LMFDB - Newform orbit 154.2.m.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [154,2,Mod(9,154)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(154, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([10, 18]))

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

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

chi := DirichletCharacter("154.9");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 154 = 2 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	154.m (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:	$1.22969619113$
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 $\left(\mathbb{Z}/154\mathbb{Z}\right)^\times$.

$n$	$45$	$57$
$\chi(n)$	$-1 - \zeta_{15}^{5}$	$-1 + \zeta_{15}^{2} - \zeta_{15}^{3} - \zeta_{15}^{6} + \zeta_{15}^{7}$

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} $

9.1

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

−0.669131

−

0.743145i

−0.348943

−

0.155360i

−0.104528

0.994522i

−2.18720

−

0.464905i

0.118034

0.363271i

−0.599960

−

2.57683i

0.809017

−

0.587785i

−1.90977

−

2.12101i

1.11803

1.93649i

25.1

−0.913545

−

0.406737i

2.56082

0.544320i

0.669131

0.743145i

0.233733

−

2.22382i

−2.11803

−

1.53884i

−1.02924

−

2.43735i

−0.309017

−

0.951057i

3.52090

1.56760i

−1.11803

1.93649i

37.1

−0.913545

0.406737i

2.56082

−

0.544320i

0.669131

−

0.743145i

0.233733

2.22382i

−2.11803

1.53884i

−1.02924

2.43735i

−0.309017

0.951057i

3.52090

−

1.56760i

−1.11803

−

1.93649i

53.1

0.978148

−

0.207912i

0.0399263

0.379874i

0.913545

−

0.406737i

1.49622

−

1.66172i

0.118034

0.363271i

−2.63611

0.225688i

0.809017

−

0.587785i

2.79173

−

0.593401i

1.11803

−

1.93649i

81.1

0.104528

−

0.994522i

−1.75181

−

1.94558i

−0.978148

−

0.207912i

−2.04275

−

0.909491i

−2.11803

1.53884i

2.26531

1.36688i

−0.309017

0.951057i

−0.402863

3.83299i

−1.11803

1.93649i

93.1

0.978148

0.207912i

0.0399263

−

0.379874i

0.913545

0.406737i

1.49622

1.66172i

0.118034

−

0.363271i

−2.63611

−

0.225688i

0.809017

0.587785i

2.79173

0.593401i

1.11803

1.93649i

135.1

0.104528

0.994522i

−1.75181

1.94558i

−0.978148

0.207912i

−2.04275

0.909491i

−2.11803

−

1.53884i

2.26531

−

1.36688i

−0.309017

−

0.951057i

−0.402863

−

3.83299i

−1.11803

−

1.93649i

137.1

−0.669131

0.743145i

−0.348943

0.155360i

−0.104528

−

0.994522i

−2.18720

0.464905i

0.118034

−

0.363271i

−0.599960

2.57683i

0.809017

0.587785i

−1.90977

2.12101i

1.11803

−

1.93649i

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	154.2.m.a	✓	8
7.c	even	3	1	inner	154.2.m.a	✓	8
11.c	even	5	1	inner	154.2.m.a	✓	8
77.m	even	15	1	inner	154.2.m.a	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
154.2.m.a	✓	8	1.a	even	1	1	trivial
154.2.m.a	✓	8	7.c	even	3	1	inner
154.2.m.a	✓	8	11.c	even	5	1	inner
154.2.m.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 $S_{2}^{\mathrm{new}}(154, [\chi])$.

