LMFDB - Newform orbit 966.2.q.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [966,2,Mod(85,966)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(966, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 0, 8]))

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

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

chi := DirichletCharacter("966.85");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 966 = 2 \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	966.q (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:	$7.71354883526$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/966\mathbb{Z}\right)^\times$.

$n$	$323$	$829$	$925$
$\chi(n)$	$1$	$1$	$\zeta_{22}^{4}$

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} $

85.1

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

0.654861

−

0.755750i

0.841254

−

0.540641i

−0.142315

−

0.989821i

0.817178

1.78937i

0.142315

−

0.989821i

−0.959493

0.281733i

−0.841254

−

0.540641i

0.415415

−

0.909632i

1.88745

0.554206i

127.1

−0.841254

0.540641i

−0.142315

0.989821i

0.415415

−

0.909632i

1.07028

−

0.314261i

−0.415415

−

0.909632i

−0.654861

−

0.755750i

0.142315

0.989821i

−0.959493

−

0.281733i

−0.730471

0.843008i

169.1

0.142315

−

0.989821i

0.415415

0.909632i

−0.959493

−

0.281733i

−1.80075

−

2.07817i

0.959493

−

0.281733i

0.841254

0.540641i

−0.415415

0.909632i

−0.654861

0.755750i

−2.31329

1.48666i

211.1

−0.415415

−

0.909632i

−0.959493

0.281733i

−0.654861

0.755750i

−0.512546

−

0.329393i

0.654861

0.755750i

−0.142315

−

0.989821i

0.959493

0.281733i

0.841254

−

0.540641i

−0.0867074

0.603063i

463.1

0.142315

0.989821i

0.415415

−

0.909632i

−0.959493

0.281733i

−1.80075

2.07817i

0.959493

0.281733i

0.841254

−

0.540641i

−0.415415

−

0.909632i

−0.654861

−

0.755750i

−2.31329

−

1.48666i

547.1

0.959493

0.281733i

−0.654861

0.755750i

0.841254

0.540641i

0.425839

−

2.96177i

−0.841254

0.540641i

0.415415

0.909632i

0.654861

0.755750i

−0.142315

−

0.989821i

1.24302

−

2.72183i

673.1

−0.415415

0.909632i

−0.959493

−

0.281733i

−0.654861

−

0.755750i

−0.512546

0.329393i

0.654861

−

0.755750i

−0.142315

0.989821i

0.959493

−

0.281733i

0.841254

0.540641i

−0.0867074

−

0.603063i

715.1

−0.841254

−

0.540641i

−0.142315

−

0.989821i

0.415415

0.909632i

1.07028

0.314261i

−0.415415

0.909632i

−0.654861

0.755750i

0.142315

−

0.989821i

−0.959493

0.281733i

−0.730471

−

0.843008i

841.1

0.654861

0.755750i

0.841254

0.540641i

−0.142315

0.989821i

0.817178

−

1.78937i

0.142315

0.989821i

−0.959493

−

0.281733i

−0.841254

0.540641i

0.415415

0.909632i

1.88745

−

0.554206i

883.1

0.959493

−

0.281733i

−0.654861

−

0.755750i

0.841254

−

0.540641i

0.425839

2.96177i

−0.841254

−

0.540641i

0.415415

−

0.909632i

0.654861

−

0.755750i

−0.142315

0.989821i

1.24302

2.72183i

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	966.2.q.c	✓	10
23.c	even	11	1	inner	966.2.q.c	✓	10

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
966.2.q.c	✓	10	1.a	even	1	1	trivial
966.2.q.c	✓	10	23.c	even	11	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{10} + 11T_{5}^{8} + 55T_{5}^{6} - 99T_{5}^{5} + 242T_{5}^{4} - 242T_{5}^{3} - 121T_{5}^{2} + 121T_{5} + 121 $ T5^10 + 11*T5^8 + 55*T5^6 - 99*T5^5 + 242*T5^4 - 242*T5^3 - 121*T5^2 + 121*T5 + 121 acting on $S_{2}^{\mathrm{new}}(966, [\chi])$.

