LMFDB - Newform orbit 1859.1.i.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1859,1,Mod(868,1859)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1859, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("1859.868");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1859 = 11 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1859.i (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:	$0.927761858485$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	12.0.17213603549184.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 $ x^12 - 5x^10 + 19x^8 - 28x^6 + 31x^4 - 6*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{7}$
Projective field:	Galois closure of 7.1.6424482779.1
Artin image:	$C_{12}\times D_7$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{84} - \cdots)$

$q$-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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\beta_{5} + \beta_{4}) q^{3} + ( - \beta_{7} + 1) q^{4} + (\beta_{11} + \beta_{10} + \cdots + \beta_{2}) q^{5}+ \cdots + (\beta_{9} + \beta_{4}) q^{9}+O(q^{10}) $ q + (b5 + b4) * q^3 + (-b7 + 1) * q^4 + (b11 + b10 + b8 - b6 + b2) * q^5 + (b9 + b4) * q^9	$ q + (\beta_{5} + \beta_{4}) q^{3} + ( - \beta_{7} + 1) q^{4} + (\beta_{11} + \beta_{10} + \cdots + \beta_{2}) q^{5}+ \cdots + ( - \beta_{11} - \beta_{8} - \beta_{2}) q^{99}+O(q^{100}) $ q + (b5 + b4) * q^3 + (-b7 + 1) * q^4 + (b11 + b10 + b8 - b6 + b2) * q^5 + (b9 + b4) * q^9 - b10 * q^11 + b5 * q^12 + (b11 + b8 - b1) * q^15 - b7 * q^16 + (-b6 + b2 + b1) * q^20 + (-b9 + b5 + b3 - 1) * q^23 + (-b5 - b3) * q^25 + (b3 - 1) * q^27 + (-b11 - b2) * q^31 - b1 * q^33 + (b9 + b4 - b3 + 1) * q^36 + (-b8 + b1) * q^37 + (-b10 + b6) * q^44 + (-b6 - b1) * q^45 + (b11 + b2) * q^47 - b4 * q^48 + b7 * q^49 - b3 * q^53 + (-b9 + b7 - b4 + b3 - 1) * q^55 - b1 * q^59 + (b11 + b8 + b2) * q^60 - q^64 + 2b10 q^67 + (b7 - b4 - 1) * q^69 + b1 * q^71 - b7 * q^75 + (-b11 - b10 - b8 + b1) * q^80 + (-b5 - b4) * q^81 - b11 * q^89 + (b5 + b3 - 1) * q^92 + (-b10 - b8 + b1) * q^93 + (b6 - b2 - b1) * q^97 + (-b11 - b8 - b2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 2 q^{3} + 6 q^{4} - 4 q^{9}+O(q^{10}) $ 12 * q + 2 * q^3 + 6 * q^4 - 4 * q^9	$ 12 q + 2 q^{3} + 6 q^{4} - 4 q^{9} + 4 q^{12} - 6 q^{16} - 2 q^{23} - 8 q^{25} - 8 q^{27} + 4 q^{36} + 2 q^{48} + 6 q^{49} - 4 q^{53} + 2 q^{55} - 12 q^{64} - 4 q^{69} - 6 q^{75} - 2 q^{81} - 4 q^{92}+O(q^{100}) $ 12 * q + 2 * q^3 + 6 * q^4 - 4 * q^9 + 4 * q^12 - 6 * q^16 - 2 * q^23 - 8 * q^25 - 8 * q^27 + 4 * q^36 + 2 * q^48 + 6 * q^49 - 4 * q^53 + 2 * q^55 - 12 * q^64 - 4 * q^69 - 6 * q^75 - 2 * q^81 - 4 * q^92

