LMFDB - Newform orbit 1183.1.t.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1183,1,Mod(699,1183)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1183.699"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1183, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 5])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 1183 = 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1183.t (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$0.590393909945$
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:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{7}$
Projective field:	Galois closure of 7.1.1655595487.1

$q$-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_{6} + \beta_{2} + \beta_1) q^{2} + ( - \beta_{9} + \beta_{7} + \beta_{5} + \cdots - 1) q^{4} + \beta_{10} q^{7} + (\beta_{11} + \beta_{10} + \cdots + \beta_{2}) q^{8} - \beta_{7} q^{9}+ \cdots + ( - \beta_{11} - \beta_{2}) q^{99}+O(q^{100}) $ q + (-b6 + b2 + b1) * q^2 + (-b9 + b7 + b5 + b3 - 1) * q^4 + b10 * q^7 + (b11 + b10 - b6 + b2) * q^8 - b7 * q^9 + b2 * q^11 - b3 * q^14 + (-b9 + b7 - b4 - 1) * q^16 + (-b11 - b10 - b8 + b6 - b2) * q^18 + (-b9 + b7 + b5 + b3 - 1) * q^22 + b4 * q^23 - q^25 + (b6 - b2) * q^28 + (b9 - b7 + b4 + 1) * q^29 + b10 * q^32 + (b9 - b7 + 1) * q^36 + b1 * q^37 + (b9 - b7 + b4 - b3 + 1) * q^43 + (b11 + 2b10 + b8 - 2b6 + b2) * q^44 + (b11 + b8 - b1) * q^46 + (-b7 + 1) * q^49 + (b6 - b2 - b1) * q^50 - b5 * q^53 + (b9 - b7 - b5 - b3 + 1) * q^56 + (-b11 - 2b10) q^58 - b6 * q^63 - b1 * q^67 + (-b8 + b1) * q^71 + (-b11 - b10) * q^72 + (b9 + b4 - b3 + 1) * q^74 + (-b5 - b3 + 1) * q^77 + (b5 + b3 - 1) * q^79 + (b7 - 1) * q^81 + (-b11 - 2b10 + 2b6 - b2) * q^86 + (-b9 + 2b7 - b4 - 2) q^88 - q^92 + (-b11 - b10 - b8 + b1) * q^98 + (-b11 - b2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{4} - 6 q^{9} - 4 q^{14} - 2 q^{16} + 4 q^{22} - 2 q^{23} - 12 q^{25} + 2 q^{29} + 4 q^{36} - 2 q^{43} + 6 q^{49} - 4 q^{53} - 4 q^{56} + 4 q^{74} + 4 q^{77} - 4 q^{79} - 6 q^{81} - 8 q^{88}+ \cdots - 12 q^{92}+O(q^{100}) $ 12 * q + 4 * q^4 - 6 * q^9 - 4 * q^14 - 2 * q^16 + 4 * q^22 - 2 * q^23 - 12 * q^25 + 2 * q^29 + 4 * q^36 - 2 * q^43 + 6 * q^49 - 4 * q^53 - 4 * q^56 + 4 * q^74 + 4 * q^77 - 4 * q^79 - 6 * q^81 - 8 * q^88 - 12 * q^92

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 $\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times$.

