Newspace parameters
Level: | \( N \) | \(=\) | \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 588.m (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(16.0218395444\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
Coefficient field: | 8.0.339738624.1 |
Defining polynomial: |
\( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 7^{2} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \)
:
\(\beta_{1}\) | \(=\) |
\( ( \nu^{7} + 76\nu ) / 14 \)
|
\(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 20 ) / 14 \)
|
\(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} - 27\nu ) / 7 \)
|
\(\beta_{4}\) | \(=\) |
\( ( -2\nu^{6} + 7\nu^{4} - 28\nu^{2} + 2 ) / 14 \)
|
\(\beta_{5}\) | \(=\) |
\( ( -2\nu^{6} + 7\nu^{4} - 21\nu^{2} + 2 ) / 7 \)
|
\(\beta_{6}\) | \(=\) |
\( ( -8\nu^{7} + 35\nu^{5} - 112\nu^{3} + 64\nu ) / 14 \)
|
\(\beta_{7}\) | \(=\) |
\( ( 11\nu^{7} - 42\nu^{5} + 154\nu^{3} - 88\nu ) / 14 \)
|
\(\nu\) | \(=\) |
\( ( \beta_{3} + 2\beta_1 ) / 7 \)
|
\(\nu^{2}\) | \(=\) |
\( \beta_{5} - 2\beta_{4} \)
|
\(\nu^{3}\) | \(=\) |
\( ( 5\beta_{7} + 6\beta_{6} + 6\beta_{3} + 5\beta_1 ) / 7 \)
|
\(\nu^{4}\) | \(=\) |
\( 4\beta_{5} - 6\beta_{4} + 4\beta_{2} - 6 \)
|
\(\nu^{5}\) | \(=\) |
\( ( 16\beta_{7} + 22\beta_{6} ) / 7 \)
|
\(\nu^{6}\) | \(=\) |
\( 14\beta_{2} - 20 \)
|
\(\nu^{7}\) | \(=\) |
\( ( -76\beta_{3} - 54\beta_1 ) / 7 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).
\(n\) | \(197\) | \(295\) | \(493\) |
\(\chi(n)\) | \(1\) | \(1\) | \(-\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
313.1 |
|
0 | 1.50000 | − | 0.866025i | 0 | −5.04718 | − | 2.91399i | 0 | 0 | 0 | 1.50000 | − | 2.59808i | 0 | ||||||||||||||||||||||||||||||||||||
313.2 | 0 | 1.50000 | − | 0.866025i | 0 | −0.416265 | − | 0.240331i | 0 | 0 | 0 | 1.50000 | − | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||||
313.3 | 0 | 1.50000 | − | 0.866025i | 0 | 0.804540 | + | 0.464502i | 0 | 0 | 0 | 1.50000 | − | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||||
313.4 | 0 | 1.50000 | − | 0.866025i | 0 | 4.65891 | + | 2.68982i | 0 | 0 | 0 | 1.50000 | − | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||||
325.1 | 0 | 1.50000 | + | 0.866025i | 0 | −5.04718 | + | 2.91399i | 0 | 0 | 0 | 1.50000 | + | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||||
325.2 | 0 | 1.50000 | + | 0.866025i | 0 | −0.416265 | + | 0.240331i | 0 | 0 | 0 | 1.50000 | + | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||||
325.3 | 0 | 1.50000 | + | 0.866025i | 0 | 0.804540 | − | 0.464502i | 0 | 0 | 0 | 1.50000 | + | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||||
325.4 | 0 | 1.50000 | + | 0.866025i | 0 | 4.65891 | − | 2.68982i | 0 | 0 | 0 | 1.50000 | + | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 588.3.m.f | 8 | |
