Newspace parameters
| Level: | \( N \) | \(=\) | \( 9072 = 2^{4} \cdot 3^{4} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9072.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(72.4402847137\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.621.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - 6x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 567) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - 6x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 4 \)
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| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{2} + \beta _1 + 8 ) / 2 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | 0 | 0 | −2.20147 | 0 | −1.00000 | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 1.45490 | 0 | −1.00000 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 3.74657 | 0 | −1.00000 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(7\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 9072.2.a.cb | 3 | |
| 3.b | odd | 2 | 1 | 9072.2.a.bu | 3 | ||
| 4.b | odd | 2 | 1 | 567.2.a.f | yes | 3 | |
| 12.b | even | 2 | 1 | 567.2.a.e | ✓ | 3 | |
| 28.d | even | 2 | 1 | 3969.2.a.n | 3 | ||
| 36.f | odd | 6 | 2 | 567.2.f.l | 6 | ||
| 36.h | even | 6 | 2 | 567.2.f.m | 6 | ||
| 84.h | odd | 2 | 1 | 3969.2.a.o | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 567.2.a.e | ✓ | 3 | 12.b | even | 2 | 1 | |
| 567.2.a.f | yes | 3 | 4.b | odd | 2 | 1 | |
| 567.2.f.l | 6 | 36.f | odd | 6 | 2 | ||
| 567.2.f.m | 6 | 36.h | even | 6 | 2 | ||
| 3969.2.a.n | 3 | 28.d | even | 2 | 1 | ||
| 3969.2.a.o | 3 | 84.h | odd | 2 | 1 | ||
| 9072.2.a.bu | 3 | 3.b | odd | 2 | 1 | ||
| 9072.2.a.cb | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9072))\):
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\( T_{5}^{3} - 3T_{5}^{2} - 6T_{5} + 12 \)
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\( T_{11}^{3} + 6T_{11}^{2} + 3T_{11} - 6 \)
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\( T_{13}^{3} - 3T_{13}^{2} - 36T_{13} + 112 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
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| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} - 3 T^{2} + \cdots + 12 \)
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| $7$ |
\( (T + 1)^{3} \)
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| $11$ |
\( T^{3} + 6 T^{2} + \cdots - 6 \)
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| $13$ |
\( T^{3} - 3 T^{2} + \cdots + 112 \)
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| $17$ |
\( T^{3} - 3 T^{2} + \cdots - 12 \)
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| $19$ |
\( T^{3} - 48T + 56 \)
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| $23$ |
\( T^{3} - 6 T^{2} + \cdots + 96 \)
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| $29$ |
\( T^{3} - 3 T^{2} + \cdots - 36 \)
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| $31$ |
\( (T + 2)^{3} \)
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| $37$ |
\( (T - 5)^{3} \)
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| $41$ |
\( T^{3} + 12T^{2} - 72 \)
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| $43$ |
\( T^{3} + 12 T^{2} + \cdots - 652 \)
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| $47$ |
\( T^{3} - 24T + 24 \)
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| $53$ |
\( T^{3} - 12 T^{2} + \cdots + 6 \)
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| $59$ |
\( T^{3} - 18 T^{2} + \cdots + 552 \)
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| $61$ |
\( T^{3} + 3 T^{2} + \cdots - 188 \)
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| $67$ |
\( T^{3} + 6 T^{2} + \cdots - 262 \)
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| $71$ |
\( T^{3} - 81T + 108 \)
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| $73$ |
\( T^{3} - 9 T^{2} + \cdots + 16 \)
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| $79$ |
\( T^{3} + 6 T^{2} + \cdots - 262 \)
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| $83$ |
\( T^{3} + 12T^{2} - 72 \)
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| $89$ |
\( T^{3} - 15 T^{2} + \cdots + 48 \)
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| $97$ |
\( T^{3} - 12T^{2} + 184 \)
|
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