$n$	$339$	$1016$
$\chi(n)$	$-1$	$\beta_{7}$

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} $

699.1

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

−1.56052

0.900969i

1.12349

−

1.94594i

0.866025

0.500000i

2.24698i

−0.500000

0.866025i

699.2

−1.07992

0.623490i

0.277479

−

0.480608i

−0.866025

−

0.500000i

−

0.554958i

−0.500000

0.866025i

699.3

−0.385418

0.222521i

−0.400969

0.694498i

0.866025

0.500000i

−

0.801938i

−0.500000

0.866025i

699.4

0.385418

−

0.222521i

−0.400969

0.694498i

−0.866025

−

0.500000i

0.801938i

−0.500000

0.866025i

699.5

1.07992

−

0.623490i

0.277479

−

0.480608i

0.866025

0.500000i

0.554958i

−0.500000

0.866025i

699.6

1.56052

−

0.900969i

1.12349

−

1.94594i

−0.866025

−

0.500000i

−

2.24698i

−0.500000

0.866025i

1161.1

−1.56052

−

0.900969i

1.12349

1.94594i

0.866025

−

0.500000i

−

2.24698i

−0.500000

−

0.866025i

1161.2

−1.07992

−

0.623490i

0.277479

0.480608i

−0.866025

0.500000i

0.554958i

−0.500000

−

0.866025i

1161.3

−0.385418

−

0.222521i

−0.400969

−

0.694498i

0.866025

−

0.500000i

0.801938i

−0.500000

−

0.866025i

1161.4

0.385418

0.222521i

−0.400969

−

0.694498i

−0.866025

0.500000i

−

0.801938i

−0.500000

−

0.866025i

1161.5

1.07992

0.623490i

0.277479

0.480608i

0.866025

−

0.500000i

−

0.554958i

−0.500000

−

0.866025i

1161.6

1.56052

0.900969i

1.12349

1.94594i

−0.866025

0.500000i

2.24698i

−0.500000

−

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by $\Q(\sqrt{-7}) $
13.b	even	2	1	inner
13.c	even	3	1	inner
13.e	even	6	1	inner
91.b	odd	2	1	inner
91.n	odd	6	1	inner
91.t	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1183.1.t.a		12
7.b	odd	2	1	CM	1183.1.t.a		12
13.b	even	2	1	inner	1183.1.t.a		12
13.c	even	3	1		1183.1.b.a		6
13.c	even	3	1	inner	1183.1.t.a		12
13.d	odd	4	1		1183.1.n.a		6
13.d	odd	4	1		1183.1.n.b		6
13.e	even	6	1		1183.1.b.a		6
13.e	even	6	1	inner	1183.1.t.a		12
13.f	odd	12	1		1183.1.d.a	✓	3
13.f	odd	12	1		1183.1.d.b	yes	3
13.f	odd	12	1		1183.1.n.a		6
13.f	odd	12	1		1183.1.n.b		6
91.b	odd	2	1	inner	1183.1.t.a		12
91.i	even	4	1		1183.1.n.a		6
91.i	even	4	1		1183.1.n.b		6
91.n	odd	6	1		1183.1.b.a		6
91.n	odd	6	1	inner	1183.1.t.a		12
91.t	odd	6	1		1183.1.b.a		6
91.t	odd	6	1	inner	1183.1.t.a		12
91.bc	even	12	1		1183.1.d.a	✓	3
91.bc	even	12	1		1183.1.d.b	yes	3
91.bc	even	12	1		1183.1.n.a		6
91.bc	even	12	1		1183.1.n.b		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1183.1.b.a		6	13.c	even	3	1
1183.1.b.a		6	13.e	even	6	1
1183.1.b.a		6	91.n	odd	6	1
1183.1.b.a		6	91.t	odd	6	1
1183.1.d.a	✓	3	13.f	odd	12	1
1183.1.d.a	✓	3	91.bc	even	12	1
1183.1.d.b	yes	3	13.f	odd	12	1
1183.1.d.b	yes	3	91.bc	even	12	1
1183.1.n.a		6	13.d	odd	4	1
1183.1.n.a		6	13.f	odd	12	1
1183.1.n.a		6	91.i	even	4	1
1183.1.n.a		6	91.bc	even	12	1
1183.1.n.b		6	13.d	odd	4	1
1183.1.n.b		6	13.f	odd	12	1
1183.1.n.b		6	91.i	even	4	1
1183.1.n.b		6	91.bc	even	12	1
1183.1.t.a		12	1.a	even	1	1	trivial
1183.1.t.a		12	7.b	odd	2	1	CM
1183.1.t.a		12	13.b	even	2	1	inner
