Newspace parameters
| Level: | \( N \) | \(=\) | \( 1760 = 2^{5} \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1760.g (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(14.0536707557\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{8})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 440) |
| 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
| \(\beta_{1}\) | \(=\) |
\( \zeta_{8}^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{8}^{3} + \zeta_{8} \)
|
| \(\beta_{3}\) | \(=\) |
\( -\zeta_{8}^{3} + \zeta_{8} \)
|
| \(\zeta_{8}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\zeta_{8}^{2}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{8}^{3}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1760\mathbb{Z}\right)^\times\).
| \(n\) | \(321\) | \(991\) | \(1057\) | \(1541\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 881.1 |
|
0 | − | 2.41421i | 0 | 1.00000i | 0 | 1.00000 | 0 | −2.82843 | 0 | |||||||||||||||||||||||||||||
| 881.2 | 0 | − | 0.414214i | 0 | − | 1.00000i | 0 | 1.00000 | 0 | 2.82843 | 0 | |||||||||||||||||||||||||||||
| 881.3 | 0 | 0.414214i | 0 | 1.00000i | 0 | 1.00000 | 0 | 2.82843 | 0 | |||||||||||||||||||||||||||||||
| 881.4 | 0 | 2.41421i | 0 | − | 1.00000i | 0 | 1.00000 | 0 | −2.82843 | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1760.2.g.a | 4 | |
| 4.b | odd | 2 | 1 | 440.2.g.a | ✓ | 4 | |
| 8.b | even | 2 | 1 | inner | 1760.2.g.a | 4 | |
| 8.d | odd | 2 | 1 | 440.2.g.a | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 440.2.g.a | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 440.2.g.a | ✓ | 4 | 8.d | odd | 2 | 1 | |
| 1760.2.g.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 1760.2.g.a | 4 | 8.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 6T_{3}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(1760, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 6T^{2} + 1 \)
|
| $5$ |
\( (T^{2} + 1)^{2} \)
|
| $7$ |
\( (T - 1)^{4} \)
|
| $11$ |
\( (T^{2} + 1)^{2} \)
|
| $13$ |
\( (T^{2} + 8)^{2} \)
|
| $17$ |
\( (T^{2} + 6 T - 9)^{2} \)
|
| $19$ |
\( (T^{2} + 9)^{2} \)
|
| $23$ |
\( (T^{2} - 8)^{2} \)
|
| $29$ |
\( T^{4} + 54T^{2} + 81 \)
|
| $31$ |
\( (T^{2} - 2 T - 17)^{2} \)
|
| $37$ |
\( T^{4} + 178T^{2} + 5329 \)
|
| $41$ |
\( (T^{2} + 12 T + 28)^{2} \)
|
| $43$ |
\( (T^{2} + 36)^{2} \)
|
| $47$ |
\( (T^{2} - 12 T + 28)^{2} \)
|
| $53$ |
\( (T^{2} + 25)^{2} \)
|
| $59$ |
\( T^{4} + 176T^{2} + 3136 \)
|
| $61$ |
\( T^{4} + 118T^{2} + 1681 \)
|
| $67$ |
\( T^{4} + 136T^{2} + 16 \)
|
| $71$ |
\( (T^{2} + 6 T - 9)^{2} \)
|
| $73$ |
\( (T^{2} - 16 T - 8)^{2} \)
|
| $79$ |
\( (T^{2} - 72)^{2} \)
|
| $83$ |
\( T^{4} + 176T^{2} + 3136 \)
|
| $89$ |
\( (T^{2} - 6 T - 23)^{2} \)
|
| $97$ |
\( (T^{2} - 4 T - 68)^{2} \)
|
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