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 $ x^12 - 5*x^10 + 19*x^8 - 28*x^6 + 31*x^4 - 6*x^2 + 1 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( -25\nu^{11} + 95\nu^{9} - 361\nu^{7} + 155\nu^{5} - 30\nu^{3} - 1563\nu ) / 559 $ (-25v^11 + 95v^9 - 361v^7 + 155v^5 - 30v^3 - 1563v) / 559
$\beta_{3}$	$=$	$ ( 25\nu^{10} - 95\nu^{8} + 361\nu^{6} - 155\nu^{4} + 30\nu^{2} + 1004 ) / 559 $ (25v^10 - 95v^8 + 361v^6 - 155v^4 + 30*v^2 + 1004) / 559
$\beta_{4}$	$=$	$ ( 3\nu^{10} - 20\nu^{8} + 76\nu^{6} - 139\nu^{4} + 124\nu^{2} - 24 ) / 43 $ (3v^10 - 20v^8 + 76v^6 - 139v^4 + 124*v^2 - 24) / 43
$\beta_{5}$	$=$	$ ( 45\nu^{10} - 171\nu^{8} + 538\nu^{6} - 279\nu^{4} + 54\nu^{2} + 242 ) / 559 $ (45v^10 - 171v^8 + 538v^6 - 279v^4 + 54*v^2 + 242) / 559
$\beta_{6}$	$=$	$ ( 70\nu^{11} - 266\nu^{9} + 899\nu^{7} - 434\nu^{5} + 84\nu^{3} + 1246\nu ) / 559 $ (70v^11 - 266v^9 + 899v^7 - 434v^5 + 84v^3 + 1246v) / 559
$\beta_{7}$	$=$	$ ( 114\nu^{10} - 545\nu^{8} + 2071\nu^{6} - 2831\nu^{4} + 3379\nu^{2} - 95 ) / 559 $ (114v^10 - 545v^8 + 2071v^6 - 2831v^4 + 3379*v^2 - 95) / 559
$\beta_{8}$	$=$	$ ( 114\nu^{11} - 545\nu^{9} + 2071\nu^{7} - 2831\nu^{5} + 3379\nu^{3} - 95\nu ) / 559 $ (114v^11 - 545v^9 + 2071v^7 - 2831v^5 + 3379v^3 - 95v) / 559
$\beta_{9}$	$=$	$ ( -128\nu^{10} + 710\nu^{8} - 2698\nu^{6} + 4483\nu^{4} - 4402\nu^{2} + 852 ) / 559 $ (-128v^10 + 710v^8 - 2698v^6 + 4483v^4 - 4402*v^2 + 852) / 559
$\beta_{10}$	$=$	$ ( -242\nu^{11} + 1255\nu^{9} - 4769\nu^{7} + 7314\nu^{5} - 7781\nu^{3} + 1506\nu ) / 559 $ (-242v^11 + 1255v^9 - 4769v^7 + 7314v^5 - 7781v^3 + 1506v) / 559
$\beta_{11}$	$=$	$ ( -317\nu^{11} + 1540\nu^{9} - 5852\nu^{7} + 8338\nu^{5} - 9548\nu^{3} + 1848\nu ) / 559 $ (-317v^11 + 1540v^9 - 5852v^7 + 8338v^5 - 9548v^3 + 1848v) / 559

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{9} + \beta_{7} + \beta_{4} - \beta_{3} + 1 $ b9 + b7 + b4 - b3 + 1
$\nu^{3}$	$=$	$ \beta_{11} + 3\beta_{8} + \beta_{2} $ b11 + 3*b8 + b2
$\nu^{4}$	$=$	$ 3\beta_{9} + 2\beta_{7} + 4\beta_{4} - 2 $ 3b9 + 2b7 + 4*b4 - 2
$\nu^{5}$	$=$	$ 4\beta_{11} - \beta_{10} + 9\beta_{8} - 9\beta_1 $ 4b11 - b10 + 9b8 - 9*b1
$\nu^{6}$	$=$	$ -5\beta_{5} + 9\beta_{3} - 14 $ -5b5 + 9b3 - 14
$\nu^{7}$	$=$	$ -5\beta_{6} - 14\beta_{2} - 28\beta_1 $ -5b6 - 14b2 - 28*b1
$\nu^{8}$	$=$	$ -28\beta_{9} - 14\beta_{7} - 19\beta_{5} - 47\beta_{4} + 28\beta_{3} - 28 $ -28b9 - 14b7 - 19b5 - 47b4 + 28*b3 - 28
$\nu^{9}$	$=$	$ -47\beta_{11} + 19\beta_{10} - 89\beta_{8} - 19\beta_{6} - 47\beta_{2} $ -47b11 + 19b10 - 89b8 - 19b6 - 47*b2
$\nu^{10}$	$=$	$ -89\beta_{9} - 42\beta_{7} - 155\beta_{4} + 42 $ -89b9 - 42b7 - 155*b4 + 42
$\nu^{11}$	$=$	$ -155\beta_{11} + 66\beta_{10} - 286\beta_{8} + 286\beta_1 $ -155b11 + 66b10 - 286b8 + 286b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1859\mathbb{Z}\right)^\times$.

