LMFDB - Newform orbit 456.2.bu.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [456,2,Mod(35,456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(456, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 9, 9, 4]))

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

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

chi := DirichletCharacter("456.35");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 456 = 2^{3} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	456.bu (of order $18$, degree $6$, 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:	$3.64117833217$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{18})$
Coefficient field:	12.0.101559956668416.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 8x^{6} + 64 $ x^12 - 8*x^6 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{18}]$

$q$-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_{11} q^{2} + (\beta_{9} + \beta_{6} - \beta_{3}) q^{3} - 2 \beta_{4} q^{4} + ( - \beta_{11} + 2 \beta_{8} + \beta_{5}) q^{6} + (2 \beta_{9} - 2 \beta_{3}) q^{8} + ( - \beta_{6} - 2 \beta_{3} + 1) q^{9}+O(q^{10}) $ q - b11 * q^2 + (b9 + b6 - b3) * q^3 - 2b4 q^4 + (-b11 + 2b8 + b5) q^6 + (2b9 - 2b3) * q^8 + (-b6 - 2b3 + 1) q^9	$ q - \beta_{11} q^{2} + (\beta_{9} + \beta_{6} - \beta_{3}) q^{3} - 2 \beta_{4} q^{4} + ( - \beta_{11} + 2 \beta_{8} + \beta_{5}) q^{6} + (2 \beta_{9} - 2 \beta_{3}) q^{8} + ( - \beta_{6} - 2 \beta_{3} + 1) q^{9} + ( - 3 \beta_{10} - 3 \beta_{8} + \cdots + 3 \beta_{2}) q^{11}+ \cdots + (5 \beta_{11} + \beta_{10} + \cdots - 7 \beta_1) q^{99}+O(q^{100}) $ q - b11 * q^2 + (b9 + b6 - b3) * q^3 - 2b4 q^4 + (-b11 + 2b8 + b5) q^6 + (2b9 - 2b3) * q^8 + (-b6 - 2b3 + 1) q^9 + (-3b10 - 3b8 - b7 + b5 + 3b4 + 3b2) * q^11 + (-2b10 + 2b1) * q^12 + 4b8 q^16 + (-2b11 - 6b8 + 3b2) q^17 + (4b8 - b5 - 4b2) * q^18 + (-3b7 + b4 + 3b1) * q^19 + (-2b10 - 3b9 - 3b7 + 2b4 - 2) * q^22 + (-4b6 - 2b3 + 4) * q^24 + 5b10 q^25 + (-b9 + 5) * q^27 + 4b1 q^32 + (4b11 + 2b7 + 5b4 + b2 - 2b1) * q^33 + (-3b7 - 4b4 - 3b1) q^34 + (2b10 + 4b7 - 2b4) q^36 + (-b9 - 6b6 + b3) q^38 + (4b7 + 3b6 - 3b4 - 4b3 - 4b1 - 3) q^41 + (-6b11 - 5b8 + 3b5 + 5b2) * q^43 + (2b11 - 2b9 - 6b2 - 6) q^44 + (4b8 - 4b5 - 4b2) q^48 - 7b6 q^49 + 5b3 q^50 + (b11 + b8 + 5b5 + 6b2) * q^51 + (-5b11 - 2b2) * q^54 + (7b10 + 2b1) * q^57 + (3b10 + 3b6 + 5b3 - 5b1 - 3) * q^59 + (-8b6 + 8) q^64 + (-5b9 - b7 + 4b6 + 8b4 + 5b3 + b1) * q^66 + (-3b11 + 7b6 + 3b3 - 7b2 - 7) * q^67 + (4b9 + 6b6 - 4b3 - 12) q^68 + (2b9 + 8) q^72 + (b10 + 6b9 + 6b7 - b4 + 1) * q^73 + (5b10 - 5b7 - 5b4) q^75 + (6b11 - 2b8 - 6b5) q^76 + (4b9 + 7b6 - 4b3) q^81 + (3b9 + 8b8 + 8b6 + 3b5 - 3b3 - 8b2) * q^82 + (b11 - 9b10 + 9b8 - b7 - b5 + 9b4) q^83 + (-6b10 - 5b7 - 6b4) q^86 + (6b11 + 6b7 + 4b4 - 4b2 - 6b1) q^88 + (9b10 - 18b4 + 2b1) q^89 + (8b10 + 4b7 - 8b4) q^96 + (-5b10 - 6b9 + 6b7 + 5b6 + 5b4 + 6b3) * q^97 + (7b11 - 7b5) * q^98 + (5b11 + b10 - 7b8 - 5b5 - 7b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 6 q^{3} + 6 q^{9}+O(q^{10}) $ 12 * q + 6 * q^3 + 6 * q^9	$ 12 q + 6 q^{3} + 6 q^{9} - 24 q^{22} + 24 q^{24} + 60 q^{27} - 36 q^{38} - 18 q^{41} - 72 q^{44} - 42 q^{49} - 18 q^{59} + 48 q^{64} + 24 q^{66} - 42 q^{67} - 108 q^{68} + 96 q^{72} + 12 q^{73} + 42 q^{81} + 48 q^{82} + 30 q^{97}+O(q^{100}) $ 12 * q + 6 * q^3 + 6 * q^9 - 24 * q^22 + 24 * q^24 + 60 * q^27 - 36 * q^38 - 18 * q^41 - 72 * q^44 - 42 * q^49 - 18 * q^59 + 48 * q^64 + 24 * q^66 - 42 * q^67 - 108 * q^68 + 96 * q^72 + 12 * q^73 + 42 * q^81 + 48 * q^82 + 30 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 8x^{6} + 64 $ x^12 - 8*x^6 + 64 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( \nu^{2} ) / 2 $ (v^2) / 2
$\beta_{3}$	$=$	$ ( \nu^{3} ) / 2 $ (v^3) / 2
$\beta_{4}$	$=$	$ ( \nu^{4} ) / 4 $ (v^4) / 4
$\beta_{5}$	$=$	$ ( \nu^{5} ) / 4 $ (v^5) / 4
$\beta_{6}$	$=$	$ ( \nu^{6} ) / 8 $ (v^6) / 8
$\beta_{7}$	$=$	$ ( \nu^{7} ) / 8 $ (v^7) / 8
$\beta_{8}$	$=$	$ ( \nu^{8} ) / 16 $ (v^8) / 16
$\beta_{9}$	$=$	$ ( \nu^{9} ) / 16 $ (v^9) / 16
$\beta_{10}$	$=$	$ ( \nu^{10} ) / 32 $ (v^10) / 32
$\beta_{11}$	$=$	$ ( \nu^{11} ) / 32 $ (v^11) / 32