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^{9} + \cdots + 1 $ T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$5$	$ T^{10} + 11 T^{8} + \cdots + 121 $ T^10 + 11T^8 + 55T^6 - 99T^5 + 242T^4 - 242T^3 - 121T^2 + 121*T + 121
$7$	$ 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
$11$	$ T^{10} + 6 T^{9} + \cdots + 1 $ T^10 + 6T^9 + 14T^8 + 18T^7 + 31T^6 + 54T^5 + 38T^4 + 52T^3 + 125T^2 - 9*T + 1
$13$	$ T^{10} + 2 T^{9} + \cdots + 978121 $ T^10 + 2T^9 - 18T^8 - 113T^7 + 346T^6 - 848T^5 + 3441T^4 + 10941T^3 + 247668T^2 + 189888*T + 978121
$17$	$ T^{10} + 5 T^{9} + \cdots + 139129 $ T^10 + 5T^9 - 8T^8 - 7T^7 + 273T^6 + 2586T^5 + 21444T^4 + 6559T^3 + 284057T^2 - 253640*T + 139129
$19$	$ T^{10} - 31 T^{9} + \cdots + 8814961 $ T^10 - 31T^9 + 488T^8 - 4975T^7 + 35909T^6 - 192402T^5 + 791008T^4 - 2556459T^3 + 6528635T^2 - 10795284*T + 8814961
$23$	$ T^{10} + T^{9} + \cdots + 6436343 $ T^10 + T^9 - 21T^8 - 219T^7 + 551T^6 + 4379T^5 + 12673T^4 - 115851T^3 - 255507T^2 + 279841T + 6436343
$29$	$ 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
$31$	$ T^{10} - 3 T^{9} + \cdots + 69172489 $ T^10 - 3T^9 + 9T^8 - 71T^7 + 1797T^6 + 17082T^5 - 72058T^4 + 195340T^3 + 4302985T^2 - 7352228*T + 69172489
$37$	$ T^{10} - 14 T^{9} + \cdots + 4489 $ T^10 - 14T^9 + 86T^8 - 60T^7 - 931T^6 + 8348T^5 + 23015T^4 - 10734T^3 + 16120T^2 - 6901*T + 4489
$41$	$ T^{10} + 28 T^{9} + \cdots + 5442889 $ T^10 + 28T^9 + 388T^8 + 3890T^7 + 31337T^6 + 190783T^5 + 836148T^4 + 2554428T^3 + 5351174T^2 + 7113317*T + 5442889
$43$	$ T^{10} - 9 T^{9} + \cdots + 4489 $ T^10 - 9T^9 + 114T^8 + 305T^7 - 523T^6 + 18809T^5 + 230085T^4 + 513278T^3 + 340541T^2 - 4757*T + 4489
$47$	$ (T^{5} + 22 T^{4} + \cdots - 4609)^{2} $ (T^5 + 22T^4 + 132T^3 - 77T^2 - 2530T - 4609)^2
$53$	$ T^{10} - T^{9} + \cdots + 18983449 $ T^10 - T^9 - 32T^8 - 595T^7 - 1682T^6 + 20987T^5 + 341254T^4 + 2059991T^3 + 7435429T^2 + 15092648T + 18983449
$59$	$ T^{10} + 11 T^{9} + \cdots + 16834609 $ T^10 + 11T^9 + 165T^8 + 550T^7 + 8206T^6 + 123442T^5 + 681351T^4 + 2365308T^3 + 8589306T^2 + 17872668*T + 16834609
$61$	$ T^{10} + \cdots + 125372809 $ T^10 + 41T^9 + 812T^8 + 9741T^7 + 69733T^6 + 256728T^5 + 503594T^4 + 2551441T^3 + 14634911T^2 - 6606230*T + 125372809
$67$	$ T^{10} + \cdots + 4866876169 $ T^10 + 12T^9 + 144T^8 - 582T^7 + 18327T^6 + 132298T^5 + 3384767T^4 + 42045290T^3 + 356317842T^2 + 1996826349*T + 4866876169
$71$	$ T^{10} + 22 T^{9} + \cdots + 543169 $ T^10 + 22T^9 + 528T^8 + 5797T^7 + 60060T^6 + 294899T^5 + 820622T^4 + 909799T^3 - 214533T^2 - 1159301*T + 543169
$73$	$ T^{10} - 19 T^{9} + \cdots + 1 $ T^10 - 19T^9 + 207T^8 - 1447T^7 + 7451T^6 - 14079T^5 + 24874T^4 + 8347T^3 + 522T^2 - 29*T + 1
$79$	$ T^{10} + \cdots + 1068832249 $ T^10 - 3T^9 + 86T^8 + 1722T^7 + 25469T^6 - 76176T^5 + 5991945T^4 - 24682590T^3 + 36147930T^2 - 319475996*T + 1068832249
$83$	$ T^{10} + \cdots + 1033043881 $ T^10 + 39T^9 + 696T^8 + 7212T^7 + 55636T^6 + 348435T^5 + 1818789T^4 + 9074986T^3 + 36553708T^2 + 142063220*T + 1033043881
$89$	$ T^{10} + \cdots + 30198445729 $ T^10 + 17T^9 + 399T^8 + 6431T^7 + 105455T^6 + 1466761T^5 + 20101779T^4 + 236714607T^3 + 2319994439T^2 + 13191238293*T + 30198445729
$97$	$ T^{10} + \cdots + 580376281 $ T^10 + 12T^9 + 375T^8 - 571T^7 + 12948T^6 - 18534T^5 + 4680875T^4 + 129435395T^3 + 624960172T^2 - 776958841*T + 580376281
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Overview	Random
Universe	Knowledge

