Newspace parameters
| Level: | \( N \) | \(=\) | \( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2178.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(59.3462015777\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{3})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 4x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{4} + 4x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 4\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 4\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2178\mathbb{Z}\right)^\times\).
| \(n\) | \(1333\) | \(1937\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 485.1 |
|
− | 1.41421i | 0 | −2.00000 | − | 8.24504i | 0 | −2.53590 | 2.82843i | 0 | −11.6603 | ||||||||||||||||||||||||||||
| 485.2 | − | 1.41421i | 0 | −2.00000 | 4.00240i | 0 | −9.46410 | 2.82843i | 0 | 5.66025 | ||||||||||||||||||||||||||||||
| 485.3 | 1.41421i | 0 | −2.00000 | − | 4.00240i | 0 | −9.46410 | − | 2.82843i | 0 | 5.66025 | |||||||||||||||||||||||||||||
| 485.4 | 1.41421i | 0 | −2.00000 | 8.24504i | 0 | −2.53590 | − | 2.82843i | 0 | −11.6603 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2178.3.c.f | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 2178.3.c.f | ✓ | 4 |
| 11.b | odd | 2 | 1 | 2178.3.c.j | yes | 4 | |
| 33.d | even | 2 | 1 | 2178.3.c.j | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2178.3.c.f | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 2178.3.c.f | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 2178.3.c.j | yes | 4 | 11.b | odd | 2 | 1 | |
| 2178.3.c.j | yes | 4 | 33.d | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2178, [\chi])\):
|
\( T_{5}^{4} + 84T_{5}^{2} + 1089 \)
|
|
\( T_{7}^{2} + 12T_{7} + 24 \)
|
|
\( T_{13} - 3 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 84T^{2} + 1089 \)
|
| $7$ |
\( (T^{2} + 12 T + 24)^{2} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( (T - 3)^{4} \)
|
| $17$ |
\( T^{4} + 756 T^{2} + 88209 \)
|
| $19$ |
\( (T^{2} - 24 T - 48)^{2} \)
|
| $23$ |
\( T^{4} + 2352 T^{2} + 1272384 \)
|
| $29$ |
\( T^{4} + 468 T^{2} + 42849 \)
|
| $31$ |
\( (T + 44)^{4} \)
|
| $37$ |
\( (T^{2} + 22 T - 1607)^{2} \)
|
| $41$ |
\( T^{4} + 3924 T^{2} + 1656369 \)
|
| $43$ |
\( (T^{2} + 36 T + 216)^{2} \)
|
| $47$ |
\( T^{4} + 7488 T^{2} + 10969344 \)
|
| $53$ |
\( T^{4} + 9972 T^{2} + 13417569 \)
|
| $59$ |
\( T^{4} + 13584 T^{2} + 30206016 \)
|
| $61$ |
\( (T^{2} + 96 T - 2496)^{2} \)
|
| $67$ |
\( (T^{2} - 20 T - 872)^{2} \)
|
| $71$ |
\( (T^{2} + 8664)^{2} \)
|
| $73$ |
\( (T^{2} - 156 T + 5892)^{2} \)
|
| $79$ |
\( (T^{2} - 120 T + 528)^{2} \)
|
| $83$ |
\( T^{4} + 2304 T^{2} + 331776 \)
|
| $89$ |
\( T^{4} + 13908 T^{2} + 28804689 \)
|
| $97$ |
\( (T^{2} - 64 T - 8723)^{2} \)
|
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