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ 2\beta_{2} $ 2*b2
$\nu^{3}$	$=$	$ 2\beta_{3} $ 2*b3
$\nu^{4}$	$=$	$ 4\beta_{4} $ 4*b4
$\nu^{5}$	$=$	$ 4\beta_{5} $ 4*b5
$\nu^{6}$	$=$	$ 8\beta_{6} $ 8*b6
$\nu^{7}$	$=$	$ 8\beta_{7} $ 8*b7
$\nu^{8}$	$=$	$ 16\beta_{8} $ 16*b8
$\nu^{9}$	$=$	$ 16\beta_{9} $ 16*b9
$\nu^{10}$	$=$	$ 32\beta_{10} $ 32*b10
$\nu^{11}$	$=$	$ 32\beta_{11} $ 32*b11

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/456\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$343$
$\chi(n)$	$\beta_{4}$	$-1$	$-1$	$-1$

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.

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} $

35.1

0.483690	−	1.32893i
−0.483690	+	1.32893i
−0.909039	−	1.08335i
0.909039	+	1.08335i
−1.39273	−	0.245576i
1.39273	+	0.245576i
−0.909039	+	1.08335i
0.909039	−	1.08335i
−1.39273	+	0.245576i
1.39273	−	0.245576i
0.483690	+	1.32893i
−0.483690	−	1.32893i