1183.1.t.a		12	13.c	even	3	1	inner
1183.1.t.a		12	13.e	even	6	1	inner
1183.1.t.a		12	91.b	odd	2	1	inner
1183.1.t.a		12	91.n	odd	6	1	inner
1183.1.t.a		12	91.t	odd	6	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(1183, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} - 5 T^{10} + \cdots + 1 $ T^12 - 5T^10 + 19T^8 - 28T^6 + 31T^4 - 6*T^2 + 1
$3$	$ T^{12} $ T^12
$5$	$ T^{12} $ T^12
$7$	$ (T^{4} - T^{2} + 1)^{3} $ (T^4 - T^2 + 1)^3
$11$	$ T^{12} - 5 T^{10} + \cdots + 1 $ T^12 - 5T^10 + 19T^8 - 28T^6 + 31T^4 - 6*T^2 + 1
$13$	$ 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^{6} - T^{5} + 3 T^{4} + \cdots + 1)^{2} $ (T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1)^2
$31$	$ T^{12} $ T^12
$37$	$ 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^{6} + T^{5} + 3 T^{4} + \cdots + 1)^{2} $ (T^6 + T^5 + 3T^4 + 5T^2 + 2*T + 1)^2
$47$	$ T^{12} $ T^12
$53$	$ (T^{3} + T^{2} - 2 T - 1)^{4} $ (T^3 + T^2 - 2*T - 1)^4
$59$	$ T^{12} $ T^12
$61$	$ T^{12} $ T^12
$67$	$ T^{12} - 5 T^{10} + \cdots + 1 $ T^12 - 5T^10 + 19T^8 - 28T^6 + 31T^4 - 6*T^2 + 1
$71$	$ 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^{3} + T^{2} - 2 T - 1)^{4} $ (T^3 + T^2 - 2*T - 1)^4
$83$	$ T^{12} $ T^12
$89$	$ T^{12} $ T^12
$97$	$ T^{12} $ T^12
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1183 = 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1183.t (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{6} + \beta_{2} + \beta_1) q^{2} + ( - \beta_{9} + \beta_{7} + \beta_{5} + \cdots - 1) q^{4} + \beta_{10} q^{7} + (\beta_{11} + \beta_{10} + \cdots + \beta_{2}) q^{8} - \beta_{7} q^{9}+ \cdots + ( - \beta_{11} - \beta_{2}) q^{99}+O(q^{100}) \) q + (-b6 + b2 + b1) * q^2 + (-b9 + b7 + b5 + b3 - 1) * q^4 + b10 * q^7 + (b11 + b10 - b6 + b2) * q^8 - b7 * q^9 + b2 * q^11 - b3 * q^14 + (-b9 + b7 - b4 - 1) * q^16 + (-b11 - b10 - b8 + b6 - b2) * q^18 + (-b9 + b7 + b5 + b3 - 1) * q^22 + b4 * q^23 - q^25 + (b6 - b2) * q^28 + (b9 - b7 + b4 + 1) * q^29 + b10 * q^32 + (b9 - b7 + 1) * q^36 + b1 * q^37 + (b9 - b7 + b4 - b3 + 1) * q^43 + (b11 + 2b10 + b8 - 2b6 + b2) * q^44 + (b11 + b8 - b1) * q^46 + (-b7 + 1) * q^49 + (b6 - b2 - b1) * q^50 - b5 * q^53 + (b9 - b7 - b5 - b3 + 1) * q^56 + (-b11 - 2b10) q^58 - b6 * q^63 - b1 * q^67 + (-b8 + b1) * q^71 + (-b11 - b10) * q^72 + (b9 + b4 - b3 + 1) * q^74 + (-b5 - b3 + 1) * q^77 + (b5 + b3 - 1) * q^79 + (b7 - 1) * q^81 + (-b11 - 2b10 + 2b6 - b2) * q^86 + (-b9 + 2b7 - b4 - 2) q^88 - q^92 + (-b11 - b10 - b8 + b1) * q^98 + (-b11 - b2) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{4} - 6 q^{9} - 4 q^{14} - 2 q^{16} + 4 q^{22} - 2 q^{23} - 12 q^{25} + 2 q^{29} + 4 q^{36} - 2 q^{43} + 6 q^{49} - 4 q^{53} - 4 q^{56} + 4 q^{74} + 4 q^{77} - 4 q^{79} - 6 q^{81} - 8 q^{88}+ \cdots - 12 q^{92}+O(q^{100}) \) 12 * q + 4 * q^4 - 6 * q^9 - 4 * q^14 - 2 * q^16 + 4 * q^22 - 2 * q^23 - 12 * q^25 + 2 * q^29 + 4 * q^36 - 2 * q^43 + 6 * q^49 - 4 * q^53 - 4 * q^56 + 4 * q^74 + 4 * q^77 - 4 * q^79 - 6 * q^81 - 8 * q^88 - 12 * 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	1183.1.t.a

