LMFDB - Newform orbit 930.2.bg.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [930,2,Mod(121,930)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(930, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([0, 0, 16]))

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

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

chi := DirichletCharacter("930.121");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 930 = 2 \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	930.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:	$7.42608738798$
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:	$\Z[a_1, a_2, a_3]$
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 $\left(\mathbb{Z}/930\mathbb{Z}\right)^\times$.

$n$	$187$	$311$	$871$
$\chi(n)$	$1$	$1$	$1 - \zeta_{15}^{2} + \zeta_{15}^{3} - \zeta_{15}^{4} + \zeta_{15}^{6} - \zeta_{15}^{7}$

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} $

121.1

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

−0.809017

0.587785i

0.104528

−

0.994522i

0.309017

−

0.951057i

0.500000

−

0.866025i

0.500000

0.866025i

−0.408977

0.0869308i

0.309017

0.951057i

−0.978148

−

0.207912i

0.104528

0.994522i

361.1

0.309017

0.951057i

−0.669131

−

0.743145i

−0.809017

0.587785i

0.500000

0.866025i

0.500000

−

0.866025i

0.279773

2.66186i

−0.809017

−

0.587785i

−0.104528

0.994522i

−0.669131

0.743145i

391.1

−0.809017

−

0.587785i

−0.913545

−

0.406737i

0.309017

0.951057i

0.500000

−

0.866025i

0.500000

0.866025i

−2.44512

2.71559i

0.309017

−

0.951057i

0.669131

0.743145i

−0.913545

0.406737i

421.1

−0.809017

0.587785i

−0.913545

0.406737i

0.309017

−

0.951057i

0.500000

0.866025i

0.500000

−

0.866025i

−2.44512

−

2.71559i

0.309017

0.951057i

0.669131

−

0.743145i

−0.913545

−

0.406737i

541.1

0.309017

−

0.951057i

−0.669131

0.743145i

−0.809017

−

0.587785i

0.500000

−

0.866025i

0.500000

0.866025i

0.279773

−

2.66186i

−0.809017

0.587785i

−0.104528

−

0.994522i

−0.669131

−

0.743145i

661.1

−0.809017

−

0.587785i

0.104528

0.994522i

0.309017

0.951057i

0.500000

0.866025i

0.500000

−

0.866025i

−0.408977

−

0.0869308i

0.309017

−

0.951057i

−0.978148

0.207912i

0.104528

−

0.994522i

691.1

0.309017

−

0.951057i

0.978148

0.207912i

−0.809017

−

0.587785i

0.500000

0.866025i

0.500000

−

0.866025i

3.57433

−

1.59139i

−0.809017

0.587785i

0.913545

0.406737i

0.978148

−

0.207912i

751.1

0.309017

0.951057i

0.978148

−

0.207912i

−0.809017

0.587785i

0.500000

−

0.866025i

0.500000

0.866025i

3.57433

1.59139i

−0.809017

−

0.587785i

0.913545

−

0.406737i

0.978148

0.207912i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
31.g	even	15	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	930.2.bg.a	✓	8
31.g	even	15	1	inner	930.2.bg.a	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
930.2.bg.a	✓	8	1.a	even	1	1	trivial
930.2.bg.a	✓	8	31.g	even	15	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{8} - 2T_{7}^{7} - 32T_{7}^{5} + 144T_{7}^{4} - 128T_{7}^{3} + 1280T_{7}^{2} + 1152T_{7} + 256 $ T7^8 - 2*T7^7 - 32*T7^5 + 144*T7^4 - 128*T7^3 + 1280*T7^2 + 1152*T7 + 256 acting on $S_{2}^{\mathrm{new}}(930, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} $ (T^4 + T^3 + T^2 + T + 1)^2
