LMFDB - Newform orbit 414.2.i.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [414,2,Mod(55,414)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(414, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 10]))

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

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

chi := DirichletCharacter("414.55");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 414 = 2 \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	414.i (of order $11$, degree $10$, 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:	$3.30580664368$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\Q(\zeta_{22})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 138)
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/414\mathbb{Z}\right)^\times$.

$n$	$47$	$235$
$\chi(n)$	$1$	$\zeta_{22}^{4}$

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} $

55.1

0.654861	−	0.755750i
0.959493	+	0.281733i
0.142315	+	0.989821i
0.142315	−	0.989821i
0.654861	+	0.755750i
−0.841254	+	0.540641i
−0.415415	+	0.909632i
−0.415415	−	0.909632i
−0.841254	−	0.540641i
0.959493	−	0.281733i

−0.959493

0.281733i

0.841254

−

0.540641i

0.0651865

0.453382i

−0.134858

0.295298i

−0.654861

0.755750i

−0.190279

−

0.416652i

73.1

0.415415

0.909632i

−0.654861

0.755750i

−2.47672

−

1.59169i

−0.151894

−

1.05645i

−0.959493

−

0.281733i

0.418986

−

2.91411i

127.1

0.841254

−

0.540641i

0.415415

−

0.909632i

−1.78074

0.522874i

−2.54487

−

2.93694i

−0.142315

−

0.989821i

−1.21537

1.40261i

163.1

0.841254

0.540641i

0.415415

0.909632i

−1.78074

−

0.522874i

−2.54487

2.93694i

−0.142315

0.989821i

−1.21537

−

1.40261i

271.1

−0.959493

−

0.281733i

0.841254

0.540641i

0.0651865

−

0.453382i

−0.134858

−

0.295298i

−0.654861

−

0.755750i

−0.190279

0.416652i

289.1

−0.654861

−

0.755750i

−0.142315

0.989821i

1.37787

−

3.01713i

3.66820

1.07708i

0.841254

−

0.540641i

−3.18251

0.934468i

307.1

−0.142315

0.989821i

−0.959493

−

0.281733i

1.81440

2.09393i

0.163423

0.105026i

0.415415

−

0.909632i

−2.33083

1.49793i

325.1

−0.142315

−

0.989821i

−0.959493

0.281733i

1.81440

−

2.09393i

0.163423

−

0.105026i

0.415415

0.909632i

−2.33083

−

1.49793i

361.1

−0.654861

0.755750i

−0.142315

−

0.989821i

1.37787

3.01713i

3.66820

−

1.07708i

0.841254

0.540641i

−3.18251

−

0.934468i

397.1

0.415415

−

0.909632i

−0.654861

−

0.755750i

−2.47672

1.59169i

−0.151894

1.05645i

−0.959493

0.281733i

0.418986

2.91411i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
23.c	even	11	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	414.2.i.a		10
3.b	odd	2	1		138.2.e.d	✓	10
23.c	even	11	1	inner	414.2.i.a		10
23.c	even	11	1		9522.2.a.bx		5
23.d	odd	22	1		9522.2.a.by		5
69.g	even	22	1		3174.2.a.w		5
69.h	odd	22	1		138.2.e.d	✓	10
69.h	odd	22	1		3174.2.a.x		5

