Newspace parameters
| Level: | \( N \) | \(=\) | \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 840.z (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(6.70743376979\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{6})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 3\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 3\nu ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{3} + 3\beta_{2} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/840\mathbb{Z}\right)^\times\).
| \(n\) | \(241\) | \(281\) | \(337\) | \(421\) | \(631\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 811.1 |
|
−1.00000 | − | 1.00000i | − | 1.00000i | 2.00000i | 1.00000 | −1.00000 | + | 1.00000i | −1.00000 | − | 2.44949i | 2.00000 | − | 2.00000i | −1.00000 | −1.00000 | − | 1.00000i | |||||||||||||||||||
| 811.2 | −1.00000 | − | 1.00000i | − | 1.00000i | 2.00000i | 1.00000 | −1.00000 | + | 1.00000i | −1.00000 | + | 2.44949i | 2.00000 | − | 2.00000i | −1.00000 | −1.00000 | − | 1.00000i | ||||||||||||||||||||
| 811.3 | −1.00000 | + | 1.00000i | 1.00000i | − | 2.00000i | 1.00000 | −1.00000 | − | 1.00000i | −1.00000 | − | 2.44949i | 2.00000 | + | 2.00000i | −1.00000 | −1.00000 | + | 1.00000i | ||||||||||||||||||||
| 811.4 | −1.00000 | + | 1.00000i | 1.00000i | − | 2.00000i | 1.00000 | −1.00000 | − | 1.00000i | −1.00000 | + | 2.44949i | 2.00000 | + | 2.00000i | −1.00000 | −1.00000 | + | 1.00000i | ||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 56.e | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 840.2.z.b | yes | 4 |
| 4.b | odd | 2 | 1 | 3360.2.z.b | 4 | ||
| 7.b | odd | 2 | 1 | 840.2.z.a | ✓ | 4 | |
| 8.b | even | 2 | 1 | 3360.2.z.a | 4 | ||
| 8.d | odd | 2 | 1 | 840.2.z.a | ✓ | 4 | |
| 28.d | even | 2 | 1 | 3360.2.z.a | 4 | ||
| 56.e | even | 2 | 1 | inner | 840.2.z.b | yes | 4 |
| 56.h | odd | 2 | 1 | 3360.2.z.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 840.2.z.a | ✓ | 4 | 7.b | odd | 2 | 1 | |
| 840.2.z.a | ✓ | 4 | 8.d | odd | 2 | 1 | |
| 840.2.z.b | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 840.2.z.b | yes | 4 | 56.e | even | 2 | 1 | inner |
| 3360.2.z.a | 4 | 8.b | even | 2 | 1 | ||
| 3360.2.z.a | 4 | 28.d | even | 2 | 1 | ||
| 3360.2.z.b | 4 | 4.b | odd | 2 | 1 | ||
| 3360.2.z.b | 4 | 56.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\):
|
\( T_{11} - 2 \)
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\( T_{13}^{2} - 4T_{13} - 20 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T + 2)^{2} \)
|
| $3$ |
\( (T^{2} + 1)^{2} \)
|
| $5$ |
\( (T - 1)^{4} \)
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| $7$ |
\( (T^{2} + 2 T + 7)^{2} \)
|
| $11$ |
\( (T - 2)^{4} \)
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| $13$ |
\( (T^{2} - 4 T - 20)^{2} \)
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| $17$ |
\( (T^{2} + 24)^{2} \)
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| $19$ |
\( T^{4} + 56T^{2} + 400 \)
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| $23$ |
\( (T^{2} + 36)^{2} \)
|
| $29$ |
\( T^{4} + 80T^{2} + 64 \)
|
| $31$ |
\( (T - 2)^{4} \)
|
| $37$ |
\( T^{4} + 200T^{2} + 8464 \)
|
| $41$ |
\( T^{4} + 56T^{2} + 400 \)
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| $43$ |
\( (T^{2} + 8 T - 8)^{2} \)
|
| $47$ |
\( (T^{2} - 12 T + 12)^{2} \)
|
| $53$ |
\( (T^{2} + 96)^{2} \)
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| $59$ |
\( T^{4} + 176T^{2} + 1600 \)
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| $61$ |
\( (T^{2} - 8 T - 8)^{2} \)
|
| $67$ |
\( (T^{2} - 24)^{2} \)
|
| $71$ |
\( T^{4} + 176T^{2} + 1600 \)
|
| $73$ |
\( T^{4} + 80T^{2} + 64 \)
|
| $79$ |
\( (T^{2} + 16)^{2} \)
|
| $83$ |
\( (T^{2} + 16)^{2} \)
|
| $89$ |
\( T^{4} + 120T^{2} + 144 \)
|
| $97$ |
\( T^{4} + 80T^{2} + 64 \)
|
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