LMFDB - Newform orbit 648.2.n.n

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,2,Mod(109,648)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("648.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	648.n (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.17430605098$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.49787136.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 $ x^8 + 3x^6 + 5x^4 + 12*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

$f(q)$	$=$	$ q + \beta_1 q^{2} + (\beta_{6} + \beta_{4} - \beta_{2} - 1) q^{4} - \beta_{3} q^{5} - \beta_{4} q^{7} + (\beta_{7} + 2 \beta_{5} + \cdots - \beta_1) q^{8}+O(q^{10}) $ q + b1 * q^2 + (b6 + b4 - b2 - 1) * q^4 - b3 * q^5 - b4 * q^7 + (b7 + 2b5 - b3 - b1) q^8	$ q + \beta_1 q^{2} + (\beta_{6} + \beta_{4} - \beta_{2} - 1) q^{4} - \beta_{3} q^{5} - \beta_{4} q^{7} + (\beta_{7} + 2 \beta_{5} + \cdots - \beta_1) q^{8}+ \cdots + ( - 6 \beta_{7} + 6 \beta_{3} + 6 \beta_1) q^{98}+O(q^{100}) $ q + b1 * q^2 + (b6 + b4 - b2 - 1) * q^4 - b3 * q^5 - b4 * q^7 + (b7 + 2b5 - b3 - b1) q^8 + (b6 + 1) * q^10 + (3b5 + 3b3) * q^11 + (4b6 - 2b4 - 4b2 + 2) q^13 + (-b7 + b3) * q^14 + (b4 + 3b2) q^16 + (-4b7 + 2b5 + 4b3 + 4b1) * q^17 + (4b6 + 2) q^19 + (2b5 + 2b3 + b1) * q^20 + (-3b6 + 3b4 + 3b2 - 3) q^22 + (4b7 - 2b3) * q^23 - 4b4 q^25 + (-2b7 + 8b5 + 2b3 + 2b1) * q^26 + (-b6 + 1) * q^28 + (6b5 + 6b3) * q^29 + (-7b4 + 7) q^31 + (b7 + 5b3) q^32 + (6b4 - 2b2) * q^34 - b5 * q^35 + (-4b6 - 2) q^37 + (8b5 + 8b3 + 2b1) q^38 + (-b6 + 3b4 + b2 - 3) q^40 + (-4b7 + 2b3) * q^41 + (-4b4 - 8b2) * q^43 + (3b7 - 6b5 - 3b3 - 3b1) * q^44 + (2b6 - 6) q^46 + (-6b4 + 6) q^49 + (-4b7 + 4b3) * q^50 + (10b4 + 6b2) * q^52 + 9b5 q^53 - 3 * q^55 + (-2b5 - 2b3 + b1) * q^56 + (-6b6 + 6b4 + 6b2 - 6) q^58 + 4b3 q^59 + (-7b7 + 7b3 + 7b1) q^62 + (-5b6 - 7) q^64 + (2b5 + 2b3 + 4b1) q^65 + (6b7 - 10b3) * q^68 + (-b4 - b2) * q^70 + (12b7 - 6b5 - 12b3 - 12b1) * q^71 + 3 * q^73 + (-8b5 - 8b3 - 2b1) q^74 + (-6b6 + 10b4 + 6b2 - 10) q^76 - 3b3 q^77 - 4b4 q^79 + (3b7 - 2b5 - 3b3 - 3b1) * q^80 + (-2b6 + 6) q^82 + (-7b5 - 7b3) * q^83 + (4b6 - 2b4 - 4b2 + 2) q^85 + (-4b7 - 12b3) * q^86 + (-9b4 - 3b2) * q^88 + (-8b7 + 4b5 + 8b3 + 8b1) * q^89 + (-4b6 - 2) q^91 + (4b5 + 4b3 - 6b1) q^92 + (4b7 - 2b3) * q^95 - 7b4 q^97 + (-6b7 + 6b3 + 6b1) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{4} - 4 q^{7}+O(q^{10}) $ 8 * q - 6 * q^4 - 4 * q^7	$ 8 q - 6 q^{4} - 4 q^{7} + 4 q^{10} - 2 q^{16} - 6 q^{22} - 16 q^{25} + 12 q^{28} + 28 q^{31} + 28 q^{34} - 10 q^{40} - 56 q^{46} + 24 q^{49} + 28 q^{52} - 24 q^{55} - 12 q^{58} - 36 q^{64} - 2 q^{70} + 24 q^{73} - 28 q^{76} - 16 q^{79} + 56 q^{82} - 30 q^{88} - 28 q^{97}+O(q^{100}) $ 8 * q - 6 * q^4 - 4 * q^7 + 4 * q^10 - 2 * q^16 - 6 * q^22 - 16 * q^25 + 12 * q^28 + 28 * q^31 + 28 * q^34 - 10 * q^40 - 56 * q^46 + 24 * q^49 + 28 * q^52 - 24 * q^55 - 12 * q^58 - 36 * q^64 - 2 * q^70 + 24 * q^73 - 28 * q^76 - 16 * q^79 + 56 * q^82 - 30 * q^88 - 28 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 $ x^8 + 3*x^6 + 5*x^4 + 12*x^2 + 16 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( -\nu^{6} + 5\nu^{4} - 5\nu^{2} - 12 ) / 20 $ (-v^6 + 5v^4 - 5v^2 - 12) / 20
$\beta_{3}$	$=$	$ ( -\nu^{7} + 5\nu^{5} - 5\nu^{3} - 12\nu ) / 40 $ (-v^7 + 5v^5 - 5v^3 - 12*v) / 40
$\beta_{4}$	$=$	$ ( 3\nu^{6} + 5\nu^{4} + 15\nu^{2} + 36 ) / 20 $ (3v^6 + 5v^4 + 15*v^2 + 36) / 20
$\beta_{5}$	$=$	$ ( -3\nu^{7} - 5\nu^{5} + 5\nu^{3} - 16\nu ) / 40 $ (-3v^7 - 5v^5 + 5v^3 - 16v) / 40
$\beta_{6}$	$=$	$ ( -\nu^{6} - 7 ) / 5 $ (-v^6 - 7) / 5
$\beta_{7}$	$=$	$ ( \nu^{7} + 3\nu^{5} + 5\nu^{3} + 12\nu ) / 8 $ (v^7 + 3v^5 + 5v^3 + 12*v) / 8

