |
\( x^{4} - 2x^{3} - 3x^{2} + 4x + 1 \)
|
| 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 a root \(\nu\) of
\( x^{4} - 2x^{3} - 3x^{2} + 4x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
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| \(\beta_{3}\) | \(=\) |
\( 2\nu^{3} - 3\nu^{2} - 5\nu + 3 \)
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| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{2} + \beta _1 + 5 ) / 2 \)
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| \(\nu^{3}\) | \(=\) |
\( ( \beta_{3} + 3\beta_{2} + 4\beta _1 + 7 ) / 2 \)
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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 | −2.73205 | 0 | −1.00000 | 0 | −2.99102 | 0 | 4.46410 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.73205 | 0 | −1.00000 | 0 | 4.99102 | 0 | 4.46410 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.732051 | 0 | −1.00000 | 0 | −0.439374 | 0 | −2.46410 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.732051 | 0 | −1.00000 | 0 | 2.43937 | 0 | −2.46410 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(13\) | \( -1 \) |
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 | 4160.2.a.bs | 4 | |
| 4.b | odd | 2 | 1 | 4160.2.a.bx | 4 | ||
| 8.b | even | 2 | 1 | 2080.2.a.t | yes | 4 | |
| 8.d | odd | 2 | 1 | 2080.2.a.o | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2080.2.a.o | ✓ | 4 | 8.d | odd | 2 | 1 | |
| 2080.2.a.t | yes | 4 | 8.b | even | 2 | 1 | |
| 4160.2.a.bs | 4 | 1.a | even | 1 | 1 | trivial | |
| 4160.2.a.bx | 4 | 4.b | odd | 2 | 1 | ||
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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\( T_{3}^{2} + 2T_{3} - 2 \)
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\( T_{7}^{4} - 4T_{7}^{3} - 12T_{7}^{2} + 32T_{7} + 16 \)
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\( T_{11}^{4} + 4T_{11}^{3} - 24T_{11}^{2} - 56T_{11} + 4 \)
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\( T_{17}^{4} - 4T_{17}^{3} - 24T_{17}^{2} + 80T_{17} + 16 \)
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\( T_{19}^{4} + 4T_{19}^{3} - 48T_{19}^{2} - 104T_{19} + 244 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( (T^{2} + 2 T - 2)^{2} \)
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| $5$ |
\( (T + 1)^{4} \)
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| $7$ |
\( T^{4} - 4 T^{3} + \cdots + 16 \)
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| $11$ |
\( T^{4} + 4 T^{3} + \cdots + 4 \)
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| $13$ |
\( (T - 1)^{4} \)
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| $17$ |
\( T^{4} - 4 T^{3} + \cdots + 16 \)
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| $19$ |
\( T^{4} + 4 T^{3} + \cdots + 244 \)
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| $23$ |
\( T^{4} - 8 T^{3} + \cdots + 52 \)
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| $29$ |
\( T^{4} - 96T^{2} + 2112 \)
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| $31$ |
\( T^{4} - 8 T^{3} + \cdots - 284 \)
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| $37$ |
\( T^{4} - 4 T^{3} + \cdots + 832 \)
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| $41$ |
\( T^{4} - 4 T^{3} + \cdots + 976 \)
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| $43$ |
\( T^{4} + 12 T^{3} + \cdots + 276 \)
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| $47$ |
\( T^{4} - 12 T^{3} + \cdots - 48 \)
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| $53$ |
\( T^{4} - 4 T^{3} + \cdots + 16 \)
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| $59$ |
\( T^{4} - 4 T^{3} + \cdots + 244 \)
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| $61$ |
\( T^{4} - 16 T^{3} + \cdots - 3008 \)
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| $67$ |
\( T^{4} + 28 T^{3} + \cdots + 976 \)
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| $71$ |
\( T^{4} - 192 T^{2} + \cdots + 996 \)
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| $73$ |
\( T^{4} + 4 T^{3} + \cdots + 832 \)
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| $79$ |
\( T^{4} - 192 T^{2} + \cdots - 192 \)
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| $83$ |
\( T^{4} + 12 T^{3} + \cdots - 48 \)
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| $89$ |
\( T^{4} - 120 T^{2} + \cdots + 1104 \)
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| $97$ |
\( T^{4} - 16 T^{3} + \cdots - 368 \)
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