Newspace parameters
| Level: | \( N \) | \(=\) | \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1008.d (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(27.4660106475\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{7})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 8x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 126) |
| 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
\( x^{4} + 8x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{3} - 16\nu ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{3} - 5\nu \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{3} - 3\beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -16\beta_{3} + 15\beta_1 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(577\) | \(757\) | \(785\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1\) |
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 | − | 8.89753i | 0 | −2.64575 | 0 | 0 | 0 | |||||||||||||||||||||||||||||
| 449.2 | 0 | 0 | 0 | − | 6.06910i | 0 | 2.64575 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 449.3 | 0 | 0 | 0 | 6.06910i | 0 | 2.64575 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 449.4 | 0 | 0 | 0 | 8.89753i | 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 | 1008.3.d.a | 4 | |
| 3.b | odd | 2 | 1 | inner | 1008.3.d.a | 4 | |
| 4.b | odd | 2 | 1 | 126.3.b.a | ✓ | 4 | |
| 8.b | even | 2 | 1 | 4032.3.d.j | 4 | ||
| 8.d | odd | 2 | 1 | 4032.3.d.i | 4 | ||
| 12.b | even | 2 | 1 | 126.3.b.a | ✓ | 4 | |
| 20.d | odd | 2 | 1 | 3150.3.e.e | 4 | ||
| 20.e | even | 4 | 2 | 3150.3.c.b | 8 | ||
| 24.f | even | 2 | 1 | 4032.3.d.i | 4 | ||
| 24.h | odd | 2 | 1 | 4032.3.d.j | 4 | ||
| 28.d | even | 2 | 1 | 882.3.b.f | 4 | ||
| 28.f | even | 6 | 2 | 882.3.s.i | 8 | ||
| 28.g | odd | 6 | 2 | 882.3.s.e | 8 | ||
| 36.f | odd | 6 | 2 | 1134.3.q.c | 8 | ||
| 36.h | even | 6 | 2 | 1134.3.q.c | 8 | ||
| 60.h | even | 2 | 1 | 3150.3.e.e | 4 | ||
| 60.l | odd | 4 | 2 | 3150.3.c.b | 8 | ||
| 84.h | odd | 2 | 1 | 882.3.b.f | 4 | ||
| 84.j | odd | 6 | 2 | 882.3.s.i | 8 | ||
| 84.n | even | 6 | 2 | 882.3.s.e | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 126.3.b.a | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 126.3.b.a | ✓ | 4 | 12.b | even | 2 | 1 | |
| 882.3.b.f | 4 | 28.d | even | 2 | 1 | ||
| 882.3.b.f | 4 | 84.h | odd | 2 | 1 | ||
| 882.3.s.e | 8 | 28.g | odd | 6 | 2 | ||
| 882.3.s.e | 8 | 84.n | even | 6 | 2 | ||
| 882.3.s.i | 8 | 28.f | even | 6 | 2 | ||
| 882.3.s.i | 8 | 84.j | odd | 6 | 2 | ||
| 1008.3.d.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 1008.3.d.a | 4 | 3.b | odd | 2 | 1 | inner | |
| 1134.3.q.c | 8 | 36.f | odd | 6 | 2 | ||
| 1134.3.q.c | 8 | 36.h | even | 6 | 2 | ||
| 3150.3.c.b | 8 | 20.e | even | 4 | 2 | ||
| 3150.3.c.b | 8 | 60.l | odd | 4 | 2 | ||
| 3150.3.e.e | 4 | 20.d | odd | 2 | 1 | ||
| 3150.3.e.e | 4 | 60.h | even | 2 | 1 | ||
| 4032.3.d.i | 4 | 8.d | odd | 2 | 1 | ||
| 4032.3.d.i | 4 | 24.f | even | 2 | 1 | ||
| 4032.3.d.j | 4 | 8.b | even | 2 | 1 | ||
| 4032.3.d.j | 4 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1008, [\chi])\):
|
\( T_{5}^{4} + 116T_{5}^{2} + 2916 \)
|
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\( T_{11}^{4} + 464T_{11}^{2} + 46656 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 116T^{2} + 2916 \)
|
| $7$ |
\( (T^{2} - 7)^{2} \)
|
| $11$ |
\( T^{4} + 464 T^{2} + 46656 \)
|
| $13$ |
\( (T^{2} + 16 T - 48)^{2} \)
|
| $17$ |
\( T^{4} + 788 T^{2} + 79524 \)
|
| $19$ |
\( (T + 20)^{4} \)
|
| $23$ |
\( T^{4} + 464 T^{2} + 46656 \)
|
| $29$ |
\( T^{4} + 1892 T^{2} + 248004 \)
|
| $31$ |
\( (T^{2} + 8 T - 432)^{2} \)
|
| $37$ |
\( (T - 38)^{4} \)
|
| $41$ |
\( T^{4} + 3924 T^{2} + 910116 \)
|
| $43$ |
\( (T^{2} + 40 T - 3632)^{2} \)
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| $47$ |
\( (T^{2} + 288)^{2} \)
|
| $53$ |
\( T^{4} + 16164 T^{2} + 64738116 \)
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| $59$ |
\( T^{4} + 3392 T^{2} + 9216 \)
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| $61$ |
\( (T^{2} - 116 T + 1572)^{2} \)
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| $67$ |
\( (T^{2} - 96 T - 4864)^{2} \)
|
| $71$ |
\( T^{4} + 464 T^{2} + 46656 \)
|
| $73$ |
\( (T^{2} + 48 T - 2224)^{2} \)
|
| $79$ |
\( (T^{2} + 152 T + 3984)^{2} \)
|
| $83$ |
\( T^{4} + 18176 T^{2} + 53231616 \)
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| $89$ |
\( T^{4} + 19476 T^{2} + 443556 \)
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| $97$ |
\( (T^{2} + 144 T - 8368)^{2} \)
|
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