LMFDB - Newform orbit 399.2.j.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [399,2,Mod(58,399)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(399, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 0]))

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

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

chi := DirichletCharacter("399.58");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$399 = 3 \cdot 7 \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	399.j (of order $3$ , degree $2$ , 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:	$3.18603104065$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.542936601.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - x^{7} + 2x^{6} - 4x^{5} + x^{4} - 8x^{3} + 8x^{2} - 8x + 16$ x^8 - x^7 + 2x^6 - 4x^5 + x^4 - 8x^3 + 8x^2 - 8*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/399\mathbb{Z}\right)^\times$ .

$n$	$115$	$134$	$211$
$\chi(n)$	$-1 + \beta_{4}$	$1$	$1$

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}

58.1

−0.139771	−	1.40729i
−1.06969	+	0.925071i
1.41379	−	0.0347146i
0.295677	+	1.38296i
−0.139771	+	1.40729i
−1.06969	−	0.925071i
1.41379	+	0.0347146i
0.295677	−	1.38296i

−1.28863

2.23198i

−0.500000

−

0.866025i

−2.32116

−

4.02036i

−1.82116

3.15434i

2.57727

0.967478

2.46252i

6.80995

−0.500000

0.866025i

−4.69361

−

8.12957i

58.2

0.266288

−

0.461225i

−0.500000

−

0.866025i

0.858181

1.48641i

1.35818

−

2.35244i

−0.532576

2.59189

0.531122i

1.97925

−0.500000

0.866025i

−0.723335

−

1.25285i

58.3

0.676830

−

1.17230i

−0.500000

−

0.866025i

0.0838024

0.145150i

0.583802

−

1.01118i

−1.35366

1.40697

2.24063i

2.93420

−0.500000

0.866025i

−0.790270

−

1.36879i

58.4

1.34552

−

2.33050i

−0.500000

−

0.866025i

−2.62083

−

4.53941i

−2.12083

3.67338i

−2.69103

−1.96634

−

1.77017i

−8.72340

−0.500000

0.866025i

5.70721

9.88518i

172.1

−1.28863

−

2.23198i

−0.500000

0.866025i

−2.32116

4.02036i

−1.82116

−

3.15434i

2.57727

0.967478

−

2.46252i

6.80995

−0.500000

−

0.866025i

−4.69361

8.12957i

172.2

0.266288

0.461225i

−0.500000

0.866025i

0.858181

−

1.48641i

1.35818

2.35244i

−0.532576

2.59189

−

0.531122i

1.97925

−0.500000

−

0.866025i

−0.723335

1.25285i

172.3

0.676830

1.17230i

−0.500000

0.866025i

0.0838024

−

0.145150i

0.583802

1.01118i

−1.35366

1.40697

−

2.24063i

2.93420

−0.500000

−

0.866025i

−0.790270

1.36879i

172.4

1.34552

2.33050i

−0.500000

0.866025i

−2.62083

4.53941i

−2.12083

−

3.67338i

−2.69103

−1.96634

1.77017i

−8.72340

−0.500000

−

0.866025i

5.70721

−

9.88518i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	399.2.j.e	✓	8
3.b	odd	2	1		1197.2.j.i		8
7.c	even	3	1	inner	399.2.j.e	✓	8
7.c	even	3	1		2793.2.a.bb		4
7.d	odd	6	1		2793.2.a.z		4
21.g	even	6	1		8379.2.a.ca		4
21.h	odd	6	1		1197.2.j.i		8
21.h	odd	6	1		8379.2.a.bz		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
399.2.j.e	✓	8	1.a	even	1	1	trivial
399.2.j.e	✓	8	7.c	even	3	1	inner
1197.2.j.i		8	3.b	odd	2	1
1197.2.j.i		8	21.h	odd	6	1
2793.2.a.z		4	7.d	odd	6	1
2793.2.a.bb		4	7.c	even	3	1
8379.2.a.bz		4	21.h	odd	6	1
8379.2.a.ca		4	21.g	even	6	1

