Newspace parameters
| Level: | \( N \) | \(=\) | \( 3871 = 7^{2} \cdot 79 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3871.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(1.93188066390\) |
| Analytic rank: | \(0\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{2}\) |
| Projective field: | Galois closure of \(\Q(\sqrt{-7}, \sqrt{-79})\) |
| Artin image: | $D_4$ |
| Artin field: | Galois closure of 4.0.27097.1 |
| Stark unit: | Root of $x^{4} - 121199354588092x^{3} + 229640528767110x^{2} - 121199354588092x + 1$ |
$q$-expansion
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3871\mathbb{Z}\right)^\times\).
| \(n\) | \(1030\) | \(2845\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2843.1 |
|
2.00000 | 0 | 3.00000 | 0 | 0 | 0 | 4.00000 | 1.00000 | 0 | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-7}) \) |
| 79.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-79}) \) |
| 553.d | even | 2 | 1 | RM by \(\Q(\sqrt{553}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3871.1.c.a | ✓ | 1 |
| 7.b | odd | 2 | 1 | CM | 3871.1.c.a | ✓ | 1 |
| 7.c | even | 3 | 2 | 3871.1.m.a | 2 | ||
| 7.d | odd | 6 | 2 | 3871.1.m.a | 2 | ||
| 79.b | odd | 2 | 1 | CM | 3871.1.c.a | ✓ | 1 |
| 553.d | even | 2 | 1 | RM | 3871.1.c.a | ✓ | 1 |
| 553.l | even | 6 | 2 | 3871.1.m.a | 2 | ||
| 553.m | odd | 6 | 2 | 3871.1.m.a | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3871.1.c.a | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
| 3871.1.c.a | ✓ | 1 | 7.b | odd | 2 | 1 | CM |
| 3871.1.c.a | ✓ | 1 | 79.b | odd | 2 | 1 | CM |
| 3871.1.c.a | ✓ | 1 | 553.d | even | 2 | 1 | RM |
| 3871.1.m.a | 2 | 7.c | even | 3 | 2 | ||
| 3871.1.m.a | 2 | 7.d | odd | 6 | 2 | ||
| 3871.1.m.a | 2 | 553.l | even | 6 | 2 | ||
| 3871.1.m.a | 2 | 553.m | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3871, [\chi])\):
|
\( T_{2} - 2 \)
|
|
\( T_{5} \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T - 2 \)
|
| $3$ |
\( T \)
|
| $5$ |
\( T \)
|
| $7$ |
\( T \)
|
| $11$ |
\( T + 2 \)
|
| $13$ |
\( T \)
|
| $17$ |
\( T \)
|
| $19$ |
\( T \)
|
| $23$ |
\( T + 2 \)
|
| $29$ |
\( T \)
|
| $31$ |
\( T \)
|
| $37$ |
\( T \)
|
| $41$ |
\( T \)
|
| $43$ |
\( T \)
|
| $47$ |
\( T \)
|
| $53$ |
\( T \)
|
| $59$ |
\( T \)
|
| $61$ |
\( T \)
|
| $67$ |
\( T + 2 \)
|
| $71$ |
\( T \)
|
| $73$ |
\( T \)
|
| $79$ |
\( T - 1 \)
|
| $83$ |
\( T \)
|
| $89$ |
\( T \)
|
| $97$ |
\( T \)
|
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