LMFDB - Newform orbit 770.2.bg.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [770,2,Mod(81,770)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(770, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([0, 20, 6]))

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

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

chi := DirichletCharacter("770.81");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 770 = 2 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	770.bg (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:	$6.14848095564$
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 $\left(\mathbb{Z}/770\mathbb{Z}\right)^\times$.

$n$	$211$	$617$	$661$
$\chi(n)$	$-\zeta_{15}^{2} - \zeta_{15}^{7}$	$1$	$-1 - \zeta_{15}^{5}$

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} $

81.1

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

0.104528

−

0.994522i

−1.75181

−

1.94558i

−0.978148

−

0.207912i

−0.913545

−

0.406737i

−2.11803

1.53884i

2.26531

1.36688i

−0.309017

0.951057i

−0.402863

3.83299i

−0.500000

0.866025i

191.1

−0.913545

0.406737i

2.56082

−

0.544320i

0.669131

−

0.743145i

0.104528

0.994522i

−2.11803

1.53884i

−1.02924

2.43735i

−0.309017

0.951057i

3.52090

−

1.56760i

−0.500000

−

0.866025i

291.1

−0.669131

0.743145i

−0.348943

0.155360i

−0.104528

−

0.994522i

0.978148

−

0.207912i

0.118034

−

0.363271i

−0.599960

2.57683i

0.809017

0.587785i

−1.90977

2.12101i

−0.500000

0.866025i

361.1

0.978148

−

0.207912i

0.0399263

0.379874i

0.913545

−

0.406737i

−0.669131

0.743145i

0.118034

0.363271i

−2.63611

0.225688i

0.809017

−

0.587785i

2.79173

−

0.593401i

−0.500000

0.866025i

401.1

0.978148

0.207912i

0.0399263

−

0.379874i

0.913545

0.406737i

−0.669131

−

0.743145i

0.118034

−

0.363271i

−2.63611

−

0.225688i

0.809017

0.587785i

2.79173

0.593401i

−0.500000

−

0.866025i

471.1

−0.669131

−

0.743145i

−0.348943

−

0.155360i

−0.104528

0.994522i

0.978148

0.207912i

0.118034

0.363271i

−0.599960

−

2.57683i

0.809017

−

0.587785i

−1.90977

−

2.12101i

−0.500000

−

0.866025i

641.1

−0.913545

−

0.406737i

2.56082

0.544320i

0.669131

0.743145i

0.104528

−

0.994522i

−2.11803

−

1.53884i

−1.02924

−

2.43735i

−0.309017

−

0.951057i

3.52090

1.56760i

−0.500000

0.866025i

751.1

0.104528

0.994522i

−1.75181

1.94558i

−0.978148

0.207912i

−0.913545

0.406737i

−2.11803

−

1.53884i

2.26531

−

1.36688i

−0.309017

−

0.951057i

−0.402863

−

3.83299i

−0.500000

−

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner
11.c	even	5	1	inner
77.m	even	15	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	770.2.bg.a	✓	8
7.c	even	3	1	inner	770.2.bg.a	✓	8
11.c	even	5	1	inner	770.2.bg.a	✓	8
77.m	even	15	1	inner	770.2.bg.a	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
770.2.bg.a	✓	8	1.a	even	1	1	trivial
770.2.bg.a	✓	8	7.c	even	3	1	inner
770.2.bg.a	✓	8	11.c	even	5	1	inner
770.2.bg.a	✓	8	77.m	even	15	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} - T_{3}^{7} - 5T_{3}^{6} - 14T_{3}^{5} + 39T_{3}^{4} + 26T_{3}^{3} + 10T_{3}^{2} + 4T_{3} + 1 $ T3^8 - T3^7 - 5*T3^6 - 14*T3^5 + 39*T3^4 + 26*T3^3 + 10*T3^2 + 4*T3 + 1 acting on $S_{2}^{\mathrm{new}}(770, [\chi])$.