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{8} - 2T_{2}^{7} + 10T_{2}^{6} - 14T_{2}^{5} + 67T_{2}^{4} - 98T_{2}^{3} + 139T_{2}^{2} - 65T_{2} + 25$ T2^8 - 2*T2^7 + 10*T2^6 - 14*T2^5 + 67*T2^4 - 98*T2^3 + 139*T2^2 - 65*T2 + 25 acting on $S_{2}^{\mathrm{new}}(399, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{8} - 2 T^{7} + \cdots + 25$ T^8 - 2T^7 + 10T^6 - 14T^5 + 67T^4 - 98T^3 + 139T^2 - 65*T + 25
$3$	$(T^{2} + T + 1)^{4}$ (T^2 + T + 1)^4
$5$	$T^{8} + 4 T^{7} + \cdots + 2401$ T^8 + 4T^7 + 28T^6 + 22T^5 + 235T^4 + 28T^3 + 1813T^2 - 1715*T + 2401
$7$	$T^{8} - 6 T^{7} + \cdots + 2401$ T^8 - 6T^7 + 19T^6 - 36T^5 + 57T^4 - 252T^3 + 931T^2 - 2058*T + 2401
$11$	$T^{8} + 42 T^{6} + \cdots + 114921$ T^8 + 42T^6 - 42T^5 + 1425T^4 - 882T^3 + 14679T^2 + 7119T + 114921
$13$	$(T^{4} - 42 T^{2} + \cdots + 339)^{2}$ (T^4 - 42T^2 - 21T + 339)^2
$17$	$(T^{2} + 3 T + 9)^{4}$ (T^2 + 3*T + 9)^4
$19$	$(T^{2} - T + 1)^{4}$ (T^2 - T + 1)^4
$23$	$T^{8} - 5 T^{7} + \cdots + 1225$ T^8 - 5T^7 + 73T^6 - 92T^5 + 3169T^4 - 8318T^3 + 25876T^2 - 5810*T + 1225
$29$	$(T^{4} + 16 T^{3} + \cdots - 1379)^{2}$ (T^4 + 16T^3 + 45T^2 - 320*T - 1379)^2
$31$	$T^{8} + 23 T^{7} + \cdots + 648025$ T^8 + 23T^7 + 340T^6 + 3041T^5 + 19897T^4 + 86387T^3 + 274264T^2 + 525665*T + 648025
$37$	$T^{8} - T^{7} + \cdots + 49$ T^8 - T^7 + 22T^6 + 47T^5 + 421T^4 + 287T^3 + 316T^2 - 91T + 49
$41$	$(T^{4} + 5 T^{3} + \cdots + 619)^{2}$ (T^4 + 5T^3 - 51T^2 - 109*T + 619)^2
$43$	$(T^{4} - 5 T^{3} + \cdots + 211)^{2}$ (T^4 - 5T^3 - 75T^2 + 379*T + 211)^2
$47$	$T^{8} + 6 T^{7} + \cdots + 1071225$ T^8 + 6T^7 + 96T^6 + 36T^5 + 3753T^4 - 540T^3 + 101304T^2 - 204930*T + 1071225
$53$	$T^{8} - 12 T^{7} + \cdots + 9801$ T^8 - 12T^7 + 114T^6 - 450T^5 + 1539T^4 - 1026T^3 + 4995T^2 - 4455*T + 9801
$59$	$T^{8} - 15 T^{7} + \cdots + 690561$ T^8 - 15T^7 + 228T^6 - 1161T^5 + 9885T^4 - 26739T^3 + 361116T^2 - 501093*T + 690561
$61$	$T^{8} + 7 T^{7} + \cdots + 49$ T^8 + 7T^7 + 58T^6 - 29T^5 + 193T^4 + 55T^3 + 352T^2 - 119*T + 49
$67$	$T^{8} + 17 T^{7} + \cdots + 67092481$ T^8 + 17T^7 + 415T^6 + 1322T^5 + 37129T^4 - 60262T^3 + 4031890T^2 - 14186812*T + 67092481
$71$	$(T^{4} + 12 T^{3} + \cdots - 3045)^{2}$ (T^4 + 12T^3 - 54T^2 - 1047*T - 3045)^2
$73$	$T^{8} - 5 T^{7} + \cdots + 737881$ T^8 - 5T^7 + 103T^6 + 286T^5 + 5485T^4 + 4534T^3 + 69706T^2 + 44668*T + 737881
$79$	$T^{8} + 3 T^{7} + \cdots + 50168889$ T^8 + 3T^7 + 186T^6 + 171T^5 + 25299T^4 + 19629T^3 + 1376892T^2 - 2486133*T + 50168889
$83$	$(T^{4} + 14 T^{3} + \cdots + 367)^{2}$ (T^4 + 14T^3 - 72T^2 - 205*T + 367)^2
$89$	$T^{8} + 7 T^{7} + \cdots + 2758921$ T^8 + 7T^7 + 142T^6 + 1063T^5 + 16309T^4 + 102955T^3 + 579976T^2 + 1423477*T + 2758921
$97$	$(T^{4} - T^{3} - 36 T^{2} + \cdots + 43)^{2}$ (T^4 - T^3 - 36T^2 + 8T + 43)^2
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Abelian varieties over $\F_{q}$

