LMFDB - Newform orbit 705.1.p.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [705,1,Mod(14,705)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(705, base_ring=CyclotomicField(46))

chi = DirichletCharacter(H, H._module([23, 23, 4]))

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

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

chi := DirichletCharacter("705.14");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 705 = 3 \cdot 5 \cdot 47 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	705.p (of order $46$, degree $22$, 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:	$0.351840833906$
Analytic rank:	$0$
Dimension:	$22$
Coefficient field:	$\Q(\zeta_{46})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} + \cdots + 1 $ x^22 - x^21 + x^20 - x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{23}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{23} - \cdots)$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/705\mathbb{Z}\right)^\times$.

$n$	$142$	$236$	$616$
$\chi(n)$	$-1$	$-1$	$-\zeta_{46}^{21}$

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} $

14.1

−0.962917	+	0.269797i
0.775711	−	0.631088i
0.576680	−	0.816970i
0.0682424	+	0.997669i
0.990686	−	0.136167i
−0.854419	+	0.519584i
−0.854419	−	0.519584i
0.917211	−	0.398401i
−0.460065	−	0.887885i
0.775711	+	0.631088i
−0.682553	−	0.730836i
0.334880	−	0.942261i
−0.460065	+	0.887885i
0.990686	+	0.136167i
0.0682424	−	0.997669i
0.917211	+	0.398401i
0.334880	+	0.942261i
0.576680	+	0.816970i
−0.962917	−	0.269797i
−0.203456	+	0.979084i