$3$	$ T^{8} + T^{7} - T^{5} + \cdots + 1 $ T^8 + T^7 - T^5 - T^4 - T^3 + T + 1
$5$	$ (T^{2} - T + 1)^{4} $ (T^2 - T + 1)^4
$7$	$ T^{8} - 2 T^{7} + \cdots + 256 $ T^8 - 2T^7 - 32T^5 + 144T^4 - 128T^3 + 1280T^2 + 1152T + 256
$11$	$ T^{8} + 23 T^{7} + \cdots + 73441 $ T^8 + 23T^7 + 265T^6 + 2068T^5 + 11589T^4 + 43292T^3 + 97300T^2 + 119782*T + 73441
$13$	$ T^{8} + 17 T^{7} + \cdots + 841 $ T^8 + 17T^7 + 135T^6 + 692T^5 + 2559T^4 + 5408T^3 + 3650T^2 - 2262*T + 841
$17$	$ T^{8} - 11 T^{7} + \cdots + 292681 $ T^8 - 11T^7 + 60T^6 - 454T^5 + 2054T^4 - 2959T^3 + 25740T^2 - 160136*T + 292681
$19$	$ T^{8} + 20 T^{6} + \cdots + 6400 $ T^8 + 20T^6 + 80T^5 + 480T^4 + 3200T^3 + 9600T^2 + 12800T + 6400
$23$	$ T^{8} + 2 T^{7} + \cdots + 1 $ T^8 + 2T^7 + 3T^6 - T^5 - T^3 + 23T^2 - 3T + 1
$29$	$ T^{8} - 10 T^{7} + \cdots + 25 $ T^8 - 10T^7 + 30T^6 + 170T^5 + 840T^4 + 1700T^3 + 1775T^2 + 150*T + 25
$31$	$ T^{8} - 14 T^{7} + \cdots + 923521 $ T^8 - 14T^7 + 165T^6 - 1306T^5 + 8279T^4 - 40486T^3 + 158565T^2 - 417074*T + 923521
$37$	$ T^{8} + T^{7} + \cdots + 961 $ T^8 + T^7 + 115T^6 + 34T^5 + 13039T^4 + 8374T^3 + 9010T^2 - 2294T + 961
$41$	$ T^{8} + 42 T^{7} + \cdots + 11397376 $ T^8 + 42T^7 + 860T^6 + 11152T^5 + 100944T^4 + 665728T^3 + 3098880T^2 + 8804608*T + 11397376
$43$	$ T^{8} + 23 T^{7} + \cdots + 3481 $ T^8 + 23T^7 + 300T^6 + 2693T^5 + 16539T^4 + 63557T^3 + 109700T^2 - 36993*T + 3481
$47$	$ T^{8} - 11 T^{7} + \cdots + 292681 $ T^8 - 11T^7 + 112T^6 - 283T^5 - 225T^4 + 8893T^3 + 75502T^2 + 135791*T + 292681
$53$	$ T^{8} - 2 T^{7} + \cdots + 246016 $ T^8 - 2T^7 + 60T^6 + 328T^5 + 1344T^4 + 23392T^3 + 150080T^2 + 333312*T + 246016
$59$	$ T^{8} - 28 T^{7} + \cdots + 436921 $ T^8 - 28T^7 + 390T^6 - 3358T^5 + 19779T^4 - 84682T^3 + 258455T^2 - 493767*T + 436921
$61$	$ (T^{4} + 2 T^{3} + \cdots + 5296)^{2} $ (T^4 + 2T^3 - 196T^2 + 88*T + 5296)^2
$67$	$ T^{8} + 20 T^{7} + \cdots + 89397025 $ T^8 + 20T^7 + 410T^6 + 3920T^5 + 50755T^4 + 398800T^3 + 4149050T^2 + 19477300*T + 89397025
$71$	$ T^{8} - 42 T^{7} + \cdots + 19927296 $ T^8 - 42T^7 + 840T^6 - 10512T^5 + 93024T^4 - 628128T^3 + 3265920T^2 - 11570688*T + 19927296
$73$	$ T^{8} - 4 T^{7} + \cdots + 256 $ T^8 - 4T^7 + 240T^6 - 176T^5 - 496T^4 + 64T^3 + 960T^2 + 896*T + 256
$79$	$ T^{8} + 42 T^{7} + \cdots + 6046681 $ T^8 + 42T^7 + 740T^6 + 6692T^5 + 36729T^4 + 81458T^3 + 184305T^2 - 3892597*T + 6046681
$83$	$ T^{8} - 6 T^{7} + \cdots + 256 $ T^8 - 6T^7 - 40T^6 + 256T^5 + 1104T^4 + 256T^3 + 1920T^2 - 896*T + 256
$89$	$ T^{8} - 16 T^{7} + \cdots + 18524416 $ T^8 - 16T^7 + 292T^6 - 1448T^5 + 2880T^4 + 21728T^3 + 404032T^2 - 1618304*T + 18524416
$97$	$ (T^{4} - 22 T^{3} + \cdots + 16)^{2} $ (T^4 - 22T^3 + 184T^2 + 32*T + 16)^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:	\( N \)	\(=\)	\( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	930.bg (of order \(15\), degree \(8\), minimal)

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

Introduction

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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	930.2.bg.a

Level	$930$
Weight	$2$
Character orbit	930.bg
Analytic conductor	$7.426$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(7.42608738798\)
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:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

