Newspace parameters
| Level: | \( N \) | \(=\) | \( 2816 = 2^{8} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2816.h (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(1.40536707557\) |
| Analytic rank: | \(0\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 704) |
| Projective image: | \(D_{2}\) |
| Projective field: | Galois closure of \(\Q(\sqrt{-2}, \sqrt{-11})\) |
| Artin image: | $D_4$ |
| Artin field: | Galois closure of 4.0.22528.2 |
| Stark unit: | Root of $x^{4} - 8431268x^{3} + 7462086x^{2} - 8431268x + 1$ |
$q$-expansion
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2816\mathbb{Z}\right)^\times\).
| \(n\) | \(1025\) | \(1541\) | \(2047\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 769.1 |
|
0 | 2.00000 | 0 | 0 | 0 | 0 | 0 | 3.00000 | 0 | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-2}) \) |
| 11.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-11}) \) |
| 88.g | even | 2 | 1 | RM by \(\Q(\sqrt{22}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2816.1.h.b | 1 | |
| 4.b | odd | 2 | 1 | 2816.1.h.a | 1 | ||
| 8.b | even | 2 | 1 | 2816.1.h.a | 1 | ||
| 8.d | odd | 2 | 1 | CM | 2816.1.h.b | 1 | |
| 11.b | odd | 2 | 1 | CM | 2816.1.h.b | 1 | |
| 16.e | even | 4 | 2 | 704.1.b.a | ✓ | 2 | |
| 16.f | odd | 4 | 2 | 704.1.b.a | ✓ | 2 | |
| 44.c | even | 2 | 1 | 2816.1.h.a | 1 | ||
| 88.b | odd | 2 | 1 | 2816.1.h.a | 1 | ||
| 88.g | even | 2 | 1 | RM | 2816.1.h.b | 1 | |
| 176.i | even | 4 | 2 | 704.1.b.a | ✓ | 2 | |
| 176.l | odd | 4 | 2 | 704.1.b.a | ✓ | 2 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 704.1.b.a | ✓ | 2 | 16.e | even | 4 | 2 | |
| 704.1.b.a | ✓ | 2 | 16.f | odd | 4 | 2 | |
| 704.1.b.a | ✓ | 2 | 176.i | even | 4 | 2 | |
| 704.1.b.a | ✓ | 2 | 176.l | odd | 4 | 2 | |
| 2816.1.h.a | 1 | 4.b | odd | 2 | 1 | ||
| 2816.1.h.a | 1 | 8.b | even | 2 | 1 | ||
| 2816.1.h.a | 1 | 44.c | even | 2 | 1 | ||
| 2816.1.h.a | 1 | 88.b | odd | 2 | 1 | ||
| 2816.1.h.b | 1 | 1.a | even | 1 | 1 | trivial | |
| 2816.1.h.b | 1 | 8.d | odd | 2 | 1 | CM | |
| 2816.1.h.b | 1 | 11.b | odd | 2 | 1 | CM | |
| 2816.1.h.b | 1 | 88.g | even | 2 | 1 | RM | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} - 2 \)
acting on \(S_{1}^{\mathrm{new}}(2816, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T \)
|
| $3$ |
\( T - 2 \)
|
| $5$ |
\( T \)
|
| $7$ |
\( T \)
|
| $11$ |
\( T + 1 \)
|
| $13$ |
\( T \)
|
| $17$ |
\( T \)
|
| $19$ |
\( T \)
|
| $23$ |
\( T \)
|
| $29$ |
\( T \)
|
| $31$ |
\( T \)
|
| $37$ |
\( T \)
|
| $41$ |
\( T \)
|
| $43$ |
\( T \)
|
| $47$ |
\( T \)
|
| $53$ |
\( T \)
|
| $59$ |
\( T + 2 \)
|
| $61$ |
\( T \)
|
| $67$ |
\( T + 2 \)
|
| $71$ |
\( T \)
|
| $73$ |
\( T \)
|
| $79$ |
\( T \)
|
| $83$ |
\( T \)
|
| $89$ |
\( T + 2 \)
|
| $97$ |
\( T - 2 \)
|
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