−0.391823

1.88555i

−0.682553

0.730836i

−2.48458

−

1.07920i

−0.962917

−

0.269797i

−1.11059

−

1.57335i

1.89782

−

2.68860i

−0.0682424

−

0.997669i

0.886009

−

1.70992i

59.1

1.49389

−

0.418569i

−0.460065

−

0.887885i

1.20209

−

0.731009i

0.775711

0.631088i

−1.05893

−

1.13384i

0.430891

−

0.461371i

−0.576680

0.816970i

1.42298

0.618088i

74.1

−0.0787081

−

1.15067i

−0.962917

−

0.269797i

−0.327165

0.0449678i

0.576680

0.816970i

−0.234658

1.12924i

−0.157164

−

0.756317i

0.854419

0.519584i

0.894675

−

0.727872i

89.1

−0.0457060

−

0.128604i

−0.203456

−

0.979084i

0.761261

−

0.619332i

0.0682424

−

0.997669i

−0.116615

0.0709153i

−0.231058

−

0.140510i

−0.917211

0.398401i

−0.131424

0.0368232i

119.1

−1.53697

1.25042i

0.917211

−

0.398401i

0.595279

−

2.86464i

0.990686

0.136167i

−0.911560

1.75923i

1.75551

3.38799i

0.682553

−

0.730836i

−1.69292

1.02949i

149.1

1.56737

0.680803i

0.0682424

0.997669i

1.31059

1.40330i

−0.854419

−

0.519584i

−0.572255

1.61017i

0.526549

1.48157i

−0.990686

0.136167i

−0.985454

−

1.39607i

194.1

1.56737

−

0.680803i

0.0682424

−

0.997669i

1.31059

−

1.40330i

−0.854419

0.519584i

−0.572255

−

1.61017i

0.526549

−

1.48157i

−0.990686

−

0.136167i

−0.985454

1.39607i

209.1

0.843954

1.62876i

0.334880

−

0.942261i

−1.36391

1.93222i

0.917211

0.398401i

1.81734

−

0.249787i

−2.48086

−

0.340986i

−0.775711

−

0.631088i

0.125185

1.83015i

224.1

−0.628038

0.672464i

0.990686

0.136167i

0.0104657

0.153003i

−0.460065

0.887885i

−0.713755

0.580683i

−0.823217

−

0.669737i

0.962917

0.269797i

−0.308133

−

0.867003i

239.1

1.49389

0.418569i

−0.460065

0.887885i

1.20209

0.731009i

0.775711

−

0.631088i

−1.05893

1.13384i

0.430891

0.461371i

−0.576680

−

0.816970i

1.42298

−

0.618088i

269.1

0.787230

1.11525i

0.775711

−

0.631088i

−0.289174

0.813657i

−0.682553

0.730836i

1.31448

0.368301i

0.179407

−

0.0502675i

0.203456

−

0.979084i

−1.35239

−

0.185882i

284.1

−0.663521

−

0.0911989i

−0.854419

0.519584i

−0.530974

−

0.148772i

0.334880

0.942261i