−0.909039

1.08335i

1.72474

−

0.158919i

−0.347296

−

1.96962i

−1.39570

2.01297i

2.44949

1.41421i

2.94949

−

0.548188i

35.2

0.909039

−

1.08335i

−0.724745

−

1.57313i

−0.347296

−

1.96962i

−2.36307

−

0.644886i

−2.44949

−

1.41421i

−1.94949

2.28024i

131.1

−1.39273

−

0.245576i

−0.724745

−

1.57313i

1.87939

0.684040i

0.623050

2.36893i

−2.44949

−

1.41421i

−1.94949

2.28024i

131.2

1.39273

0.245576i

1.72474

−

0.158919i

1.87939

0.684040i

2.44113

0.202225i

2.44949

1.41421i

2.94949

−

0.548188i

251.1

−0.483690

1.32893i

1.72474

0.158919i

−1.53209

−

1.28558i

−1.04543

2.21519i

2.44949

−

1.41421i

2.94949

0.548188i

251.2

0.483690

−

1.32893i

−0.724745

1.57313i

−1.53209

−

1.28558i

1.74002

1.72404i

−2.44949

1.41421i

−1.94949

−

2.28024i

275.1

−1.39273

0.245576i

−0.724745

1.57313i

1.87939

−

0.684040i

0.623050

−

2.36893i

−2.44949

1.41421i

−1.94949

−

2.28024i

275.2

1.39273

−

0.245576i

1.72474

0.158919i

1.87939

−

0.684040i

2.44113

−

0.202225i

2.44949

−

1.41421i

2.94949

0.548188i

347.1

−0.483690

−

1.32893i

1.72474

−

0.158919i

−1.53209

1.28558i

−1.04543

−

2.21519i

2.44949

1.41421i

2.94949

−

0.548188i

347.2

0.483690

1.32893i

−0.724745

−

1.57313i

−1.53209

1.28558i

1.74002

−

1.72404i

−2.44949

−

1.41421i

−1.94949

2.28024i

443.1

−0.909039

−

1.08335i

1.72474

0.158919i

−0.347296

1.96962i

−1.39570

−

2.01297i

2.44949

−

1.41421i

2.94949

0.548188i

443.2

0.909039

1.08335i

−0.724745

1.57313i

−0.347296

1.96962i

−2.36307

0.644886i

−2.44949

1.41421i

−1.94949

−

2.28024i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2}) $
57.l	odd	18	1	inner
456.bu	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	456.2.bu.b	yes	12
3.b	odd	2	1		456.2.bu.a	✓	12
8.d	odd	2	1	CM	456.2.bu.b	yes	12
19.e	even	9	1		456.2.bu.a	✓	12
24.f	even	2	1		456.2.bu.a	✓	12
57.l	odd	18	1	inner	456.2.bu.b	yes	12
152.u	odd	18	1		456.2.bu.a	✓	12
456.bu	even	18	1	inner	456.2.bu.b	yes	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
456.2.bu.a	✓	12	3.b	odd	2	1
456.2.bu.a	✓	12	19.e	even	9	1
456.2.bu.a	✓	12	24.f	even	2	1
456.2.bu.a	✓	12	152.u	odd	18	1
456.2.bu.b	yes	12	1.a	even	1	1	trivial
456.2.bu.b	yes	12	8.d	odd	2	1	CM
456.2.bu.b	yes	12	57.l	odd	18	1	inner
456.2.bu.b	yes	12	456.bu	even	18	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(456, [\chi])$:

$ T_{5} $

$ T_{11}^{12} - 66 T_{11}^{10} + 3267 T_{11}^{8} + 1782 T_{11}^{7} - 68888 T_{11}^{6} - 58806 T_{11}^{5} + \cdots + 1014049 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} - 8T^{6} + 64 $ T^12 - 8*T^6 + 64
$3$	$ (T^{4} - 2 T^{3} + T^{2} + \cdots + 9)^{3} $ (T^4 - 2T^3 + T^2 - 6T + 9)^3
$5$	$ T^{12} $ T^12
$7$	$ T^{12} $ T^12
$11$	$ T^{12} - 66 T^{10} + \cdots + 1014049 $ T^12 - 66T^10 + 3267T^8 + 1782T^7 - 68888T^6 - 58806T^5 + 1087383T^4 + 1940598T^3 - 38115T^2 - 1794474*T + 1014049
$13$	$ T^{12} $ T^12
$17$	$ T^{12} + 918 T^{9} + \cdots + 47045881 $ T^12 + 918T^9 + 287767T^6 + 6296562*T^3 + 47045881
$19$	$ T^{12} - 106 T^{9} + \cdots + 47045881 $ T^12 - 106T^9 + 4377T^6 - 727054*T^3 + 47045881
$23$	$ T^{12} $ T^12
$29$	$ T^{12} $ T^12
$31$	$ T^{12} $ T^12
$37$	$ T^{12} $ T^12
$41$	$ T^{12} + \cdots + 41437894969 $ T^12 + 18T^11 + 93T^10 + 1296T^9 + 28488T^8 + 351576T^7 + 2202202T^6 - 1888506T^5 - 71266908T^4 - 623389716T^3 + 830094720T^2 + 2345045760*T + 41437894969
$43$	$ T^{12} + \cdots + 594823321 $ T^12 + 1870T^9 + 3521289T^6 - 45607430*T^3 + 594823321
$47$	$ T^{12} $ T^12
$53$	$ T^{12} $ T^12
$59$	$ T^{12} + \cdots + 260443853569 $ T^12 + 18T^11 + 39T^10 + 1296T^9 + 52032T^8 + 669096T^7 + 1827118T^6 - 45627534T^5 - 340620204T^4 - 1916313012T^3 + 17399909400T^2 + 42255903600*T + 260443853569
$61$	$ T^{12} $ T^12
$67$	$ T^{12} + \cdots + 805307017321 $ T^12 + 42T^11 + 975T^10 + 16324T^9 + 190170T^8 + 1615572T^7 + 17172516T^6 + 203605290T^5 + 2498333517T^4 + 23838564274T^3 + 147790636341T^2 + 519259786626*T + 805307017321
$71$	$ T^{12} $ T^12
$73$	$ T^{12} + \cdots + 964620586801 $ T^12 - 12T^11 + 498T^10 - 4970T^9 + 71757T^8 - 451728T^7 + 3008127T^6 + 26934210T^5 + 650190528T^4 - 7953790574T^3 - 20405021655T^2 + 265610952138*T + 964620586801
$79$	$ T^{12} $ T^12
$83$	$ T^{12} + \cdots + 1564283008369 $ T^12 - 498T^10 + 186003T^8 - 1008450T^7 - 27910424T^6 + 251104050T^5 + 3105571575T^4 - 62524908450T^3 + 416535924213T^2 - 1261281524850*T + 1564283008369
$89$	$ T^{12} + \cdots + 168425239515625 $ T^12 + 14418T^9 + 82270783T^6 + 187115001750*T^3 + 168425239515625
$97$	$ T^{12} + \cdots + 828247426561 $ T^12 - 30T^11 + 309T^10 - 2180T^9 + 84762T^8 - 2488860T^7 + 53007348T^6 - 592298010T^5 + 2713546629T^4 + 19656113290T^3 + 48112833039T^2 + 280477863390*T + 828247426561
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	456.bu (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{11} q^{2} + (\beta_{9} + \beta_{6} - \beta_{3}) q^{3} - 2 \beta_{4} q^{4} + ( - \beta_{11} + 2 \beta_{8} + \beta_{5}) q^{6} + (2 \beta_{9} - 2 \beta_{3}) q^{8} + ( - \beta_{6} - 2 \beta_{3} + 1) q^{9}+O(q^{10}) \) q - b11 * q^2 + (b9 + b6 - b3) * q^3 - 2b4 q^4 + (-b11 + 2b8 + b5) q^6 + (2b9 - 2b3) * q^8 + (-b6 - 2b3 + 1) q^9	\( q - \beta_{11} q^{2} + (\beta_{9} + \beta_{6} - \beta_{3}) q^{3} - 2 \beta_{4} q^{4} + ( - \beta_{11} + 2 \beta_{8} + \beta_{5}) q^{6} + (2 \beta_{9} - 2 \beta_{3}) q^{8} + ( - \beta_{6} - 2 \beta_{3} + 1) q^{9} + ( - 3 \beta_{10} - 3 \beta_{8} + \cdots + 3 \beta_{2}) q^{11}+ \cdots + (5 \beta_{11} + \beta_{10} + \cdots - 7 \beta_1) q^{99}+O(q^{100}) \) q - b11 * q^2 + (b9 + b6 - b3) * q^3 - 2b4 q^4 + (-b11 + 2b8 + b5) q^6 + (2b9 - 2b3) * q^8 + (-b6 - 2b3 + 1) q^9 + (-3b10 - 3b8 - b7 + b5 + 3b4 + 3b2) * q^11 + (-2b10 + 2b1) * q^12 + 4b8 q^16 + (-2b11 - 6b8 + 3b2) q^17 + (4b8 - b5 - 4b2) * q^18 + (-3b7 + b4 + 3b1) * q^19 + (-2b10 - 3b9 - 3b7 + 2b4 - 2) * q^22 + (-4b6 - 2b3 + 4) * q^24 + 5b10 q^25 + (-b9 + 5) * q^27 + 4b1 q^32 + (4b11 + 2b7 + 5b4 + b2 - 2b1) * q^33 + (-3b7 - 4b4 - 3b1) q^34 + (2b10 + 4b7 - 2b4) q^36 + (-b9 - 6b6 + b3) q^38 + (4b7 + 3b6 - 3b4 - 4b3 - 4b1 - 3) q^41 + (-6b11 - 5b8 + 3b5 + 5b2) * q^43 + (2b11 - 2b9 - 6b2 - 6) q^44 + (4b8 - 4b5 - 4b2) q^48 - 7b6 q^49 + 5b3 q^50 + (b11 + b8 + 5b5 + 6b2) * q^51 + (-5b11 - 2b2) * q^54 + (7b10 + 2b1) * q^57 + (3b10 + 3b6 + 5b3 - 5b1 - 3) * q^59 + (-8b6 + 8) q^64 + (-5b9 - b7 + 4b6 + 8b4 + 5b3 + b1) * q^66 + (-3b11 + 7b6 + 3b3 - 7b2 - 7) * q^67 + (4b9 + 6b6 - 4b3 - 12) q^68 + (2b9 + 8) q^72 + (b10 + 6b9 + 6b7 - b4 + 1) * q^73 + (5b10 - 5b7 - 5b4) q^75 + (6b11 - 2b8 - 6b5) q^76 + (4b9 + 7b6 - 4b3) q^81 + (3b9 + 8b8 + 8b6 + 3b5 - 3b3 - 8b2) * q^82 + (b11 - 9b10 + 9b8 - b7 - b5 + 9b4) q^83 + (-6b10 - 5b7 - 6b4) q^86 + (6b11 + 6b7 + 4b4 - 4b2 - 6b1) q^88 + (9b10 - 18b4 + 2b1) q^89 + (8b10 + 4b7 - 8b4) q^96 + (-5b10 - 6b9 + 6b7 + 5b6 + 5b4 + 6b3) * q^97 + (7b11 - 7b5) * q^98 + (5b11 + b10 - 7b8 - 5b5 - 7b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{3} + 6 q^{9}+O(q^{10}) \) 12 * q + 6 * q^3 + 6 * q^9	\( 12 q + 6 q^{3} + 6 q^{9} - 24 q^{22} + 24 q^{24} + 60 q^{27} - 36 q^{38} - 18 q^{41} - 72 q^{44} - 42 q^{49} - 18 q^{59} + 48 q^{64} + 24 q^{66} - 42 q^{67} - 108 q^{68} + 96 q^{72} + 12 q^{73} + 42 q^{81} + 48 q^{82} + 30 q^{97}+O(q^{100}) \) 12 * q + 6 * q^3 + 6 * q^9 - 24 * q^22 + 24 * q^24 + 60 * q^27 - 36 * q^38 - 18 * q^41 - 72 * q^44 - 42 * q^49 - 18 * q^59 + 48 * q^64 + 24 * q^66 - 42 * q^67 - 108 * q^68 + 96 * q^72 + 12 * q^73 + 42 * q^81 + 48 * q^82 + 30 * q^97

