Newspace parameters
| Level: | \( N \) | \(=\) | \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 588.i (of order \(3\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(94.3056860500\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{5569})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 1393x^{2} + 1392x + 1937664 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 84) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{4} - x^{3} + 1393x^{2} + 1392x + 1937664 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 1393\nu^{2} - 1393\nu + 1937664 ) / 1939056 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 1393\nu^{2} + 3879505\nu - 1937664 ) / 1939056 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{3} + 4177 ) / 1393 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 2785\beta _1 - 2785 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 1393\beta_{3} - 4177 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).
| \(n\) | \(197\) | \(295\) | \(493\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 361.1 |
|
0 | 4.50000 | − | 7.79423i | 0 | −35.8129 | − | 62.0297i | 0 | 0 | 0 | −40.5000 | − | 70.1481i | 0 | ||||||||||||||||||||||||
| 361.2 | 0 | 4.50000 | − | 7.79423i | 0 | 38.8129 | + | 67.2259i | 0 | 0 | 0 | −40.5000 | − | 70.1481i | 0 | |||||||||||||||||||||||||
| 373.1 | 0 | 4.50000 | + | 7.79423i | 0 | −35.8129 | + | 62.0297i | 0 | 0 | 0 | −40.5000 | + | 70.1481i | 0 | |||||||||||||||||||||||||
| 373.2 | 0 | 4.50000 | + | 7.79423i | 0 | 38.8129 | − | 67.2259i | 0 | 0 | 0 | −40.5000 | + | 70.1481i | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 588.6.i.l | 4 | |
| 7.b | odd | 2 | 1 | 588.6.i.i | 4 | ||
| 7.c | even | 3 | 1 | 84.6.a.c | ✓ | 2 | |
| 7.c | even | 3 | 1 | inner | 588.6.i.l | 4 | |
| 7.d | odd | 6 | 1 | 588.6.a.k | 2 | ||
| 7.d | odd | 6 | 1 | 588.6.i.i | 4 | ||
| 21.h | odd | 6 | 1 | 252.6.a.h | 2 | ||
| 28.g | odd | 6 | 1 | 336.6.a.x | 2 | ||
| 84.n | even | 6 | 1 | 1008.6.a.bo | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 84.6.a.c | ✓ | 2 | 7.c | even | 3 | 1 | |
| 252.6.a.h | 2 | 21.h | odd | 6 | 1 | ||
| 336.6.a.x | 2 | 28.g | odd | 6 | 1 | ||
| 588.6.a.k | 2 | 7.d | odd | 6 | 1 | ||
| 588.6.i.i | 4 | 7.b | odd | 2 | 1 | ||
| 588.6.i.i | 4 | 7.d | odd | 6 | 1 | ||
| 588.6.i.l | 4 | 1.a | even | 1 | 1 | trivial | |
| 588.6.i.l | 4 | 7.c | even | 3 | 1 | inner | |
| 1008.6.a.bo | 2 | 84.n | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 6T_{5}^{3} + 5596T_{5}^{2} + 33360T_{5} + 30913600 \)
acting on \(S_{6}^{\mathrm{new}}(588, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} - 9 T + 81)^{2} \)
|
| $5$ |
\( T^{4} - 6 T^{3} + \cdots + 30913600 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 73362972736 \)
|
| $13$ |
\( (T^{2} - 768 T - 53028)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 849715240000 \)
|
| $19$ |
\( T^{4} + \cdots + 3780566919424 \)
|
| $23$ |
\( T^{4} + \cdots + 101762901817600 \)
|
| $29$ |
\( (T^{2} - 10572 T + 19031396)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 682638940569600 \)
|
| $37$ |
\( T^{4} + \cdots + 15\!\cdots\!16 \)
|
| $41$ |
\( (T^{2} - 1266 T - 5663952)^{2} \)
|
| $43$ |
\( (T^{2} + 5768 T - 312456944)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 72\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots + 56\!\cdots\!84 \)
|
| $59$ |
\( T^{4} + \cdots + 855117957760000 \)
|
| $61$ |
\( T^{4} + \cdots + 14\!\cdots\!00 \)
|
| $67$ |
\( T^{4} + \cdots + 16\!\cdots\!84 \)
|
| $71$ |
\( (T^{2} - 11022 T - 4430537104)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 27\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots + 38\!\cdots\!16 \)
|
| $83$ |
\( (T^{2} + 44424 T - 76892656)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 10\!\cdots\!44 \)
|
| $97$ |
\( (T^{2} - 44864 T - 14004028100)^{2} \)
|
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