0.614311

−

0.266833i

0.953056

0.413970i

0.460065

−

0.887885i

−0.136267

−

0.655751i

299.1

−0.628038

−

0.672464i

0.990686

−

0.136167i

0.0104657

−

0.153003i

−0.460065

−

0.887885i

−0.713755

−

0.580683i

−0.823217

0.669737i

0.962917

−

0.269797i

−0.308133

0.867003i

314.1

−1.53697

−

1.25042i

0.917211

0.398401i

0.595279

2.86464i

0.990686

−

0.136167i

−0.911560

−

1.75923i

1.75551

−

3.38799i

0.682553

0.730836i

−1.69292

−

1.02949i

404.1

−0.0457060

0.128604i

−0.203456

0.979084i

0.761261

0.619332i

0.0682424

0.997669i

−0.116615

−

0.0709153i

−0.231058

0.140510i

−0.917211

−

0.398401i

−0.131424

−

0.0368232i

479.1

0.843954

−

1.62876i

0.334880

0.942261i

−1.36391

−

1.93222i

0.917211

−

0.398401i

1.81734

0.249787i

−2.48086

0.340986i

−0.775711

0.631088i

0.125185

−

1.83015i

494.1

−0.663521

0.0911989i

−0.854419

−

0.519584i

−0.530974

0.148772i

0.334880

−

0.942261i

0.614311

0.266833i

0.953056

−

0.413970i

0.460065

0.887885i

−0.136267

0.655751i

524.1

−0.0787081

1.15067i

−0.962917

0.269797i

−0.327165

−

0.0449678i

0.576680

−

0.816970i

−0.234658

−

1.12924i

−0.157164

0.756317i

0.854419

−

0.519584i

0.894675

0.727872i

554.1

−0.391823

−

1.88555i

−0.682553

−

0.730836i

−2.48458

1.07920i

−0.962917

0.269797i

−1.11059

1.57335i

1.89782

2.68860i

−0.0682424

0.997669i

0.886009

1.70992i

614.1

−0.347674

0.211425i

0.576680

−

0.816970i

−0.383889

0.740871i

−0.203456

−

0.979084i

−0.0277687

0.405963i

−0.0509395

−

0.744708i

−0.334880

−

0.942261i

0.277739

0.297386i

See all 22 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by $\Q(\sqrt{-15}) $
47.c	even	23	1	inner
705.p	odd	46	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	705.1.p.b	yes	22
3.b	odd	2	1		705.1.p.a	✓	22
5.b	even	2	1		705.1.p.a	✓	22
5.c	odd	4	2		3525.1.bd.a		44
15.d	odd	2	1	CM	705.1.p.b	yes	22
15.e	even	4	2		3525.1.bd.a		44
47.c	even	23	1	inner	705.1.p.b	yes	22
141.h	odd	46	1		705.1.p.a	✓	22
235.i	even	46	1		705.1.p.a	✓	22
235.k	odd	92	2		3525.1.bd.a		44
705.p	odd	46	1	inner	705.1.p.b	yes	22
705.w	even	92	2		3525.1.bd.a		44