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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	456.2.bu.b

Level	$456$
Weight	$2$
Character orbit	456.bu
Analytic conductor	$3.641$
Analytic rank	$0$
Dimension	$12$
CM discriminant	-8
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(3.64117833217\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{18})\)
Coefficient field:	12.0.101559956668416.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 8x^{6} + 64 \) x^12 - 8*x^6 + 64
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{18}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( \nu^{2} ) / 2 \) (v^2) / 2
\(\beta_{3}\)	\(=\)	\( ( \nu^{3} ) / 2 \) (v^3) / 2
\(\beta_{4}\)	\(=\)	\( ( \nu^{4} ) / 4 \) (v^4) / 4
\(\beta_{5}\)	\(=\)	\( ( \nu^{5} ) / 4 \) (v^5) / 4
\(\beta_{6}\)	\(=\)	\( ( \nu^{6} ) / 8 \) (v^6) / 8
\(\beta_{7}\)	\(=\)	\( ( \nu^{7} ) / 8 \) (v^7) / 8
\(\beta_{8}\)	\(=\)	\( ( \nu^{8} ) / 16 \) (v^8) / 16
\(\beta_{9}\)	\(=\)	\( ( \nu^{9} ) / 16 \) (v^9) / 16
\(\beta_{10}\)	\(=\)	\( ( \nu^{10} ) / 32 \) (v^10) / 32
\(\beta_{11}\)	\(=\)	\( ( \nu^{11} ) / 32 \) (v^11) / 32