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} - T^{7} - 5 T^{6} + \cdots + 1 $ T^8 - T^7 - 5T^6 - 14T^5 + 39T^4 + 26T^3 + 10T^2 + 4T + 1
$5$	$ T^{8} + T^{7} - T^{5} + \cdots + 1 $ T^8 + T^7 - T^5 - T^4 - T^3 + T + 1
$7$	$ T^{8} + 4 T^{7} + \cdots + 2401 $ T^8 + 4T^7 + 9T^6 + 8T^5 - 31T^4 + 56T^3 + 441T^2 + 1372*T + 2401
$11$	$ T^{8} - T^{7} + \cdots + 14641 $ T^8 - T^7 - 20T^6 + T^5 + 309T^4 + 11T^3 - 2420T^2 - 1331*T + 14641
$13$	$ (T^{4} - 5 T^{3} + \cdots + 625)^{2} $ (T^4 - 5T^3 + 25T^2 - 125*T + 625)^2
$17$	$ T^{8} + 5 T^{7} + \cdots + 625 $ T^8 + 5T^7 + 15T^6 + 50T^5 + 75T^4 - 250T^3 - 250T^2 + 625
$19$	$ T^{8} - 7 T^{7} + \cdots + 5764801 $ T^8 - 7T^7 + 343T^5 - 2401T^4 + 16807T^3 - 823543*T + 5764801
$23$	$ (T^{4} - 3 T^{3} + 18 T^{2} + \cdots + 81)^{2} $ (T^4 - 3T^3 + 18T^2 + 27*T + 81)^2
$29$	$ (T^{4} - 3 T^{3} + 19 T^{2} + \cdots + 1)^{2} $ (T^4 - 3T^3 + 19T^2 - 7*T + 1)^2
$31$	$ T^{8} - 3 T^{7} + \cdots + 707281 $ T^8 - 3T^7 - 25T^6 + 362T^5 - 381T^4 - 2842T^3 + 25230T^2 - 195112*T + 707281
$37$	$ T^{8} + 9 T^{7} + \cdots + 130321 $ T^8 + 9T^7 + 20T^6 + 131T^5 + 1479T^4 + 6251T^3 + 21660T^2 + 75449*T + 130321
$41$	$ (T^{4} - 10 T^{3} + \cdots + 625)^{2} $ (T^4 - 10T^3 + 100T^2 - 375*T + 625)^2
$43$	$ (T^{2} + 6 T - 36)^{4} $ (T^2 + 6*T - 36)^4
$47$	$ T^{8} + 15 T^{7} + \cdots + 9150625 $ T^8 + 15T^7 + 35T^6 + 650T^5 + 16575T^4 + 118250T^3 + 635250T^2 + 3327500*T + 9150625
$53$	$ T^{8} + 21 T^{7} + \cdots + 38950081 $ T^8 + 21T^7 + 200T^6 + 2059T^5 + 20319T^4 + 99619T^3 + 748920T^2 + 9367741*T + 38950081
$59$	$ T^{8} + 27 T^{7} + \cdots + 214358881 $ T^8 + 27T^7 + 350T^6 + 4667T^5 + 53859T^4 + 264143T^3 + 2196150T^2 + 40745903*T + 214358881
$61$	$ T^{8} + 3 T^{7} + \cdots + 1 $ T^8 + 3T^7 - 10T^6 + 43T^5 + 339T^4 + 127T^3 + 30T^2 + 7*T + 1
$67$	$ (T^{4} + 4 T^{3} + \cdots + 14641)^{2} $ (T^4 + 4T^3 + 137T^2 - 484*T + 14641)^2
$71$	$ (T^{4} - 18 T^{3} + \cdots + 9801)^{2} $ (T^4 - 18T^3 + 144T^2 - 297*T + 9801)^2
$73$	$ T^{8} - 12 T^{7} + \cdots + 1679616 $ T^8 - 12T^7 - 432T^5 + 11664T^4 - 62208T^3 + 233280T^2 - 839808T + 1679616
$79$	$ T^{8} + 9 T^{7} + \cdots + 14641 $ T^8 + 9T^7 - 175T^6 + 2876T^5 + 67989T^4 - 75394T^3 + 50820T^2 - 34606*T + 14641