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} + 5 T^{7} + \cdots + 625 $ T^8 + 5T^7 + 10T^6 + 25T^5 + 75T^4 + 125T^3 + 250T^2 + 625*T + 625
$7$	$ 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} + 11 T^{7} + \cdots + 14641 $ T^8 + 11T^7 + 70T^6 + 319T^5 + 1149T^4 + 3509T^3 + 8470T^2 + 14641*T + 14641
$13$	$ (T^{4} + 9 T^{3} + 36 T^{2} + \cdots + 81)^{2} $ (T^4 + 9T^3 + 36T^2 + 54*T + 81)^2
$17$	$ T^{8} - 9 T^{7} + \cdots + 14641 $ T^8 - 9T^7 + 50T^6 - 301T^5 + 939T^4 + 2519T^3 - 3630T^2 + 1331*T + 14641
$19$	$ T^{8} + 3 T^{7} + \cdots + 707281 $ T^8 + 3T^7 - 25T^6 - 362T^5 - 381T^4 + 2842T^3 + 25230T^2 + 195112*T + 707281
$23$	$ (T^{4} - 2 T^{3} + 8 T^{2} + \cdots + 16)^{2} $ (T^4 - 2T^3 + 8T^2 + 8*T + 16)^2
$29$	$ (T^{4} + 9 T^{3} + 36 T^{2} + \cdots + 81)^{2} $ (T^4 + 9T^3 + 36T^2 + 54*T + 81)^2
$31$	$ T^{8} - T^{7} + \cdots + 923521 $ T^8 - T^7 - 75T^6 - 944T^5 + 5249T^4 + 34906T^3 + 115320T^2 + 417074T + 923521
$37$	$ T^{8} - 2 T^{7} + \cdots + 3748096 $ T^8 - 2T^7 - 60T^6 + 928T^5 + 1104T^4 - 26048T^3 + 154880T^2 - 1022208*T + 3748096
$41$	$ (T^{4} + 14 T^{3} + \cdots + 841)^{2} $ (T^4 + 14T^3 + 96T^2 + 319*T + 841)^2
$43$	$ (T^{2} - 17 T + 61)^{4} $ (T^2 - 17*T + 61)^4
$47$	$ T^{8} - 9 T^{7} + \cdots + 6561 $ T^8 - 9T^7 + 45T^6 - 216T^5 + 729T^4 - 486T^3 - 4374T + 6561
$53$	$ T^{8} - 5 T^{7} + \cdots + 625 $ T^8 - 5T^7 + 10T^6 - 25T^5 + 75T^4 - 125T^3 + 250T^2 - 625*T + 625
$59$	$ T^{8} + 3 T^{7} + \cdots + 6561 $ T^8 + 3T^7 - 27T^5 - 81T^4 - 243T^3 + 2187*T + 6561
$61$	$ T^{8} + \cdots + 1026625681 $ T^8 + 29T^7 + 300T^6 + 4591T^5 + 99719T^4 + 1143631T^3 + 13457220T^2 + 177795509*T + 1026625681
$67$	$ (T^{2} + T + 1)^{4} $ (T^2 + T + 1)^4
$71$	$ (T^{4} - 5 T^{3} + \cdots + 9025)^{2} $ (T^4 - 5T^3 + 115T^2 - 1425*T + 9025)^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} - 15 T^{7} + \cdots + 9150625 $ T^8 - 15T^7 + 35T^6 - 650T^5 + 16575T^4 - 118250T^3 + 635250T^2 - 3327500*T + 9150625
$83$	$ (T^{4} - 16 T^{3} + \cdots + 4096)^{2} $ (T^4 - 16T^3 + 256T^2 - 1536*T + 4096)^2
$89$	$ (T^{4} + 9 T^{3} + \cdots + 6561)^{2} $ (T^4 + 9T^3 + 162T^2 - 729*T + 6561)^2
$97$	$ (T^{4} - 23 T^{3} + \cdots + 14641)^{2} $ (T^4 - 23T^3 + 379T^2 - 3267*T + 14641)^2
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 154 = 2 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	154.m (of order \(15\), degree \(8\), minimal)

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

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Hecke kernels

Hecke characteristic polynomials

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Label	154.2.m.a

Level	$154$
Weight	$2$
Character orbit	154.m
Analytic conductor	$1.230$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.22969619113\)
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:	$\mathrm{SU}(2)[C_{15}]$

