Newspace parameters
| Level: | \( N \) | \(=\) | \( 164 = 2^{2} \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 164.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(4.46867633551\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.83968.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 20x^{2} + 82 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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
\( x^{4} - 20x^{2} + 82 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 10 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 13\nu ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} + 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{3} + 13\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/164\mathbb{Z}\right)^\times\).
| \(n\) | \(83\) | \(129\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 163.1 |
|
2.00000 | −5.79278 | 4.00000 | −4.24264 | −11.5856 | 13.9850 | 8.00000 | 24.5563 | −8.48528 | ||||||||||||||||||||||||||||||
| 163.2 | 2.00000 | −1.56322 | 4.00000 | 4.24264 | −3.12644 | −0.647506 | 8.00000 | −6.55635 | 8.48528 | |||||||||||||||||||||||||||||||
| 163.3 | 2.00000 | 1.56322 | 4.00000 | 4.24264 | 3.12644 | 0.647506 | 8.00000 | −6.55635 | 8.48528 | |||||||||||||||||||||||||||||||
| 163.4 | 2.00000 | 5.79278 | 4.00000 | −4.24264 | 11.5856 | −13.9850 | 8.00000 | 24.5563 | −8.48528 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 164.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-41}) \) |
| 4.b | odd | 2 | 1 | inner |
| 41.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 164.3.d.b | ✓ | 4 |
| 4.b | odd | 2 | 1 | inner | 164.3.d.b | ✓ | 4 |
| 41.b | even | 2 | 1 | inner | 164.3.d.b | ✓ | 4 |
| 164.d | odd | 2 | 1 | CM | 164.3.d.b | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 164.3.d.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 164.3.d.b | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 164.3.d.b | ✓ | 4 | 41.b | even | 2 | 1 | inner |
| 164.3.d.b | ✓ | 4 | 164.d | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 36T_{3}^{2} + 82 \)
acting on \(S_{3}^{\mathrm{new}}(164, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{4} \)
|
| $3$ |
\( T^{4} - 36T^{2} + 82 \)
|
| $5$ |
\( (T^{2} - 18)^{2} \)
|
| $7$ |
\( T^{4} - 196T^{2} + 82 \)
|
| $11$ |
\( T^{4} - 484T^{2} + 82 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( T^{4} - 1444 T^{2} + 511762 \)
|
| $23$ |
\( T^{4} \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( T^{4} \)
|
| $37$ |
\( (T^{2} - 1458)^{2} \)
|
| $41$ |
\( (T + 41)^{4} \)
|
| $43$ |
\( T^{4} \)
|
| $47$ |
\( T^{4} - 8836 T^{2} + 2125522 \)
|
| $53$ |
\( T^{4} \)
|
| $59$ |
\( T^{4} \)
|
| $61$ |
\( (T - 40)^{4} \)
|
| $67$ |
\( T^{4} - 17956 T^{2} + 75413842 \)
|
| $71$ |
\( T^{4} - 20164 T^{2} + 33692242 \)
|
| $73$ |
\( (T^{2} - 17298)^{2} \)
|
| $79$ |
\( T^{4} - 24964 T^{2} + 134138962 \)
|
| $83$ |
\( T^{4} \)
|
| $89$ |
\( T^{4} \)
|
| $97$ |
\( T^{4} \)
|
| show more | |
| show less |