Newspace parameters
| Level: | \( N \) | \(=\) | \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 588.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(183.682394985\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 36766x + 1071280 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | yes |
| 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
\( x^{3} - x^{2} - 36766x + 1071280 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4\nu^{2} + 168\nu - 98095 ) / 15 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 15\beta_{2} - 84\beta _1 + 98011 ) / 4 \)
|
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 | 27.0000 | 0 | −324.907 | 0 | 0 | 0 | 729.000 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 27.0000 | 0 | 143.672 | 0 | 0 | 0 | 729.000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 27.0000 | 0 | 435.235 | 0 | 0 | 0 | 729.000 | 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 | 588.8.a.h | yes | 3 |
| 7.b | odd | 2 | 1 | 588.8.a.g | ✓ | 3 | |
| 7.c | even | 3 | 2 | 588.8.i.m | 6 | ||
| 7.d | odd | 6 | 2 | 588.8.i.n | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 588.8.a.g | ✓ | 3 | 7.b | odd | 2 | 1 | |
| 588.8.a.h | yes | 3 | 1.a | even | 1 | 1 | trivial |
| 588.8.i.m | 6 | 7.c | even | 3 | 2 | ||
| 588.8.i.n | 6 | 7.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} - 254T_{5}^{2} - 125560T_{5} + 20316800 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(588))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( (T - 27)^{3} \)
|
| $5$ |
\( T^{3} - 254 T^{2} + \cdots + 20316800 \)
|
| $7$ |
\( T^{3} \)
|
| $11$ |
\( T^{3} + \cdots - 5161667200 \)
|
| $13$ |
\( T^{3} + \cdots - 712475748032 \)
|
| $17$ |
\( T^{3} + \cdots + 4712803143552 \)
|
| $19$ |
\( T^{3} + \cdots - 5022321610112 \)
|
| $23$ |
\( T^{3} + \cdots + 185763011482368 \)
|
| $29$ |
\( T^{3} + \cdots - 69099970871800 \)
|
| $31$ |
\( T^{3} + \cdots + 364445294210816 \)
|
| $37$ |
\( T^{3} + \cdots - 73\!\cdots\!00 \)
|
| $41$ |
\( T^{3} + \cdots - 17\!\cdots\!44 \)
|
| $43$ |
\( T^{3} + \cdots - 26\!\cdots\!08 \)
|
| $47$ |
\( T^{3} + \cdots - 10\!\cdots\!00 \)
|
| $53$ |
\( T^{3} + \cdots + 61\!\cdots\!76 \)
|
| $59$ |
\( T^{3} + \cdots + 14\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots + 11\!\cdots\!32 \)
|
| $67$ |
\( T^{3} + \cdots - 17\!\cdots\!96 \)
|
| $71$ |
\( T^{3} + \cdots - 52\!\cdots\!80 \)
|
| $73$ |
\( T^{3} + \cdots - 19\!\cdots\!28 \)
|
| $79$ |
\( T^{3} + \cdots + 28\!\cdots\!04 \)
|
| $83$ |
\( T^{3} + \cdots - 73\!\cdots\!04 \)
|
| $89$ |
\( T^{3} + \cdots - 63\!\cdots\!04 \)
|
| $97$ |
\( T^{3} + \cdots - 18\!\cdots\!00 \)
|
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