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:	\( N \)	\(=\)	\( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	966.q (of order \(11\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{22}^{4} q^{2} + (\zeta_{22}^{9} - \zeta_{22}^{8} + \cdots - 1) q^{3} + \cdots - \zeta_{22}^{9} q^{9} +O(q^{10}) \) q - z^4 * q^2 + (z^9 - z^8 + z^7 - z^6 + z^5 - z^4 + z^3 - z^2 + z - 1) * q^3 + z^8 * q^4 + (-z^9 + z^8 - z^7 + z^6 + z^4 - 2z^3 + z^2 - 2z + 1) * q^5 + z^3 * q^6 - z^5 * q^7 + z * q^8 - z^9 * q^9	\( q - \zeta_{22}^{4} q^{2} + (\zeta_{22}^{9} - \zeta_{22}^{8} + \cdots - 1) q^{3} + \cdots + (\zeta_{22}^{9} - \zeta_{22}^{6} + \cdots + \zeta_{22}) q^{99} +O(q^{100}) \) q - z^4 * q^2 + (z^9 - z^8 + z^7 - z^6 + z^5 - z^4 + z^3 - z^2 + z - 1) * q^3 + z^8 * q^4 + (-z^9 + z^8 - z^7 + z^6 + z^4 - 2z^3 + z^2 - 2z + 1) * q^5 + z^3 * q^6 - z^5 * q^7 + z * q^8 - z^9 * q^9 + (-z^9 + z^7 + z^5 - z^3) * q^10 + (-z^8 + z^7 - z^6 + z^3 - 1) * q^11 - z^7 * q^12 + (-z^9 - z^8 + 3z^7 - z^6 + 3z^5 - z^4 - z^3 + z - 1) * q^13 + z^9 * q^14 + (z^9 - z^4 + z^2 + 1) * q^15 - z^5 * q^16 + (-4z^9 + 4z^8 + z^6 - z^5 + z^4 - 2z^3 + z^2 - z + 1) q^17 - z^2 * q^18 + (-2z^9 + z^8 - 2z^7 - 4z^5 + z^4 + z^3 - z^2 - z + 4) q^19 + (-z^9 + z^7 - z^2 + 1) * q^20 + z^4 * q^21 + (z^9 - z^8 - z^6 + z^5 + z^3 - z^2) * q^22 + (2z^9 - 4z^8 + 4z^7 + 3z^5 - 2z^4 + 2z^3 - 2z^2 - 2) q^23 - q^24 + (-3z^8 + z^7 - 2z^6 + z^5 - 2z^4 + z^3 - 2z^2 + z - 3) * q^25 + (-2z^9 + 2z^7 - z^6 + z^3 - 2z^2 + 2) q^26 + z^8 * q^27 + z^2 * q^28 + (-2z^9 + 2z^8 + z^6 - 2z^5 + 2z^4 - 2z^3 + 2z^2 - 2z + 1) q^29 + (z^8 - z^6 - z^4 + z^2) * q^30 + (4z^9 - 3z^8 + 6z^7 - 6z^6 + 3z^5 - 4z^4 - 3z^2 + 4z - 3) * q^31 + z^9 * q^32 + (-z^9 + z^8 + z^4 - z^3 - z + 1) * q^33 + (z^7 - z^3 - 3z^2 + 3z + 1) * q^34 + (-z^9 + 2z^8 - z^7 + 2z^6 - z^5 - z^3 + z^2 - z + 1) * q^35 + z^6 * q^36 + (-4z^9 - 2z^7 - z^5 - z^4 + z^3 + z^2 + 2) * q^37 + (4z^9 - z^8 - z^7 + z^6 + z^5 - 4z^4 - 2z^2 + z - 2) q^38 + (-z^9 + 2z^8 - 2z^6 - 2z^4 + 2z^2 - z) * q^39 + (z^6 - z^4 - z^2 + 1) * q^40 + (-3z^9 + 3z^6 - 2z^4 - 2z^3 - 2z - 2) q^41 - z^8 * q^42 + (-z^9 - z^8 + 3z^7 + 2z^6 - 2z^5 + 2z^4 - 2z^3 - 3z^2 + z + 1) * q^43 + (-z^8 + z^5 - z^4 + z^3 - 1) * q^44 + (z^9 - 2z^8 + z^7 - z^6 + z^5 - z^4 + 2z^3 - z^2 - 1) * q^45 + (-3z^9 + 2z^8 - 2z^7 + 2z^6 + 2z^4 + 