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
705.1.p.a	✓	22	3.b	odd	2	1
705.1.p.a	✓	22	5.b	even	2	1
705.1.p.a	✓	22	141.h	odd	46	1
705.1.p.a	✓	22	235.i	even	46	1
705.1.p.b	yes	22	1.a	even	1	1	trivial
705.1.p.b	yes	22	15.d	odd	2	1	CM
705.1.p.b	yes	22	47.c	even	23	1	inner
705.1.p.b	yes	22	705.p	odd	46	1	inner
3525.1.bd.a		44	5.c	odd	4	2
3525.1.bd.a		44	15.e	even	4	2
3525.1.bd.a		44	235.k	odd	92	2
3525.1.bd.a		44	705.w	even	92	2

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{22} - 2 T_{2}^{21} + 4 T_{2}^{20} - 8 T_{2}^{19} + 16 T_{2}^{18} - 32 T_{2}^{17} + 64 T_{2}^{16} + \cdots + 1 $ T2^22 - 2*T2^21 + 4*T2^20 - 8*T2^19 + 16*T2^18 - 32*T2^17 + 64*T2^16 - 128*T2^15 + 233*T2^14 - 144*T2^13 + 12*T2^12 - T2^11 + 2*T2^10 - 4*T2^9 + 8*T2^8 - 16*T2^7 + 124*T2^6 + 718*T2^5 + 841*T2^4 + 411*T2^3 + 98*T2^2 + 11*T2 + 1 acting on $S_{1}^{\mathrm{new}}(705, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{22} - 2 T^{21} + \cdots + 1 $ T^22 - 2T^21 + 4T^20 - 8T^19 + 16T^18 - 32T^17 + 64T^16 - 128T^15 + 233T^14 - 144T^13 + 12T^12 - T^11 + 2T^10 - 4T^9 + 8T^8 - 16T^7 + 124T^6 + 718T^5 + 841T^4 + 411T^3 + 98T^2 + 11T + 1
$3$	$ T^{22} - T^{21} + \cdots + 1 $ T^22 - T^21 + T^20 - T^19 + T^18 - T^17 + T^16 - T^15 + T^14 - T^13 + T^12 - T^11 + T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$5$	$ T^{22} - T^{21} + \cdots + 1 $ T^22 - T^21 + T^20 - T^19 + T^18 - T^17 + T^16 - T^15 + T^14 - T^13 + T^12 - T^11 + T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$7$	$ T^{22} $ T^22
$11$	$ T^{22} $ T^22
$13$	$ T^{22} $ T^22
$17$	$ T^{22} - 2 T^{21} + \cdots + 1 $ T^22 - 2T^21 + 4T^20 - 8T^19 - 7T^18 + 14T^17 - 28T^16 + 56T^15 + 72T^14 - 121T^13 + 242T^12 - 484T^11 + 370T^10 - 4T^9 + 8T^8 - 16T^7 + 722T^6 + 5T^5 + 59T^4 - 118T^3 + 75T^2 - 12*T + 1
$19$	$ T^{22} + 2 T^{21} + \cdots + 1 $ T^22 + 2T^21 + 4T^20 + 8T^19 + 16T^18 + 32T^17 + 41T^16 + 13T^15 + 26T^14 + 52T^13 + 104T^12 + 208T^11 + 554T^10 - 341T^9 + 54T^8 + 85T^7 + 170T^6 + 340T^5 + 519T^4 + 348T^3 + 98T^2 + 12*T + 1
$23$	$ T^{22} + 21 T^{21} + \cdots + 1 $ T^22 + 21T^21 + 211T^20 + 1349T^19 + 6157T^18 + 21335T^17 + 58277T^16 + 128603T^15 + 233108T^14 + 350974T^13 + 442118T^12 + 467842T^11 + 416394T^10 + 311278T^9 + 194634T^8 + 101046T^7 + 43065T^6 + 14817T^5 + 4015T^4 + 825T^3 + 121T^2 + 11*T + 1
$29$	$ T^{22} $ T^22
$31$	$ T^{22} + 2 T^{21} + \cdots + 1 $ T^22 + 2T^21 + 4T^20 + 8T^19 + 16T^18 + 32T^17 + 41T^16 + 13T^15 + 26T^14 + 52T^13 + 104T^12 + 208T^11 + 554T^10 - 341T^9 + 54T^8 + 85T^7 + 170T^6 + 340T^5 + 519T^4 + 348T^3 + 98T^2 + 12*T + 1