$83$	$ (T^{4} + 9 T^{3} + \cdots + 361)^{2} $ (T^4 + 9T^3 + 61T^2 + 209*T + 361)^2
$89$	$ (T^{4} + 18 T^{3} + 323 T^{2} + \cdots + 1)^{2} $ (T^4 + 18T^3 + 323T^2 + 18*T + 1)^2
$97$	$ (T^{4} - 25 T^{3} + \cdots + 3025)^{2} $ (T^4 - 25T^3 + 235T^2 + 275*T + 3025)^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:	\( N \)	\(=\)	\( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	770.bg (of order \(15\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{15} q^{2} + ( - 2 \zeta_{15}^{7} + \zeta_{15}^{6} + \cdots + 2) q^{3} + \cdots + ( - 3 \zeta_{15}^{7} + 3 \zeta_{15}^{5} + \cdots + 3) q^{9} +O(q^{10}) \) q - z * q^2 + (-2z^7 + z^6 + z^5 - 2z^4 + 2z^3 - z + 2) q^3 + z^2 * q^4 - z^4 * q^5 + (z^7 - z^6 - 2z^3 + z^2 - 2) q^6 + (-z^7 - 3z^2) q^7 - z^3 * q^8 + (-3z^7 + 3z^5 - z + 3) * q^9	\( q - \zeta_{15} q^{2} + ( - 2 \zeta_{15}^{7} + \zeta_{15}^{6} + \cdots + 2) q^{3} + \cdots + (7 \zeta_{15}^{7} - 5 \zeta_{15}^{6} + \cdots - 6) q^{99} +O(q^{100}) \) q - z * q^2 + (-2z^7 + z^6 + z^5 - 2z^4 + 2z^3 - z + 2) q^3 + z^2 * q^4 - z^4 * q^5 + (z^7 - z^6 - 2z^3 + z^2 - 2) q^6 + (-z^7 - 3z^2) q^7 - z^3 * q^8 + (-3z^7 + 3z^5 - z + 3) * q^9 + z^5 * q^10 + (3z^7 - 4z^6 - z^5 + 3z^4 - 3z^3 + 2z^2 - z - 3) q^11 + (z^5 + z^4 + z + 1) * q^12 - 5z^6 q^13 + (z^7 - z^5 + z^4 + 2z^3 + z - 1) q^14 + (-z^7 - z^6 - z^3 - z^2) * q^15 + z^4 * q^16 + (3z^7 - 2z^6 + 3z^4 - 3z^3 + 2z^2 + z - 3) q^17 + (3z^7 - 3z^6 - 3z^5 + 3z^4 - 3z^3 + z^2 - 3) q^18 + (-7z^7 + 7z^5 - 7z^4 - 7z + 7) * q^19 - z^6 * q^20 + (-z^7 - 3z^5 - 3z^4 + z^3 - z^2 - 3z - 1) q^21 + (z^7 + z^6 + z^3 + z^2 + 3) * q^22 + (3z^4 + 3z) * q^23 + (-z^6 - z^5 - z^2 - z) * q^24 + (z^7 - z^5 + z^4 - z^3 + z - 1) * q^25 + 5z^7 q^26 + (z^7 - z^6 + z^3 + z^2 + 1) * q^27 + (-z^7 + z^6 - 3z^4 + z^3 - z^2 + 1) q^28 + (-z^7 - 2z^6 - 2z^3 - z^2) * q^29 + (2z^7 - z^5 + 2z^4 + z - 1) * q^30 + (-6z^6 + z^5 + z^2 - 6z) * q^31 - z^5 * q^32 + (3z^7 - 3z^5 - 2z^4 - 3z - 3) * q^33 + (-z^7 + z^3 - z^2 + 3) * q^34 + (2z^6 - z) q^35 + (3z^6 + 2z^3 + 3) * q^36 + (3z^7 - 4z^5 + 4z^4 - 4) q^37 + (7z^7 - 7z^6 + 7z^4 - 7z^3 + 7z^2 - 7) q^38 + (-10z^7 + 5z^6 - 10z^4 + 10z^3 - 5z^2 - 5z + 10) * q^39 + z^7 * q^40 + (5z^6 + 5z^3 + 5) * q^41 + (z^7 + 3z^6 + 2z^5 + 3z^2 + 2z - 1) * q^42 + (-6z^7 + 6z^3 - 6z^2) q^43 + (-2z^7 + z^5 - 2z^4 - 4z + 1) q^44 + (-3z^7 + z^5 - 3z^4 + 3z^3 - 3z^2 - 3z + 3) q^45 + (-3z^5 - 3z^2) * q^46 + (-6z^7 - z^5 - 6z^4 + z - 1) * q^47 + (z^7 + z^6 + z^3 + z^2) * q^48 + (5z^7 - 5z^6 + 8z^4 - 5z^3 + 5z^2 - 5) q^49 + (-z^7 + z^6 + z^3 - z^2 + 1) * q^50 + (7z^7 - 4z^5 + 4z^4 - 4) q^51 + (-5z^7 + 5z^5 - 5z^4 + 5z^3 - 5z + 5) q^52 + (-3z^6 + 8z^5 + 8z^2 - 3z) * q^53 + (z^5 - 2z^4 - 2z + 1) * q^54 + (z^7 + 2z^3 + z^2 - 2) q^55 + (2z^5 - 1) q^56 + (-7z^7 - 7z^6 - 7z^2 + 7) q^57 + (3z^7 - z^5 + 3z^4 + z - 1) * q^58 + (-10z^7 + 10z^6 + 10z^5 - 10z^4 + 10z^3 - 7z^2 + 10) * q^59 + (-2z^7 + z^6 - 2z^4 + 2z^3 - z^2 - z + 2) q^60 + (-2z^7 + z^4 - 2z) * q^61 + (6z^7 - z^6 - z^3 + 6z^2) * q^62 + (-2z^7 - 6z^6 - z^5 - 2z^4 - 4z^3 + z - 7) * q^63 + z^6 * q^64 + (-5z^5 - 5) q^65 + (-3z^7 + 3z^6 + 5z^5 - 3z^4 + 3z^3 + 3z^2 + 3) * q^66 + (10z^7 - 3z^5 + 10z^4 - 10z^3 + 10z^2 + 10z - 10) * q^67 + (z^7 - z^5 - 2z - 1) q^68 + (6z^6 + 9z^3 + 6) * q^69 + (-2z^7 + z^2) q^70 + (-9z^7 + 9z^6 + 12z^3 - 9z^2 + 12) * q^71 + (-3z^7 - 2z^4 - 3z) q^72 + (6z^7 - 6z^6 - 6z^5 + 6z^4 - 6z^3 - 6) q^73 + (-3z^7 + 4z^6 - z^5 - 3z^4 + 3z^3 + z + 3) * q^74 + (z^7 - z^5 - z - 1) * q^75 + 7 * q^76 + (6z^7 + 2z^6 - 3z^5 + 6z^4 - 2z^3 + 12z - 4) * q^77 + (5z^7 - 5z^3 + 5z^2 - 10) q^78 + (7z^7 - 7z^5 + 12z - 7) q^79 + (-z^7 + z^5 - z^4 + z^3 - z + 1) * q^80 + (6z^7 - 6z^6 - 6z^5 + 6z^4 - 6z^3 + 4z^2 - 6) * q^81 + (-5z^7 - 5z^4 - 5z) q^82 + (-3z^7 + 3z^6 - z^3 - 3z^2 - 1) q^83 + (-4z^7 - 2z^6 + z^5 - z^4 - 2z^3 - 2z^2 + 1) * q^84 + (z^6 + 3z^3 + 1) q^85 + (6z^7 - 6z^5 + 6z - 6) q^86 + (-5z^7 - 3z^5 - 5z^4 + 5z^3 - 5z^2 - 5z + 5) * q^87 + (2z^7 - z^6 + 2z^4 - 2z^3 + 4z^2 + z - 2) * q^88 + (-13z^5 + 8z^4 + 8z - 13) q^89 + (3z^7 - z^6 + 3z^2 - 3) * q^90 + (10z^7 - 10z^5 + 10z^4 - 15z^3 + 10z - 10) q^91 + (3z^6 + 3z^3) * q^92 + (-4z^7 - 9z^4 - 4z) q^93 + (6z^7 + z^6 + 6z^4 - 6z^3 - z^2 + 7z - 6) * q^94 - 7z^2 q^95 + (-2z^7 + z^5 - 2z^4 - z + 1) * q^96 + (-6z^7 - 7z^6 - 6z^2 + 6) q^97 + (-3z^5 + 5) q^98 + (7z^7 - 5z^6 - 14z^3 + 7z^2 - 6) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + q^{3} + q^{4} - q^{5} - 8 q^{6} - 4 q^{7} + 2 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q - q^2 + q^3 + q^4 - q^5 - 8 * q^6 - 4 * q^7 + 2 * q^8 + 8 * q^9	\( 8 q - q^{2} + q^{3} + q^{4} - q^{5} - 8 q^{6} - 4 q^{7} + 2 q^{8} + 8 q^{9} - 4 q^{10} + q^{11} + 6 q^{12} + 10 q^{13} - 5 q^{14} + 2 q^{15} + q^{16} - 5 q^{17} + 7 q^{18} + 7 q^{19} + 2 q^{20} - 6 q^{21} + 22 q^{22} + 6 q^{23} + 4 q^{24} + q^{25} + 5 q^{26} + 10 q^{27} - q^{28} + 6 q^{29} + q^{30} + 3 q^{31} + 4 q^{32} - 14 q^{33} + 20 q^{34} - 5 q^{35} + 14 q^{36} - 9 q^{37} - 7 q^{38} + 20 q^{39} + q^{40} + 20 q^{41} - 16 q^{42} - 24 q^{43} - 4 q^{44} + 2 q^{45} + 9 q^{46} - 15 q^{47} - 2 q^{48} - 2 q^{49} + 2 q^{50} - 5 q^{51} - 5 q^{52} - 21 q^{53} - 18 q^{55} - 16 q^{56} + 56 q^{57} + 3 q^{58} - 27 q^{59} + 4 q^{60} - 3 q^{61} + 16 q^{62} - 35 q^{63} - 2 q^{64} - 20 q^{65} - 11 q^{66} - 8 q^{67} - 5 q^{68} + 18 q^{69} - q^{70} + 36 q^{71} - 8 q^{72} + 12 q^{73} + 9 q^{74} - 4 q^{75} + 56 q^{76} + 4 q^{77} - 60 q^{78} - 9 q^{79} - q^{80} + 16 q^{81} - 15 q^{82} - 18 q^{83} + 5 q^{84} - 12 q^{86} + 22 q^{87} - q^{88} - 36 q^{89} - 16 q^{90} + 20 q^{91} - 12 q^{92} - 17 q^{93} - 20 q^{94} - 7 q^{95} - q^{96} + 50 q^{97} + 52 q^{98} + 4 q^{99}+O(q^{100}) \) 8 * q - q^2 + q^3 + q^4 - q^5 - 8 * q^6 - 4 * q^7 + 2 * q^8 + 8 * q^9 - 4 * q^10 + q^11 + 6 * q^12 + 10 * q^13 - 5 * q^14 + 2 * q^15 + q^16 - 5 * q^17 + 7 * q^18 + 7 * q^19 + 2 * q^20 - 6 * q^21 + 22 * q^22 + 6 * q^23 + 4 * q^24 + q^25 + 5 * q^26 + 10 * q^27 - q^28 + 6 * q^29 + q^30 + 3 * q^31 + 4 * q^32 - 14 * q^33 + 20 * q^34 - 5 * q^35 + 14 * q^36 - 9 * q^37 - 7 * q^38 + 20 * q^39 + q^40 + 20 * q^41 - 16 * q^42 - 24 * q^43 - 4 * q^44 + 2 * q^45 + 9 * q^46 - 15 * q^47 - 2 * q^48 - 2 * q^49 + 2 * q^50 - 5 * q^51 - 5 * q^52 - 21 * q^53 - 18 * q^55 - 16 * q^56 + 56 * q^57 + 3 * q^58 - 27 * q^59 + 4 * q^60 - 3 * q^61 + 16 * q^62 - 35 * q^63 - 2 * q^64 - 20 * q^65 - 11 * q^66 - 8 * q^67 - 5 * q^68 + 18 * q^69 - q^70 + 36 * q^71 - 8 * q^72 + 12 * q^73 + 9 * q^74 - 4 * q^75 + 56 * q^76 + 4 * q^77 - 60 * q^78 - 9 * q^79 - q^80 + 16 * q^81 - 15 * q^82 - 18 * q^83 + 5 * q^84 - 12 * q^86 + 22 * q^87 - q^88 - 36 * q^89 - 16 * q^90 + 20 * q^91 - 12 * q^92 - 17 * q^93 - 20 * q^94 - 7 * q^95 - q^96 + 50 * q^97 + 52 * q^98 + 4 * 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	770.2.bg.a

