Newspace parameters
| Level: | \( N \) | \(=\) | \( 1344 = 2^{6} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1344.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(79.2985670477\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.37341.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - 57x - 148 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 672) |
| 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
\( x^{3} - 57x - 148 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( 4\nu^{2} - 16\nu - 152 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 8\beta _1 + 152 ) / 4 \)
|
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 | −3.00000 | 0 | −12.9475 | 0 | 7.00000 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −3.00000 | 0 | −8.27848 | 0 | 7.00000 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | −3.00000 | 0 | 15.2260 | 0 | 7.00000 | 0 | 9.00000 | 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 | 1344.4.a.bs | 3 | |
| 4.b | odd | 2 | 1 | 1344.4.a.bu | 3 | ||
| 8.b | even | 2 | 1 | 672.4.a.r | yes | 3 | |
| 8.d | odd | 2 | 1 | 672.4.a.p | ✓ | 3 | |
| 24.f | even | 2 | 1 | 2016.4.a.u | 3 | ||
| 24.h | odd | 2 | 1 | 2016.4.a.v | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 672.4.a.p | ✓ | 3 | 8.d | odd | 2 | 1 | |
| 672.4.a.r | yes | 3 | 8.b | even | 2 | 1 | |
| 1344.4.a.bs | 3 | 1.a | even | 1 | 1 | trivial | |
| 1344.4.a.bu | 3 | 4.b | odd | 2 | 1 | ||
| 2016.4.a.u | 3 | 24.f | even | 2 | 1 | ||
| 2016.4.a.v | 3 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1344))\):
|
\( T_{5}^{3} + 6T_{5}^{2} - 216T_{5} - 1632 \)
|
|
\( T_{11}^{3} + 48T_{11}^{2} - 3060T_{11} - 95488 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( (T + 3)^{3} \)
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| $5$ |
\( T^{3} + 6 T^{2} + \cdots - 1632 \)
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| $7$ |
\( (T - 7)^{3} \)
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| $11$ |
\( T^{3} + 48 T^{2} + \cdots - 95488 \)
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| $13$ |
\( T^{3} + 6 T^{2} + \cdots + 63656 \)
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| $17$ |
\( T^{3} - 54 T^{2} + \cdots + 24736 \)
|
| $19$ |
\( T^{3} + 84 T^{2} + \cdots - 200256 \)
|
| $23$ |
\( T^{3} - 48 T^{2} + \cdots + 187648 \)
|
| $29$ |
\( T^{3} + 18 T^{2} + \cdots + 4848088 \)
|
| $31$ |
\( T^{3} - 72 T^{2} + \cdots + 2342912 \)
|
| $37$ |
\( T^{3} + 210 T^{2} + \cdots - 53576 \)
|
| $41$ |
\( T^{3} - 414 T^{2} + \cdots + 4957664 \)
|
| $43$ |
\( T^{3} + 168 T^{2} + \cdots - 722944 \)
|
| $47$ |
\( T^{3} - 72 T^{2} + \cdots - 17124736 \)
|
| $53$ |
\( T^{3} + 402 T^{2} + \cdots - 7840072 \)
|
| $59$ |
\( T^{3} + 540 T^{2} + \cdots + 11538688 \)
|
| $61$ |
\( T^{3} + 798 T^{2} + \cdots - 170037272 \)
|
| $67$ |
\( T^{3} + 48 T^{2} + \cdots - 44046592 \)
|
| $71$ |
\( T^{3} + 456 T^{2} + \cdots - 19646112 \)
|
| $73$ |
\( T^{3} - 1230 T^{2} + \cdots + 640250616 \)
|
| $79$ |
\( T^{3} + 1368 T^{2} + \cdots - 182717824 \)
|
| $83$ |
\( T^{3} + 60 T^{2} + \cdots - 90712000 \)
|
| $89$ |
\( T^{3} - 2742 T^{2} + \cdots - 524337504 \)
|
| $97$ |
\( T^{3} - 1950 T^{2} + \cdots - 50085448 \)
|
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