$n$	$508$	$1354$
$\chi(n)$	$-1$	$1 - \beta_{7}$

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} $

868.1

−1.07992	−	0.623490i
1.07992	+	0.623490i
0.385418	+	0.222521i
−0.385418	−	0.222521i
−1.56052	−	0.900969i
1.56052	+	0.900969i
1.07992	−	0.623490i
−1.07992	+	0.623490i
−0.385418	+	0.222521i
0.385418	−	0.222521i
1.56052	−	0.900969i
−1.56052	+	0.900969i

−0.623490

−

1.07992i

0.500000

−

0.866025i

−

0.445042i

−0.277479

0.480608i

868.2

−0.623490

−

1.07992i

0.500000

−

0.866025i

0.445042i

−0.277479

0.480608i

868.3

0.222521

0.385418i

0.500000

−

0.866025i

−

1.80194i

0.400969

−

0.694498i

868.4

0.222521

0.385418i

0.500000

−

0.866025i

1.80194i

0.400969

−

0.694498i

868.5

0.900969

1.56052i

0.500000

−

0.866025i

−

1.24698i

−1.12349

1.94594i

868.6

0.900969

1.56052i

0.500000

−

0.866025i

1.24698i

−1.12349

1.94594i

1330.1

−0.623490

1.07992i

0.500000

0.866025i

−

0.445042i

−0.277479

−

0.480608i

1330.2

−0.623490

1.07992i

0.500000

0.866025i

0.445042i

−0.277479

−

0.480608i

1330.3

0.222521

−

0.385418i

0.500000

0.866025i

−

1.80194i

0.400969

0.694498i

1330.4

0.222521

−

0.385418i

0.500000

0.866025i

1.80194i

0.400969

0.694498i

1330.5

0.900969

−

1.56052i

0.500000

0.866025i

−

1.24698i

−1.12349

−

1.94594i

1330.6

0.900969

−

1.56052i

0.500000

0.866025i

1.24698i

−1.12349

−

1.94594i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	CM by $\Q(\sqrt{-11}) $
13.b	even	2	1	inner
13.c	even	3	1	inner
13.e	even	6	1	inner
143.d	odd	2	1	inner
143.i	odd	6	1	inner
143.k	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1859.1.i.c		12
11.b	odd	2	1	CM	1859.1.i.c		12
13.b	even	2	1	inner	1859.1.i.c		12
13.c	even	3	1		1859.1.d.a		6
13.c	even	3	1	inner	1859.1.i.c		12
13.d	odd	4	1		1859.1.k.a		6
13.d	odd	4	1		1859.1.k.b		6
13.e	even	6	1		1859.1.d.a		6
13.e	even	6	1	inner	1859.1.i.c		12
13.f	odd	12	1		1859.1.c.a	✓	3
13.f	odd	12	1		1859.1.c.b	yes	3
13.f	odd	12	1		1859.1.k.a		6
13.f	odd	12	1		1859.1.k.b		6
143.d	odd	2	1	inner	1859.1.i.c		12
143.g	even	4	1		1859.1.k.a		6
143.g	even	4	1		1859.1.k.b		6
143.i	odd	6	1		1859.1.d.a		6
143.i	odd	6	1	inner	1859.1.i.c		12