Level	$1183$
Weight	$1$
Character orbit	1183.t
Analytic conductor	$0.590$
Analytic rank	$0$
Dimension	$12$
Projective image	$D_{7}$
CM discriminant	-7
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(0.590393909945\)
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:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{7}\)
Projective field:	Galois closure of 7.1.1655595487.1

\(\beta_{1}\)	\(=\)	\( \nu \) 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

\(\nu\)	\(=\)	\( \beta_1 \) 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 699.6
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by \(\Q(\sqrt{-7}) \)
13.b	even	2	1	inner
13.c	even	3	1	inner
13.e	even	6	1	inner
91.b	odd	2	1	inner
91.n	odd	6	1	inner
91.t	odd	6	1	inner

$p$	$F_p(T)$
$2$	\( T^{12} - 5 T^{10} + \cdots + 1 \) T^12 - 5T^10 + 19T^8 - 28T^6 + 31T^4 - 6*T^2 + 1
$3$	\( T^{12} \) T^12
$5$	\( T^{12} \) T^12
$7$	\( (T^{4} - T^{2} + 1)^{3} \) (T^4 - T^2 + 1)^3
$11$	\( T^{12} - 5 T^{10} + \cdots + 1 \) T^12 - 5T^10 + 19T^8 - 28T^6 + 31T^4 - 6*T^2 + 1
$13$	\( T^{12} \) 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
$29$	\( (T^{6} - T^{5} + 3 T^{4} + \cdots + 1)^{2} \) (T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1)^2
$31$	\( T^{12} \) T^12
$37$	\( T^{12} - 5 T^{10} + \cdots + 1 \) T^12 - 5T^10 + 19T^8 - 28T^6 + 31T^4 - 6*T^2 + 1
$41$	\( T^{12} \) T^12
$43$	\( (T^{6} + T^{5} + 3 T^{4} + \cdots + 1)^{2} \) (T^6 + T^5 + 3T^4 + 5T^2 + 2*T + 1)^2
$47$	\( T^{12} \) T^12
$53$	\( (T^{3} + T^{2} - 2 T - 1)^{4} \) (T^3 + T^2 - 2*T - 1)^4
$59$	\( T^{12} \) T^12
$61$	\( T^{12} \) T^12
$67$	\( T^{12} - 5 T^{10} + \cdots + 1 \) T^12 - 5T^10 + 19T^8 - 28T^6 + 31T^4 - 6*T^2 + 1
$71$	\( T^{12} - 5 T^{10} + \cdots + 1 \) T^12 - 5T^10 + 19T^8 - 28T^6 + 31T^4 - 6*T^2 + 1
$73$	\( T^{12} \) T^12
$79$	\( (T^{3} + T^{2} - 2 T - 1)^{4} \) (T^3 + T^2 - 2*T - 1)^4
$83$	\( T^{12} \) T^12
$89$	\( T^{12} \) T^12
$97$	\( T^{12} \) T^12
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\(n\)	\(339\)	\(1016\)
\(\chi(n)\)	\(-1\)	\(\beta_{7}\)