LMFDB - Newform orbit 350.2.q.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,2,Mod(11,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([24, 20]))

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

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

chi := DirichletCharacter("350.11");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	350.q (of order $15$, degree $8$, 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:	$2.79476407074$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{15})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 $ x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/350\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1 - \zeta_{15}^{5}$	$-\zeta_{15}^{2} - \zeta_{15}^{7}$

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} $

11.1

0.913545	+	0.406737i
0.669131	−	0.743145i
0.669131	+	0.743145i
0.913545	−	0.406737i
−0.978148	+	0.207912i
−0.104528	−	0.994522i
−0.104528	+	0.994522i
−0.978148	−	0.207912i

−0.913545

−

0.406737i

0.413545

0.459289i

0.669131

0.743145i

1.11803

1.93649i

−0.190983

−

0.587785i

−0.500000

−

2.59808i

−0.309017

−

0.951057i

0.273659

−

2.60369i

−0.233733

−

2.22382i

81.1

−0.669131

0.743145i

0.169131

1.60917i

−0.104528

−

0.994522i

−1.11803

−

1.93649i

−1.30902

−

0.951057i

−0.500000

−

2.59808i

0.809017

0.587785i

0.373619

−

0.0794152i

2.18720

0.464905i

121.1

−0.669131

−

0.743145i

0.169131

−

1.60917i

−0.104528

0.994522i

−1.11803

1.93649i

−1.30902

0.951057i

−0.500000

2.59808i

0.809017

−

0.587785i

0.373619

0.0794152i

2.18720

−

0.464905i

191.1

−0.913545

0.406737i

0.413545

−

0.459289i

0.669131

−

0.743145i

1.11803

−

1.93649i

−0.190983

0.587785i

−0.500000

2.59808i

−0.309017

0.951057i

0.273659

2.60369i

−0.233733

2.22382i

221.1

0.978148

−

0.207912i

−1.47815

0.658114i

0.913545

−

0.406737i

−1.11803

−

1.93649i

−1.30902

0.951057i

−0.500000

−

2.59808i

0.809017

−

0.587785i

−0.255585

0.283856i

−1.49622

−

1.66172i

261.1

0.104528

0.994522i

−0.604528

0.128496i

−0.978148

0.207912i

1.11803

−

1.93649i

−0.190983

−

0.587785i

−0.500000

2.59808i

−0.309017

−

0.951057i

−2.39169

1.06485i

2.04275

0.909491i

291.1

0.104528

−

0.994522i

−0.604528

−

0.128496i

−0.978148

−

0.207912i

1.11803

1.93649i

−0.190983

0.587785i

−0.500000

−

2.59808i

−0.309017

0.951057i

−2.39169

−

1.06485i

2.04275

−

0.909491i

331.1

0.978148

0.207912i

−1.47815

−

0.658114i

0.913545

0.406737i

−1.11803

1.93649i

−1.30902

−

0.951057i

−0.500000

2.59808i

0.809017

0.587785i

−0.255585

−

0.283856i

−1.49622

1.66172i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner
25.d	even	5	1	inner
175.q	even	15	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	350.2.q.a	✓	8
7.c	even	3	1	inner	350.2.q.a	✓	8
25.d	even	5	1	inner	350.2.q.a	✓	8
175.q	even	15	1	inner	350.2.q.a	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
350.2.q.a	✓	8	1.a	even	1	1	trivial
350.2.q.a	✓	8	7.c	even	3	1	inner
350.2.q.a	✓	8	25.d	even	5	1	inner
