Newspace parameters
| Level: | \( N \) | \(=\) | \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 588.i (of order \(3\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(183.682394985\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
|
|
|
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 12) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).
| \(n\) | \(197\) | \(295\) | \(493\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-\zeta_{6}\) |
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 | −13.5000 | + | 23.3827i | 0 | −189.000 | − | 327.358i | 0 | 0 | 0 | −364.500 | − | 631.333i | 0 | ||||||||||||||||||
| 373.1 | 0 | −13.5000 | − | 23.3827i | 0 | −189.000 | + | 327.358i | 0 | 0 | 0 | −364.500 | + | 631.333i | 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 | 588.8.i.a | 2 | |
| 7.b | odd | 2 | 1 | 588.8.i.h | 2 | ||
| 7.c | even | 3 | 1 | 588.8.a.d | 1 | ||
| 7.c | even | 3 | 1 | inner | 588.8.i.a | 2 | |
| 7.d | odd | 6 | 1 | 12.8.a.a | ✓ | 1 | |
| 7.d | odd | 6 | 1 | 588.8.i.h | 2 | ||
| 21.g | even | 6 | 1 | 36.8.a.c | 1 | ||
| 28.f | even | 6 | 1 | 48.8.a.e | 1 | ||
| 35.i | odd | 6 | 1 | 300.8.a.g | 1 | ||
| 35.k | even | 12 | 2 | 300.8.d.c | 2 | ||
| 56.j | odd | 6 | 1 | 192.8.a.o | 1 | ||
| 56.m | even | 6 | 1 | 192.8.a.g | 1 | ||
| 63.i | even | 6 | 1 | 324.8.e.a | 2 | ||
| 63.k | odd | 6 | 1 | 324.8.e.f | 2 | ||
| 63.s | even | 6 | 1 | 324.8.e.a | 2 | ||
| 63.t | odd | 6 | 1 | 324.8.e.f | 2 | ||
| 84.j | odd | 6 | 1 | 144.8.a.j | 1 | ||
| 168.ba | even | 6 | 1 | 576.8.a.d | 1 | ||
| 168.be | odd | 6 | 1 | 576.8.a.e | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 12.8.a.a | ✓ | 1 | 7.d | odd | 6 | 1 | |
| 36.8.a.c | 1 | 21.g | even | 6 | 1 | ||
| 48.8.a.e | 1 | 28.f | even | 6 | 1 | ||
| 144.8.a.j | 1 | 84.j | odd | 6 | 1 | ||
| 192.8.a.g | 1 | 56.m | even | 6 | 1 | ||
| 192.8.a.o | 1 | 56.j | odd | 6 | 1 | ||
| 300.8.a.g | 1 | 35.i | odd | 6 | 1 | ||
| 300.8.d.c | 2 | 35.k | even | 12 | 2 | ||
| 324.8.e.a | 2 | 63.i | even | 6 | 1 | ||
| 324.8.e.a | 2 | 63.s | even | 6 | 1 | ||
| 324.8.e.f | 2 | 63.k | odd | 6 | 1 | ||
| 324.8.e.f | 2 | 63.t | odd | 6 | 1 | ||
| 576.8.a.d | 1 | 168.ba | even | 6 | 1 | ||
| 576.8.a.e | 1 | 168.be | odd | 6 | 1 | ||
| 588.8.a.d | 1 | 7.c | even | 3 | 1 | ||
| 588.8.i.a | 2 | 1.a | even | 1 | 1 | trivial | |
| 588.8.i.a | 2 | 7.c | even | 3 | 1 | inner | |
| 588.8.i.h | 2 | 7.b | odd | 2 | 1 | ||
| 588.8.i.h | 2 | 7.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 378T_{5} + 142884 \)
acting on \(S_{8}^{\mathrm{new}}(588, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 27T + 729 \)
|
| $5$ |
\( T^{2} + 378T + 142884 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} - 2484 T + 6170256 \)
|
| $13$ |
\( (T + 14870)^{2} \)
|
| $17$ |
\( T^{2} + 22302 T + 497379204 \)
|
| $19$ |
\( T^{2} + 16300 T + 265690000 \)
|
| $23$ |
\( T^{2} + \cdots + 13254456384 \)
|
| $29$ |
\( (T - 157086)^{2} \)
|
| $31$ |
\( T^{2} + 16456 T + 270799936 \)
|
| $37$ |
\( T^{2} + \cdots + 22280338756 \)
|
| $41$ |
\( (T - 241110)^{2} \)
|
| $43$ |
\( (T + 443188)^{2} \)
|
| $47$ |
\( T^{2} + \cdots + 851471253504 \)
|
| $53$ |
\( T^{2} + \cdots + 486682035876 \)
|
| $59$ |
\( T^{2} + \cdots + 757171464336 \)
|
| $61$ |
\( T^{2} + \cdots + 4272745311844 \)
|
| $67$ |
\( T^{2} + \cdots + 2824913839504 \)
|
| $71$ |
\( (T + 1070280)^{2} \)
|
| $73$ |
\( T^{2} + \cdots + 5776014315556 \)
|
| $79$ |
\( T^{2} + \cdots + 5296957486144 \)
|
| $83$ |
\( (T + 4708692)^{2} \)
|
| $89$ |
\( T^{2} + \cdots + 17170166816100 \)
|
| $97$ |
\( (T - 1622974)^{2} \)
|
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