Newspace parameters
| Level: | \( N \) | \(=\) | \( 5328 = 2^{4} \cdot 3^{2} \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5328.h (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(42.5442941969\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.27648.1 |
|
|
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| Defining polynomial: |
\( x^{4} + 6x^{2} + 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 111) |
| 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
\( x^{4} + 6x^{2} + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} + 5\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} + 6 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 6 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{2} - 5\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5328\mathbb{Z}\right)^\times\).
| \(n\) | \(1297\) | \(1333\) | \(1999\) | \(2369\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2737.1 |
|
0 | 0 | 0 | − | 3.30136i | 0 | 2.00000 | 0 | 0 | 0 | |||||||||||||||||||||||||||||
| 2737.2 | 0 | 0 | 0 | − | 1.04930i | 0 | 2.00000 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 2737.3 | 0 | 0 | 0 | 1.04930i | 0 | 2.00000 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 2737.4 | 0 | 0 | 0 | 3.30136i | 0 | 2.00000 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 37.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 5328.2.h.l | 4 | |
| 3.b | odd | 2 | 1 | 1776.2.h.e | 4 | ||
| 4.b | odd | 2 | 1 | 333.2.c.c | 4 | ||
| 12.b | even | 2 | 1 | 111.2.c.b | ✓ | 4 | |
| 37.b | even | 2 | 1 | inner | 5328.2.h.l | 4 | |
| 111.d | odd | 2 | 1 | 1776.2.h.e | 4 | ||
| 148.b | odd | 2 | 1 | 333.2.c.c | 4 | ||
| 444.g | even | 2 | 1 | 111.2.c.b | ✓ | 4 | |
| 444.j | odd | 4 | 2 | 4107.2.a.g | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 111.2.c.b | ✓ | 4 | 12.b | even | 2 | 1 | |
| 111.2.c.b | ✓ | 4 | 444.g | even | 2 | 1 | |
| 333.2.c.c | 4 | 4.b | odd | 2 | 1 | ||
| 333.2.c.c | 4 | 148.b | odd | 2 | 1 | ||
| 1776.2.h.e | 4 | 3.b | odd | 2 | 1 | ||
| 1776.2.h.e | 4 | 111.d | odd | 2 | 1 | ||
| 4107.2.a.g | 4 | 444.j | odd | 4 | 2 | ||
| 5328.2.h.l | 4 | 1.a | even | 1 | 1 | trivial | |
| 5328.2.h.l | 4 | 37.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5328, [\chi])\):
|
\( T_{5}^{4} + 12T_{5}^{2} + 12 \)
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\( T_{7} - 2 \)
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\( T_{13}^{4} + 48T_{13}^{2} + 192 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 12T^{2} + 12 \)
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| $7$ |
\( (T - 2)^{4} \)
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| $11$ |
\( (T^{2} - 24)^{2} \)
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| $13$ |
\( T^{4} + 48T^{2} + 192 \)
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| $17$ |
\( T^{4} + 12T^{2} + 12 \)
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| $19$ |
\( T^{4} + 24T^{2} + 48 \)
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| $23$ |
\( T^{4} + 36T^{2} + 300 \)
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| $29$ |
\( T^{4} + 108T^{2} + 12 \)
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| $31$ |
\( T^{4} + 72T^{2} + 1200 \)
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| $37$ |
\( T^{4} - 4 T^{3} + \cdots + 1369 \)
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| $41$ |
\( (T^{2} - 12 T + 12)^{2} \)
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| $43$ |
\( T^{4} + 264 T^{2} + 17328 \)
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| $47$ |
\( (T^{2} - 24)^{2} \)
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| $53$ |
\( (T^{2} + 12 T + 12)^{2} \)
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| $59$ |
\( T^{4} + 36T^{2} + 300 \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( (T - 2)^{4} \)
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| $71$ |
\( (T^{2} - 24)^{2} \)
|
| $73$ |
\( (T^{2} + 16 T + 40)^{2} \)
|
| $79$ |
\( T^{4} + 24T^{2} + 48 \)
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| $83$ |
\( (T^{2} - 96)^{2} \)
|
| $89$ |
\( T^{4} + 108T^{2} + 12 \)
|
| $97$ |
\( T^{4} + 144T^{2} + 4800 \)
|
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