LMFDB - Newform orbit 414.2.i.b

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

0.592229

4.11904i

−1.22301

2.67803i

−0.654861

0.755750i

−1.72871

−

3.78534i

73.1

0.415415

0.909632i

−0.654861

0.755750i

−2.43450

−

1.56456i

0.394306

2.74246i

−0.959493

−

0.281733i

0.411844

−

2.86444i

127.1

0.841254

−

0.540641i

0.415415

−

0.909632i

4.24593

−

1.24672i

−2.17208

−

2.50672i

−0.142315

−

0.989821i

2.89788

−

3.34433i

163.1

0.841254

0.540641i

0.415415

0.909632i

4.24593

1.24672i

−2.17208

2.50672i

−0.142315

0.989821i

2.89788

3.34433i

271.1

−0.959493

−

0.281733i

0.841254

0.540641i

0.592229

−

4.11904i

−1.22301

−

2.67803i

−0.654861

−

0.755750i

−1.72871

3.78534i

289.1

−0.654861

−

0.755750i

−0.142315

0.989821i

−0.810827

1.77546i

0.439490

0.129046i

0.841254

−

0.540641i

1.87279

−

0.549899i

307.1

−0.142315

0.989821i

−0.959493

−

0.281733i

−1.59283

−

1.83823i

1.56130

1.00339i

0.415415

−

0.909632i

2.04620

−

1.31501i

325.1

−0.142315

−

0.989821i

−0.959493

0.281733i

−1.59283

1.83823i

1.56130

−

1.00339i

0.415415

0.909632i

2.04620

1.31501i

361.1

−0.654861

0.755750i

−0.142315

−

0.989821i

−0.810827

−

1.77546i

0.439490

−

0.129046i

0.841254

0.540641i

1.87279

0.549899i

397.1

0.415415

−

0.909632i

−0.654861

−

0.755750i

−2.43450

1.56456i

0.394306

−

2.74246i

−0.959493

0.281733i

0.411844

2.86444i

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.b		10
3.b	odd	2	1		138.2.e.c	✓	10
23.c	even	11	1	inner	414.2.i.b		10
23.c	even	11	1		9522.2.a.bv		5
23.d	odd	22	1		9522.2.a.ca		5
69.g	even	22	1		3174.2.a.y		5
69.h	odd	22	1		138.2.e.c	✓	10
69.h	odd	22	1		3174.2.a.z		5