$37$	$ T^{22} $ T^22
$41$	$ T^{22} $ T^22
$43$	$ T^{22} $ T^22
$47$	$ T^{22} - T^{21} + \cdots + 1 $ T^22 - T^21 + T^20 - T^19 + T^18 - T^17 + T^16 - T^15 + T^14 - T^13 + T^12 - T^11 + T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$53$	$ T^{22} - 2 T^{21} + \cdots + 1 $ T^22 - 2T^21 + 4T^20 - 8T^19 + 16T^18 + 14T^17 - 28T^16 + 56T^15 - 112T^14 + 224T^13 + 127T^12 - 139T^11 + 278T^10 - 556T^9 + 1089T^8 - 499T^7 + 9T^6 + 5T^5 - 10T^4 + 135T^3 + 75T^2 + 11*T + 1
$59$	$ T^{22} $ T^22
$61$	$ T^{22} + 2 T^{21} + \cdots + 1 $ T^22 + 2T^21 + 4T^20 - 15T^19 - 30T^18 - 60T^17 + 87T^16 + 174T^15 + 348T^14 - 224T^13 - 448T^12 - 896T^11 + 301T^10 + 579T^9 + 1158T^8 + 39T^7 - 198T^6 - 396T^5 + 174T^4 + 26T^3 + 52T^2 + 12*T + 1
$67$	$ T^{22} $ T^22
$71$	$ T^{22} $ T^22
$73$	$ T^{22} $ T^22
$79$	$ T^{22} + 2 T^{21} + \cdots + 1 $ T^22 + 2T^21 + 4T^20 + 8T^19 + 16T^18 + 32T^17 + 64T^16 + 128T^15 + 256T^14 + 512T^13 + 1024T^12 + 2048T^11 + 4073T^10 + 7640T^9 + 11991T^8 + 14115T^7 + 11785T^6 + 6826T^5 + 2704T^4 + 716T^3 + 121T^2 + 12*T + 1
$83$	$ T^{22} - 2 T^{21} + \cdots + 1 $ T^22 - 2T^21 + 4T^20 - 8T^19 + 16T^18 - 32T^17 + 64T^16 - 128T^15 + 256T^14 - 420T^13 + 840T^12 - 714T^11 + 1428T^10 - 579T^9 + 1158T^8 - 223T^7 + 423T^6 + 74T^5 + 174T^4 - 141T^3 + 6T^2 + 11*T + 1
$89$	$ T^{22} $ T^22
$97$	$ T^{22} $ T^22
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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:	\( N \)	\(=\)	\( 705 = 3 \cdot 5 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	705.p (of order \(46\), degree \(22\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{46}^{6} - \zeta_{46}^{4}) q^{2} + \zeta_{46}^{3} q^{3} + (\zeta_{46}^{12} + \cdots + \zeta_{46}^{8}) q^{4}+ \cdots + \zeta_{46}^{6} q^{9}+O(q^{10}) \) q + (-z^6 - z^4) * q^2 + z^3 * q^3 + (z^12 + z^10 + z^8) * q^4 - z^22 * q^5 + (-z^9 - z^7) * q^6 + (-z^18 - z^16 - z^14 - z^12) * q^8 + z^6 * q^9	\( q + ( - \zeta_{46}^{6} - \zeta_{46}^{4}) q^{2} + \zeta_{46}^{3} q^{3} + (\zeta_{46}^{12} + \cdots + \zeta_{46}^{8}) q^{4}+ \cdots + (\zeta_{46}^{11} + \zeta_{46}^{9}) q^{98}+O(q^{100}) \) q + (-z^6 - z^4) * q^2 + z^3 * q^3 + (z^12 + z^10 + z^8) * q^4 - z^22 * q^5 + (-z^9 - z^7) * q^6 + (-z^18 - z^16 - z^14 - z^12) * q^8 + z^6 * q^9 + (-z^5 - z^3) * q^10 + (z^15 + z^13 + z^11) * q^12 + z^2 * q^15 + (z^22 + z^20 + z^18 + z^16 - z) * q^16 + (z^21 - z^16) * q^17 + (-z^12 - z^10) * q^18 + (z^14 - z^11) * q^19 + (z^11 + z^9 + z^7) * q^20 + (z^13 - 1) * q^23 + (-z^21 - z^19 - z^17 - z^15) * q^24 - z^21 * q^25 + z^9 * q^27 + (-z^8 - z^6) * q^30 + (-z^13 + z^4) * q^31 + (-z^22 - z^20 + z^7 - z^5 - z^3 - z) * q^32 + (z^22 + z^20 + z^4 + z^2) * q^34 + (z^18 + z^16 + z^14) * q^36 + (-z^20 - z^18 + z^17 + z^15) * q^38 + (-z^17 - z^15 - z^13 - z^11) * q^40 + z^5 * q^45 + (-z^19 - z^17 + z^6 + z^4) * q^46 + z^19 * q^47 + (z^21 + z^19 - z^4 - z^2 - 1) * q^48 - z^5 * q^49 + (-z^4 - z^2) * q^50 + (-z^19 - z) * q^51 + (-z^14 - z^2) * q^53 + (-z^15 - z^13) * q^54 + (z^17 - z^14) * q^57 + (z^14 + z^12 + z^10) * q^60 + (z^8 - z^7) * q^61 + (z^19 + z^17 - z^10 - z^8) * q^62 + (-z^13 - z^11 + z^9 + z^7 + z^5 + z^3 - z) * q^64 + (-z^10 - z^8 - z^6 + z^5 + z^3 + z) * q^68 + (z^16 - z^3) * q^69 + (-z^22 - z^20 - z^18 + z) * q^72 + z * q^75 + (z^22 - z^21 - z^19 - z^3 - z + 1) * q^76 + (z^18 - z^9) * q^79 + (z^21 + z^19 + z^17 + z^15 - 1) * q^80 + z^12 * q^81 + (-z^8 + z) * q^83 + (z^20 - z^15) * q^85 + (-z^11 - z^9) * q^90 + (z^21 - z^12 - z^10 - z^8 - z^2 - 1) * q^92 + (-z^16 + z^7) * q^93 + (z^2 + 1) * q^94 + (z^13 - z^10) * q^95 + (z^10 + z^8 + z^6 + z^4 + z^2 + 1) * q^96 + (z^11 + z^9) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 2 q^{2} + q^{3} - 3 q^{4} + q^{5} - 2 q^{6} + 4 q^{8} - q^{9}+O(q^{10}) \) 22 * q + 2 * q^2 + q^3 - 3 * q^4 + q^5 - 2 * q^6 + 4 * q^8 - q^9	\( 22 q + 2 q^{2} + q^{3} - 3 q^{4} + q^{5} - 2 q^{6} + 4 q^{8} - q^{9} - 2 q^{10} + 3 q^{12} - q^{15} - 5 q^{16} + 2 q^{17} + 2 q^{18} - 2 q^{19} + 3 q^{20} - 21 q^{23} - 4 q^{24} - q^{25} + q^{27} + 2 q^{30} - 2 q^{31} + 6 q^{32} - 4 q^{34} - 3 q^{36} + 4 q^{38} - 4 q^{40} + q^{45} - 4 q^{46} + q^{47} - 18 q^{48} - q^{49} + 2 q^{50} - 2 q^{51} + 2 q^{53} - 2 q^{54} + 2 q^{57} - 3 q^{60} - 2 q^{61} + 4 q^{62} - 7 q^{64} + 6 q^{68} - 2 q^{69} + 4 q^{72} + q^{75} + 17 q^{76} - 2 q^{79} - 18 q^{80} - q^{81} + 2 q^{83} - 2 q^{85} - 2 q^{90} - 17 q^{92} + 2 q^{93} + 21 q^{94} + 2 q^{95} + 17 q^{96} + 2 q^{98}+O(q^{100}) \) 22 * q + 2 * q^2 + q^3 - 3 * q^4 + q^5 - 2 * q^6 + 4 * q^8 - q^9 - 2 * q^10 + 3 * q^12 - q^15 - 5 * q^16 + 2 * q^17 + 2 * q^18 - 2 * q^19 + 3 * q^20 - 21 * q^23 - 4 * q^24 - q^25 + q^27 + 2 * q^30 - 2 * q^31 + 6 * q^32 - 4 * q^34 - 3 * q^36 + 4 * q^38 - 4 * q^40 + q^45 - 4 * q^46 + q^47 - 18 * q^48 - q^49 + 2 * q^50 - 2 * q^51 + 2 * q^53 - 2 * q^54 + 2 * q^57 - 3 * q^60 - 2 * q^61 + 4 * q^62 - 7 * q^64 + 6 * q^68 - 2 * q^69 + 4 * q^72 + q^75 + 17 * q^76 - 2 * q^79 - 18 * q^80 - q^81 + 2 * q^83 - 2 * q^85 - 2 * q^90 - 17 * q^92 + 2 * q^93 + 21 * q^94 + 2 * q^95 + 17 * q^96 + 2 * q^98