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
138.2.e.d	✓	10	3.b	odd	2	1
138.2.e.d	✓	10	69.h	odd	22	1
414.2.i.a		10	1.a	even	1	1	trivial
414.2.i.a		10	23.c	even	11	1	inner
3174.2.a.w		5	69.g	even	22	1
3174.2.a.x		5	69.h	odd	22	1
9522.2.a.bx		5	23.c	even	11	1
9522.2.a.by		5	23.d	odd	22	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{10} + 2 T_{5}^{9} + 4 T_{5}^{8} + 41 T_{5}^{7} + 137 T_{5}^{6} + 76 T_{5}^{5} + 460 T_{5}^{4} + \cdots + 529 $ T5^10 + 2*T5^9 + 4*T5^8 + 41*T5^7 + 137*T5^6 + 76*T5^5 + 460*T5^4 + 2185*T5^3 + 2324*T5^2 + 138*T5 + 529 acting on $S_{2}^{\mathrm{new}}(414, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{10} + T^{9} + \cdots + 1 $ T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$3$	$ T^{10} $ T^10
$5$	$ T^{10} + 2 T^{9} + \cdots + 529 $ T^10 + 2T^9 + 4T^8 + 41T^7 + 137T^6 + 76T^5 + 460T^4 + 2185T^3 + 2324T^2 + 138*T + 529
$7$	$ T^{10} - 2 T^{9} + \cdots + 1 $ T^10 - 2T^9 - 7T^8 - 41T^7 + 203T^6 + 12T^5 + 262T^4 - 18T^3 + 14T^2 - 6*T + 1
$11$	$ T^{10} - 5 T^{9} + \cdots + 529 $ T^10 - 5T^9 + 25T^8 - 37T^7 + 20T^6 + 10T^5 - 138T^4 - 1290T^3 + 3689T^2 + 1817*T + 529
$13$	$ T^{10} - 13 T^{9} + \cdots + 529 $ T^10 - 13T^9 + 125T^8 - 646T^7 + 2854T^6 - 5026T^5 + 5575T^4 - 2020T^3 - 1218T^2 + 1380*T + 529
$17$	$ T^{10} + 55 T^{8} + \cdots + 7745089 $ T^10 + 55T^8 + 121T^7 + 726T^6 + 4235T^5 + 11979T^4 + 166375T^3 + 58564T^2 - 2693944T + 7745089
$19$	$ T^{10} - 33 T^{7} + \cdots + 64009 $ T^10 - 33T^7 - 165T^6 - 506T^5 + 2178T^4 + 14399T^3 + 45012T^2 + 66792*T + 64009
$23$	$ T^{10} - 32 T^{9} + \cdots + 6436343 $ T^10 - 32T^9 + 496T^8 - 4971T^7 + 35982T^6 - 197097T^5 + 827586T^4 - 2629659T^3 + 6034832T^2 - 8954912*T + 6436343
$29$	$ T^{10} + 27 T^{9} + \cdots + 7921 $ T^10 + 27T^9 + 322T^8 + 2039T^7 + 6532T^6 + 9527T^5 + 9080T^4 - 234253T^3 + 680409T^2 - 62834*T + 7921
$31$	$ T^{10} + 8 T^{9} + \cdots + 8300161 $ T^10 + 8T^9 + 108T^8 + 699T^7 + 3513T^6 + 8337T^5 - 26298T^4 + 42077T^3 + 1661379T^2 + 5398994*T + 8300161
$37$	$ T^{10} + 11 T^{9} + \cdots + 64009 $ T^10 + 11T^9 + 66T^8 + 33T^7 + 462T^6 + 583T^5 + 84579T^4 - 285197T^3 + 256157T^2 - 30613*T + 64009
$41$	$ T^{10} - 10 T^{9} + \cdots + 1515361 $ T^10 - 10T^9 + 23T^8 - 461T^7 + 4940T^6 - 8370T^5 + 48291T^4 - 590523T^3 + 2279850T^2 - 922019*T + 1515361
$43$	$ T^{10} - 34 T^{9} + \cdots + 109561 $ T^10 - 34T^9 + 595T^8 - 6403T^7 + 46080T^6 - 215634T^5 + 623217T^4 - 769715T^3 + 778922T^2 - 466379*T + 109561
