Newspace parameters
| Level: | \( N \) | \(=\) | \( 164 = 2^{2} \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 164.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(9.67631324094\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.4344.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 16x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 16x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} + \nu - 12 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{2} + 3\nu + 10 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{2} + 3\beta _1 + 23 ) / 2 \)
|
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 | −6.96442 | 0 | 1.09445 | 0 | 22.6332 | 0 | 21.5032 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 0.118522 | 0 | 3.09161 | 0 | −16.7758 | 0 | −26.9860 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 4.84590 | 0 | −14.1861 | 0 | −5.85740 | 0 | −3.51724 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(41\) | \(1\) |
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 | 164.4.a.a | ✓ | 3 |
| 3.b | odd | 2 | 1 | 1476.4.a.d | 3 | ||
| 4.b | odd | 2 | 1 | 656.4.a.d | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 164.4.a.a | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 656.4.a.d | 3 | 4.b | odd | 2 | 1 | ||
| 1476.4.a.d | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 2T_{3}^{2} - 34T_{3} + 4 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(164))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + 2 T^{2} + \cdots + 4 \)
|
| $5$ |
\( T^{3} + 10 T^{2} + \cdots + 48 \)
|
| $7$ |
\( T^{3} - 414T - 2224 \)
|
| $11$ |
\( T^{3} + 42 T^{2} + \cdots - 92 \)
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| $13$ |
\( T^{3} + 76 T^{2} + \cdots + 9728 \)
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| $17$ |
\( T^{3} + 78 T^{2} + \cdots - 132728 \)
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| $19$ |
\( T^{3} + 162 T^{2} + \cdots - 337628 \)
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| $23$ |
\( T^{3} + 48 T^{2} + \cdots - 160128 \)
|
| $29$ |
\( T^{3} + 88 T^{2} + \cdots - 2234112 \)
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| $31$ |
\( T^{3} + 16 T^{2} + \cdots + 130176 \)
|
| $37$ |
\( T^{3} + 406 T^{2} + \cdots - 3955632 \)
|
| $41$ |
\( (T + 41)^{3} \)
|
| $43$ |
\( T^{3} + 236 T^{2} + \cdots - 21644864 \)
|
| $47$ |
\( T^{3} - 528 T^{2} + \cdots + 1976112 \)
|
| $53$ |
\( T^{3} + 92 T^{2} + \cdots - 8586432 \)
|
| $59$ |
\( T^{3} - 712 T^{2} + \cdots - 6652608 \)
|
| $61$ |
\( T^{3} + 394 T^{2} + \cdots + 3975704 \)
|
| $67$ |
\( T^{3} - 446 T^{2} + \cdots + 773172 \)
|
| $71$ |
\( T^{3} - 804 T^{2} + \cdots + 234302328 \)
|
| $73$ |
\( T^{3} - 298 T^{2} + \cdots + 138642432 \)
|
| $79$ |
\( T^{3} - 936 T^{2} + \cdots + 6872832 \)
|
| $83$ |
\( T^{3} - 796 T^{2} + \cdots + 38462784 \)
|
| $89$ |
\( T^{3} + 314 T^{2} + \cdots + 245723832 \)
|
| $97$ |
\( T^{3} + \cdots - 1402773656 \)
|
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