Newspace parameters
| Level: | \( N \) | \(=\) | \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 588.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(94.3056860500\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{177 +28 \sqrt{2}})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 87x^{2} + 88x + 1838 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 7 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 87x^{2} + 88x + 1838 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu - 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4\nu^{3} - 7\nu^{2} - 173\nu + 132 ) / 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - \nu - 44 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 2 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{3} + \beta _1 + 178 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 7\beta_{3} + 7\beta_{2} + 45\beta _1 + 266 ) / 4 \)
|
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 | −9.00000 | 0 | −35.3849 | 0 | 0 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −9.00000 | 0 | −33.4054 | 0 | 0 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | −9.00000 | 0 | −9.02102 | 0 | 0 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | −9.00000 | 0 | 77.8114 | 0 | 0 | 0 | 81.0000 | 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 | 588.6.a.m | ✓ | 4 |
| 7.b | odd | 2 | 1 | 588.6.a.o | yes | 4 | |
| 7.c | even | 3 | 2 | 588.6.i.q | 8 | ||
| 7.d | odd | 6 | 2 | 588.6.i.p | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 588.6.a.m | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 588.6.a.o | yes | 4 | 7.b | odd | 2 | 1 | |
| 588.6.i.p | 8 | 7.d | odd | 6 | 2 | ||
| 588.6.i.q | 8 | 7.c | even | 3 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 4252T_{5}^{2} - 129600T_{5} - 829724 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(588))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T + 9)^{4} \)
|
| $5$ |
\( T^{4} - 4252 T^{2} + \cdots - 829724 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + \cdots - 1088220656 \)
|
| $13$ |
\( T^{4} + \cdots + 9699012900 \)
|
| $17$ |
\( T^{4} + \cdots - 188238630524 \)
|
| $19$ |
\( T^{4} + \cdots + 1579617420352 \)
|
| $23$ |
\( T^{4} + \cdots - 323086549232 \)
|
| $29$ |
\( T^{4} + \cdots + 6692607711808 \)
|
| $31$ |
\( T^{4} + \cdots + 54042275262528 \)
|
| $37$ |
\( T^{4} + \cdots - 10\!\cdots\!84 \)
|
| $41$ |
\( T^{4} + \cdots + 41\!\cdots\!56 \)
|
| $43$ |
\( T^{4} + \cdots - 26\!\cdots\!28 \)
|
| $47$ |
\( T^{4} + \cdots + 289010761502272 \)
|
| $53$ |
\( T^{4} + \cdots - 22\!\cdots\!72 \)
|
| $59$ |
\( T^{4} + \cdots - 30\!\cdots\!84 \)
|
| $61$ |
\( T^{4} + \cdots + 53\!\cdots\!00 \)
|
| $67$ |
\( T^{4} + \cdots + 33\!\cdots\!84 \)
|
| $71$ |
\( T^{4} + \cdots - 817160501542896 \)
|
| $73$ |
\( T^{4} + \cdots + 21\!\cdots\!12 \)
|
| $79$ |
\( T^{4} + \cdots + 37\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots - 84\!\cdots\!24 \)
|
| $89$ |
\( T^{4} + \cdots - 13\!\cdots\!48 \)
|
| $97$ |
\( T^{4} + \cdots + 26\!\cdots\!76 \)
|
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