Newspace parameters
| Level: | \( N \) | \(=\) | \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 504.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(13.7330053238\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{7})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 8x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{4} + 8x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 5\nu ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 11\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -5\beta_{2} + 11\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).
| \(n\) | \(73\) | \(127\) | \(253\) | \(281\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 449.1 |
|
0 | 0 | 0 | − | 7.98430i | 0 | 2.64575 | 0 | 0 | 0 | |||||||||||||||||||||||||||||
| 449.2 | 0 | 0 | 0 | − | 0.500983i | 0 | −2.64575 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 449.3 | 0 | 0 | 0 | 0.500983i | 0 | −2.64575 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 449.4 | 0 | 0 | 0 | 7.98430i | 0 | 2.64575 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
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 | 504.3.d.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 504.3.d.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | 1008.3.d.b | 4 | ||
| 7.b | odd | 2 | 1 | 3528.3.d.d | 4 | ||
| 8.b | even | 2 | 1 | 4032.3.d.f | 4 | ||
| 8.d | odd | 2 | 1 | 4032.3.d.g | 4 | ||
| 12.b | even | 2 | 1 | 1008.3.d.b | 4 | ||
| 21.c | even | 2 | 1 | 3528.3.d.d | 4 | ||
| 24.f | even | 2 | 1 | 4032.3.d.g | 4 | ||
| 24.h | odd | 2 | 1 | 4032.3.d.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 504.3.d.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 504.3.d.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 1008.3.d.b | 4 | 4.b | odd | 2 | 1 | ||
| 1008.3.d.b | 4 | 12.b | even | 2 | 1 | ||
| 3528.3.d.d | 4 | 7.b | odd | 2 | 1 | ||
| 3528.3.d.d | 4 | 21.c | even | 2 | 1 | ||
| 4032.3.d.f | 4 | 8.b | even | 2 | 1 | ||
| 4032.3.d.f | 4 | 24.h | odd | 2 | 1 | ||
| 4032.3.d.g | 4 | 8.d | odd | 2 | 1 | ||
| 4032.3.d.g | 4 | 24.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 64T_{5}^{2} + 16 \)
acting on \(S_{3}^{\mathrm{new}}(504, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 64T^{2} + 16 \)
|
| $7$ |
\( (T^{2} - 7)^{2} \)
|
| $11$ |
\( T^{4} + 116T^{2} + 2916 \)
|
| $13$ |
\( (T^{2} + 4 T - 108)^{2} \)
|
| $17$ |
\( T^{4} + 32T^{2} + 144 \)
|
| $19$ |
\( (T^{2} - 12 T + 8)^{2} \)
|
| $23$ |
\( T^{4} + 788 T^{2} + 79524 \)
|
| $29$ |
\( T^{4} + 116T^{2} + 2916 \)
|
| $31$ |
\( (T^{2} - 36 T + 296)^{2} \)
|
| $37$ |
\( (T^{2} + 32 T - 2544)^{2} \)
|
| $41$ |
\( T^{4} + 3712 T^{2} + 169744 \)
|
| $43$ |
\( (T^{2} + 56 T - 224)^{2} \)
|
| $47$ |
\( T^{4} + 9136 T^{2} + 20286016 \)
|
| $53$ |
\( T^{4} + 5156 T^{2} + 617796 \)
|
| $59$ |
\( T^{4} + 9648 T^{2} + 18045504 \)
|
| $61$ |
\( (T^{2} - 40 T - 300)^{2} \)
|
| $67$ |
\( (T^{2} - 68 T - 1644)^{2} \)
|
| $71$ |
\( T^{4} + 2900 T^{2} + 1822500 \)
|
| $73$ |
\( (T^{2} - 200 T + 9748)^{2} \)
|
| $79$ |
\( (T + 14)^{4} \)
|
| $83$ |
\( T^{4} + 23552 T^{2} + 51151104 \)
|
| $89$ |
\( T^{4} + 10592 T^{2} + 7817616 \)
|
| $97$ |
\( (T^{2} + 80 T + 900)^{2} \)
|
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