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{6} + \beta_{4} - \beta_{2} - 1 $ b6 + b4 - b2 - 1
$\nu^{3}$	$=$	$ \beta_{7} + 2\beta_{5} - \beta_{3} - \beta_1 $ b7 + 2*b5 - b3 - b1
$\nu^{4}$	$=$	$ \beta_{4} + 3\beta_{2} $ b4 + 3*b2
$\nu^{5}$	$=$	$ \beta_{7} + 5\beta_{3} $ b7 + 5*b3
$\nu^{6}$	$=$	$ -5\beta_{6} - 7 $ -5*b6 - 7
$\nu^{7}$	$=$	$ -10\beta_{5} - 10\beta_{3} - 7\beta_1 $ -10b5 - 10b3 - 7*b1

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$.

$n$	$325$	$487$	$569$
$\chi(n)$	$-1$	$1$	$-1 + \beta_{4}$

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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

109.1

−1.09445	−	0.895644i
−0.228425	−	1.39564i
0.228425	+	1.39564i
1.09445	+	0.895644i
−1.09445	+	0.895644i
−0.228425	+	1.39564i
0.228425	−	1.39564i
1.09445	−	0.895644i

−1.09445

−

0.895644i

0.395644

1.96048i

−0.866025

−

0.500000i

−0.500000

−

0.866025i

1.32288

−

2.50000i

0.500000

1.32288i

109.2

−0.228425

−

1.39564i

−1.89564

0.637600i

0.866025

0.500000i

−0.500000

−

0.866025i

1.32288

2.50000i

0.500000

−

1.32288i

109.3

0.228425

1.39564i

−1.89564

0.637600i

−0.866025

−

0.500000i

−0.500000

−

0.866025i

−1.32288

−

2.50000i

0.500000

−

1.32288i

109.4

1.09445

0.895644i

0.395644

1.96048i

0.866025

0.500000i

−0.500000

−

0.866025i

−1.32288

2.50000i

0.500000

1.32288i

541.1

−1.09445

0.895644i

0.395644

−

1.96048i

−0.866025

0.500000i

−0.500000

0.866025i

1.32288

2.50000i

0.500000

−

1.32288i

541.2

−0.228425

1.39564i

−1.89564

−

0.637600i

0.866025

−

0.500000i

−0.500000

0.866025i

1.32288

−

2.50000i

0.500000

1.32288i

541.3

0.228425

−

1.39564i

−1.89564

−

0.637600i

−0.866025

0.500000i

−0.500000

0.866025i

−1.32288

2.50000i

0.500000

1.32288i

541.4

1.09445

−

0.895644i

0.395644

−

1.96048i

0.866025

−

0.500000i

−0.500000

0.866025i

−1.32288

−

2.50000i

0.500000

−

1.32288i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
8.b	even	2	1	inner
9.c	even	3	1	inner
9.d	odd	6	1	inner
24.h	odd	2	1	inner
72.j	odd	6	1	inner