350.2.q.a	✓	8	175.q	even	15	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} + 3T_{3}^{7} + 5T_{3}^{6} + 8T_{3}^{5} + 9T_{3}^{4} + 2T_{3}^{3} + 2T_{3} + 1 $ T3^8 + 3*T3^7 + 5*T3^6 + 8*T3^5 + 9*T3^4 + 2*T3^3 + 2*T3 + 1 acting on $S_{2}^{\mathrm{new}}(350, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} + T^{7} - T^{5} + \cdots + 1 $ T^8 + T^7 - T^5 - T^4 - T^3 + T + 1
$3$	$ T^{8} + 3 T^{7} + \cdots + 1 $ T^8 + 3T^7 + 5T^6 + 8T^5 + 9T^4 + 2T^3 + 2T + 1
$5$	$ (T^{4} + 5 T^{2} + 25)^{2} $ (T^4 + 5*T^2 + 25)^2
$7$	$ (T^{2} + T + 7)^{4} $ (T^2 + T + 7)^4
$11$	$ T^{8} - 8 T^{7} + \cdots + 14641 $ T^8 - 8T^7 + 30T^6 - 118T^5 + 419T^4 - 682T^3 + 1815T^2 - 9317*T + 14641
$13$	$ (T^{4} - 10 T^{3} + \cdots + 25)^{2} $ (T^4 - 10T^3 + 40T^2 - 25*T + 25)^2
$17$	$ T^{8} - 14 T^{7} + \cdots + 3748096 $ T^8 - 14T^7 + 60T^6 - 496T^5 + 6704T^4 - 41536T^3 + 232320T^2 - 1362944*T + 3748096
$19$	$ T^{8} + 6 T^{7} + \cdots + 1679616 $ T^8 + 6T^7 - 216T^5 - 1296T^4 - 7776T^3 + 279936*T + 1679616
$23$	$ T^{8} - 7 T^{7} + \cdots + 14641 $ T^8 - 7T^7 - 20T^6 - 197T^5 + 3639T^4 - 8173T^3 + 12100T^2 - 17303*T + 14641
$29$	$ (T^{4} - 11 T^{3} + \cdots + 361)^{2} $ (T^4 - 11T^3 + 61T^2 - 171*T + 361)^2
$31$	$ T^{8} - 6 T^{7} + \cdots + 6561 $ T^8 - 6T^7 - 54T^5 + 729T^4 - 1944T^3 + 3645T^2 - 6561T + 6561
$37$	$ T^{8} + T^{7} - 5 T^{6} + \cdots + 1 $ T^8 + T^7 - 5T^6 + 14T^5 + 39T^4 - 26T^3 + 10T^2 - 4T + 1
$41$	$ (T^{4} + 8 T^{3} + \cdots + 11881)^{2} $ (T^4 + 8T^3 + 114T^2 + 1417*T + 11881)^2
$43$	$ (T^{2} - 5)^{4} $ (T^2 - 5)^4
$47$	$ T^{8} - 2 T^{7} + \cdots + 3748096 $ T^8 - 2T^7 - 60T^6 + 928T^5 + 1104T^4 - 26048T^3 + 154880T^2 - 1022208*T + 3748096
$53$	$ T^{8} - 7 T^{7} + \cdots + 1 $ T^8 - 7T^7 + 30T^6 - 127T^5 + 339T^4 - 43T^3 - 10T^2 - 3*T + 1
$59$	$ T^{8} + 15 T^{7} + \cdots + 9150625 $ T^8 + 15T^7 + 35T^6 + 650T^5 + 16575T^4 + 118250T^3 + 635250T^2 + 3327500*T + 9150625
$61$	$ T^{8} - 11 T^{7} + \cdots + 923521 $ T^8 - 11T^7 + 70T^6 - 499T^5 + 1299T^4 + 19561T^3 - 48050T^2 - 29791*T + 923521
$67$	$ T^{8} + T^{7} - T^{5} + \cdots + 1 $ T^8 + T^7 - T^5 - T^4 - T^3 + T + 1
$71$	$ (T^{4} - 13 T^{3} + \cdots + 961)^{2} $ (T^4 - 13T^3 + 289T^2 - 837*T + 961)^2
$73$	$ T^{8} - 4 T^{7} + \cdots + 2825761 $ T^8 - 4T^7 - 30T^6 + 554T^5 - 1041T^4 - 3526T^3 + 58835T^2 - 620289*T + 2825761
$79$	$ T^{8} + 20 T^{7} + \cdots + 40960000 $ T^8 + 20T^7 + 160T^6 + 1600T^5 + 19200T^4 + 128000T^3 + 1024000T^2 + 10240000*T + 40960000
$83$	$ (T^{4} - 13 T^{3} + \cdots + 3481)^{2} $ (T^4 - 13T^3 + 79T^2 - 177*T + 3481)^2
$89$	$ T^{8} - T^{7} + \cdots + 104060401 $ T^8 - T^7 - 225T^6 - 5074T^5 + 43299T^4 + 568226T^3 + 3570350T^2 + 24727224T + 104060401
$97$	$ (T^{4} - 4 T^{3} + \cdots + 5776)^{2} $ (T^4 - 4T^3 + 96T^2 - 1064*T + 5776)^2
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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:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	350.q (of order \(15\), degree \(8\), minimal)

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

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Newspace parameters

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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	350.2.q.a