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

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{10} - 33T_{5}^{7} - 165T_{5}^{6} - 506T_{5}^{5} + 2178T_{5}^{4} + 14399T_{5}^{3} + 45012T_{5}^{2} + 66792T_{5} + 64009 $ T5^10 - 33*T5^7 - 165*T5^6 - 506*T5^5 + 2178*T5^4 + 14399*T5^3 + 45012*T5^2 + 66792*T5 + 64009 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} - 33 T^{7} + \cdots + 64009 $ T^10 - 33T^7 - 165T^6 - 506T^5 + 2178T^4 + 14399T^3 + 45012T^2 + 66792*T + 64009
$7$	$ T^{10} + 2 T^{9} + \cdots + 529 $ T^10 + 2T^9 + 15T^8 - 3T^7 + 93T^6 - 210T^5 + 504T^4 - 1192T^3 + 3336T^2 - 2392*T + 529
$11$	$ T^{10} + 11 T^{9} + \cdots + 64009 $ T^10 + 11T^9 + 99T^8 + 605T^7 + 2882T^6 + 10384T^5 + 21780T^4 + 17908T^3 + 4961T^2 + 30613*T + 64009
$13$	$ T^{10} + 13 T^{9} + \cdots + 1 $ T^10 + 13T^9 + 59T^8 + 96T^7 - 6T^6 - 254T^5 + 317T^4 + 40T^3 + 58T^2 + 6*T + 1
$17$	$ T^{10} - 11 T^{8} + \cdots + 33860761 $ T^10 - 11T^8 - 253T^7 - 110T^6 - 7117T^5 + 69575T^4 + 457743T^3 + 2630056T^2 - 3328468T + 33860761
$19$	$ T^{10} + 2 T^{9} + \cdots + 978121 $ T^10 + 2T^9 + 26T^8 - 355T^7 + 313T^6 - 3532T^5 + 50158T^4 - 155533T^3 + 333732T^2 - 658674*T + 978121
$23$	$ T^{10} - 10 T^{9} + \cdots + 6436343 $ T^10 - 10T^9 + 100T^8 - 593T^7 + 3532T^6 - 16301T^5 + 81236T^4 - 313697T^3 + 1216700T^2 - 2798410*T + 6436343
$29$	$ T^{10} - 27 T^{9} + \cdots + 591361 $ T^10 - 27T^9 + 344T^8 - 2589T^7 + 14122T^6 - 54231T^5 + 141938T^4 - 241123T^3 + 288149T^2 - 424488*T + 591361
$31$	$ T^{10} + 18 T^{9} + \cdots + 529 $ T^10 + 18T^9 + 148T^8 + 959T^7 + 5327T^6 + 17885T^5 + 42420T^4 + 59637T^3 + 54569T^2 + 8786*T + 529
$37$	$ T^{10} + T^{9} + \cdots + 157609 $ T^10 + T^9 + 34T^8 - 285T^7 + 1530T^6 - 10185T^5 + 75439T^4 - 161391T^3 + 88947T^2 - 43273T + 157609
$41$	$ T^{10} - 16 T^{9} + \cdots + 38809 $ T^10 - 16T^9 + 223T^8 - 1511T^7 + 8380T^6 - 25686T^5 + 26757T^4 + 89889T^3 - 9368T^2 - 24231*T + 38809
$43$	$ T^{10} - 20 T^{9} + \cdots + 978121 $ T^10 - 20T^9 + 191T^8 - 1675T^7 + 15900T^6 - 106294T^5 + 407097T^4 - 892863T^3 + 1577436T^2 - 1561631*T + 978121
$47$	$ (T^{5} - 55 T^{3} + \cdots - 253)^{2} $ (T^5 - 55T^3 + 22T^2 + 352*T - 253)^2
$53$	$ T^{10} - T^{9} + \cdots + 31956409 $ T^10 - T^9 + 67T^8 + 1143T^7 + 13003T^6 + 117281T^5 + 886183T^4 + 4558619T^3 + 16282839T^2 + 34703767T + 31956409