2z^2 - 4z + 4) q^46 + (z^9 - 2z^8 + 2z^7 + 2z^6 - 2z^5 - 2z^4 + 2z^3 - z^2 - 5) * q^47 + z^4 * q^48 + (z^9 - z^8 + z^7 - z^6 + z^5 - z^4 + z^3 - z^2 + z - 1) * q^49 + (z^9 + z^7 + z^5 + z^4 + 2z^3 - 2z^2 - z - 1) * q^50 + (z^9 + 3z^8 - 3z^7 - z^6 + z^2) * q^51 + (z^9 - z^8 + z^6 + z^5 - 3z^4 + z^3 - 3z^2 + z + 1) * q^52 + (-3z^9 - 3z^6 + z^5 - 3z^4 - 3z) * q^53 + z * q^54 + (2z^9 - 2z^5 - z^4 + z^3 + 2z^2) q^55 - z^6 * q^56 + (4z^9 - 2z^8 + 3z^7 - 2z^6 + 4z^5 + 3z^3 - 5z^2 + 5z - 3) * q^57 + (z^9 - z^8 + z^7 - z^6 + z^5 - z^3 - z^2 + z + 1) * q^58 + (2z^9 + 2z^8 - 3z^7 - 4z^6 - 3z^5 + 2z^4 + 2z^3 + z - 1) q^59 + (z^9 + z^7 - 2z^6 + z^5 - z^4 + z^3 - z^2 + 2z - 1) * q^60 + (3z^9 - z^8 + 4z^7 - 4z^6 + z^5 - 3z^4 - 7z^2 + 6z - 7) * q^61 + (3z^9 - 2z^8 + 6z^7 - 3z^6 + 2z^5 - 3z^4 + 6z^3 - 2z^2 + 3z) q^62 - z^3 * q^63 + z^2 * q^64 + (3z^9 - 10z^8 + 3z^7 + z^5 + z^4 + 3z^3 - 3z^2 - z - 1) q^65 + (-z^8 + z^7 + z^5 - z^4 - z^2 + z) * q^66 + (4z^8 - 9z^7 + 4z^6 - 7z^5 + 8z^4 - 7z^3 + 4z^2 - 9z + 4) * q^67 + (z^7 + 3z^6 - 3z^5 - z^4 + 1) * q^68 + (-2z^9 + 2z^7 - 2z^6 - 2z^5 - z^4 + 2) * q^69 + (-z^9 + 2z^8 - z^7 + z^6 - z^5 + z^4 - 2z^3 + z^2 + 1) * q^70 + (-4z^8 + 3z^7 - 5z^6 - 2z^5 - 2z^4 - 2z^3 - 5z^2 + 3z - 4) * q^71 + (-z^9 + z^8 - z^7 + z^6 - z^5 + z^4 - z^3 + z^2 - z + 1) * q^72 + (z^9 + 2z^8 + z^7 - 3z^5 + z^4 - 3z^3 + 3z^2 - z + 3) * q^73 + (z^9 + z^8 - z^7 - z^6 - 2z^4 - 4z^2 - 2) * q^74 + (-3z^9 + 3z^8 + 2z^6 - z^5 + 2z^4 - z^3 + 2z^2 - z + 2) q^75 + (-2z^9 + 5z^8 - z^7 + 3z^6 - 2z^5 + 3z^4 - z^3 + 5z^2 - 2z) q^76 + (-z^8 + z^5 - z^2 + z - 1) * q^77 + (2z^9 + 2z^7 - 4z^6 + 3z^5 - 2z^4 + 2z^3 - 3z^2 + 4z - 2) * q^78 + (-4z^9 + 4z^8 - 3z^7 - 7z^6 - 3z^5 + 4z^4 - 4z^3 - 2z + 2) * q^79 + (-z^9 + 2z^8 - z^7 + 2z^6 - z^5 - z^3 + z^2 - z + 1) * q^80 - z^7 * q^81 + (-3z^9 + 5z^8 - z^7 + 3z^6 - z^5 + 5z^4 - 3z^3 - 3z + 3) * q^82 + (6z^9 + 5z^7 + z^6 - z^5 - 2z^4 + 2z^3 + z^2 - z - 5) * q^83 - z * q^84 + (-2z^9 - 7z^8 + 5z^7 + 2z^5 + 5z^3 - 7z^2 - 2z) q^85 + (5z^6 - 3z^5 + z^4 - 2z^3 + z^2 - 3z + 5) * q^86 + (z^9 + z^8 - z^7 - z^6 + z^4 - z^3 + z^2 - z + 1) * q^87 + (-z^9 + z^8 - z^7 + z^4 - z) * q^88 + (-7z^9 + 13z^8 - 7z^7 + 11z^6 - 9z^5 + 9z^4 - 11z^3 + 7z^2 - 13z + 7) q^89 + (-z^7 + z^5 + z^3 - z) * q^90 + (-2z^9 + 4z^8 - 3z^7 + 2z^6 - 2z^5 + 3z^4 - 4z^3 + 2z^2 + 2) * q^91 + (-2z^9 - 2z^7 + 2z^5 - 2z^4 - 2z^3 - z^2) q^92 + (-3z^9 - z^8 - 3z^6 + 3z^5 + z^3 + 3z^2 - 1) * q^93 + (4z^8 - 4z^7 + 3z^6 - 2z^5 + 7z^4 - 2z^3 + 3z^2 - 4z + 4) * q^94 + (-4z^9 + 9z^8 - 5z^7 + 4z^6 - 4z^3 + 5z^2 - 9z + 4) q^95 - z^8 * q^96 + (z^8 - z^7 - 3z^4 + 9z^3 + z^2 + 9z - 3) q^97 + z^3 * q^98 + (z^9 - z^6 + z^5 - z^4 + z) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + q^{2} - q^{3} - q^{4} + q^{6} - q^{7} + q^{8} - q^{9}+O(q^{10}) \) 10 * q + q^2 - q^3 - q^4 + q^6 - q^7 + q^8 - q^9	\( 10 q + q^{2} - q^{3} - q^{4} + q^{6} - q^{7} + q^{8} - q^{9} - 6 q^{11} - q^{12} - 2 q^{13} + q^{14} + 11 q^{15} - q^{16} - 5 q^{17} + q^{18} + 31 q^{19} + 11 q^{20} - q^{21} + 6 q^{22} - q^{23} - 10 q^{24} - 17 q^{25} + 24 q^{26} - q^{27} - q^{28} - 5 q^{29} + 3 q^{31} + q^{32} + 5 q^{33} + 16 q^{34} - q^{36} + 14 q^{37} - 9 q^{38} - 2 q^{39} + 11 q^{40} - 28 q^{41} + q^{42} + 9 q^{43} - 6 q^{44} + 23 q^{46} - 44 q^{47} - q^{48} - q^{49} - 5 q^{50} - 5 q^{51} + 20 q^{52} + q^{53} + q^{54} + q^{56} - 2 q^{57} + 16 q^{58} - 11 q^{59} - 41 q^{61} + 30 q^{62} - q^{63} - q^{64} + 11 q^{65} + 6 q^{66} - 12 q^{67} + 6 q^{68} + 21 q^{69} - 22 q^{71} + q^{72} + 19 q^{73} - 14 q^{74} + 5 q^{75} - 24 q^{76} - 6 q^{77} + 2 q^{78} + 3 q^{79} - q^{81} + 6 q^{82} - 39 q^{83} - q^{84} + 22 q^{85} + 35 q^{86} + 6 q^{87} - 5 q^{88} - 17 q^{89} - 2 q^{91} - q^{92} - 8 q^{93} + 11 q^{94} + q^{96} - 12 q^{97} + q^{98} + 5 q^{99}+O(q^{100}) \) 10 * q + q^2 - q^3 - q^4 + q^6 - q^7 + q^8 - q^9 - 6 * q^11 - q^12 - 2 * q^13 + q^14 + 11 * q^15 - q^16 - 5 * q^17 + q^18 + 31 * q^19 + 11 * q^20 - q^21 + 6 * q^22 - q^23 - 10 * q^24 - 17 * q^25 + 24 * q^26 - q^27 - q^28 - 5 * q^29 + 3 * q^31 + q^32 + 5 * q^33 + 16 * q^34 - q^36 + 14 * q^37 - 9 * q^38 - 2 * q^39 + 11 * q^40 - 28 * q^41 + q^42 + 9 * q^43 - 6 * q^44 + 23 * q^46 - 44 * q^47 - q^48 - q^49 - 5 * q^50 - 5 * q^51 + 20 * q^52 + q^53 + q^54 + q^56 - 2 * q^57 + 16 * q^58 - 11 * q^59 - 41 * q^61 + 30 * q^62 - q^63 - q^64 + 11 * q^65 + 6 * q^66 - 12 * q^67 + 6 * q^68 + 21 * q^69 - 22 * q^71 + q^72 + 19 * q^73 - 14 * q^74 + 5 * q^75 - 24 * q^76 - 6 * q^77 + 2 * q^78 + 3 * q^79 - q^81 + 6 * q^82 - 39 * q^83 - q^84 + 22 * q^85 + 35 * q^86 + 6 * q^87 - 5 * q^88 - 17 * q^89 - 2 * q^91 - q^92 - 8 * q^93 + 11 * q^94 + q^96 - 12 * q^97 + q^98 + 5 * q^99

