Newspace parameters
| Level: | \( N \) | \(=\) | \( 448 = 2^{6} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 448.g (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(12.2071158433\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{7})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 3x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{4} - 3x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - \nu ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 5\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} - 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 3 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{2} + 5\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(129\) | \(197\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 351.1 |
|
0 | −1.64575 | 0 | − | 7.64575i | 0 | 2.64575i | 0 | −6.29150 | 0 | |||||||||||||||||||||||||||||
| 351.2 | 0 | −1.64575 | 0 | 7.64575i | 0 | − | 2.64575i | 0 | −6.29150 | 0 | ||||||||||||||||||||||||||||||
| 351.3 | 0 | 3.64575 | 0 | − | 2.35425i | 0 | − | 2.64575i | 0 | 4.29150 | 0 | |||||||||||||||||||||||||||||
| 351.4 | 0 | 3.64575 | 0 | 2.35425i | 0 | 2.64575i | 0 | 4.29150 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 448.3.g.b | yes | 4 |
| 4.b | odd | 2 | 1 | 448.3.g.a | ✓ | 4 | |
| 8.b | even | 2 | 1 | 448.3.g.a | ✓ | 4 | |
| 8.d | odd | 2 | 1 | inner | 448.3.g.b | yes | 4 |
| 16.e | even | 4 | 1 | 1792.3.d.a | 4 | ||
| 16.e | even | 4 | 1 | 1792.3.d.c | 4 | ||
| 16.f | odd | 4 | 1 | 1792.3.d.a | 4 | ||
| 16.f | odd | 4 | 1 | 1792.3.d.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 448.3.g.a | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 448.3.g.a | ✓ | 4 | 8.b | even | 2 | 1 | |
| 448.3.g.b | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 448.3.g.b | yes | 4 | 8.d | odd | 2 | 1 | inner |
| 1792.3.d.a | 4 | 16.e | even | 4 | 1 | ||
| 1792.3.d.a | 4 | 16.f | odd | 4 | 1 | ||
| 1792.3.d.c | 4 | 16.e | even | 4 | 1 | ||
| 1792.3.d.c | 4 | 16.f | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 2T_{3} - 6 \)
acting on \(S_{3}^{\mathrm{new}}(448, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} - 2 T - 6)^{2} \)
|
| $5$ |
\( T^{4} + 64T^{2} + 324 \)
|
| $7$ |
\( (T^{2} + 7)^{2} \)
|
| $11$ |
\( (T^{2} - 4 T - 24)^{2} \)
|
| $13$ |
\( T^{4} + 112T^{2} + 1764 \)
|
| $17$ |
\( (T^{2} + 4 T - 108)^{2} \)
|
| $19$ |
\( (T^{2} - 26 T + 162)^{2} \)
|
| $23$ |
\( T^{4} + 904 T^{2} + 197136 \)
|
| $29$ |
\( T^{4} + 2752T^{2} + 576 \)
|
| $31$ |
\( T^{4} + 928 T^{2} + 186624 \)
|
| $37$ |
\( T^{4} + 1408 T^{2} + 484416 \)
|
| $41$ |
\( (T^{2} - 4 T - 2796)^{2} \)
|
| $43$ |
\( (T^{2} + 140 T + 4872)^{2} \)
|
| $47$ |
\( T^{4} + 5632 T^{2} + 589824 \)
|
| $53$ |
\( T^{4} + 9472 T^{2} + 746496 \)
|
| $59$ |
\( (T^{2} + 138 T + 4194)^{2} \)
|
| $61$ |
\( T^{4} + 10656 T^{2} + 23794884 \)
|
| $67$ |
\( (T^{2} + 16 T - 11136)^{2} \)
|
| $71$ |
\( T^{4} + 4608 T^{2} + 82944 \)
|
| $73$ |
\( (T^{2} + 52 T - 332)^{2} \)
|
| $79$ |
\( T^{4} + 5632 T^{2} + 6718464 \)
|
| $83$ |
\( (T^{2} - 86 T - 4038)^{2} \)
|
| $89$ |
\( (T^{2} - 188 T + 7044)^{2} \)
|
| $97$ |
\( (T^{2} - 76 T - 14684)^{2} \)
|
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