$47$	$ (T^{5} + 4 T^{4} + \cdots - 857)^{2} $ (T^5 + 4T^4 - 75T^3 + 28T^2 + 658T - 857)^2
$53$	$ T^{10} + 9 T^{9} + \cdots + 8427409 $ T^10 + 9T^9 + 103T^8 - 481T^7 + 1765T^6 - 36211T^5 + 340085T^4 - 1481817T^3 + 4374571T^2 - 8828023*T + 8427409
$59$	$ T^{10} - 21 T^{9} + \cdots + 978121 $ T^10 - 21T^9 + 199T^8 - 2023T^7 + 27677T^6 - 182236T^5 + 194382T^4 + 1009152T^3 + 5151323T^2 - 447028*T + 978121
$61$	$ T^{10} + 4 T^{9} + \cdots + 9078169 $ T^10 + 4T^9 + 5T^8 + 845T^7 + 21255T^6 - 8810T^5 + 1007054T^4 - 92098T^3 + 4660676T^2 - 13630812*T + 9078169
$67$	$ T^{10} + 32 T^{9} + \cdots + 83375161 $ T^10 + 32T^9 + 628T^8 + 10438T^7 + 128921T^6 + 1069540T^5 + 6033270T^4 + 22918928T^3 + 58716219T^2 + 95409819*T + 83375161
$71$	$ T^{10} + 22 T^{9} + \cdots + 64009 $ T^10 + 22T^9 + 198T^8 + 3476T^7 + 32824T^6 - 80564T^5 + 1834723T^4 - 940775T^3 + 31097T^2 + 61226*T + 64009
$73$	$ T^{10} - 43 T^{9} + \cdots + 21132409 $ T^10 - 43T^9 + 1013T^8 - 13804T^7 + 112740T^6 - 340900T^5 - 1509815T^4 + 14290552T^3 + 45389554T^2 + 15822874*T + 21132409
$79$	$ T^{10} + 16 T^{9} + \cdots + 72471169 $ T^10 + 16T^9 + 14T^8 + 1665T^7 + 48519T^6 + 439847T^5 + 3749476T^4 - 23797177T^3 + 57311049T^2 - 55606916*T + 72471169
$83$	$ T^{10} - 3 T^{9} + \cdots + 16752649 $ T^10 - 3T^9 - 79T^8 - 236T^7 + 9860T^6 - 148413T^5 + 1658407T^4 - 5063364T^3 + 19306721T^2 + 22495128*T + 16752649
$89$	$ T^{10} + \cdots + 517972081 $ T^10 - 11T^9 + 55T^8 - 1034T^7 + 16566T^6 - 117711T^5 + 813241T^4 - 6520206T^3 + 33689667T^2 - 136189856*T + 517972081
$97$	$ T^{10} - 39 T^{9} + \cdots + 5650129 $ T^10 - 39T^9 + 773T^8 - 10204T^7 + 116972T^6 - 1250688T^5 + 10535838T^4 - 53199165T^3 + 116212529T^2 - 9809879*T + 5650129
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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:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	414.i (of order \(11\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{22}^{4} q^{2} + \zeta_{22}^{8} q^{4} + (\zeta_{22}^{9} - 2 \zeta_{22}^{8} + \cdots - 1) q^{5}+ \cdots - \zeta_{22} q^{8} +O(q^{10}) \) q + z^4 * q^2 + z^8 * q^4 + (z^9 - 2z^8 + 2z^7 - z^6 - z^4 - z^2 - 1) * q^5 + (-z^9 + z^7 - z^6 - z^4 + z^3 - z) * q^7 - z * q^8	\( q + \zeta_{22}^{4} q^{2} + \zeta_{22}^{8} q^{4} + (\zeta_{22}^{9} - 2 \zeta_{22}^{8} + \cdots - 1) q^{5}+ \cdots + (5 \zeta_{22}^{9} + \cdots + 2 \zeta_{22}) q^{98}+O(q^{100}) \) q + z^4 * q^2 + z^8 * q^4 + (z^9 - 2z^8 + 