143.k	odd	6	1		1859.1.d.a		6
143.k	odd	6	1	inner	1859.1.i.c		12
143.o	even	12	1		1859.1.c.a	✓	3
143.o	even	12	1		1859.1.c.b	yes	3
143.o	even	12	1		1859.1.k.a		6
143.o	even	12	1		1859.1.k.b		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1859.1.c.a	✓	3	13.f	odd	12	1
1859.1.c.a	✓	3	143.o	even	12	1
1859.1.c.b	yes	3	13.f	odd	12	1
1859.1.c.b	yes	3	143.o	even	12	1
1859.1.d.a		6	13.c	even	3	1
1859.1.d.a		6	13.e	even	6	1
1859.1.d.a		6	143.i	odd	6	1
1859.1.d.a		6	143.k	odd	6	1
1859.1.i.c		12	1.a	even	1	1	trivial
1859.1.i.c		12	11.b	odd	2	1	CM
1859.1.i.c		12	13.b	even	2	1	inner
1859.1.i.c		12	13.c	even	3	1	inner
1859.1.i.c		12	13.e	even	6	1	inner
1859.1.i.c		12	143.d	odd	2	1	inner
1859.1.i.c		12	143.i	odd	6	1	inner
1859.1.i.c		12	143.k	odd	6	1	inner
1859.1.k.a		6	13.d	odd	4	1
1859.1.k.a		6	13.f	odd	12	1
1859.1.k.a		6	143.g	even	4	1
1859.1.k.a		6	143.o	even	12	1
1859.1.k.b		6	13.d	odd	4	1
1859.1.k.b		6	13.f	odd	12	1
1859.1.k.b		6	143.g	even	4	1
1859.1.k.b		6	143.o	even	12	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2} $ T2 acting on $S_{1}^{\mathrm{new}}(1859, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ (T^{6} - T^{5} + 3 T^{4} + \cdots + 1)^{2} $ (T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1)^2
$5$	$ (T^{6} + 5 T^{4} + 6 T^{2} + 1)^{2} $ (T^6 + 5T^4 + 6T^2 + 1)^2
$7$	$ T^{12} $ T^12
$11$	$ (T^{4} - T^{2} + 1)^{3} $ (T^4 - T^2 + 1)^3
$13$	$ T^{12} $ T^12
$17$	$ T^{12} $ T^12
$19$	$ T^{12} $ T^12
$23$	$ (T^{6} + T^{5} + 3 T^{4} + \cdots + 1)^{2} $ (T^6 + T^5 + 3T^4 + 5T^2 + 2*T + 1)^2
$29$	$ T^{12} $ T^12
$31$	$ (T^{6} + 5 T^{4} + 6 T^{2} + 1)^{2} $ (T^6 + 5T^4 + 6T^2 + 1)^2
$37$	$ T^{12} - 5 T^{10} + \cdots + 1 $ T^12 - 5T^10 + 19T^8 - 28T^6 + 31T^4 - 6*T^2 + 1
$41$	$ T^{12} $ T^12
$43$	$ T^{12} $ T^12
$47$	$ (T^{6} + 5 T^{4} + 6 T^{2} + 1)^{2} $ (T^6 + 5T^4 + 6T^2 + 1)^2
$53$	$ (T^{3} + T^{2} - 2 T - 1)^{4} $ (T^3 + T^2 - 2*T - 1)^4
$59$	$ T^{12} - 5 T^{10} + \cdots + 1 $ T^12 - 5T^10 + 19T^8 - 28T^6 + 31T^4 - 6*T^2 + 1
$61$	$ T^{12} $ T^12
$67$	$ (T^{4} - 4 T^{2} + 16)^{3} $ (T^4 - 4*T^2 + 16)^3
$71$	$ T^{12} - 5 T^{10} + \cdots + 1 $ T^12 - 5T^10 + 19T^8 - 28T^6 + 31T^4 - 6*T^2 + 1
$73$	$ T^{12} $ T^12
