Newspace parameters
| Level: | \( N \) | \(=\) | \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1050.d (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(8.38429221223\) |
| 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: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 42) |
| 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
\( x^{4} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(451\) | \(701\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1049.1 |
|
1.00000 | −1.22474 | − | 1.22474i | 1.00000 | 0 | −1.22474 | − | 1.22474i | −2.44949 | + | 1.00000i | 1.00000 | 3.00000i | 0 | ||||||||||||||||||||||||
| 1049.2 | 1.00000 | −1.22474 | + | 1.22474i | 1.00000 | 0 | −1.22474 | + | 1.22474i | −2.44949 | − | 1.00000i | 1.00000 | − | 3.00000i | 0 | ||||||||||||||||||||||||
| 1049.3 | 1.00000 | 1.22474 | − | 1.22474i | 1.00000 | 0 | 1.22474 | − | 1.22474i | 2.44949 | − | 1.00000i | 1.00000 | − | 3.00000i | 0 | ||||||||||||||||||||||||
| 1049.4 | 1.00000 | 1.22474 | + | 1.22474i | 1.00000 | 0 | 1.22474 | + | 1.22474i | 2.44949 | + | 1.00000i | 1.00000 | 3.00000i | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
| 15.d | odd | 2 | 1 | inner |
| 105.g | even | 2 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1050, [\chi])\):
|
\( T_{11} \)
|
|
\( T_{13}^{2} - 6 \)
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\( T_{23} + 6 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{4} \)
|
| $3$ |
\( T^{4} + 9 \)
|
| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} - 10T^{2} + 49 \)
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| $11$ |
\( T^{4} \)
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| $13$ |
\( (T^{2} - 6)^{2} \)
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| $17$ |
\( (T^{2} + 24)^{2} \)
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| $19$ |
\( (T^{2} + 6)^{2} \)
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| $23$ |
\( (T + 6)^{4} \)
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| $29$ |
\( (T^{2} + 36)^{2} \)
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| $31$ |
\( T^{4} \)
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| $37$ |
\( (T^{2} + 4)^{2} \)
|
| $41$ |
\( (T^{2} - 24)^{2} \)
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| $43$ |
\( (T^{2} + 16)^{2} \)
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| $47$ |
\( (T^{2} + 24)^{2} \)
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| $53$ |
\( (T + 6)^{4} \)
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| $59$ |
\( (T^{2} - 150)^{2} \)
|
| $61$ |
\( (T^{2} + 150)^{2} \)
|
| $67$ |
\( (T^{2} + 64)^{2} \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( (T^{2} - 96)^{2} \)
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| $79$ |
\( (T - 10)^{4} \)
|
| $83$ |
\( (T^{2} + 6)^{2} \)
|
| $89$ |
\( T^{4} \)
|
| $97$ |
\( (T^{2} - 24)^{2} \)
|
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