$59$	$ T^{10} - T^{9} + \cdots + 1 $ T^10 - T^9 - 21T^8 + 65T^7 + 199T^6 - 672T^5 + 1046T^4 - 188T^3 + 177T^2 + 10T + 1
$61$	$ T^{10} + 34 T^{9} + \cdots + 39601 $ T^10 + 34T^9 + 507T^8 + 4291T^7 + 23189T^6 + 88298T^5 + 245136T^4 + 406440T^3 + 409388T^2 + 133728*T + 39601
$67$	$ T^{10} - 8 T^{9} + \cdots + 10042561 $ T^10 - 8T^9 + 64T^8 + 1644T^7 + 10311T^6 + 50942T^5 + 273716T^4 + 1122064T^3 + 3290875T^2 + 7228489*T + 10042561
$71$	$ T^{10} - 22 T^{9} + \cdots + 64009 $ T^10 - 22T^9 + 374T^8 - 6028T^7 + 56584T^6 - 257224T^5 + 589633T^4 - 573661T^3 + 431607T^2 + 200376*T + 64009
$73$	$ T^{10} - 31 T^{9} + \cdots + 20151121 $ T^10 - 31T^9 + 389T^8 - 2852T^7 + 17924T^6 - 103896T^5 + 500817T^4 - 1763172T^3 + 5198262T^2 - 11428994*T + 20151121
$79$	$ T^{10} + \cdots + 650199001 $ T^10 - 32T^9 + 496T^8 - 4751T^7 + 34783T^6 - 259159T^5 + 2343870T^4 - 17829723T^3 + 84028209T^2 - 253256068*T + 650199001
$83$	$ T^{10} + \cdots + 6146089609 $ T^10 + 33T^9 + 627T^8 + 7876T^7 + 92510T^6 + 836693T^5 + 10096119T^4 + 87644172T^3 + 728881131T^2 + 3440844330*T + 6146089609
$89$	$ T^{10} + \cdots + 4197614521 $ T^10 - 23T^9 + 221T^8 - 1772T^7 + 25884T^6 - 244289T^5 - 49675T^4 + 8214260T^3 + 149279439T^2 + 717732542*T + 4197614521
$97$	$ T^{10} + \cdots + 16124682289 $ T^10 + T^9 + 221T^8 + 6172T^7 + 27380T^6 + 621204T^5 + 9899638T^4 + 4919091T^3 + 618026861T^2 + 5369222189T + 16124682289
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Elliptic curves over $\Q$
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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 - 2z^3 - z^2 - 2z + 1) * q^5 + (-z^9 + z^7 + z^6 + z^4 + z^3 - z) * q^7 - z * q^8	\( q + \zeta_{22}^{4} q^{2} + \zeta_{22}^{8} q^{4} + ( - \zeta_{22}^{9} + 2 \zeta_{22}^{8} + \cdots + 1) q^{5}+ \cdots + (\zeta_{22}^{9} - \zeta_{22}^{8} + \cdots + 2 \zeta_{22}) q^{98}+O(q^{100}) \) q + z^4 * q^2 + z^8 * q^4 + (-z^9 + 2z^8 - 2z^7 + z^6 + z^4 - 2z^3 - z^2 - 2z + 1) * q^5 + (-z^9 + z^7 + z^6 + z^4 + z^3 - z) * q^7 - z * q^8 + (z^9 - z^7 - 2z^6 - z^5 + z^3 - z + 1) q^10 + (2z^9 - 2z^8 + 4z^7 - 2z^6 + 2z^5 + 3z^3 - 2z^2 + 2z - 3) * q^11 + (-z^7 + 2z^6 - z^5 + z - 1) q^13 + (z^9 + 2z^7 - z^6 - z^4 + z^3 + z - 2) q^14 - z^5 * q^16 + (4z^9 - 4z^8 - 3z^6 + 2z^5 - 