$79$	$ T^{12} $ T^12
$83$	$ T^{12} $ T^12
$89$	$ T^{12} - 5 T^{10} + \cdots + 1 $ T^12 - 5T^10 + 19T^8 - 28T^6 + 31T^4 - 6*T^2 + 1
$97$	$ T^{12} - 5 T^{10} + \cdots + 1 $ T^12 - 5T^10 + 19T^8 - 28T^6 + 31T^4 - 6*T^2 + 1
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Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1859 = 11 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1859.i (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{5} + \beta_{4}) q^{3} + ( - \beta_{7} + 1) q^{4} + (\beta_{11} + \beta_{10} + \cdots + \beta_{2}) q^{5}+ \cdots + (\beta_{9} + \beta_{4}) q^{9}+O(q^{10}) \) q + (b5 + b4) * q^3 + (-b7 + 1) * q^4 + (b11 + b10 + b8 - b6 + b2) * q^5 + (b9 + b4) * q^9	\( q + (\beta_{5} + \beta_{4}) q^{3} + ( - \beta_{7} + 1) q^{4} + (\beta_{11} + \beta_{10} + \cdots + \beta_{2}) q^{5}+ \cdots + ( - \beta_{11} - \beta_{8} - \beta_{2}) q^{99}+O(q^{100}) \) q + (b5 + b4) * q^3 + (-b7 + 1) * q^4 + (b11 + b10 + b8 - b6 + b2) * q^5 + (b9 + b4) * q^9 - b10 * q^11 + b5 * q^12 + (b11 + b8 - b1) * q^15 - b7 * q^16 + (-b6 + b2 + b1) * q^20 + (-b9 + b5 + b3 - 1) * q^23 + (-b5 - b3) * q^25 + (b3 - 1) * q^27 + (-b11 - b2) * q^31 - b1 * q^33 + (b9 + b4 - b3 + 1) * q^36 + (-b8 + b1) * q^37 + (-b10 + b6) * q^44 + (-b6 - b1) * q^45 + (b11 + b2) * q^47 - b4 * q^48 + b7 * q^49 - b3 * q^53 + (-b9 + b7 - b4 + b3 - 1) * q^55 - b1 * q^59 + (b11 + b8 + b2) * q^60 - q^64 + 2b10 q^67 + (b7 - b4 - 1) * q^69 + b1 * q^71 - b7 * q^75 + (-b11 - b10 - b8 + b1) * q^80 + (-b5 - b4) * q^81 - b11 * q^89 + (b5 + b3 - 1) * q^92 + (-b10 - b8 + b1) * q^93 + (b6 - b2 - b1) * q^97 + (-b11 - b8 - b2) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2 q^{3} + 6 q^{4} - 4 q^{9}+O(q^{10}) \) 12 * q + 2 * q^3 + 6 * q^4 - 4 * q^9	\( 12 q + 2 q^{3} + 6 q^{4} - 4 q^{9} + 4 q^{12} - 6 q^{16} - 2 q^{23} - 8 q^{25} - 8 q^{27} + 4 q^{36} + 2 q^{48} + 6 q^{49} - 4 q^{53} + 2 q^{55} - 12 q^{64} - 4 q^{69} - 6 q^{75} - 2 q^{81} - 4 q^{92}+O(q^{100}) \) 12 * q + 2 * q^3 + 6 * q^4 - 4 * q^9 + 4 * q^12 - 6 * q^16 - 2 * q^23 - 8 * q^25 - 8 * q^27 + 4 * q^36 + 2 * q^48 + 6 * q^49 - 4 * q^53 + 2 * q^55 - 12 * q^64 - 4 * q^69 - 6 * q^75 - 2 * q^81 - 4 * q^92