$f(q)$	$=$	$q + (\beta_{5} + \beta_{3}) q^{2} - \beta_{4} q^{3} + ( - \beta_{7} - \beta_{5} - 3 \beta_{4} + \cdots + 2) q^{4} + (2 \beta_{7} - \beta_{6} + \beta_{5} + \cdots - 3) q^{5} + (\beta_{6} - \beta_{5} - \beta_{3} - \beta_1) q^{6}+ \cdots + ( - \beta_{6} + \beta_{5} + 3 \beta_{3} + \cdots - 1) q^{99}+O(q^{100})$ q + (b5 + b3) * q^2 - b4 * q^3 + (-b7 - b5 - 3b4 - b3 + 2b2 - b1 + 2) * q^4 + (2b7 - b6 + b5 + 2b4 + b3 - b2 - 3) * q^5 + (b6 - b5 - b3 - b1) * q^6 + (-b5 + b4 + b2 - 2b1 + 1) q^7 + (b7 + 2b6 - 3b5 - 2b3 + b2 - 3b1 + 2) * q^8 + (b4 - 1) * q^9 + (b7 - 3b6 + 2b5 - b4 - 2b2 + 5b1 - 2) * q^10 + (3b6 + 2b5 + b4 - 2b3 - b1) q^11 + (2b7 - b6 + b5 + 3b4 + b3 - b2 - 4) * q^12 + (-b6 + b5 + 3b3 + 3b1 - 1) * q^13 + (2b7 - 2b6 + b3 + b2 - 3) * q^14 + (-b7 + b6 - b2 + b1 + 1) * q^15 + (4b7 + 3b5 + 7b4 + b3 - 2b2 + 2b1 - 9) q^16 - 3b4 q^17 + (-b6 + b1) * q^18 + (-b4 + 1) * q^19 + (-3b7 + 2b6 + b5 - b3 - 3b2 + 2b1 + 11) * q^20 + (b7 - b4 - b2 + b1) * q^21 + (b7 - 2b6 + b5 - b3 + b2 - 2b1 - 6) * q^22 + (-3b6 - 3b5 - 2b4 - 3b1 + 2) * q^23 + (b7 - 2b6 + 2b5 - b4 - 2b2 + 4b1 - 2) * q^24 + (-2b7 - b6 - 4b5 - 7b4 + 4b2 - 3b1 + 4) q^25 + (-2b7 + 3b6 - 7b4 - 2b3 + b2 + 2b1 + 8) q^26 + q^27 + (b7 + b6 - 2b5 + b4 - 5b3 - b2 + 2b1 + 1) q^28 + (b7 - 2b6 + b5 - b3 + b2 - 2b1 - 4) * q^29 + (-2b7 + 2b5 + b4 + 3b3 + b2 - b1) q^30 + (b5 - 6b4 - b3 + b1) q^31 + (b7 - 3b6 + 3b5 + 3b4 - b3 - 2b2 + 6b1 - 2) q^32 + (-2b6 - 3b5 - b4 - b3 - 2b1 + 1) q^33 + (3b6 - 3b5 - 3b3 - 3b1) * q^34 + (-5b6 + 3b5 - 3b4 + 4b3 + 2b1 + 2) q^35 + (-b7 + b6 - b2 + b1 + 2) * q^36 + (-2b7 - b5 - b4 + b2 - b1 + 2) q^37 + (b6 - b1) * q^38 + (3b6 + 2b5 + b4 - 2b3 - b1) q^39 + (-4b7 - b6 + 7b5 - 2b4 + 10b3 + 2b2 - 3b1 + 4) * q^40 + (2b6 - 2b5 + b3 + b1 - 1) * q^41 + (b7 + b6 + 3b5 + 4b4 + b3 - 3b2 + 2b1 - 3) * q^42 + (-2b7 + 2b6 + b3 - 2b2 + 3b1 + 1) * q^43 + (3b6 - 4b5 - 2b4 - 7b3 + 3b1 + 2) q^44 + (-b7 - b5 - 2b4 - b3 + 2b2 - b1 + 2) * q^45 + (-b6 - 3b5 + 3b4 + 3b3 - 2b1) * q^46 + (2b6 + 4b5 + 3b4 + 2b3 + 2b1 - 3) q^47 + (-2b7 + b6 + b5 - b3 - 2b2 + b1 + 5) * q^48 + (b7 - 3b6 + b5 + 6b4 - b1 - 4) * q^49 + (3b7 + 8b6 - 11b5 - 7b3 + 3b2 - 10b1 + 2) * q^50 + (3b4 - 3) q^51 + (4b6 - 3b5 + 2b4 + 3b3 - 7b1) q^52 + (-b6 - 2b5 + 3b4 + 2b3 - b1) q^53 + (b5 + b3) * q^54 + (-6b6 + 6b5 + b3 + b1 - 1) * q^55 + (2b7 - 2b6 + 4b5 + 14b4 + 4b3 - 5b2 + 3b1 - 3) q^56 - q^57 + (-b6 - 8b5 - 4b4 - 7b3 - b1 + 4) q^58 + (-4b6 - b5 + 2b4 + b3 + 3b1) q^59 + (-3b7 - b6 - 5b5 - 14b4 - b3 + 6b2 - 4b1 + 6) q^60 + (-2b7 + 2b6 + b4 - b3 + b2 + b1) * q^61 + (-b7 + 7b6 - 6b5 - 7b3 - b2 - 6b1 + 3) * q^62 + (-b7 + b5 + b1 - 1) * q^63 + (-2b7 - b6 + 3b5 + 4b3 - 2b2 + 6b1 + 6) q^64 + (5b6 - b5 - b4 - 6b3 + 5b1 + 1) q^65 + (b7 - b6 - b5 + 7b4 + 3b3 - 2b2 - 2) q^66 + (-5b5 - 3b4 + 5b3 - 5b1) * q^67 + (6b7 - 3b6 + 3b5 + 9b4 + 3b3 - 3b2 - 12) * q^68 + (3b3 + 3b1 - 2) * q^69 + (-8b7 + 5b6 - 7b5 - 20b4 - 4b3 + 5b2 - 4b1 + 23) q^70 + (-2b7 - b6 + 3b5 + b3 - 2b2 + 3b1 - 4) * q^71 + (-2b7 + b5 + b4 + 2b3 + b2 - b1) * q^72 + (4b6 + b5 + 3b4 - b3 - 3b1) q^73 + (-2b7 + b6 - 4b5 - 3b4 + 4b2 - 5b1 + 4) q^74 + (4b7 + 3b5 + 7b4 + b3 - 2b2 + 2b1 - 9) q^75 + (b7 - b6 + b2 - b1 - 2) * q^76 + (4b7 + 3b6 + 9b5 + 7b4 - b2 + 5b1 - 6) q^77 + (b7 - 2b6 + b5 - b3 + b2 - 2b1 - 6) * q^78 + (-6b7 + 2b6 - 5b5 - 3b4 - 4b3 + 3b2 - b1 + 6) * q^79 + (-8b7 + 3b6 - 9b5 - 27b4 - 7b3 + 16b2 - 12b1 + 16) q^80 - b4 * q^81 + (4b7 + b6 + 4b5 + 7b4 + b3 - 2b2 + 3b1 - 9) q^82 + (b7 - 5b6 + 4b5 + 2b3 + b2 + b1 - 5) q^83 + (-b7 - 5b6 + 4b5 - 3b4 + 4b3 + 3b1 + 1) q^84 + (-3b7 + 3b6 - 3b2 + 3b1 + 3) * q^85 + (-4b7 + b6 + 4b5 + b4 + 5b3 + 2b2 - b1 + 1) * q^86 + (b7 - b6 - b5 + 5b4 + 3b3 - 2b2 - 2) q^87 + (5b7 + 7b6 + 12b5 + 18b4 - 2b3 - 10b2 + 5b1 - 10) q^88 + (-2b7 - 5b5 - b4 - 4b3 + b2 - b1 + 