Newspace parameters
| Level: | \( N \) | \(=\) | \( 371 = 7 \cdot 53 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 371.g (of order \(4\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(2.96244991499\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{7})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 3x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{4}]$ |
$q$-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 a root \(\nu\) of
\( x^{4} - 3x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} + \nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{2} + \nu + 1 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + \beta _1 + 2 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{3} + 4\beta_{2} + \beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/371\mathbb{Z}\right)^\times\).
| \(n\) | \(213\) | \(267\) |
| \(\chi(n)\) | \(-1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 76.1 |
|
−0.822876 | + | 0.822876i | 0 | 0.645751i | 0 | 0 | − | 2.64575i | −2.17712 | − | 2.17712i | − | 3.00000i | 0 | ||||||||||||||||||||||||
| 76.2 | 1.82288 | − | 1.82288i | 0 | − | 4.64575i | 0 | 0 | 2.64575i | −4.82288 | − | 4.82288i | − | 3.00000i | 0 | |||||||||||||||||||||||||
| 83.1 | −0.822876 | − | 0.822876i | 0 | − | 0.645751i | 0 | 0 | 2.64575i | −2.17712 | + | 2.17712i | 3.00000i | 0 | ||||||||||||||||||||||||||
| 83.2 | 1.82288 | + | 1.82288i | 0 | 4.64575i | 0 | 0 | − | 2.64575i | −4.82288 | + | 4.82288i | 3.00000i | 0 | ||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-7}) \) |
| 53.c | odd | 4 | 1 | inner |
| 371.g | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 371.2.g.d | ✓ | 4 |
| 7.b | odd | 2 | 1 | CM | 371.2.g.d | ✓ | 4 |
| 53.c | odd | 4 | 1 | inner | 371.2.g.d | ✓ | 4 |
| 371.g | even | 4 | 1 | inner | 371.2.g.d | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 371.2.g.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 371.2.g.d | ✓ | 4 | 7.b | odd | 2 | 1 | CM |
| 371.2.g.d | ✓ | 4 | 53.c | odd | 4 | 1 | inner |
| 371.2.g.d | ✓ | 4 | 371.g | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(371, [\chi])\):
|
\( T_{2}^{4} - 2T_{2}^{3} + 2T_{2}^{2} + 6T_{2} + 9 \)
|
|
\( T_{5} \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 2 T^{3} + \cdots + 9 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} + 7)^{2} \)
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| $11$ |
\( (T^{2} + 16)^{2} \)
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| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} - 16 T^{3} + \cdots + 324 \)
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| $29$ |
\( (T^{2} + 112)^{2} \)
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| $31$ |
\( T^{4} \)
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| $37$ |
\( (T^{2} + 36)^{2} \)
|
| $41$ |
\( T^{4} \)
|
| $43$ |
\( (T^{2} + 144)^{2} \)
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| $47$ |
\( T^{4} \)
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| $53$ |
\( (T^{2} - 10 T + 53)^{2} \)
|
| $59$ |
\( T^{4} \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( T^{4} - 8 T^{3} + \cdots + 13924 \)
|
| $71$ |
\( T^{4} + 32 T^{3} + \cdots + 12996 \)
|
| $73$ |
\( T^{4} \)
|
| $79$ |
\( T^{4} + 16 T^{3} + \cdots + 8836 \)
|
| $83$ |
\( T^{4} \)
|
| $89$ |
\( T^{4} \)
|
| $97$ |
\( T^{4} \)
|
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