Newspace parameters
| Level: | \( N \) | \(=\) | \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2880.l (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(78.4743161358\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-5})\) |
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|
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| Defining polynomial: |
\( x^{4} - 4x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 360) |
| 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
\( x^{4} - 4x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - \nu ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 7\nu ) / 3 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( ( \beta_{3} + 7\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times\).
| \(n\) | \(577\) | \(641\) | \(901\) | \(2431\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1601.1 |
|
0 | 0 | 0 | − | 2.23607i | 0 | −3.16228 | 0 | 0 | 0 | |||||||||||||||||||||||||||||
| 1601.2 | 0 | 0 | 0 | − | 2.23607i | 0 | 3.16228 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1601.3 | 0 | 0 | 0 | 2.23607i | 0 | −3.16228 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1601.4 | 0 | 0 | 0 | 2.23607i | 0 | 3.16228 | 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 | 2880.3.l.e | 4 | |
| 3.b | odd | 2 | 1 | inner | 2880.3.l.e | 4 | |
| 4.b | odd | 2 | 1 | 2880.3.l.d | 4 | ||
| 8.b | even | 2 | 1 | 360.3.l.b | ✓ | 4 | |
| 8.d | odd | 2 | 1 | 720.3.l.b | 4 | ||
| 12.b | even | 2 | 1 | 2880.3.l.d | 4 | ||
| 24.f | even | 2 | 1 | 720.3.l.b | 4 | ||
| 24.h | odd | 2 | 1 | 360.3.l.b | ✓ | 4 | |
| 40.e | odd | 2 | 1 | 3600.3.l.q | 4 | ||
| 40.f | even | 2 | 1 | 1800.3.l.d | 4 | ||
| 40.i | odd | 4 | 2 | 1800.3.c.d | 8 | ||
| 40.k | even | 4 | 2 | 3600.3.c.h | 8 | ||
| 120.i | odd | 2 | 1 | 1800.3.l.d | 4 | ||
| 120.m | even | 2 | 1 | 3600.3.l.q | 4 | ||
| 120.q | odd | 4 | 2 | 3600.3.c.h | 8 | ||
| 120.w | even | 4 | 2 | 1800.3.c.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 360.3.l.b | ✓ | 4 | 8.b | even | 2 | 1 | |
| 360.3.l.b | ✓ | 4 | 24.h | odd | 2 | 1 | |
| 720.3.l.b | 4 | 8.d | odd | 2 | 1 | ||
| 720.3.l.b | 4 | 24.f | even | 2 | 1 | ||
| 1800.3.c.d | 8 | 40.i | odd | 4 | 2 | ||
| 1800.3.c.d | 8 | 120.w | even | 4 | 2 | ||
| 1800.3.l.d | 4 | 40.f | even | 2 | 1 | ||
| 1800.3.l.d | 4 | 120.i | odd | 2 | 1 | ||
| 2880.3.l.d | 4 | 4.b | odd | 2 | 1 | ||
| 2880.3.l.d | 4 | 12.b | even | 2 | 1 | ||
| 2880.3.l.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 2880.3.l.e | 4 | 3.b | odd | 2 | 1 | inner | |
| 3600.3.c.h | 8 | 40.k | even | 4 | 2 | ||
| 3600.3.c.h | 8 | 120.q | odd | 4 | 2 | ||
| 3600.3.l.q | 4 | 40.e | odd | 2 | 1 | ||
| 3600.3.l.q | 4 | 120.m | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2880, [\chi])\):
|
\( T_{7}^{2} - 10 \)
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\( T_{19}^{2} - 16T_{19} - 296 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T^{2} + 5)^{2} \)
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| $7$ |
\( (T^{2} - 10)^{2} \)
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| $11$ |
\( T^{4} + 44T^{2} + 324 \)
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| $13$ |
\( (T^{2} + 4 T - 6)^{2} \)
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| $17$ |
\( (T^{2} + 320)^{2} \)
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| $19$ |
\( (T^{2} - 16 T - 296)^{2} \)
|
| $23$ |
\( T^{4} + 376 T^{2} + 29584 \)
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| $29$ |
\( T^{4} + 2216 T^{2} + 725904 \)
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| $31$ |
\( (T^{2} - 4 T - 996)^{2} \)
|
| $37$ |
\( (T^{2} + 12 T - 54)^{2} \)
|
| $41$ |
\( T^{4} + 2564 T^{2} + 1633284 \)
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| $43$ |
\( (T^{2} + 72 T - 664)^{2} \)
|
| $47$ |
\( T^{4} + 304T^{2} + 64 \)
|
| $53$ |
\( T^{4} + 7184 T^{2} + 1065024 \)
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| $59$ |
\( T^{4} + 5804 T^{2} + 887364 \)
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| $61$ |
\( (T^{2} + 76 T - 516)^{2} \)
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| $67$ |
\( (T^{2} - 8 T - 5744)^{2} \)
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| $71$ |
\( T^{4} + 6624 T^{2} + 3504384 \)
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| $73$ |
\( (T^{2} - 76 T - 3396)^{2} \)
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| $79$ |
\( (T^{2} + 188 T + 7836)^{2} \)
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| $83$ |
\( T^{4} + 10256 T^{2} + 26132544 \)
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| $89$ |
\( T^{4} + 12260 T^{2} + 35640900 \)
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| $97$ |
\( (T^{2} - 60 T - 3940)^{2} \)
|
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