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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	966.2.q.c

Level	$966$
Weight	$2$
Character orbit	966.q
Analytic conductor	$7.714$
Analytic rank	$0$
Dimension	$10$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(7.71354883526\)
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

\(n\)	\(323\)	\(829\)	\(925\)
\(\chi(n)\)	\(1\)	\(1\)	\(\zeta_{22}^{4}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 85.1
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
$3$	\( T^{10} + T^{9} + \cdots + 1 \) T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$5$	\( T^{10} + 11 T^{8} + \cdots + 121 \) T^10 + 11T^8 + 55T^6 - 99T^5 + 242T^4 - 242T^3 - 121T^2 + 121*T + 121
$7$	\( T^{10} + T^{9} + \cdots + 1 \) T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$11$	\( T^{10} + 6 T^{9} + \cdots + 1 \) T^10 + 6T^9 + 14T^8 + 18T^7 + 31T^6 + 54T^5 + 38T^4 + 52T^3 + 125T^2 - 9*T + 1
$13$	\( T^{10} + 2 T^{9} + \cdots + 978121 \) T^10 + 2T^9 - 18T^8 - 113T^7 + 346T^6 - 848T^5 + 3441T^4 + 10941T^3 + 247668T^2 + 189888*T + 978121
$17$	\( T^{10} + 5 T^{9} + \cdots + 139129 \) T^10 + 5T^9 - 8T^8 - 7T^7 + 273T^6 + 2586T^5 + 21444T^4 + 6559T^3 + 284057T^2 - 253640*T + 139129
$19$	\( T^{10} - 31 T^{9} + \cdots + 8814961 \) T^10 - 31T^9 + 488T^8 - 4975T^7 + 35909T^6 - 192402T^5 + 791008T^4 - 2556459T^3 + 6528635T^2 - 10795284*T + 8814961
$23$	\( T^{10} + T^{9} + \cdots + 6436343 \) T^10 + T^9 - 21T^8 - 219T^7 + 551T^6 + 4379T^5 + 12673T^4 - 115851T^3 - 255507T^2 + 279841T + 6436343
$29$	\( T^{10} + 5 T^{9} + \cdots + 529 \) T^10 + 5T^9 + 25T^8 + 37T^7 + 20T^6 - 10T^5 - 138T^4 + 1290T^3 + 3689T^2 - 1817*T + 529
$31$	\( T^{10} - 3 T^{9} + \cdots + 69172489 \) T^10 - 3T^9 + 9T^8 - 71T^7 + 1797T^6 + 17082T^5 - 72058T^4 + 195340T^3 + 4302985T^2 - 7352228*T + 69172489
$37$	\( T^{10} - 14 T^{9} + \cdots + 4489 \) T^10 - 14T^9 + 86T^8 - 60T^7 - 931T^6 + 8348T^5 + 23015T^4 - 10734T^3 + 16120T^2 - 6901*T + 4489