2z^7 - z^6 - z^4 - z^2 - 1) * q^5 + (-z^9 + z^7 - z^6 - z^4 + z^3 - z) * q^7 - z * q^8 + (-z^9 - z^7 - z^5 - z^3 + z - 1) * q^10 + (-2z^9 - 2z^5 - z^3 + 1) * q^11 + (-2z^9 + 2z^8 - 3z^7 - 3z^5 + 2z^4 - 2z^3 - 3z + 3) q^13 + (-z^9 + z^6 - 2z^5 + z^4 - z^3 + 2z^2 - z) * q^14 - z^5 * q^16 + (-4z^9 + 4z^8 + z^6 + z^4 + z^3 + z^2 + 1) * q^17 + (2z^9 - 3z^8 + 2z^7 + z^5 + z^3 - z^2 - 1) q^19 + (-z^9 - z^7 + z^5 - z^4 + z^2 + 1) * q^20 + (-2z^9 - z^7 + z^4 + 2z^2) * q^22 + (-z^8 - z^7 + z^6 - 2z^5 + z^4 - 3z^3 - z + 4) * q^23 + (z^8 + z^7 + 2z^6 - 2z^5 + 2z^4 - 2z^3 + 2z^2 + z + 1) q^25 + (-3z^9 + 2z^8 - 2z^7 - 3z^5 + 3z^4 + 2z^2 - 2z + 3) q^26 + (-z^9 + z^6 - z^4 + z^3 + z - 1) * q^28 + (-z^9 + z^8 - 2z^6 - z^5 - 5z^3 - z - 2) * q^29 + (3z^9 - 5z^8 + 4z^7 - 4z^6 + 5z^5 - 3z^4 - 4z^2 + 4z - 4) * q^31 - z^9 * q^32 + (z^9 + 2z^7 + z^5 + z^3 + 3z^2 - 3z - 1) q^34 + (4z^9 - 3z^8 + 3z^7 - 3z^6 + 4z^5 - z^3 + 1) q^35 + (4z^9 + z^7 + 2z^6 + z^4 - z^3 - 2z - 1) q^37 + (z^9 + z^7 - z^6 - z^4 - 2z^2 + 3z - 2) * q^38 + (z^9 - z^8 + z^6 + z^4 + z^2 + 1) * q^40 + (-2z^9 + 5z^8 - 5z^7 + 2z^6 + 3z^4 - z^3 + z^2 - z + 3) q^41 + (-2z^9 - z^8 + 4z^7 - z^6 + 2z^5 - 2z^4 + z^3 - 4z^2 + z + 2) q^43 + (z^8 + 2z^6 + 2z^2 + 1) * q^44 + (-z^9 - 2z^7 - z^6 + 3z^4 + z^3 - z^2 + 2z) q^46 + (-z^9 + z^8 - 2z^7 + 5z^6 - 5z^5 + 2z^4 - z^3 + z^2 + 1) * q^47 + (-3z^9 - 2z^7 + 2z^6 + 3z^5 - 3z^4 - 2z^3 + 2z^2 + 3) q^49 + (3z^5 - z^4 + 2z^3 - 2z^2 + z - 3) q^50 + (-3z^9 + 3z^8 + 2z^6 - 2z^5 + 3z^4 + 3z^2 - 2z + 2) q^52 + (-4z^9 + 2z^7 + 2z^6 - z^5 + 2z^4 + 2z^3 - 4z) * q^53 + (4z^9 + 5z^7 - 3z^6 + 5z^5 - z^4 + z^3 - 5z^2 + 3z - 5) * q^55 + (z^9 - 2z^8 + 2z^7 - z^6 + 2z^5 - 2z^4 + z^3 + z - 1) * q^56 + (-3z^9 + 2z^8 - 7z^7 + 2z^6 - 3z^5 - 2z^3 + 3z^2 - 3z + 2) * q^58 + (6z^9 - 5z^8 + 2z^7 - 4z^6 + 2z^5 - 5z^4 + 6z^3 + z - 1) q^59 + (2z^9 + 2z^8 - z^7 + z^6 - 2z^5 - 2z^4 - z^2 + 7z - 1) q^61 + (z^9 + z^8 - 4z^7 - 4z^3 + z^2 + z) * q^62 + z^2 * q^64 + (8z^9 - 5z^8 + 8z^7 + 8z^5 - 6z^4 + 5z^3 - 5z^2 + 6z - 8) * q^65 + (-z^8 - 6z^7 + 4z^6 - z^5 + z^4 - z^3 + 4z^2 - 6z - 1) * q^67 + (z^9 + z^7 + 3z^6 - 3z^5 - z^4 - z^2 - 2) * q^68 + (z^9 + 3z^8 - 4z^7 + 3z^6 - 3z^5 + 4z^4 - 3z^3 - z^2) * q^70 + (-z^8 - 5z^7 + 3z^6 + 3z^5 + 3z^4 + 3z^3 + 3z^2 - 5z - 1) q^71 + (-6z^9 + 8z^8 - 6z^7 - 9z^5 + 4z^4 - 5z^3 + 5z^2 - 4z + 9) * q^73 + (2z^9 - z^8 + z^7 - 2z^6 - 3z^4 + 2z^3 - 6z^2 + 2z - 3) * q^74 + (-z^9 - z^7 - z^6 + 2z^5 - z^4 - z^3 - z) q^76 + (3z^9 - 3z^8 + z^7 - z^6 + 3z^5 - 3z^4 - 3z^2 + 3z - 3) * q^77 + (z^9 + 6z^8 - 7z^7 + z^6 - 7z^5 + 6z^4 + z^3 - z + 1) * q^79 + (z^9 + z^7 + z^5 + z^3 - 2z^2 + 2z - 1) * q^80 + (2z^9 + z^8 + z^7 - z^6 + z^5 + z^4 + 2z^3 - 3z + 3) q^82 + (9z^9 + 4z^7 - 5z^6 + 6z^5 - 4z^4 + 4z^3 - 6z^2 + 5z - 4) * q^83 + (-3z^8 - z^7 - 10z^6 + 9z^5 - 10z^4 - z^3 - 3z^2) q^85 + (z^9 - z^8 - 3z^6 + 3z^4 - z^3 + 3z^2 - 3) q^86 + (2z^9 - 2z^8 + 2z^7 + 2z^5 - z^4 + 2z^3 - 2z^2 + z - 2) * q^88 + (-z^9 + z^8 - 3z^7 - 4z^6 + z^5 - z^4 + 4z^3 + 3z^2 - z + 1) * q^89 + (-2z^9 - 3z^8 + 6z^7 + 2z^6 - 2z^5 - 6z^4 + 3z^3 + 2z^2 - 8) * q^91 + (-z^9 + 4z^8 + z^5 + z^4 - z^3 + 2z^2 - z + 3) * q^92 + (-3z^8 + 4z^7 - 4z^6 + 5z^5 - 4z^4 + 5z^3 - 4z^2 + 4z - 3) * q^94 + (-2z^9 - z^8 - z^7 + 2z^6 - 7z^5 + 7z^4 - 2z^3 + z^2 + z + 2) q^95 + (-z^9 + z^6 + 3z^4 + 5z^3 - 4z^2 + 5z + 3) * q^97 + (5z^9 - 5z^8 + 2z^5 + z^4 + 2z^3 + z^2 + 2z) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - q^{2} - q^{4} - 2 q^{5} + 2 q^{7} - q^{8}+O(q^{10}) \) 10 * q - q^2 - q^4 - 2 * q^5 + 2 * q^7 - q^8	\( 10 q - q^{2} - q^{4} - 2 q^{5} + 2 q^{7} - q^{8} - 13 q^{10} + 5 q^{11} + 13 q^{13} - 9 q^{14} - q^{16} + 9 q^{20} - 6 q^{22} + 32 q^{23} + q^{25} + 13 q^{26} - 9 q^{28} - 27 q^{29} - 8 q^{31} - q^{32} - 11 q^{34} + 26 q^{35} - 11 q^{37} - 11 q^{38} + 9 q^{40} + 10 q^{41} + 34 q^{43} + 5 q^{44} - q^{46} - 8 q^{47} + 25 q^{49} - 21 q^{50} + 2 q^{52} - 9 q^{53} - 23 q^{55} + 2 q^{56} - 5 q^{58} + 21 q^{59} - 4 q^{61} - 8 q^{62} - q^{64} - 29 q^{65} - 32 q^{67} - 22 q^{68} - 18 q^{70} - 22 q^{71} + 43 q^{73} - 11 q^{74} - 10 q^{77} - 16 q^{79} - 2 q^{80} + 32 q^{82} + 3 q^{83} + 33 q^{85} - 32 q^{86} - 6 q^{88} + 11 q^{89} - 70 q^{91} + 21 q^{92} + 3 q^{94} + 39 q^{97} + 14 q^{98}+O(q^{100}) \) 10 * q - q^2 - q^4 - 2 * q^5 + 2 * q^7 - q^8 - 13 * q^10 + 5 * q^11 + 13 * q^13 - 9 * q^14 - q^16 + 9 * q^20 - 6 * q^22 + 32 * q^23 + q^25 + 13 * q^26 - 9 * q^28 - 27 * q^29 - 8 * q^31 - q^32 - 11 * q^34 + 26 * q^35 - 11 * q^37 - 11 * q^38 + 9 * q^40 + 10 * q^41 + 34 * q^43 + 5 * q^44 - q^46 - 8 * q^47 + 25 * q^49 - 21 * q^50 + 2 * q^52 - 9 * q^53 - 23 * q^55 + 2 * q^56 - 5 * q^58 + 21 * q^59 - 4 * q^61 - 8 * q^62 - q^64 - 29 * q^65 - 32 * q^67 - 22 * q^68 - 18 * q^70 - 22 * q^71 + 43 * q^73 - 11 * q^74 - 10 * q^77 - 16 * q^79 - 2 * q^80 + 32 * q^82 + 3 * q^83 + 33 * q^85 - 32 * q^86 - 6 * q^88 + 11 * q^89 - 70 * q^91 + 21 * q^92 + 3 * q^94 + 39 * q^97 + 14 * q^98

