Newspace parameters
| Level: | \( N \) | \(=\) | \( 656 = 2^{4} \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 656.d (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(38.7052529638\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-10}, \sqrt{-66})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 38x^{2} + 196 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 82) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} + 38x^{2} + 196 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 31\nu ) / 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 19 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 19 \)
|
| \(\nu^{3}\) | \(=\) |
\( 7\beta_{2} - 31\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/656\mathbb{Z}\right)^\times\).
| \(n\) | \(129\) | \(165\) | \(575\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 81.1 |
|
0 | − | 5.64316i | 0 | 13.8452 | 0 | 4.28036i | 0 | −4.84523 | 0 | |||||||||||||||||||||||||||||
| 81.2 | 0 | − | 2.48088i | 0 | −11.8452 | 0 | 20.0918i | 0 | 20.8452 | 0 | ||||||||||||||||||||||||||||||
| 81.3 | 0 | 2.48088i | 0 | −11.8452 | 0 | − | 20.0918i | 0 | 20.8452 | 0 | ||||||||||||||||||||||||||||||
| 81.4 | 0 | 5.64316i | 0 | 13.8452 | 0 | − | 4.28036i | 0 | −4.84523 | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 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 | 656.4.d.b | 4 | |
| 4.b | odd | 2 | 1 | 82.4.b.a | ✓ | 4 | |
| 12.b | even | 2 | 1 | 738.4.d.a | 4 | ||
| 41.b | even | 2 | 1 | inner | 656.4.d.b | 4 | |
| 164.d | odd | 2 | 1 | 82.4.b.a | ✓ | 4 | |
| 492.d | even | 2 | 1 | 738.4.d.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 82.4.b.a | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 82.4.b.a | ✓ | 4 | 164.d | odd | 2 | 1 | |
| 656.4.d.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 656.4.d.b | 4 | 41.b | even | 2 | 1 | inner | |
| 738.4.d.a | 4 | 12.b | even | 2 | 1 | ||
| 738.4.d.a | 4 | 492.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 38T_{3}^{2} + 196 \)
acting on \(S_{4}^{\mathrm{new}}(656, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 38T^{2} + 196 \)
|
| $5$ |
\( (T^{2} - 2 T - 164)^{2} \)
|
| $7$ |
\( T^{4} + 422T^{2} + 7396 \)
|
| $11$ |
\( T^{4} + 1230 T^{2} + 44100 \)
|
| $13$ |
\( T^{4} + 5648 T^{2} + 5271616 \)
|
| $17$ |
\( T^{4} + 4992 T^{2} + 147456 \)
|
| $19$ |
\( T^{4} + 13622 T^{2} + 26977636 \)
|
| $23$ |
\( (T^{2} - 4 T - 656)^{2} \)
|
| $29$ |
\( T^{4} + 43472 T^{2} + 288456256 \)
|
| $31$ |
\( (T^{2} - 156 T + 5424)^{2} \)
|
| $37$ |
\( (T^{2} + 22 T - 72644)^{2} \)
|
| $41$ |
\( (T^{2} - 82 T + 68921)^{2} \)
|
| $43$ |
\( (T^{2} + 276 T - 34416)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 59353627876 \)
|
| $53$ |
\( T^{4} + 66768 T^{2} + 621804096 \)
|
| $59$ |
\( (T^{2} - 20 T - 290960)^{2} \)
|
| $61$ |
\( (T^{2} + 16 T - 190676)^{2} \)
|
| $67$ |
\( T^{4} + 869342 T^{2} + 291248356 \)
|
| $71$ |
\( T^{4} + \cdots + 255277562500 \)
|
| $73$ |
\( (T^{2} + 74 T - 877916)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 18590231716 \)
|
| $83$ |
\( (T^{2} - 744 T - 379056)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 801497629696 \)
|
| $97$ |
\( T^{4} + \cdots + 29342319616 \)
|
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