Introduction

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Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1859.1.i.c

Level	$1859$
Weight	$1$
Character orbit	1859.i
Analytic conductor	$0.928$
Analytic rank	$0$
Dimension	$12$
Projective image	$D_{7}$
CM discriminant	-11
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(0.927761858485\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{6})\)
Coefficient field:	12.0.17213603549184.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 \) x^12 - 5x^10 + 19x^8 - 28x^6 + 31x^4 - 6*x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{7}\)
Projective field:	Galois closure of 7.1.6424482779.1
Artin image:	$C_{12}\times D_7$
Artin field:	Galois closure of \(\mathbb{Q}[x]/(x^{84} - \cdots)\)

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( -25\nu^{11} + 95\nu^{9} - 361\nu^{7} + 155\nu^{5} - 30\nu^{3} - 1563\nu ) / 559 \) (-25v^11 + 95v^9 - 361v^7 + 155v^5 - 30v^3 - 1563v) / 559
\(\beta_{3}\)	\(=\)	\( ( 25\nu^{10} - 95\nu^{8} + 361\nu^{6} - 155\nu^{4} + 30\nu^{2} + 1004 ) / 559 \) (25v^10 - 95v^8 + 361v^6 - 155v^4 + 30*v^2 + 1004) / 559
\(\beta_{4}\)	\(=\)	\( ( 3\nu^{10} - 20\nu^{8} + 76\nu^{6} - 139\nu^{4} + 124\nu^{2} - 24 ) / 43 \) (3v^10 - 20v^8 + 76v^6 - 139v^4 + 124*v^2 - 24) / 43
\(\beta_{5}\)	\(=\)	\( ( 45\nu^{10} - 171\nu^{8} + 538\nu^{6} - 279\nu^{4} + 54\nu^{2} + 242 ) / 559 \) (45v^10 - 171v^8 + 538v^6 - 279v^4 + 54*v^2 + 242) / 559
\(\beta_{6}\)	\(=\)	\( ( 70\nu^{11} - 266\nu^{9} + 899\nu^{7} - 434\nu^{5} + 84\nu^{3} + 1246\nu ) / 559 \) (70v^11 - 266v^9 + 899v^7 - 434v^5 + 84v^3 + 1246v) / 559
\(\beta_{7}\)	\(=\)	\( ( 114\nu^{10} - 545\nu^{8} + 2071\nu^{6} - 2831\nu^{4} + 3379\nu^{2} - 95 ) / 559 \) (114v^10 - 545v^8 + 2071v^6 - 2831v^4 + 3379*v^2 - 95) / 559
\(\beta_{8}\)	\(=\)	\( ( 114\nu^{11} - 545\nu^{9} + 2071\nu^{7} - 2831\nu^{5} + 3379\nu^{3} - 95\nu ) / 559 \) (114v^11 - 545v^9 + 2071v^7 - 2831v^5 + 3379v^3 - 95v) / 559
\(\beta_{9}\)	\(=\)	\( ( -128\nu^{10} + 710\nu^{8} - 2698\nu^{6} + 4483\nu^{4} - 4402\nu^{2} + 852 ) / 559 \) (-128v^10 + 710v^8 - 2698v^6 + 4483v^4 - 4402*v^2 + 852) / 559
\(\beta_{10}\)	\(=\)	\( ( -242\nu^{11} + 1255\nu^{9} - 4769\nu^{7} + 7314\nu^{5} - 7781\nu^{3} + 1506\nu ) / 559 \) (-242v^11 + 1255v^9 - 4769v^7 + 7314v^5 - 7781v^3 + 1506v) / 559
\(\beta_{11}\)	\(=\)	\( ( -317\nu^{11} + 1540\nu^{9} - 5852\nu^{7} + 8338\nu^{5} - 9548\nu^{3} + 1848\nu ) / 559 \) (-317v^11 + 1540v^9 - 5852v^7 + 8338v^5 - 9548v^3 + 1848v) / 559