\(n\)	\(45\)	\(57\)
\(\chi(n)\)	\(-1 - \zeta_{15}^{5}\)	\(-1 + \zeta_{15}^{2} - \zeta_{15}^{3} - \zeta_{15}^{6} + \zeta_{15}^{7}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 9.1
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
$5$	\( T^{8} + 5 T^{7} + \cdots + 625 \) T^8 + 5T^7 + 10T^6 + 25T^5 + 75T^4 + 125T^3 + 250T^2 + 625*T + 625
$7$	\( T^{8} + 4 T^{7} + \cdots + 2401 \) T^8 + 4T^7 + 9T^6 + 8T^5 - 31T^4 + 56T^3 + 441T^2 + 1372*T + 2401
$11$	\( T^{8} + 11 T^{7} + \cdots + 14641 \) T^8 + 11T^7 + 70T^6 + 319T^5 + 1149T^4 + 3509T^3 + 8470T^2 + 14641*T + 14641
$13$	\( (T^{4} + 9 T^{3} + 36 T^{2} + \cdots + 81)^{2} \) (T^4 + 9T^3 + 36T^2 + 54*T + 81)^2
$17$	\( T^{8} - 9 T^{7} + \cdots + 14641 \) T^8 - 9T^7 + 50T^6 - 301T^5 + 939T^4 + 2519T^3 - 3630T^2 + 1331*T + 14641
$19$	\( T^{8} + 3 T^{7} + \cdots + 707281 \) T^8 + 3T^7 - 25T^6 - 362T^5 - 381T^4 + 2842T^3 + 25230T^2 + 195112*T + 707281
$23$	\( (T^{4} - 2 T^{3} + 8 T^{2} + \cdots + 16)^{2} \) (T^4 - 2T^3 + 8T^2 + 8*T + 16)^2
$29$	\( (T^{4} + 9 T^{3} + 36 T^{2} + \cdots + 81)^{2} \) (T^4 + 9T^3 + 36T^2 + 54*T + 81)^2
$31$	\( T^{8} - T^{7} + \cdots + 923521 \) T^8 - T^7 - 75T^6 - 944T^5 + 5249T^4 + 34906T^3 + 115320T^2 + 417074T + 923521
$37$	\( T^{8} - 2 T^{7} + \cdots + 3748096 \) T^8 - 2T^7 - 60T^6 + 928T^5 + 1104T^4 - 26048T^3 + 154880T^2 - 1022208*T + 3748096
$41$	\( (T^{4} + 14 T^{3} + \cdots + 841)^{2} \) (T^4 + 14T^3 + 96T^2 + 319*T + 841)^2
$43$	\( (T^{2} - 17 T + 61)^{4} \) (T^2 - 17*T + 61)^4
$47$	\( T^{8} - 9 T^{7} + \cdots + 6561 \) T^8 - 9T^7 + 45T^6 - 216T^5 + 729T^4 - 486T^3 - 4374T + 6561
$53$	\( T^{8} - 5 T^{7} + \cdots + 625 \) T^8 - 5T^7 + 10T^6 - 25T^5 + 75T^4 - 125T^3 + 250T^2 - 625*T + 625
$59$	\( T^{8} + 3 T^{7} + \cdots + 6561 \) T^8 + 3T^7 - 27T^5 - 81T^4 - 243T^3 + 2187*T + 6561
$61$	\( T^{8} + \cdots + 1026625681 \) T^8 + 29T^7 + 300T^6 + 4591T^5 + 99719T^4 + 1143631T^3 + 13457220T^2 + 177795509*T + 1026625681
$67$	\( (T^{2} + T + 1)^{4} \) (T^2 + T + 1)^4
$71$	\( (T^{4} - 5 T^{3} + \cdots + 9025)^{2} \) (T^4 - 5T^3 + 115T^2 - 1425*T + 9025)^2
$73$	\( T^{8} - 12 T^{7} + \cdots + 1679616 \) T^8 - 12T^7 - 432T^5 + 11664T^4 - 62208T^3 + 233280T^2 - 839808T + 1679616
$79$	\( T^{8} - 15 T^{7} + \cdots + 9150625 \) T^8 - 15T^7 + 35T^6 - 650T^5 + 16575T^4 - 118250T^3 + 635250T^2 - 3327500*T + 9150625
$83$	\( (T^{4} - 16 T^{3} + \cdots + 4096)^{2} \) (T^4 - 16T^3 + 256T^2 - 1536*T + 4096)^2
$89$	\( (T^{4} + 9 T^{3} + \cdots + 6561)^{2} \) (T^4 + 9T^3 + 162T^2 - 729*T + 6561)^2
$97$	\( (T^{4} - 23 T^{3} + \cdots + 14641)^{2} \) (T^4 - 23T^3 + 379T^2 - 3267*T + 14641)^2
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