Newspace parameters
| Level: | \( N \) | \(=\) | \( 324 = 2^{2} \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 324.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(19.1166188419\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{3})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 4x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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
\( x^{4} + 4x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} + 3\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( 5\nu^{3} + 24\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( 9\nu^{2} + 18 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 5\beta_1 ) / 9 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 18 ) / 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{2} + 8\beta_1 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).
| \(n\) | \(163\) | \(245\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 323.1 |
|
− | 2.82843i | 0 | −8.00000 | − | 1.83032i | 0 | 0 | 22.6274i | 0 | −5.17691 | ||||||||||||||||||||||||||||
| 323.2 | − | 2.82843i | 0 | −8.00000 | 20.2151i | 0 | 0 | 22.6274i | 0 | 57.1769 | ||||||||||||||||||||||||||||||
| 323.3 | 2.82843i | 0 | −8.00000 | − | 20.2151i | 0 | 0 | − | 22.6274i | 0 | 57.1769 | |||||||||||||||||||||||||||||
| 323.4 | 2.82843i | 0 | −8.00000 | 1.83032i | 0 | 0 | − | 22.6274i | 0 | −5.17691 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-1}) \) |
| 3.b | odd | 2 | 1 | inner |
| 12.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 324.4.b.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 324.4.b.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | CM | 324.4.b.a | ✓ | 4 |
| 12.b | even | 2 | 1 | inner | 324.4.b.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 324.4.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 324.4.b.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 324.4.b.a | ✓ | 4 | 4.b | odd | 2 | 1 | CM |
| 324.4.b.a | ✓ | 4 | 12.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 412T_{5}^{2} + 1369 \)
acting on \(S_{4}^{\mathrm{new}}(324, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 8)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 412T^{2} + 1369 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( (T^{2} + 92 T + 1873)^{2} \)
|
| $17$ |
\( T^{4} + 29428 T^{2} + 215766721 \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} \)
|
| $29$ |
\( T^{4} + \cdots + 3758793481 \)
|
| $31$ |
\( T^{4} \)
|
| $37$ |
\( (T^{2} - 214 T - 106163)^{2} \)
|
| $41$ |
\( (T^{2} + 29282)^{2} \)
|
| $43$ |
\( T^{4} \)
|
| $47$ |
\( T^{4} \)
|
| $53$ |
\( (T^{2} + 594050)^{2} \)
|
| $59$ |
\( T^{4} \)
|
| $61$ |
\( (T^{2} + 830 T + 7957)^{2} \)
|
| $67$ |
\( T^{4} \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( (T^{2} - 592 T - 816587)^{2} \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( T^{4} \)
|
| $89$ |
\( T^{4} + \cdots + 168961280401 \)
|
| $97$ |
\( (T + 1816)^{4} \)
|
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