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{9} + \beta_{7} + \beta_{4} - \beta_{3} + 1 \) b9 + b7 + b4 - b3 + 1
\(\nu^{3}\)	\(=\)	\( \beta_{11} + 3\beta_{8} + \beta_{2} \) b11 + 3*b8 + b2
\(\nu^{4}\)	\(=\)	\( 3\beta_{9} + 2\beta_{7} + 4\beta_{4} - 2 \) 3b9 + 2b7 + 4*b4 - 2
\(\nu^{5}\)	\(=\)	\( 4\beta_{11} - \beta_{10} + 9\beta_{8} - 9\beta_1 \) 4b11 - b10 + 9b8 - 9*b1
\(\nu^{6}\)	\(=\)	\( -5\beta_{5} + 9\beta_{3} - 14 \) -5b5 + 9b3 - 14
\(\nu^{7}\)	\(=\)	\( -5\beta_{6} - 14\beta_{2} - 28\beta_1 \) -5b6 - 14b2 - 28*b1
\(\nu^{8}\)	\(=\)	\( -28\beta_{9} - 14\beta_{7} - 19\beta_{5} - 47\beta_{4} + 28\beta_{3} - 28 \) -28b9 - 14b7 - 19b5 - 47b4 + 28*b3 - 28
\(\nu^{9}\)	\(=\)	\( -47\beta_{11} + 19\beta_{10} - 89\beta_{8} - 19\beta_{6} - 47\beta_{2} \) -47b11 + 19b10 - 89b8 - 19b6 - 47*b2
\(\nu^{10}\)	\(=\)	\( -89\beta_{9} - 42\beta_{7} - 155\beta_{4} + 42 \) -89b9 - 42b7 - 155*b4 + 42
\(\nu^{11}\)	\(=\)	\( -155\beta_{11} + 66\beta_{10} - 286\beta_{8} + 286\beta_1 \) -155b11 + 66b10 - 286b8 + 286b1

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 868.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( (T^{6} - T^{5} + 3 T^{4} + \cdots + 1)^{2} \) (T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1)^2
$5$	\( (T^{6} + 5 T^{4} + 6 T^{2} + 1)^{2} \) (T^6 + 5T^4 + 6T^2 + 1)^2
$7$	\( T^{12} \) T^12
$11$	\( (T^{4} - T^{2} + 1)^{3} \) (T^4 - T^2 + 1)^3
$13$	\( T^{12} \) T^12
$17$	\( T^{12} \) T^12
$19$	\( T^{12} \) T^12
$23$	\( (T^{6} + T^{5} + 3 T^{4} + \cdots + 1)^{2} \) (T^6 + T^5 + 3T^4 + 5T^2 + 2*T + 1)^2
$29$	\( T^{12} \) T^12
$31$	\( (T^{6} + 5 T^{4} + 6 T^{2} + 1)^{2} \) (T^6 + 5T^4 + 6T^2 + 1)^2
$37$	\( T^{12} - 5 T^{10} + \cdots + 1 \) T^12 - 5T^10 + 19T^8 - 28T^6 + 31T^4 - 6*T^2 + 1
$41$	\( T^{12} \) T^12
$43$	\( T^{12} \) T^12
$47$	\( (T^{6} + 5 T^{4} + 6 T^{2} + 1)^{2} \) (T^6 + 5T^4 + 6T^2 + 1)^2
$53$	\( (T^{3} + T^{2} - 2 T - 1)^{4} \) (T^3 + T^2 - 2*T - 1)^4
$59$	\( T^{12} - 5 T^{10} + \cdots + 1 \) T^12 - 5T^10 + 19T^8 - 28T^6 + 31T^4 - 6*T^2 + 1
$61$	\( T^{12} \) T^12
$67$	\( (T^{4} - 4 T^{2} + 16)^{3} \) (T^4 - 4*T^2 + 16)^3
$71$	\( T^{12} - 5 T^{10} + \cdots + 1 \) T^12 - 5T^10 + 19T^8 - 28T^6 + 31T^4 - 6*T^2 + 1
$73$	\( T^{12} \) T^12
$79$	\( T^{12} \) T^12
$83$	\( T^{12} \) T^12
$89$	\( T^{12} - 5 T^{10} + \cdots + 1 \) T^12 - 5T^10 + 19T^8 - 28T^6 + 31T^4 - 6*T^2 + 1
$97$	\( T^{12} - 5 T^{10} + \cdots + 1 \) T^12 - 5T^10 + 19T^8 - 28T^6 + 31T^4 - 6*T^2 + 1
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\(n\)	\(508\)	\(1354\)
\(\chi(n)\)	\(-1\)	\(1 - \beta_{7}\)