Newspace parameters
| Level: | \( N \) | \(=\) | \( 1344 = 2^{6} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1344.k (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(10.7318940317\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{6})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| 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
\( x^{4} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(449\) | \(577\) | \(1093\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1217.1 |
|
0 | −1.22474 | − | 1.22474i | 0 | −2.44949 | 0 | 1.00000 | − | 2.44949i | 0 | 3.00000i | 0 | ||||||||||||||||||||||||||
| 1217.2 | 0 | −1.22474 | + | 1.22474i | 0 | −2.44949 | 0 | 1.00000 | + | 2.44949i | 0 | − | 3.00000i | 0 | ||||||||||||||||||||||||||
| 1217.3 | 0 | 1.22474 | − | 1.22474i | 0 | 2.44949 | 0 | 1.00000 | − | 2.44949i | 0 | − | 3.00000i | 0 | ||||||||||||||||||||||||||
| 1217.4 | 0 | 1.22474 | + | 1.22474i | 0 | 2.44949 | 0 | 1.00000 | + | 2.44949i | 0 | 3.00000i | 0 | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 21.c | 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 \(S_{2}^{\mathrm{new}}(1344, [\chi])\):
|
\( T_{5}^{2} - 6 \)
|
|
\( T_{43} - 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 9 \)
|
| $5$ |
\( (T^{2} - 6)^{2} \)
|
| $7$ |
\( (T^{2} - 2 T + 7)^{2} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( (T^{2} + 6)^{2} \)
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| $17$ |
\( (T^{2} - 24)^{2} \)
|
| $19$ |
\( (T^{2} + 6)^{2} \)
|
| $23$ |
\( (T^{2} + 36)^{2} \)
|
| $29$ |
\( (T^{2} + 36)^{2} \)
|
| $31$ |
\( T^{4} \)
|
| $37$ |
\( (T - 2)^{4} \)
|
| $41$ |
\( (T^{2} - 24)^{2} \)
|
| $43$ |
\( (T - 4)^{4} \)
|
| $47$ |
\( (T^{2} - 24)^{2} \)
|
| $53$ |
\( (T^{2} + 36)^{2} \)
|
| $59$ |
\( (T^{2} - 150)^{2} \)
|
| $61$ |
\( (T^{2} + 150)^{2} \)
|
| $67$ |
\( (T - 8)^{4} \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( (T^{2} + 96)^{2} \)
|
| $79$ |
\( (T - 10)^{4} \)
|
| $83$ |
\( (T^{2} - 6)^{2} \)
|
| $89$ |
\( T^{4} \)
|
| $97$ |
\( (T^{2} + 24)^{2} \)
|
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