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: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{20})^+\) |
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| Defining polynomial: |
\( x^{4} - 5x^{2} + 5 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{10}\) |
| Projective field: | Galois closure of 10.0.654634011367.1 |
$q$-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 \(\nu = \zeta_{20} + \zeta_{20}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_1 \)
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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 | −1.90211 | 0 | 0 | −1.00000 | 1.00000 | 3.07768 | ||||||||||||||||||||||||||||||
| 2843.2 | −1.61803 | 0 | 1.61803 | 1.90211 | 0 | 0 | −1.00000 | 1.00000 | −3.07768 | |||||||||||||||||||||||||||||||
| 2843.3 | 0.618034 | 0 | −0.618034 | −1.17557 | 0 | 0 | −1.00000 | 1.00000 | −0.726543 | |||||||||||||||||||||||||||||||
| 2843.4 | 0.618034 | 0 | −0.618034 | 1.17557 | 0 | 0 | −1.00000 | 1.00000 | 0.726543 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 79.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-79}) \) |
| 7.b | odd | 2 | 1 | inner |
| 553.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3871.1.c.d | ✓ | 4 |
| 7.b | odd | 2 | 1 | inner | 3871.1.c.d | ✓ | 4 |
| 7.c | even | 3 | 2 | 3871.1.m.e | 8 | ||
| 7.d | odd | 6 | 2 | 3871.1.m.e | 8 | ||
| 79.b | odd | 2 | 1 | CM | 3871.1.c.d | ✓ | 4 |
| 553.d | even | 2 | 1 | inner | 3871.1.c.d | ✓ | 4 |
| 553.l | even | 6 | 2 | 3871.1.m.e | 8 | ||
| 553.m | odd | 6 | 2 | 3871.1.m.e | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3871.1.c.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 3871.1.c.d | ✓ | 4 | 7.b | odd | 2 | 1 | inner |
| 3871.1.c.d | ✓ | 4 | 79.b | odd | 2 | 1 | CM |
| 3871.1.c.d | ✓ | 4 | 553.d | even | 2 | 1 | inner |
| 3871.1.m.e | 8 | 7.c | even | 3 | 2 | ||
| 3871.1.m.e | 8 | 7.d | odd | 6 | 2 | ||
| 3871.1.m.e | 8 | 553.l | even | 6 | 2 | ||
| 3871.1.m.e | 8 | 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 \(S_{1}^{\mathrm{new}}(3871, [\chi])\):
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\( T_{2}^{2} + T_{2} - 1 \)
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\( T_{5}^{4} - 5T_{5}^{2} + 5 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T - 1)^{2} \)
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| $3$ |
\( T^{4} \)
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| $5$ |
\( T^{4} - 5T^{2} + 5 \)
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| $7$ |
\( T^{4} \)
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| $11$ |
\( (T^{2} - T - 1)^{2} \)
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| $13$ |
\( T^{4} - 5T^{2} + 5 \)
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| $17$ |
\( T^{4} \)
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| $19$ |
\( T^{4} - 5T^{2} + 5 \)
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| $23$ |
\( (T^{2} - T - 1)^{2} \)
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| $29$ |
\( T^{4} \)
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| $31$ |
\( T^{4} - 5T^{2} + 5 \)
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| $37$ |
\( T^{4} \)
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| $41$ |
\( T^{4} \)
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| $43$ |
\( T^{4} \)
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| $47$ |
\( T^{4} \)
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| $53$ |
\( T^{4} \)
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| $59$ |
\( T^{4} \)
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| $61$ |
\( T^{4} \)
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| $67$ |
\( (T^{2} - T - 1)^{2} \)
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| $71$ |
\( T^{4} \)
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| $73$ |
\( T^{4} - 5T^{2} + 5 \)
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| $79$ |
\( (T - 1)^{4} \)
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| $83$ |
\( T^{4} \)
|
| $89$ |
\( T^{4} - 5T^{2} + 5 \)
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| $97$ |
\( T^{4} - 5T^{2} + 5 \)
|
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