72.n	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	648.2.n.n		8
3.b	odd	2	1	inner	648.2.n.n		8
4.b	odd	2	1		2592.2.r.p		8
8.b	even	2	1	inner	648.2.n.n		8
8.d	odd	2	1		2592.2.r.p		8
9.c	even	3	1		216.2.d.b	✓	4
9.c	even	3	1	inner	648.2.n.n		8
9.d	odd	6	1		216.2.d.b	✓	4
9.d	odd	6	1	inner	648.2.n.n		8
12.b	even	2	1		2592.2.r.p		8
24.f	even	2	1		2592.2.r.p		8
24.h	odd	2	1	inner	648.2.n.n		8
36.f	odd	6	1		864.2.d.a		4
36.f	odd	6	1		2592.2.r.p		8
36.h	even	6	1		864.2.d.a		4
36.h	even	6	1		2592.2.r.p		8
72.j	odd	6	1		216.2.d.b	✓	4
72.j	odd	6	1	inner	648.2.n.n		8
72.l	even	6	1		864.2.d.a		4
72.l	even	6	1		2592.2.r.p		8
72.n	even	6	1		216.2.d.b	✓	4
72.n	even	6	1	inner	648.2.n.n		8
72.p	odd	6	1		864.2.d.a		4
72.p	odd	6	1		2592.2.r.p		8
144.u	even	12	1		6912.2.a.bd		2
144.u	even	12	1		6912.2.a.bv		2
144.v	odd	12	1		6912.2.a.bd		2
144.v	odd	12	1		6912.2.a.bv		2
144.w	odd	12	1		6912.2.a.bc		2
144.w	odd	12	1		6912.2.a.bu		2
144.x	even	12	1		6912.2.a.bc		2
144.x	even	12	1		6912.2.a.bu		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
216.2.d.b	✓	4	9.c	even	3	1
216.2.d.b	✓	4	9.d	odd	6	1
216.2.d.b	✓	4	72.j	odd	6	1
216.2.d.b	✓	4	72.n	even	6	1
648.2.n.n		8	1.a	even	1	1	trivial
648.2.n.n		8	3.b	odd	2	1	inner
648.2.n.n		8	8.b	even	2	1	inner
648.2.n.n		8	9.c	even	3	1	inner
648.2.n.n		8	9.d	odd	6	1	inner
648.2.n.n		8	24.h	odd	2	1	inner
648.2.n.n		8	72.j	odd	6	1	inner
648.2.n.n		8	72.n	even	6	1	inner
864.2.d.a		4	36.f	odd	6	1
864.2.d.a		4	36.h	even	6	1
864.2.d.a		4	72.l	even	6	1
864.2.d.a		4	72.p	odd	6	1
2592.2.r.p		8	4.b	odd	2	1
2592.2.r.p		8	8.d	odd	2	1
2592.2.r.p		8	12.b	even	2	1
2592.2.r.p		8	24.f	even	2	1
2592.2.r.p		8	36.f	odd	6	1
2592.2.r.p		8	36.h	even	6	1
2592.2.r.p		8	72.l	even	6	1
2592.2.r.p		8	72.p	odd	6	1
6912.2.a.bc		2	144.w	odd	12	1
6912.2.a.bc		2	144.x	even	12	1
6912.2.a.bd		2	144.u	even	12	1
6912.2.a.bd		2	144.v	odd	12	1
6912.2.a.bu		2	144.w	odd	12	1
6912.2.a.bu		2	144.x	even	12	1
6912.2.a.bv		2	144.u	even	12	1
6912.2.a.bv		2	144.v	odd	12	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(648, [\chi])$:

$ T_{5}^{4} - T_{5}^{2} + 1 $

$ T_{13}^{4} - 28T_{13}^{2} + 784 $

$ T_{17}^{2} - 28 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} + 3 T^{6} + \cdots + 16 $ T^8 + 3T^6 + 5T^4 + 12*T^2 + 16
$3$	$ T^{8} $ T^8
$5$	$ (T^{4} - T^{2} + 1)^{2} $ (T^4 - T^2 + 1)^2
$7$	$ (T^{2} + T + 1)^{4} $ (T^2 + T + 1)^4
$11$	$ (T^{4} - 9 T^{2} + 81)^{2} $ (T^4 - 9*T^2 + 81)^2
$13$	$ (T^{4} - 28 T^{2} + 784)^{2} $ (T^4 - 28*T^2 + 784)^2
$17$	$ (T^{2} - 28)^{4} $ (T^2 - 28)^4
$19$	$ (T^{2} + 28)^{4} $ (T^2 + 28)^4
$23$	$ (T^{4} + 28 T^{2} + 784)^{2} $ (T^4 + 28*T^2 + 784)^2
$29$	$ (T^{4} - 36 T^{2} + 1296)^{2} $ (T^4 - 36*T^2 + 1296)^2
$31$	$ (T^{2} - 7 T + 49)^{4} $ (T^2 - 7*T + 49)^4
$37$	$ (T^{2} + 28)^{4} $ (T^2 + 28)^4
$41$	$ (T^{4} + 28 T^{2} + 784)^{2} $ (T^4 + 28*T^2 + 784)^2
$43$	$ (T^{4} - 112 T^{2} + 12544)^{2} $ (T^4 - 112*T^2 + 12544)^2
$47$	$ T^{8} $ T^8
$53$	$ (T^{2} + 81)^{4} $ (T^2 + 81)^4
$59$	$ (T^{4} - 16 T^{2} + 256)^{2} $ (T^4 - 16*T^2 + 256)^2
$61$	$ T^{8} $ T^8
$67$	$ T^{8} $ T^8
$71$	$ (T^{2} - 252)^{4} $ (T^2 - 252)^4
$73$	$ (T - 3)^{8} $ (T - 3)^8
$79$	$ (T^{2} + 4 T + 16)^{4} $ (T^2 + 4*T + 16)^4
$83$	$ (T^{4} - 49 T^{2} + 2401)^{2} $ (T^4 - 49*T^2 + 2401)^2
$89$	$ (T^{2} - 112)^{4} $ (T^2 - 112)^4
$97$	$ (T^{2} + 7 T + 49)^{4} $ (T^2 + 7*T + 49)^4
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Elliptic curves over $\Q$
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Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	648.n (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + (\beta_{6} + \beta_{4} - \beta_{2} - 1) q^{4} - \beta_{3} q^{5} - \beta_{4} q^{7} + (\beta_{7} + 2 \beta_{5} + \cdots - \beta_1) q^{8}+O(q^{10}) \) q + b1 * q^2 + (b6 + b4 - b2 - 1) * q^4 - b3 * q^5 - b4 * q^7 + (b7 + 2b5 - b3 - b1) q^8	\( q + \beta_1 q^{2} + (\beta_{6} + \beta_{4} - \beta_{2} - 1) q^{4} - \beta_{3} q^{5} - \beta_{4} q^{7} + (\beta_{7} + 2 \beta_{5} + \cdots - \beta_1) q^{8}+ \cdots + ( - 6 \beta_{7} + 6 \beta_{3} + 6 \beta_1) q^{98}+O(q^{100}) \) q + b1 * q^2 + (b6 + b4 - b2 - 1) * q^4 - b3 * q^5 - b4 * q^7 + (b7 + 2b5 - b3 - b1) q^8 + (b6 + 1) * q^10 + (3b5 + 3b3) * q^11 + (4b6 - 2b4 - 4b2 + 2) q^13 + (-b7 + b3) * q^14 + (b4 + 3b2) q^16 + (-4b7 + 2b5 + 4b3 + 4b1) * q^17 + (4b6 + 2) q^19 + (2b5 + 2b3 + b1) * q^20 + (-3b6 + 3b4 + 3b2 - 3) q^22 + (4b7 - 2b3) * q^23 - 4b4 q^25 + (-2b7 + 8b5 + 2b3 + 2b1) * q^26 + (-b6 + 1) * q^28 + (6b5 + 6b3) * q^29 + (-7b4 + 7) q^31 + (b7 + 5b3) q^32 + (6b4 - 2b2) * q^34 - b5 * q^35 + (-4b6 - 2) q^37 + (8b5 + 8b3 + 2b1) q^38 + (-b6 + 3b4 + b2 - 3) q^40 + (-4b7 + 2b3) * q^41 + (-4b4 - 8b2) * q^43 + (3b7 - 6b5 - 3b3 - 3b1) * q^44 + (2b6 - 6) q^46 + (-6b4 + 6) q^49 + (-4b7 + 4b3) * q^50 + (10b4 + 6b2) * q^52 + 9b5 q^53 - 3 * q^55 + (-2b5 - 2b3 + b1) * q^56 + (-6b6 + 6b4 + 6b2 - 6) q^58 + 4b3 q^59 + (-7b7 + 7b3 + 7b1) q^62 + (-5b6 - 7) q^64 + (2b5 + 2b3 + 4b1) q^65 + (6b7 - 10b3) * q^68 + (-b4 - b2) * q^70 + (12b7 - 6b5 - 12b3 - 12b1) * q^71 + 3 * q^73 + (-8b5 - 8b3 - 2b1) q^74 + (-6b6 + 10b4 + 6b2 - 10) q^76 - 3b3 q^77 - 4b4 q^79 + (3b7 - 2b5 - 3b3 - 3b1) * q^80 + (-2b6 + 6) q^82 + (-7b5 - 7b3) * q^83 + (4b6 - 2b4 - 4b2 + 2) q^85 + (-4b7 - 12b3) * q^86 + (-9b4 - 3b2) * q^88 + (-8b7 + 4b5 + 8b3 + 8b1) * q^89 + (-4b6 - 2) q^91 + (4b5 + 4b3 - 6b1) q^92 + (4b7 - 2b3) * q^95 - 7b4 q^97 + (-6b7 + 6b3 + 6b1) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{4} - 4 q^{7}+O(q^{10}) \) 8 * q - 6 * q^4 - 4 * q^7	\( 8 q - 6 q^{4} - 4 q^{7} + 4 q^{10} - 2 q^{16} - 6 q^{22} - 16 q^{25} + 12 q^{28} + 28 q^{31} + 28 q^{34} - 10 q^{40} - 56 q^{46} + 24 q^{49} + 28 q^{52} - 24 q^{55} - 12 q^{58} - 36 q^{64} - 2 q^{70} + 24 q^{73} - 28 q^{76} - 16 q^{79} + 56 q^{82} - 30 q^{88} - 28 q^{97}+O(q^{100}) \) 8 * q - 6 * q^4 - 4 * q^7 + 4 * q^10 - 2 * q^16 - 6 * q^22 - 16 * q^25 + 12 * q^28 + 28 * q^31 + 28 * q^34 - 10 * q^40 - 56 * q^46 + 24 * q^49 + 28 * q^52 - 24 * q^55 - 12 * q^58 - 36 * q^64 - 2 * q^70 + 24 * q^73 - 28 * q^76 - 16 * q^79 + 56 * q^82 - 30 * q^88 - 28 * q^97

