Newspace parameters
| Level: | \( N \) | \(=\) | \( 3328 = 2^{8} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3328.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(26.5742137927\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.13448.1 |
|
|
|
| Defining polynomial: |
\( x^{4} - 7x^{2} + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 1664) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{4} - 7x^{2} + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 6\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 6\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −3.11473 | 0 | 3.70156 | 0 | −0.546295 | 0 | 6.70156 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.81616 | 0 | −2.70156 | 0 | 2.58874 | 0 | 0.298438 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.81616 | 0 | −2.70156 | 0 | −2.58874 | 0 | 0.298438 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 3.11473 | 0 | 3.70156 | 0 | 0.546295 | 0 | 6.70156 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(13\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3328.2.a.bl | 4 | |
| 4.b | odd | 2 | 1 | inner | 3328.2.a.bl | 4 | |
| 8.b | even | 2 | 1 | 3328.2.a.bk | 4 | ||
| 8.d | odd | 2 | 1 | 3328.2.a.bk | 4 | ||
| 16.e | even | 4 | 2 | 1664.2.b.j | ✓ | 8 | |
| 16.f | odd | 4 | 2 | 1664.2.b.j | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1664.2.b.j | ✓ | 8 | 16.e | even | 4 | 2 | |
| 1664.2.b.j | ✓ | 8 | 16.f | odd | 4 | 2 | |
| 3328.2.a.bk | 4 | 8.b | even | 2 | 1 | ||
| 3328.2.a.bk | 4 | 8.d | odd | 2 | 1 | ||
| 3328.2.a.bl | 4 | 1.a | even | 1 | 1 | trivial | |
| 3328.2.a.bl | 4 | 4.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3328))\):
|
\( T_{3}^{4} - 13T_{3}^{2} + 32 \)
|
|
\( T_{5}^{2} - T_{5} - 10 \)
|
|
\( T_{7}^{4} - 7T_{7}^{2} + 2 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 13T^{2} + 32 \)
|
| $5$ |
\( (T^{2} - T - 10)^{2} \)
|
| $7$ |
\( T^{4} - 7T^{2} + 2 \)
|
| $11$ |
\( T^{4} - 14T^{2} + 8 \)
|
| $13$ |
\( (T - 1)^{4} \)
|
| $17$ |
\( (T^{2} - 5 T - 4)^{2} \)
|
| $19$ |
\( T^{4} - 26T^{2} + 128 \)
|
| $23$ |
\( T^{4} - 52T^{2} + 512 \)
|
| $29$ |
\( (T - 2)^{4} \)
|
| $31$ |
\( T^{4} - 58T^{2} + 800 \)
|
| $37$ |
\( (T^{2} + 7 T + 2)^{2} \)
|
| $41$ |
\( (T^{2} - 10 T - 16)^{2} \)
|
| $43$ |
\( T^{4} - 261 T^{2} + 16200 \)
|
| $47$ |
\( T^{4} - 167T^{2} + 6962 \)
|
| $53$ |
\( (T^{2} + 10 T - 16)^{2} \)
|
| $59$ |
\( T^{4} - 106T^{2} + 800 \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( T^{4} - 334 T^{2} + 27848 \)
|
| $71$ |
\( T^{4} - 47T^{2} + 50 \)
|
| $73$ |
\( (T^{2} - 2 T - 40)^{2} \)
|
| $79$ |
\( T^{4} - 188T^{2} + 800 \)
|
| $83$ |
\( T^{4} - 94T^{2} + 200 \)
|
| $89$ |
\( (T^{2} - 10 T - 16)^{2} \)
|
| $97$ |
\( (T - 10)^{4} \)
|
| show more | |
| show less |