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( 2\beta_{2} \) 2*b2
\(\nu^{3}\)	\(=\)	\( 2\beta_{3} \) 2*b3
\(\nu^{4}\)	\(=\)	\( 4\beta_{4} \) 4*b4
\(\nu^{5}\)	\(=\)	\( 4\beta_{5} \) 4*b5
\(\nu^{6}\)	\(=\)	\( 8\beta_{6} \) 8*b6
\(\nu^{7}\)	\(=\)	\( 8\beta_{7} \) 8*b7
\(\nu^{8}\)	\(=\)	\( 16\beta_{8} \) 16*b8
\(\nu^{9}\)	\(=\)	\( 16\beta_{9} \) 16*b9
\(\nu^{10}\)	\(=\)	\( 32\beta_{10} \) 32*b10
\(\nu^{11}\)	\(=\)	\( 32\beta_{11} \) 32*b11

\(n\)	\(97\)	\(229\)	\(305\)	\(343\)
\(\chi(n)\)	\(\beta_{4}\)	\(-1\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 35.2
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
57.l	odd	18	1	inner
456.bu	even	18	1	inner

$p$	$F_p(T)$
$2$	\( T^{12} - 8T^{6} + 64 \) T^12 - 8*T^6 + 64
$3$	\( (T^{4} - 2 T^{3} + T^{2} + \cdots + 9)^{3} \) (T^4 - 2T^3 + T^2 - 6T + 9)^3
$5$	\( T^{12} \) T^12
$7$	\( T^{12} \) T^12
$11$	\( T^{12} - 66 T^{10} + \cdots + 1014049 \) T^12 - 66T^10 + 3267T^8 + 1782T^7 - 68888T^6 - 58806T^5 + 1087383T^4 + 1940598T^3 - 38115T^2 - 1794474*T + 1014049
$13$	\( T^{12} \) T^12
$17$	\( T^{12} + 918 T^{9} + \cdots + 47045881 \) T^12 + 918T^9 + 287767T^6 + 6296562*T^3 + 47045881
$19$	\( T^{12} - 106 T^{9} + \cdots + 47045881 \) T^12 - 106T^9 + 4377T^6 - 727054*T^3 + 47045881
$23$	\( T^{12} \) T^12
$29$	\( T^{12} \) T^12
$31$	\( T^{12} \) T^12
$37$	\( T^{12} \) T^12
$41$	\( T^{12} + \cdots + 41437894969 \) T^12 + 18T^11 + 93T^10 + 1296T^9 + 28488T^8 + 351576T^7 + 2202202T^6 - 1888506T^5 - 71266908T^4 - 623389716T^3 + 830094720T^2 + 2345045760*T + 41437894969
$43$	\( T^{12} + \cdots + 594823321 \) T^12 + 1870T^9 + 3521289T^6 - 45607430*T^3 + 594823321
$47$	\( T^{12} \) T^12
$53$	\( T^{12} \) T^12
$59$	\( T^{12} + \cdots + 260443853569 \) T^12 + 18T^11 + 39T^10 + 1296T^9 + 52032T^8 + 669096T^7 + 1827118T^6 - 45627534T^5 - 340620204T^4 - 1916313012T^3 + 17399909400T^2 + 42255903600*T + 260443853569
$61$	\( T^{12} \) T^12
$67$	\( T^{12} + \cdots + 805307017321 \) T^12 + 42T^11 + 975T^10 + 16324T^9 + 190170T^8 + 1615572T^7 + 17172516T^6 + 203605290T^5 + 2498333517T^4 + 23838564274T^3 + 147790636341T^2 + 519259786626*T + 805307017321
$71$	\( T^{12} \) T^12
$73$	\( T^{12} + \cdots + 964620586801 \) T^12 - 12T^11 + 498T^10 - 4970T^9 + 71757T^8 - 451728T^7 + 3008127T^6 + 26934210T^5 + 650190528T^4 - 7953790574T^3 - 20405021655T^2 + 265610952138*T + 964620586801
$79$	\( T^{12} \) T^12
$83$	\( T^{12} + \cdots + 1564283008369 \) T^12 - 498T^10 + 186003T^8 - 1008450T^7 - 27910424T^6 + 251104050T^5 + 3105571575T^4 - 62524908450T^3 + 416535924213T^2 - 1261281524850*T + 1564283008369
$89$	\( T^{12} + \cdots + 168425239515625 \) T^12 + 14418T^9 + 82270783T^6 + 187115001750*T^3 + 168425239515625
$97$	\( T^{12} + \cdots + 828247426561 \) T^12 - 30T^11 + 309T^10 - 2180T^9 + 84762T^8 - 2488860T^7 + 53007348T^6 - 592298010T^5 + 2713546629T^4 + 19656113290T^3 + 48112833039T^2 + 280477863390*T + 828247426561
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