Level	$350$
Weight	$2$
Character orbit	350.q
Analytic conductor	$2.795$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(2.79476407074\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\Q(\zeta_{15})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(-1 - \zeta_{15}^{5}\)	\(-\zeta_{15}^{2} - \zeta_{15}^{7}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 11.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} + T^{7} - T^{5} + \cdots + 1 \) T^8 + T^7 - T^5 - T^4 - T^3 + T + 1
$3$	\( T^{8} + 3 T^{7} + \cdots + 1 \) T^8 + 3T^7 + 5T^6 + 8T^5 + 9T^4 + 2T^3 + 2T + 1
$5$	\( (T^{4} + 5 T^{2} + 25)^{2} \) (T^4 + 5*T^2 + 25)^2
$7$	\( (T^{2} + T + 7)^{4} \) (T^2 + T + 7)^4
$11$	\( T^{8} - 8 T^{7} + \cdots + 14641 \) T^8 - 8T^7 + 30T^6 - 118T^5 + 419T^4 - 682T^3 + 1815T^2 - 9317*T + 14641
$13$	\( (T^{4} - 10 T^{3} + \cdots + 25)^{2} \) (T^4 - 10T^3 + 40T^2 - 25*T + 25)^2
$17$	\( T^{8} - 14 T^{7} + \cdots + 3748096 \) T^8 - 14T^7 + 60T^6 - 496T^5 + 6704T^4 - 41536T^3 + 232320T^2 - 1362944*T + 3748096
$19$	\( T^{8} + 6 T^{7} + \cdots + 1679616 \) T^8 + 6T^7 - 216T^5 - 1296T^4 - 7776T^3 + 279936*T + 1679616
$23$	\( T^{8} - 7 T^{7} + \cdots + 14641 \) T^8 - 7T^7 - 20T^6 - 197T^5 + 3639T^4 - 8173T^3 + 12100T^2 - 17303*T + 14641
$29$	\( (T^{4} - 11 T^{3} + \cdots + 361)^{2} \) (T^4 - 11T^3 + 61T^2 - 171*T + 361)^2
$31$	\( T^{8} - 6 T^{7} + \cdots + 6561 \) T^8 - 6T^7 - 54T^5 + 729T^4 - 1944T^3 + 3645T^2 - 6561T + 6561
$37$	\( T^{8} + T^{7} - 5 T^{6} + \cdots + 1 \) T^8 + T^7 - 5T^6 + 14T^5 + 39T^4 - 26T^3 + 10T^2 - 4T + 1
$41$	\( (T^{4} + 8 T^{3} + \cdots + 11881)^{2} \) (T^4 + 8T^3 + 114T^2 + 1417*T + 11881)^2
$43$	\( (T^{2} - 5)^{4} \) (T^2 - 5)^4
$47$	\( T^{8} - 2 T^{7} + \cdots + 3748096 \) T^8 - 2T^7 - 60T^6 + 928T^5 + 1104T^4 - 26048T^3 + 154880T^2 - 1022208*T + 3748096
$53$	\( T^{8} - 7 T^{7} + \cdots + 1 \) T^8 - 7T^7 + 30T^6 - 127T^5 + 339T^4 - 43T^3 - 10T^2 - 3*T + 1
$59$	\( T^{8} + 15 T^{7} + \cdots + 9150625 \) T^8 + 15T^7 + 35T^6 + 650T^5 + 16575T^4 + 118250T^3 + 635250T^2 + 3327500*T + 9150625
$61$	\( T^{8} - 11 T^{7} + \cdots + 923521 \) T^8 - 11T^7 + 70T^6 - 499T^5 + 1299T^4 + 19561T^3 - 48050T^2 - 29791*T + 923521
$67$	\( T^{8} + T^{7} - T^{5} + \cdots + 1 \) T^8 + T^7 - T^5 - T^4 - T^3 + T + 1
$71$	\( (T^{4} - 13 T^{3} + \cdots + 961)^{2} \) (T^4 - 13T^3 + 289T^2 - 837*T + 961)^2
$73$	\( T^{8} - 4 T^{7} + \cdots + 2825761 \) T^8 - 4T^7 - 30T^6 + 554T^5 - 1041T^4 - 3526T^3 + 58835T^2 - 620289*T + 2825761
$79$	\( T^{8} + 20 T^{7} + \cdots + 40960000 \) T^8 + 20T^7 + 160T^6 + 1600T^5 + 19200T^4 + 128000T^3 + 1024000T^2 + 10240000*T + 40960000
$83$	\( (T^{4} - 13 T^{3} + \cdots + 3481)^{2} \) (T^4 - 13T^3 + 79T^2 - 177*T + 3481)^2
$89$	\( T^{8} - T^{7} + \cdots + 104060401 \) T^8 - T^7 - 225T^6 - 5074T^5 + 43299T^4 + 568226T^3 + 3570350T^2 + 24727224T + 104060401
$97$	\( (T^{4} - 4 T^{3} + \cdots + 5776)^{2} \) (T^4 - 4T^3 + 96T^2 - 1064*T + 5776)^2
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