3.b | odd | 2 | 1 | 1764.3.z.l | 8 | ||
7.b | odd | 2 | 1 | 588.3.m.e | 8 | ||
7.c | even | 3 | 1 | 588.3.d.c | ✓ | 8 | |
7.c | even | 3 | 1 | 588.3.m.e | 8 | ||
7.d | odd | 6 | 1 | 588.3.d.c | ✓ | 8 | |
7.d | odd | 6 | 1 | inner | 588.3.m.f | 8 | |
21.c | even | 2 | 1 | 1764.3.z.m | 8 | ||
21.g | even | 6 | 1 | 1764.3.d.h | 8 | ||
21.g | even | 6 | 1 | 1764.3.z.l | 8 | ||
21.h | odd | 6 | 1 | 1764.3.d.h | 8 | ||
21.h | odd | 6 | 1 | 1764.3.z.m | 8 | ||
28.f | even | 6 | 1 | 2352.3.f.j | 8 | ||
28.g | odd | 6 | 1 | 2352.3.f.j | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
588.3.d.c | ✓ | 8 | 7.c | even | 3 | 1 | |
588.3.d.c | ✓ | 8 | 7.d | odd | 6 | 1 | |
588.3.m.e | 8 | 7.b | odd | 2 | 1 | ||
588.3.m.e | 8 | 7.c | even | 3 | 1 | ||
588.3.m.f | 8 | 1.a | even | 1 | 1 | trivial | |
588.3.m.f | 8 | 7.d | odd | 6 | 1 | inner | |
1764.3.d.h | 8 | 21.g | even | 6 | 1 | ||
1764.3.d.h | 8 | 21.h | odd | 6 | 1 | ||
1764.3.z.l | 8 | 3.b | odd | 2 | 1 | ||
1764.3.z.l | 8 | 21.g | even | 6 | 1 | ||
1764.3.z.m | 8 | 21.c | even | 2 | 1 | ||
1764.3.z.m | 8 | 21.h | odd | 6 | 1 | ||
2352.3.f.j | 8 | 28.f | even | 6 | 1 | ||
2352.3.f.j | 8 | 28.g | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} - 32T_{5}^{6} + 1010T_{5}^{4} - 768T_{5}^{3} - 256T_{5}^{2} + 336T_{5} + 196 \)
acting on \(S_{3}^{\mathrm{new}}(588, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( (T^{2} - 3 T + 3)^{4} \)
$5$
\( T^{8} - 32 T^{6} + 1010 T^{4} + \cdots + 196 \)
$7$
\( T^{8} \)
$11$
\( T^{8} + 124 T^{6} + \cdots + 10837264 \)
$13$
\( T^{8} + 712 T^{6} + 123284 T^{4} + \cdots + 2979076 \)
$17$
\( T^{8} - 48 T^{7} + \cdots + 168428484 \)
$19$
\( T^{8} + 96 T^{7} + \cdots + 285745216 \)
$23$
\( T^{8} - 8 T^{7} + \cdots + 1443088144 \)
$29$
\( (T^{4} - 40 T^{3} - 1924 T^{2} + \cdots + 468892)^{2} \)
$31$
\( T^{8} - 48 T^{7} + \cdots + 2052452416 \)
$37$
\( T^{8} + 64 T^{7} + \cdots + 298373767696 \)
$41$
\( T^{8} + 9808 T^{6} + \cdots + 26697826996036 \)
$43$
\( (T^{4} + 56 T^{3} - 3784 T^{2} + \cdots + 192784)^{2} \)
$47$
\( T^{8} + 264 T^{7} + \cdots + 8881401308224 \)
$53$
\( T^{8} - 72 T^{7} + \cdots + 71641191948544 \)
$59$
\( T^{8} + \cdots + 372481662446656 \)
$61$
\( T^{8} + 144 T^{7} + 3772 T^{6} + \cdots + 454276 \)
$67$
\( T^{8} - 32 T^{7} + \cdots + 306756468736 \)
$71$
\( (T^{4} - 112 T^{3} - 6460 T^{2} + \cdots - 7722596)^{2} \)
$73$
\( T^{8} - 336 T^{7} + \cdots + 3712242251524 \)
$79$
\( T^{8} - 216 T^{7} + \cdots + 49705658450176 \)
$83$
\( T^{8} + 19712 T^{6} + \cdots + 61585579131904 \)
$89$
\( T^{8} - 96 T^{7} + \cdots + 17\!\cdots\!76 \)
$97$
\( T^{8} + 13640 T^{6} + \cdots + 5315948141956 \)
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