Newspace parameters
Level: | \( N \) | \(=\) | \( 567 = 3^{4} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 567.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.52751779461\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | 8.0.12745506816.3 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} + 16x^{6} + 79x^{4} + 120x^{2} + 9 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 3^{5} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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 \( x^{8} + 16x^{6} + 79x^{4} + 120x^{2} + 9 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} + 4 \) |
\(\beta_{3}\) | \(=\) | \( \nu^{3} + 6\nu \) |
\(\beta_{4}\) | \(=\) | \( ( \nu^{6} + 12\nu^{4} + 36\nu^{2} + 16 ) / 5 \) |
\(\beta_{5}\) | \(=\) | \( ( -\nu^{7} - 12\nu^{5} - 36\nu^{3} - 21\nu ) / 5 \) |
\(\beta_{6}\) | \(=\) | \( \nu^{4} + 8\nu^{2} + 7 \) |
\(\beta_{7}\) | \(=\) | \( ( \nu^{7} + 17\nu^{5} + 91\nu^{3} + 151\nu ) / 5 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} - 4 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{3} - 6\beta_1 \) |
\(\nu^{4}\) | \(=\) | \( \beta_{6} - 8\beta_{2} + 25 \) |
\(\nu^{5}\) | \(=\) | \( \beta_{7} + \beta_{5} - 11\beta_{3} + 40\beta_1 \) |
\(\nu^{6}\) | \(=\) | \( -12\beta_{6} + 5\beta_{4} + 60\beta_{2} - 172 \) |
\(\nu^{7}\) | \(=\) | \( -12\beta_{7} - 17\beta_{5} + 96\beta_{3} - 285\beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).
\(n\) | \(325\) | \(407\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
566.1 |
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− | 2.81442i | 0 | −5.92095 | 0 | 0 | −2.64575 | 11.0352i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
566.2 | − | 2.29678i | 0 | −3.27520 | 0 | 0 | 2.64575 | 2.92886i | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
566.3 | − | 1.65070i | 0 | −0.724799 | 0 | 0 | −2.64575 | − | 2.10497i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
566.4 | − | 0.281155i | 0 | 1.92095 | 0 | 0 | 2.64575 | − | 1.10240i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
566.5 | 0.281155i | 0 | 1.92095 | 0 | 0 | 2.64575 | 1.10240i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
566.6 | 1.65070i | 0 | −0.724799 | 0 | 0 | −2.64575 | 2.10497i | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
566.7 | 2.29678i | 0 | −3.27520 | 0 | 0 | 2.64575 | − | 2.92886i | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
566.8 | 2.81442i | 0 | −5.92095 | 0 | 0 | −2.64575 | − | 11.0352i | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-7}) \) |
3.b | odd | 2 | 1 | inner |
21.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 567.2.c.a | ✓ | 8 |
3.b | odd | 2 | 1 | inner | 567.2.c.a | ✓ | 8 |
7.b | odd | 2 | 1 | CM | 567.2.c.a | ✓ | 8 |
9.c | even | 3 | 2 | 567.2.o.h | 16 | ||
9.d | odd | 6 | 2 | 567.2.o.h | 16 | ||
21.c | even | 2 | 1 | inner | 567.2.c.a | ✓ | 8 |
63.l | odd | 6 | 2 | 567.2.o.h | 16 | ||
63.o | even | 6 | 2 | 567.2.o.h | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
567.2.c.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
567.2.c.a | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
567.2.c.a | ✓ | 8 | 7.b | odd | 2 | 1 | CM |
567.2.c.a | ✓ | 8 | 21.c | even | 2 | 1 | inner |
567.2.o.h | 16 | 9.c | even | 3 | 2 | ||
567.2.o.h | 16 | 9.d | odd | 6 | 2 | ||
567.2.o.h | 16 | 63.l | odd | 6 | 2 | ||
567.2.o.h | 16 | 63.o | even | 6 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 16T_{2}^{6} + 79T_{2}^{4} + 120T_{2}^{2} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(567, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} + 16 T^{6} + 79 T^{4} + 120 T^{2} + \cdots + 9 \)
$3$
\( T^{8} \)
$5$
\( T^{8} \)
$7$
\( (T^{2} - 7)^{4} \)
$11$
\( T^{8} + 88 T^{6} + 2626 T^{4} + \cdots + 106929 \)
$13$
\( T^{8} \)
$17$
\( T^{8} \)
$19$
\( T^{8} \)
$23$
\( (T^{4} + 92 T^{2} + 324)^{2} \)
$29$
\( (T^{4} + 116 T^{2} + 2916)^{2} \)
$31$
\( T^{8} \)
$37$
\( (T^{4} - 110 T^{2} + 1)^{2} \)
$41$
\( T^{8} \)
$43$
\( (T^{4} - 230 T^{2} + 10201)^{2} \)
$47$
\( T^{8} \)
$53$
\( T^{8} + 424 T^{6} + \cdots + 70408881 \)
$59$
\( T^{8} \)
$61$
\( T^{8} \)
$67$
\( (T^{2} - 4 T - 185)^{4} \)
$71$
\( T^{8} + 568 T^{6} + 97906 T^{4} + \cdots + 4524129 \)
$73$
\( T^{8} \)
$79$
\( (T^{2} + 8 T - 173)^{4} \)
$83$
\( T^{8} \)
$89$
\( T^{8} \)
$97$
\( T^{8} \)
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