\(n\)	\(187\)	\(311\)	\(871\)
\(\chi(n)\)	\(1\)	\(1\)	\(1 - \zeta_{15}^{2} + \zeta_{15}^{3} - \zeta_{15}^{4} + \zeta_{15}^{6} - \zeta_{15}^{7}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 121.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) (T^4 + T^3 + T^2 + T + 1)^2
$3$	\( T^{8} + T^{7} - T^{5} + \cdots + 1 \) T^8 + T^7 - T^5 - T^4 - T^3 + T + 1
$5$	\( (T^{2} - T + 1)^{4} \) (T^2 - T + 1)^4
$7$	\( T^{8} - 2 T^{7} + \cdots + 256 \) T^8 - 2T^7 - 32T^5 + 144T^4 - 128T^3 + 1280T^2 + 1152T + 256
$11$	\( T^{8} + 23 T^{7} + \cdots + 73441 \) T^8 + 23T^7 + 265T^6 + 2068T^5 + 11589T^4 + 43292T^3 + 97300T^2 + 119782*T + 73441
$13$	\( T^{8} + 17 T^{7} + \cdots + 841 \) T^8 + 17T^7 + 135T^6 + 692T^5 + 2559T^4 + 5408T^3 + 3650T^2 - 2262*T + 841
$17$	\( T^{8} - 11 T^{7} + \cdots + 292681 \) T^8 - 11T^7 + 60T^6 - 454T^5 + 2054T^4 - 2959T^3 + 25740T^2 - 160136*T + 292681
$19$	\( T^{8} + 20 T^{6} + \cdots + 6400 \) T^8 + 20T^6 + 80T^5 + 480T^4 + 3200T^3 + 9600T^2 + 12800T + 6400
$23$	\( T^{8} + 2 T^{7} + \cdots + 1 \) T^8 + 2T^7 + 3T^6 - T^5 - T^3 + 23T^2 - 3T + 1
$29$	\( T^{8} - 10 T^{7} + \cdots + 25 \) T^8 - 10T^7 + 30T^6 + 170T^5 + 840T^4 + 1700T^3 + 1775T^2 + 150*T + 25
$31$	\( T^{8} - 14 T^{7} + \cdots + 923521 \) T^8 - 14T^7 + 165T^6 - 1306T^5 + 8279T^4 - 40486T^3 + 158565T^2 - 417074*T + 923521
$37$	\( T^{8} + T^{7} + \cdots + 961 \) T^8 + T^7 + 115T^6 + 34T^5 + 13039T^4 + 8374T^3 + 9010T^2 - 2294T + 961
$41$	\( T^{8} + 42 T^{7} + \cdots + 11397376 \) T^8 + 42T^7 + 860T^6 + 11152T^5 + 100944T^4 + 665728T^3 + 3098880T^2 + 8804608*T + 11397376
$43$	\( T^{8} + 23 T^{7} + \cdots + 3481 \) T^8 + 23T^7 + 300T^6 + 2693T^5 + 16539T^4 + 63557T^3 + 109700T^2 - 36993*T + 3481
$47$	\( T^{8} - 11 T^{7} + \cdots + 292681 \) T^8 - 11T^7 + 112T^6 - 283T^5 - 225T^4 + 8893T^3 + 75502T^2 + 135791*T + 292681
$53$	\( T^{8} - 2 T^{7} + \cdots + 246016 \) T^8 - 2T^7 + 60T^6 + 328T^5 + 1344T^4 + 23392T^3 + 150080T^2 + 333312*T + 246016
$59$	\( T^{8} - 28 T^{7} + \cdots + 436921 \) T^8 - 28T^7 + 390T^6 - 3358T^5 + 19779T^4 - 84682T^3 + 258455T^2 - 493767*T + 436921
$61$	\( (T^{4} + 2 T^{3} + \cdots + 5296)^{2} \) (T^4 + 2T^3 - 196T^2 + 88*T + 5296)^2
$67$	\( T^{8} + 20 T^{7} + \cdots + 89397025 \) T^8 + 20T^7 + 410T^6 + 3920T^5 + 50755T^4 + 398800T^3 + 4149050T^2 + 19477300*T + 89397025
$71$	\( T^{8} - 42 T^{7} + \cdots + 19927296 \) T^8 - 42T^7 + 840T^6 - 10512T^5 + 93024T^4 - 628128T^3 + 3265920T^2 - 11570688*T + 19927296
$73$	\( T^{8} - 4 T^{7} + \cdots + 256 \) T^8 - 4T^7 + 240T^6 - 176T^5 - 496T^4 + 64T^3 + 960T^2 + 896*T + 256
$79$	\( T^{8} + 42 T^{7} + \cdots + 6046681 \) T^8 + 42T^7 + 740T^6 + 6692T^5 + 36729T^4 + 81458T^3 + 184305T^2 - 3892597*T + 6046681
$83$	\( T^{8} - 6 T^{7} + \cdots + 256 \) T^8 - 6T^7 - 40T^6 + 256T^5 + 1104T^4 + 256T^3 + 1920T^2 - 896*T + 256
$89$	\( T^{8} - 16 T^{7} + \cdots + 18524416 \) T^8 - 16T^7 + 292T^6 - 1448T^5 + 2880T^4 + 21728T^3 + 404032T^2 - 1618304*T + 18524416
$97$	\( (T^{4} - 22 T^{3} + \cdots + 16)^{2} \) (T^4 - 22T^3 + 184T^2 + 32*T + 16)^2
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