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Twists

Hecke kernels

Hecke characteristic polynomials

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Label	414.2.i.a

Level	$414$
Weight	$2$
Character orbit	414.i
Analytic conductor	$3.306$
Analytic rank	$0$
Dimension	$10$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(3.30580664368\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Coefficient field:	\(\Q(\zeta_{22})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 138)
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 55.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{10} + T^{9} + \cdots + 1 \) T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$3$	\( T^{10} \) T^10
$5$	\( T^{10} + 2 T^{9} + \cdots + 529 \) T^10 + 2T^9 + 4T^8 + 41T^7 + 137T^6 + 76T^5 + 460T^4 + 2185T^3 + 2324T^2 + 138*T + 529
$7$	\( T^{10} - 2 T^{9} + \cdots + 1 \) T^10 - 2T^9 - 7T^8 - 41T^7 + 203T^6 + 12T^5 + 262T^4 - 18T^3 + 14T^2 - 6*T + 1
$11$	\( T^{10} - 5 T^{9} + \cdots + 529 \) T^10 - 5T^9 + 25T^8 - 37T^7 + 20T^6 + 10T^5 - 138T^4 - 1290T^3 + 3689T^2 + 1817*T + 529
$13$	\( T^{10} - 13 T^{9} + \cdots + 529 \) T^10 - 13T^9 + 125T^8 - 646T^7 + 2854T^6 - 5026T^5 + 5575T^4 - 2020T^3 - 1218T^2 + 1380*T + 529
$17$	\( T^{10} + 55 T^{8} + \cdots + 7745089 \) T^10 + 55T^8 + 121T^7 + 726T^6 + 4235T^5 + 11979T^4 + 166375T^3 + 58564T^2 - 2693944T + 7745089
$19$	\( T^{10} - 33 T^{7} + \cdots + 64009 \) T^10 - 33T^7 - 165T^6 - 506T^5 + 2178T^4 + 14399T^3 + 45012T^2 + 66792*T + 64009
$23$	\( T^{10} - 32 T^{9} + \cdots + 6436343 \) T^10 - 32T^9 + 496T^8 - 4971T^7 + 35982T^6 - 197097T^5 + 827586T^4 - 2629659T^3 + 6034832T^2 - 8954912*T + 6436343
$29$	\( T^{10} + 27 T^{9} + \cdots + 7921 \) T^10 + 27T^9 + 322T^8 + 2039T^7 + 6532T^6 + 9527T^5 + 9080T^4 - 234253T^3 + 680409T^2 - 62834*T + 7921
$31$	\( T^{10} + 8 T^{9} + \cdots + 8300161 \) T^10 + 8T^9 + 108T^8 + 699T^7 + 3513T^6 + 8337T^5 - 26298T^4 + 42077T^3 + 1661379T^2 + 5398994*T + 8300161
$37$	\( T^{10} + 11 T^{9} + \cdots + 64009 \) T^10 + 11T^9 + 66T^8 + 33T^7 + 462T^6 + 583T^5 + 84579T^4 - 285197T^3 + 256157T^2 - 30613*T + 64009