5z^4 + 5z^3 - 5z^2 + 2z - 3) * q^17 + (4z^9 - 3z^8 + 4z^7 + 3z^5 - 4z^4 + 3z^3 - 3z^2 + 4z - 3) * q^19 + (-3z^9 + 2z^8 - z^7 + 2z^6 - 3z^5 + 3z^4 - 2z^3 + z^2 - 2z + 3) q^20 + (2z^8 + z^7 - z^4 - 2z^3 - 2) * q^22 + (2z^9 + z^8 + z^7 - z^6 + z^4 + 3z^3 - 2z^2 + 3z) * q^23 + (z^8 - 3z^7 - 2z^6 + 2z^5 + 6z^4 + 2z^3 - 2z^2 - 3z + 1) q^25 + (z^9 - 2z^8 + 2z^7 - 2z^6 + 3z^5 - 3z^4 + 2z^3 - 2z^2 + 2z - 1) * q^26 + (-z^9 + z^6 - z^4 - z^3 - z - 1) * q^28 + (-3z^9 + 3z^8 + 4z^6 - z^5 - z^3 - z + 4) q^29 + (3z^9 - 3z^8 + 2z^7 - 2z^6 + 3z^5 - 3z^4 - 4z^2 + 2z - 4) * q^31 - z^9 * q^32 + (-z^9 - 2z^8 + 2z^7 - 2z^6 - z^5 - 3z^3 - z^2 + z + 3) * q^34 + (-4z^9 - z^8 - 3z^7 - z^6 - 4z^5 + 3z^3 + 4z^2 - 4z - 3) * q^35 + (-4z^9 - z^7 + 3z^4 - 3z^3 + 1) q^37 + (3z^9 - 4z^8 + 3z^7 - 3z^6 + 4z^5 - 3z^4 - 4z^2 + 3z - 4) * q^38 + (-z^9 + z^8 - z^6 + z^4 + 2z^3 + z^2 - 1) q^40 + (-2z^9 - z^8 + z^7 + 2z^6 + 3z^4 - 3z^3 + 3z^2 - 3z + 3) * q^41 + (-4z^9 + 3z^8 - 4z^7 + z^6 - z^3 + 4z^2 - 3z + 4) q^43 + (-z^8 - 2z^7 - 2z^4 - 2z - 1) q^44 + (-z^9 + 2z^8 + 2z^7 - z^6 + 2z^5 + z^4 - z^3 - z^2 - 2z) * q^46 + (-3z^9 + z^8 - 2z^7 - z^6 + z^5 + 2z^4 - z^3 + 3z^2 + 1) * q^47 + (z^9 - 2z^7 + 2z^6 - z^5 + z^4 - 2z^3 + 2z^2 - 1) * q^49 + (8z^8 - 5z^5 + 3z^4 - 2z^3 + 2z^2 - 3z + 5) * q^50 + (z^9 - z^8 + z^4 - 2z^3 + z^2) q^52 + (2z^9 - 2z^7 + 2z^6 + 5z^5 + 2z^4 - 2z^3 + 2z) q^53 + (-2z^9 - 5z^7 + z^6 - 7z^5 - 3z^4 + 3z^3 + 7z^2 - z + 5) * q^55 + (z^9 - 2z^8 - z^6 - 2z^4 + z^3 + z - 1) * q^56 + (3z^9 - 4z^8 + 3z^7 - 4z^6 + 3z^5 + 4z^3 - z^2 + z - 4) * q^58 + (z^8 - 2z^7 + 2z^6 - 2z^5 + z^4 - z + 1) q^59 + (2z^8 - z^7 + z^6 - 2z^5 - 3z^2 - z - 3) q^61 + (z^9 - z^8 - 2z^7 - 2z^6 - 2z^4 - 2z^3 - z^2 + z) * q^62 + z^2 * q^64 + (2z^9 - 5z^8 + 2z^7 + 2z^5 - 2z^4 - 3z^3 + 3z^2 + 2z - 2) * q^65 + (-3z^8 + 4z^7 - 4z^6 + 7z^5 - 5z^4 + 7z^3 - 4z^2 + 4z - 3) * q^67 + (-3z^9 + 2z^8 - 5z^7 + z^6 - z^5 + 5z^4 - 2z^3 + 3z^2) * q^68 + (-5z^9 + z^8 + 2z^7 + 5z^6 - 5z^5 - 2z^4 - z^3 + 5z^2 + 4) * q^70 + (z^8 + 5z^7 - 3z^6 - z^5 + z^4 - z^3 - 3z^2 + 5z + 1) * q^71 + (2z^9 - 4z^8 + 2z^7 - z^5 - 4z^4 + 3z^3 - 3z^2 + 4z + 1) q^73 + (3z^8 - 3z^7 + z^4 + 4z^2 + 1) q^74 + (z^9 - 3z^7 - z^6 - z^4 - 3z^3 + z) * q^76 + (7z^9 - 3z^8 + 7z^7 - 7z^6 + 3z^5 - 7z^4 - 7z^2 + 7z - 7) * q^77 + (-z^9 - 4z^8 + z^7 + 3z^6 + z^5 - 4z^4 - z^3 - 3z + 3) * q^79 + (-z^9 + 2z^8 + z^7 + 2z^6 - 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 + (3z^9 + 2z^7 + 7z^6 + 2z^4 - 2z^3 - 7z - 2) * q^83 + (8z^9 + 9z^8 - 3z^7 + 2z^6 + z^5 + 2z^4 - 3z^3 + 9z^2 + 8z) * q^85 + (z^9 - z^8 + 3z^6 - 2z^5 + 3z^4 + z^3 + 3z^2 - 2z + 3) q^86 + (-2z^8 - 2z^5 - z^4 + z + 2) * q^88 + (-9z^9 + 5z^8 - 9z^7 + 10z^6 - 5z^5 + 5z^4 - 10z^3 + 9z^2 - 5z + 9) q^89 + (2z^9 + z^8 - 2z^7 + 2z^4 - z^3 - 2z^2 + 2) * q^91 + (z^9 + 2z^8 - 2z^7 - 3z^5 + z^4 - z^3 + 2z^2 - 3z - 1) q^92 + (3z^8 - 2z^7 + 4z^6 - z^5 + 2z^4 - z^3 + 4z^2 - 2z + 3) * q^94 + (-4z^9 - 5z^8 - 7z^7 + 8z^6 + 3z^5 - 3z^4 - 8z^3 + 7z^2 + 5z + 4) q^95 + (-z^9 - 8z^8 + 8z^7 + z^6 - z^4 - 3z^3 - 3z - 1) * q^97 + (z^9 - z^8 + 2z^5 - 3z^4 + 2z^3 - 3z^2 + 2z) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - q^{2} - q^{4} - 2 q^{7} - q^{8}+O(q^{10}) \) 10 * q - q^2 - q^4 - 2 * q^7 - q^8	\( 10 q - q^{2} - q^{4} - 2 q^{7} - q^{8} + 11 q^{10} - 11 q^{11} - 13 q^{13} - 13 q^{14} - q^{16} - 2 q^{19} + 11 q^{20} - 22 q^{22} + 10 q^{23} + 5 q^{25} + 9 q^{26} - 13 q^{28} + 27 q^{29} - 18 q^{31} - q^{32} + 33 q^{34} - 44 q^{35} - q^{37} - 13 q^{38} - 11 q^{40} + 16 q^{41} + 20 q^{43} - 11 q^{44} - q^{46} - 19 q^{49} + 27 q^{50} - 2 q^{52} + q^{53} + 33 q^{55} - 2 q^{56} - 17 q^{58} + q^{59} - 34 q^{61} + 4 q^{62} - q^{64} - 11 q^{65} + 8 q^{67} - 22 q^{68} + 22 q^{70} + 22 q^{71} + 31 q^{73} - q^{74} - 2 q^{76} - 22 q^{77} + 32 q^{79} - 28 q^{82} - 33 q^{83} - 11 q^{85} + 20 q^{86} + 22 q^{88} + 23 q^{89} + 18 q^{91} - 23 q^{92} + 11 q^{94} + 22 q^{95} - q^{97} + 14 q^{98}+O(q^{100}) \) 10 * q - q^2 - q^4 - 2 * q^7 - q^8 + 11 * q^10 - 11 * q^11 - 13 * q^13 - 13 * q^14 - q^16 - 2 * q^19 + 11 * q^20 - 22 * q^22 + 10 * q^23 + 5 * q^25 + 9 * q^26 - 13 * q^28 + 27 * q^29 - 18 * q^31 - q^32 + 33 * q^34 - 44 * q^35 - q^37 - 13 * q^38 - 11 * q^40 + 16 * q^41 + 20 * q^43 - 11 * q^44 - q^46 - 19 * q^49 + 27 * q^50 - 2 * q^52 + q^53 + 33 * q^55 - 2 * q^56 - 17 * q^58 + q^59 - 34 * q^61 + 4 * q^62 - q^64 - 11 * q^65 + 8 * q^67 - 22 * q^68 + 22 * q^70 + 22 * q^71 + 31 * q^73 - q^74 - 2 * q^76 - 22 * q^77 + 32 * q^79 - 28 * q^82 - 33 * q^83 - 11 * q^85 + 20 * q^86 + 22 * q^88 + 23 * q^89 + 18 * q^91 - 23 * q^92 + 11 * q^94 + 22 * q^95 - q^97 + 14 * q^98

