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: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{5}) \) |
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| Defining polynomial: |
\( x^{2} - x - 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 79) |
| Projective image: | \(D_{5}\) |
| Projective field: | Galois closure of 5.1.6241.1 |
| Artin image: | $D_{10}$ |
| Artin field: | Galois closure of 10.0.654634011367.1 |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
−1.61803 | 0 | 1.61803 | −0.618034 | 0 | 0 | −1.00000 | 1.00000 | 1.00000 | ||||||||||||||||||||||||
| 2843.2 | 0.618034 | 0 | −0.618034 | 1.61803 | 0 | 0 | −1.00000 | 1.00000 | 1.00000 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 79.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-79}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3871.1.c.c | 2 | |
| 7.b | odd | 2 | 1 | 79.1.b.a | ✓ | 2 | |
| 7.c | even | 3 | 2 | 3871.1.m.b | 4 | ||
| 7.d | odd | 6 | 2 | 3871.1.m.c | 4 | ||
| 21.c | even | 2 | 1 | 711.1.d.b | 2 | ||
| 28.d | even | 2 | 1 | 1264.1.e.a | 2 | ||
| 35.c | odd | 2 | 1 | 1975.1.d.c | 2 | ||
| 35.f | even | 4 | 2 | 1975.1.c.a | 4 | ||
| 79.b | odd | 2 | 1 | CM | 3871.1.c.c | 2 | |
| 553.d | even | 2 | 1 | 79.1.b.a | ✓ | 2 | |
| 553.l | even | 6 | 2 | 3871.1.m.c | 4 | ||
| 553.m | odd | 6 | 2 | 3871.1.m.b | 4 | ||
| 1659.d | odd | 2 | 1 | 711.1.d.b | 2 | ||
| 2212.b | odd | 2 | 1 | 1264.1.e.a | 2 | ||
| 2765.c | even | 2 | 1 | 1975.1.d.c | 2 | ||
| 2765.p | odd | 4 | 2 | 1975.1.c.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 79.1.b.a | ✓ | 2 | 7.b | odd | 2 | 1 | |
| 79.1.b.a | ✓ | 2 | 553.d | even | 2 | 1 | |
| 711.1.d.b | 2 | 21.c | even | 2 | 1 | ||
| 711.1.d.b | 2 | 1659.d | odd | 2 | 1 | ||
| 1264.1.e.a | 2 | 28.d | even | 2 | 1 | ||
| 1264.1.e.a | 2 | 2212.b | odd | 2 | 1 | ||
| 1975.1.c.a | 4 | 35.f | even | 4 | 2 | ||
| 1975.1.c.a | 4 | 2765.p | odd | 4 | 2 | ||
| 1975.1.d.c | 2 | 35.c | odd | 2 | 1 | ||
| 1975.1.d.c | 2 | 2765.c | even | 2 | 1 | ||
| 3871.1.c.c | 2 | 1.a | even | 1 | 1 | trivial | |
| 3871.1.c.c | 2 | 79.b | odd | 2 | 1 | CM | |
| 3871.1.m.b | 4 | 7.c | even | 3 | 2 | ||
| 3871.1.m.b | 4 | 553.m | odd | 6 | 2 | ||
| 3871.1.m.c | 4 | 7.d | odd | 6 | 2 | ||
| 3871.1.m.c | 4 | 553.l | even | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3871, [\chi])\):
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\( T_{2}^{2} + T_{2} - 1 \)
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\( T_{5}^{2} - T_{5} - 1 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + T - 1 \)
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| $3$ |
\( T^{2} \)
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| $5$ |
\( T^{2} - T - 1 \)
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| $7$ |
\( T^{2} \)
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| $11$ |
\( T^{2} + T - 1 \)
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| $13$ |
\( T^{2} - T - 1 \)
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| $17$ |
\( T^{2} \)
|
| $19$ |
\( T^{2} - T - 1 \)
|
| $23$ |
\( T^{2} + T - 1 \)
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| $29$ |
\( T^{2} \)
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| $31$ |
\( T^{2} - T - 1 \)
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| $37$ |
\( T^{2} \)
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| $41$ |
\( T^{2} \)
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| $43$ |
\( T^{2} \)
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| $47$ |
\( T^{2} \)
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| $53$ |
\( T^{2} \)
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| $59$ |
\( T^{2} \)
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| $61$ |
\( T^{2} \)
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| $67$ |
\( T^{2} + T - 1 \)
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| $71$ |
\( T^{2} \)
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| $73$ |
\( T^{2} - T - 1 \)
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| $79$ |
\( (T - 1)^{2} \)
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| $83$ |
\( (T + 2)^{2} \)
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| $89$ |
\( T^{2} - T - 1 \)
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| $97$ |
\( T^{2} - T - 1 \)
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