Introduction

L-functions

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Label	648.2.n.n

Level	$648$
Weight	$2$
Character orbit	648.n
Analytic conductor	$5.174$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(5.17430605098\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{6})\)
Coefficient field:	8.0.49787136.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) x^8 + 3x^6 + 5x^4 + 12*x^2 + 16
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( -\nu^{6} + 5\nu^{4} - 5\nu^{2} - 12 ) / 20 \) (-v^6 + 5v^4 - 5v^2 - 12) / 20
\(\beta_{3}\)	\(=\)	\( ( -\nu^{7} + 5\nu^{5} - 5\nu^{3} - 12\nu ) / 40 \) (-v^7 + 5v^5 - 5v^3 - 12*v) / 40
\(\beta_{4}\)	\(=\)	\( ( 3\nu^{6} + 5\nu^{4} + 15\nu^{2} + 36 ) / 20 \) (3v^6 + 5v^4 + 15*v^2 + 36) / 20
\(\beta_{5}\)	\(=\)	\( ( -3\nu^{7} - 5\nu^{5} + 5\nu^{3} - 16\nu ) / 40 \) (-3v^7 - 5v^5 + 5v^3 - 16v) / 40
\(\beta_{6}\)	\(=\)	\( ( -\nu^{6} - 7 ) / 5 \) (-v^6 - 7) / 5
\(\beta_{7}\)	\(=\)	\( ( \nu^{7} + 3\nu^{5} + 5\nu^{3} + 12\nu ) / 8 \) (v^7 + 3v^5 + 5v^3 + 12*v) / 8

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{6} + \beta_{4} - \beta_{2} - 1 \) b6 + b4 - b2 - 1
\(\nu^{3}\)	\(=\)	\( \beta_{7} + 2\beta_{5} - \beta_{3} - \beta_1 \) b7 + 2*b5 - b3 - b1
\(\nu^{4}\)	\(=\)	\( \beta_{4} + 3\beta_{2} \) b4 + 3*b2
\(\nu^{5}\)	\(=\)	\( \beta_{7} + 5\beta_{3} \) b7 + 5*b3
\(\nu^{6}\)	\(=\)	\( -5\beta_{6} - 7 \) -5*b6 - 7
\(\nu^{7}\)	\(=\)	\( -10\beta_{5} - 10\beta_{3} - 7\beta_1 \) -10b5 - 10b3 - 7*b1

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1 + \beta_{4}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 109.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} + 3 T^{6} + \cdots + 16 \) T^8 + 3T^6 + 5T^4 + 12*T^2 + 16
$3$	\( T^{8} \) T^8
$5$	\( (T^{4} - T^{2} + 1)^{2} \) (T^4 - T^2 + 1)^2
$7$	\( (T^{2} + T + 1)^{4} \) (T^2 + T + 1)^4
$11$	\( (T^{4} - 9 T^{2} + 81)^{2} \) (T^4 - 9*T^2 + 81)^2
$13$	\( (T^{4} - 28 T^{2} + 784)^{2} \) (T^4 - 28*T^2 + 784)^2
$17$	\( (T^{2} - 28)^{4} \) (T^2 - 28)^4
$19$	\( (T^{2} + 28)^{4} \) (T^2 + 28)^4
$23$	\( (T^{4} + 28 T^{2} + 784)^{2} \) (T^4 + 28*T^2 + 784)^2
$29$	\( (T^{4} - 36 T^{2} + 1296)^{2} \) (T^4 - 36*T^2 + 1296)^2
$31$	\( (T^{2} - 7 T + 49)^{4} \) (T^2 - 7*T + 49)^4
$37$	\( (T^{2} + 28)^{4} \) (T^2 + 28)^4
$41$	\( (T^{4} + 28 T^{2} + 784)^{2} \) (T^4 + 28*T^2 + 784)^2
$43$	\( (T^{4} - 112 T^{2} + 12544)^{2} \) (T^4 - 112*T^2 + 12544)^2
$47$	\( T^{8} \) T^8
$53$	\( (T^{2} + 81)^{4} \) (T^2 + 81)^4
$59$	\( (T^{4} - 16 T^{2} + 256)^{2} \) (T^4 - 16*T^2 + 256)^2
$61$	\( T^{8} \) T^8
$67$	\( T^{8} \) T^8
$71$	\( (T^{2} - 252)^{4} \) (T^2 - 252)^4
$73$	\( (T - 3)^{8} \) (T - 3)^8
$79$	\( (T^{2} + 4 T + 16)^{4} \) (T^2 + 4*T + 16)^4
$83$	\( (T^{4} - 49 T^{2} + 2401)^{2} \) (T^4 - 49*T^2 + 2401)^2
$89$	\( (T^{2} - 112)^{4} \) (T^2 - 112)^4
$97$	\( (T^{2} + 7 T + 49)^{4} \) (T^2 + 7*T + 49)^4
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