Level	$770$
Weight	$2$
Character orbit	770.bg
Analytic conductor	$6.148$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(6.14848095564\)
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:	$\mathrm{SU}(2)[C_{15}]$

\(n\)	\(211\)	\(617\)	\(661\)
\(\chi(n)\)	\(-\zeta_{15}^{2} - \zeta_{15}^{7}\)	\(1\)	\(-1 - \zeta_{15}^{5}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 81.1
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
$3$	\( T^{8} - T^{7} - 5 T^{6} + \cdots + 1 \) T^8 - T^7 - 5T^6 - 14T^5 + 39T^4 + 26T^3 + 10T^2 + 4T + 1
$5$	\( T^{8} + T^{7} - T^{5} + \cdots + 1 \) T^8 + T^7 - T^5 - T^4 - T^3 + T + 1
$7$	\( T^{8} + 4 T^{7} + \cdots + 2401 \) T^8 + 4T^7 + 9T^6 + 8T^5 - 31T^4 + 56T^3 + 441T^2 + 1372*T + 2401
$11$	\( T^{8} - T^{7} + \cdots + 14641 \) T^8 - T^7 - 20T^6 + T^5 + 309T^4 + 11T^3 - 2420T^2 - 1331*T + 14641
$13$	\( (T^{4} - 5 T^{3} + \cdots + 625)^{2} \) (T^4 - 5T^3 + 25T^2 - 125*T + 625)^2
$17$	\( T^{8} + 5 T^{7} + \cdots + 625 \) T^8 + 5T^7 + 15T^6 + 50T^5 + 75T^4 - 250T^3 - 250T^2 + 625
$19$	\( T^{8} - 7 T^{7} + \cdots + 5764801 \) T^8 - 7T^7 + 343T^5 - 2401T^4 + 16807T^3 - 823543*T + 5764801
$23$	\( (T^{4} - 3 T^{3} + 18 T^{2} + \cdots + 81)^{2} \) (T^4 - 3T^3 + 18T^2 + 27*T + 81)^2
$29$	\( (T^{4} - 3 T^{3} + 19 T^{2} + \cdots + 1)^{2} \) (T^4 - 3T^3 + 19T^2 - 7*T + 1)^2
$31$	\( T^{8} - 3 T^{7} + \cdots + 707281 \) T^8 - 3T^7 - 25T^6 + 362T^5 - 381T^4 - 2842T^3 + 25230T^2 - 195112*T + 707281
$37$	\( T^{8} + 9 T^{7} + \cdots + 130321 \) T^8 + 9T^7 + 20T^6 + 131T^5 + 1479T^4 + 6251T^3 + 21660T^2 + 75449*T + 130321
$41$	\( (T^{4} - 10 T^{3} + \cdots + 625)^{2} \) (T^4 - 10T^3 + 100T^2 - 375*T + 625)^2
$43$	\( (T^{2} + 6 T - 36)^{4} \) (T^2 + 6*T - 36)^4
$47$	\( T^{8} + 15 T^{7} + \cdots + 9150625 \) T^8 + 15T^7 + 35T^6 + 650T^5 + 16575T^4 + 118250T^3 + 635250T^2 + 3327500*T + 9150625
$53$	\( T^{8} + 21 T^{7} + \cdots + 38950081 \) T^8 + 21T^7 + 200T^6 + 2059T^5 + 20319T^4 + 99619T^3 + 748920T^2 + 9367741*T + 38950081
$59$	\( T^{8} + 27 T^{7} + \cdots + 214358881 \) T^8 + 27T^7 + 350T^6 + 4667T^5 + 53859T^4 + 264143T^3 + 2196150T^2 + 40745903*T + 214358881
$61$	\( T^{8} + 3 T^{7} + \cdots + 1 \) T^8 + 3T^7 - 10T^6 + 43T^5 + 339T^4 + 127T^3 + 30T^2 + 7*T + 1
$67$	\( (T^{4} + 4 T^{3} + \cdots + 14641)^{2} \) (T^4 + 4T^3 + 137T^2 - 484*T + 14641)^2
$71$	\( (T^{4} - 18 T^{3} + \cdots + 9801)^{2} \) (T^4 - 18T^3 + 144T^2 - 297*T + 9801)^2
$73$	\( T^{8} - 12 T^{7} + \cdots + 1679616 \) T^8 - 12T^7 - 432T^5 + 11664T^4 - 62208T^3 + 233280T^2 - 839808T + 1679616
$79$	\( T^{8} + 9 T^{7} + \cdots + 14641 \) T^8 + 9T^7 - 175T^6 + 2876T^5 + 67989T^4 - 75394T^3 + 50820T^2 - 34606*T + 14641
$83$	\( (T^{4} + 9 T^{3} + \cdots + 361)^{2} \) (T^4 + 9T^3 + 61T^2 + 209*T + 361)^2
$89$	\( (T^{4} + 18 T^{3} + 323 T^{2} + \cdots + 1)^{2} \) (T^4 + 18T^3 + 323T^2 + 18*T + 1)^2
$97$	\( (T^{4} - 25 T^{3} + \cdots + 3025)^{2} \) (T^4 - 25T^3 + 235T^2 + 275*T + 3025)^2
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