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

Character values

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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	705.1.p.b

Level	$705$
Weight	$1$
Character orbit	705.p
Analytic conductor	$0.352$
Analytic rank	$0$
Dimension	$22$
Projective image	$D_{23}$
CM discriminant	-15
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.351840833906\)
Analytic rank:	\(0\)
Dimension:	\(22\)
Coefficient field:	\(\Q(\zeta_{46})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} + \cdots + 1 \) x^22 - x^21 + x^20 - x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{23}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{23} - \cdots)\)

\(n\)	\(142\)	\(236\)	\(616\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-\zeta_{46}^{21}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 14.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{22} - 2 T^{21} + \cdots + 1 \) T^22 - 2T^21 + 4T^20 - 8T^19 + 16T^18 - 32T^17 + 64T^16 - 128T^15 + 233T^14 - 144T^13 + 12T^12 - T^11 + 2T^10 - 4T^9 + 8T^8 - 16T^7 + 124T^6 + 718T^5 + 841T^4 + 411T^3 + 98T^2 + 11T + 1
$3$	\( T^{22} - T^{21} + \cdots + 1 \) T^22 - T^21 + T^20 - T^19 + T^18 - T^17 + T^16 - T^15 + T^14 - T^13 + T^12 - T^11 + T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$5$	\( T^{22} - T^{21} + \cdots + 1 \) T^22 - T^21 + T^20 - T^19 + T^18 - T^17 + T^16 - T^15 + T^14 - T^13 + T^12 - T^11 + T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$7$	\( T^{22} \) T^22
$11$	\( T^{22} \) T^22
$13$	\( T^{22} \) T^22
$17$	\( T^{22} - 2 T^{21} + \cdots + 1 \) T^22 - 2T^21 + 4T^20 - 8T^19 - 7T^18 + 14T^17 - 28T^16 + 56T^15 + 72T^14 - 121T^13 + 242T^12 - 484T^11 + 370T^10 - 4T^9 + 8T^8 - 16T^7 + 722T^6 + 5T^5 + 59T^4 - 118T^3 + 75T^2 - 12*T + 1
$19$	\( T^{22} + 2 T^{21} + \cdots + 1 \) T^22 + 2T^21 + 4T^20 + 8T^19 + 16T^18 + 32T^17 + 41T^16 + 13T^15 + 26T^14 + 52T^13 + 104T^12 + 208T^11 + 554T^10 - 341T^9 + 54T^8 + 85T^7 + 170T^6 + 340T^5 + 519T^4 + 348T^3 + 98T^2 + 12*T + 1
$23$	\( T^{22} + 21 T^{21} + \cdots + 1 \) T^22 + 21T^21 + 211T^20 + 1349T^19 + 6157T^18 + 21335T^17 + 58277T^16 + 128603T^15 + 233108T^14 + 350974T^13 + 442118T^12 + 467842T^11 + 416394T^10 + 311278T^9 + 194634T^8 + 101046T^7 + 43065T^6 + 14817T^5 + 4015T^4 + 825T^3 + 121T^2 + 11*T + 1
$29$	\( T^{22} \) T^22
$31$	\( T^{22} + 2 T^{21} + \cdots + 1 \) T^22 + 2T^21 + 4T^20 + 8T^19 + 16T^18 + 32T^17 + 41T^16 + 13T^15 + 26T^14 + 52T^13 + 104T^12 + 208T^11 + 554T^10 - 341T^9 + 54T^8 + 85T^7 + 170T^6 + 340T^5 + 519T^4 + 348T^3 + 98T^2 + 12*T + 1
$37$	\( T^{22} \) T^22
$41$	\( T^{22} \) T^22
$43$	\( T^{22} \) T^22
$47$	\( T^{22} - T^{21} + \cdots + 1 \) T^22 - T^21 + T^20 - T^19 + T^18 - T^17 + T^16 - T^15 + T^14 - T^13 + T^12 - T^11 + T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$53$	\( T^{22} - 2 T^{21} + \cdots + 1 \) T^22 - 2T^21 + 4T^20 - 8T^19 + 16T^18 + 14T^17 - 28T^16 + 56T^15 - 112T^14 + 224T^13 + 127T^12 - 139T^11 + 278T^10 - 556T^9 + 1089T^8 - 499T^7 + 9T^6 + 5T^5 - 10T^4 + 135T^3 + 75T^2 + 11*T + 1
$59$	\( T^{22} \) T^22
$61$	\( T^{22} + 2 T^{21} + \cdots + 1 \) T^22 + 2T^21 + 4T^20 - 15T^19 - 30T^18 - 60T^17 + 87T^16 + 174T^15 + 348T^14 - 224T^13 - 448T^12 - 896T^11 + 301T^10 + 579T^9 + 1158T^8 + 39T^7 - 198T^6 - 396T^5 + 174T^4 + 26T^3 + 52T^2 + 12*T + 1
$67$	\( T^{22} \) T^22
$71$	\( T^{22} \) T^22
$73$	\( T^{22} \) T^22
$79$	\( T^{22} + 2 T^{21} + \cdots + 1 \) T^22 + 2T^21 + 4T^20 + 8T^19 + 16T^18 + 32T^17 + 64T^16 + 128T^15 + 256T^14 + 512T^13 + 1024T^12 + 2048T^11 + 4073T^10 + 7640T^9 + 11991T^8 + 14115T^7 + 11785T^6 + 6826T^5 + 2704T^4 + 716T^3 + 121T^2 + 12*T + 1
$83$	\( T^{22} - 2 T^{21} + \cdots + 1 \) T^22 - 2T^21 + 4T^20 - 8T^19 + 16T^18 - 32T^17 + 64T^16 - 128T^15 + 256T^14 - 420T^13 + 840T^12 - 714T^11 + 1428T^10 - 579T^9 + 1158T^8 - 223T^7 + 423T^6 + 74T^5 + 174T^4 - 141T^3 + 6T^2 + 11*T + 1
$89$	\( T^{22} \) T^22
$97$	\( T^{22} \) T^22
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