$41$	\( T^{10} - 10 T^{9} + \cdots + 1515361 \) T^10 - 10T^9 + 23T^8 - 461T^7 + 4940T^6 - 8370T^5 + 48291T^4 - 590523T^3 + 2279850T^2 - 922019*T + 1515361
$43$	\( T^{10} - 34 T^{9} + \cdots + 109561 \) T^10 - 34T^9 + 595T^8 - 6403T^7 + 46080T^6 - 215634T^5 + 623217T^4 - 769715T^3 + 778922T^2 - 466379*T + 109561
$47$	\( (T^{5} + 4 T^{4} + \cdots - 857)^{2} \) (T^5 + 4T^4 - 75T^3 + 28T^2 + 658T - 857)^2
$53$	\( T^{10} + 9 T^{9} + \cdots + 8427409 \) T^10 + 9T^9 + 103T^8 - 481T^7 + 1765T^6 - 36211T^5 + 340085T^4 - 1481817T^3 + 4374571T^2 - 8828023*T + 8427409
$59$	\( T^{10} - 21 T^{9} + \cdots + 978121 \) T^10 - 21T^9 + 199T^8 - 2023T^7 + 27677T^6 - 182236T^5 + 194382T^4 + 1009152T^3 + 5151323T^2 - 447028*T + 978121
$61$	\( T^{10} + 4 T^{9} + \cdots + 9078169 \) T^10 + 4T^9 + 5T^8 + 845T^7 + 21255T^6 - 8810T^5 + 1007054T^4 - 92098T^3 + 4660676T^2 - 13630812*T + 9078169
$67$	\( T^{10} + 32 T^{9} + \cdots + 83375161 \) T^10 + 32T^9 + 628T^8 + 10438T^7 + 128921T^6 + 1069540T^5 + 6033270T^4 + 22918928T^3 + 58716219T^2 + 95409819*T + 83375161
$71$	\( T^{10} + 22 T^{9} + \cdots + 64009 \) T^10 + 22T^9 + 198T^8 + 3476T^7 + 32824T^6 - 80564T^5 + 1834723T^4 - 940775T^3 + 31097T^2 + 61226*T + 64009
$73$	\( T^{10} - 43 T^{9} + \cdots + 21132409 \) T^10 - 43T^9 + 1013T^8 - 13804T^7 + 112740T^6 - 340900T^5 - 1509815T^4 + 14290552T^3 + 45389554T^2 + 15822874*T + 21132409
$79$	\( T^{10} + 16 T^{9} + \cdots + 72471169 \) T^10 + 16T^9 + 14T^8 + 1665T^7 + 48519T^6 + 439847T^5 + 3749476T^4 - 23797177T^3 + 57311049T^2 - 55606916*T + 72471169
$83$	\( T^{10} - 3 T^{9} + \cdots + 16752649 \) T^10 - 3T^9 - 79T^8 - 236T^7 + 9860T^6 - 148413T^5 + 1658407T^4 - 5063364T^3 + 19306721T^2 + 22495128*T + 16752649
$89$	\( T^{10} + \cdots + 517972081 \) T^10 - 11T^9 + 55T^8 - 1034T^7 + 16566T^6 - 117711T^5 + 813241T^4 - 6520206T^3 + 33689667T^2 - 136189856*T + 517972081
$97$	\( T^{10} - 39 T^{9} + \cdots + 5650129 \) T^10 - 39T^9 + 773T^8 - 10204T^7 + 116972T^6 - 1250688T^5 + 10535838T^4 - 53199165T^3 + 116212529T^2 - 9809879*T + 5650129
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\(n\)	\(47\)	\(235\)
\(\chi(n)\)	\(1\)	\(\zeta_{22}^{4}\)