$41$	\( T^{10} + 28 T^{9} + \cdots + 5442889 \) T^10 + 28T^9 + 388T^8 + 3890T^7 + 31337T^6 + 190783T^5 + 836148T^4 + 2554428T^3 + 5351174T^2 + 7113317*T + 5442889
$43$	\( T^{10} - 9 T^{9} + \cdots + 4489 \) T^10 - 9T^9 + 114T^8 + 305T^7 - 523T^6 + 18809T^5 + 230085T^4 + 513278T^3 + 340541T^2 - 4757*T + 4489
$47$	\( (T^{5} + 22 T^{4} + \cdots - 4609)^{2} \) (T^5 + 22T^4 + 132T^3 - 77T^2 - 2530T - 4609)^2
$53$	\( T^{10} - T^{9} + \cdots + 18983449 \) T^10 - T^9 - 32T^8 - 595T^7 - 1682T^6 + 20987T^5 + 341254T^4 + 2059991T^3 + 7435429T^2 + 15092648T + 18983449
$59$	\( T^{10} + 11 T^{9} + \cdots + 16834609 \) T^10 + 11T^9 + 165T^8 + 550T^7 + 8206T^6 + 123442T^5 + 681351T^4 + 2365308T^3 + 8589306T^2 + 17872668*T + 16834609
$61$	\( T^{10} + \cdots + 125372809 \) T^10 + 41T^9 + 812T^8 + 9741T^7 + 69733T^6 + 256728T^5 + 503594T^4 + 2551441T^3 + 14634911T^2 - 6606230*T + 125372809
$67$	\( T^{10} + \cdots + 4866876169 \) T^10 + 12T^9 + 144T^8 - 582T^7 + 18327T^6 + 132298T^5 + 3384767T^4 + 42045290T^3 + 356317842T^2 + 1996826349*T + 4866876169
$71$	\( T^{10} + 22 T^{9} + \cdots + 543169 \) T^10 + 22T^9 + 528T^8 + 5797T^7 + 60060T^6 + 294899T^5 + 820622T^4 + 909799T^3 - 214533T^2 - 1159301*T + 543169
$73$	\( T^{10} - 19 T^{9} + \cdots + 1 \) T^10 - 19T^9 + 207T^8 - 1447T^7 + 7451T^6 - 14079T^5 + 24874T^4 + 8347T^3 + 522T^2 - 29*T + 1
$79$	\( T^{10} + \cdots + 1068832249 \) T^10 - 3T^9 + 86T^8 + 1722T^7 + 25469T^6 - 76176T^5 + 5991945T^4 - 24682590T^3 + 36147930T^2 - 319475996*T + 1068832249
$83$	\( T^{10} + \cdots + 1033043881 \) T^10 + 39T^9 + 696T^8 + 7212T^7 + 55636T^6 + 348435T^5 + 1818789T^4 + 9074986T^3 + 36553708T^2 + 142063220*T + 1033043881
$89$	\( T^{10} + \cdots + 30198445729 \) T^10 + 17T^9 + 399T^8 + 6431T^7 + 105455T^6 + 1466761T^5 + 20101779T^4 + 236714607T^3 + 2319994439T^2 + 13191238293*T + 30198445729
$97$	\( T^{10} + \cdots + 580376281 \) T^10 + 12T^9 + 375T^8 - 571T^7 + 12948T^6 - 18534T^5 + 4680875T^4 + 129435395T^3 + 624960172T^2 - 776958841*T + 580376281
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