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Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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Label	414.2.i.b

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
$5$	\( T^{10} - 33 T^{7} + \cdots + 64009 \) T^10 - 33T^7 - 165T^6 - 506T^5 + 2178T^4 + 14399T^3 + 45012T^2 + 66792*T + 64009
$7$	\( T^{10} + 2 T^{9} + \cdots + 529 \) T^10 + 2T^9 + 15T^8 - 3T^7 + 93T^6 - 210T^5 + 504T^4 - 1192T^3 + 3336T^2 - 2392*T + 529
$11$	\( T^{10} + 11 T^{9} + \cdots + 64009 \) T^10 + 11T^9 + 99T^8 + 605T^7 + 2882T^6 + 10384T^5 + 21780T^4 + 17908T^3 + 4961T^2 + 30613*T + 64009
$13$	\( T^{10} + 13 T^{9} + \cdots + 1 \) T^10 + 13T^9 + 59T^8 + 96T^7 - 6T^6 - 254T^5 + 317T^4 + 40T^3 + 58T^2 + 6*T + 1
$17$	\( T^{10} - 11 T^{8} + \cdots + 33860761 \) T^10 - 11T^8 - 253T^7 - 110T^6 - 7117T^5 + 69575T^4 + 457743T^3 + 2630056T^2 - 3328468T + 33860761
$19$	\( T^{10} + 2 T^{9} + \cdots + 978121 \) T^10 + 2T^9 + 26T^8 - 355T^7 + 313T^6 - 3532T^5 + 50158T^4 - 155533T^3 + 333732T^2 - 658674*T + 978121
$23$	\( T^{10} - 10 T^{9} + \cdots + 6436343 \) T^10 - 10T^9 + 100T^8 - 593T^7 + 3532T^6 - 16301T^5 + 81236T^4 - 313697T^3 + 1216700T^2 - 2798410*T + 6436343
$29$	\( T^{10} - 27 T^{9} + \cdots + 591361 \) T^10 - 27T^9 + 344T^8 - 2589T^7 + 14122T^6 - 54231T^5 + 141938T^4 - 241123T^3 + 288149T^2 - 424488*T + 591361
$31$	\( T^{10} + 18 T^{9} + \cdots + 529 \) T^10 + 18T^9 + 148T^8 + 959T^7 + 5327T^6 + 17885T^5 + 42420T^4 + 59637T^3 + 54569T^2 + 8786*T + 529
$37$	\( T^{10} + T^{9} + \cdots + 157609 \) T^10 + T^9 + 34T^8 - 285T^7 + 1530T^6 - 10185T^5 + 75439T^4 - 161391T^3 + 88947T^2 - 43273T + 157609
$41$	\( T^{10} - 16 T^{9} + \cdots + 38809 \) T^10 - 16T^9 + 223T^8 - 1511T^7 + 8380T^6 - 25686T^5 + 26757T^4 + 89889T^3 - 9368T^2 - 24231*T + 38809
$43$	\( T^{10} - 20 T^{9} + \cdots + 978121 \) T^10 - 20T^9 + 191T^8 - 1675T^7 + 15900T^6 - 106294T^5 + 407097T^4 - 892863T^3 + 1577436T^2 - 1561631*T + 978121
$47$	\( (T^{5} - 55 T^{3} + \cdots - 253)^{2} \) (T^5 - 55T^3 + 22T^2 + 352*T - 253)^2
$53$	\( T^{10} - T^{9} + \cdots + 31956409 \) T^10 - T^9 + 67T^8 + 1143T^7 + 13003T^6 + 117281T^5 + 886183T^4 + 4558619T^3 + 16282839T^2 + 34703767T + 31956409
$59$	\( T^{10} - T^{9} + \cdots + 1 \) T^10 - T^9 - 21T^8 + 65T^7 + 199T^6 - 672T^5 + 1046T^4 - 188T^3 + 177T^2 + 10T + 1
$61$	\( T^{10} + 34 T^{9} + \cdots + 39601 \) T^10 + 34T^9 + 507T^8 + 4291T^7 + 23189T^6 + 88298T^5 + 245136T^4 + 406440T^3 + 409388T^2 + 133728*T + 39601
$67$	\( T^{10} - 8 T^{9} + \cdots + 10042561 \) T^10 - 8T^9 + 64T^8 + 1644T^7 + 10311T^6 + 50942T^5 + 273716T^4 + 1122064T^3 + 3290875T^2 + 7228489*T + 10042561
$71$	\( T^{10} - 22 T^{9} + \cdots + 64009 \) T^10 - 22T^9 + 374T^8 - 6028T^7 + 56584T^6 - 257224T^5 + 589633T^4 - 573661T^3 + 431607T^2 + 200376*T + 64009
$73$	\( T^{10} - 31 T^{9} + \cdots + 20151121 \) T^10 - 31T^9 + 389T^8 - 2852T^7 + 17924T^6 - 103896T^5 + 500817T^4 - 1763172T^3 + 5198262T^2 - 11428994*T + 20151121
$79$	\( T^{10} + \cdots + 650199001 \) T^10 - 32T^9 + 496T^8 - 4751T^7 + 34783T^6 - 259159T^5 + 2343870T^4 - 17829723T^3 + 84028209T^2 - 253256068*T + 650199001
$83$	\( T^{10} + \cdots + 6146089609 \) T^10 + 33T^9 + 627T^8 + 7876T^7 + 92510T^6 + 836693T^5 + 10096119T^4 + 87644172T^3 + 728881131T^2 + 3440844330*T + 6146089609
$89$	\( T^{10} + \cdots + 4197614521 \) T^10 - 23T^9 + 221T^8 - 1772T^7 + 25884T^6 - 244289T^5 - 49675T^4 + 8214260T^3 + 149279439T^2 + 717732542*T + 4197614521
$97$	\( T^{10} + \cdots + 16124682289 \) T^10 + T^9 + 221T^8 + 6172T^7 + 27380T^6 + 621204T^5 + 9899638T^4 + 4919091T^3 + 618026861T^2 + 5369222189T + 16124682289
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\(n\)	\(47\)	\(235\)
\(\chi(n)\)	\(1\)	\(\zeta_{22}^{4}\)