2) q^89 + (b7 + 3b6 - 4b5 - 3b3 + b2 - 4b1 + 2) * q^90 + (-3b7 - 3b6 - 8b5 - 5b4 + 3b3 + 4b2 + 2) * q^91 + (2b7 - 5b6 + 3b5 + 2b2 - 2b1 - 1) q^92 + (-b6 + 6b4 + b3 - b1 - 6) q^93 + (-2b7 - b6 - 12b4 - 4b3 + 4b2 + b1 + 4) * q^94 + (b7 + b5 + 2b4 + b3 - 2b2 + b1 - 2) * q^95 + (-2b7 - b6 + 2b5 - 3b4 + 4b3 + b2 - 2b1 + 4) q^96 + (b7 - b5 - 2b3 + b2 - 3b1 + 1) * q^97 + (-2b7 - 7b6 - b5 - 5b4 + b3 + 6b1 + 7) * q^98 + (-b6 + b5 + 3b3 + 3b1 - 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$8 q + 2 q^{2} - 4 q^{3} - 8 q^{4} - 4 q^{5} - 4 q^{6} + 6 q^{7} + 6 q^{8} - 4 q^{9} - q^{10} - 8 q^{12} - 18 q^{14} + 8 q^{15} - 20 q^{16} - 12 q^{17} + 2 q^{18} + 4 q^{19} + 88 q^{20} + 3 q^{21} - 48 q^{22}+ \cdots + 43 q^{98}+O(q^{100})$ 8 * q + 2 * q^2 - 4 * q^3 - 8 * q^4 - 4 * q^5 - 4 * q^6 + 6 * q^7 + 6 * q^8 - 4 * q^9 - q^10 - 8 * q^12 - 18 * q^14 + 8 * q^15 - 20 * q^16 - 12 * q^17 + 2 * q^18 + 4 * q^19 + 88 * q^20 + 3 * q^21 - 48 * q^22 + 5 * q^23 - 3 * q^24 - 20 * q^25 + 24 * q^26 + 8 * q^27 + 12 * q^28 - 32 * q^29 - q^30 - 23 * q^31 + 16 * q^32 - 12 * q^34 + 18 * q^35 + 16 * q^36 + q^37 - 2 * q^38 + 21 * q^40 - 10 * q^41 + 9 * q^42 + 10 * q^43 - 3 * q^44 - 4 * q^45 + 11 * q^46 - 6 * q^47 + 40 * q^48 - 2 * q^49 - 20 * q^50 - 12 * q^51 - 3 * q^52 + 12 * q^53 + 2 * q^54 + 6 * q^55 + 66 * q^56 - 8 * q^57 + q^58 + 15 * q^59 - 44 * q^60 - 7 * q^61 - 2 * q^62 - 9 * q^63 + 62 * q^64 - 3 * q^65 + 24 * q^66 - 17 * q^67 - 24 * q^68 - 10 * q^69 + 45 * q^70 - 24 * q^71 - 3 * q^72 + 5 * q^73 - 8 * q^74 - 20 * q^75 - 16 * q^76 + 6 * q^77 - 48 * q^78 - 3 * q^79 - 83 * q^80 - 4 * q^81 - 19 * q^82 - 28 * q^83 + 9 * q^84 + 24 * q^85 + q^86 + 16 * q^87 + 45 * q^88 - 7 * q^89 + 2 * q^90 - 27 * q^91 - 2 * q^92 - 23 * q^93 - 36 * q^94 + 4 * q^95 + 16 * q^96 + 2 * q^97 + 43 * q^98

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( -\nu^{7} - \nu^{6} + 7\nu^{3} + 6\nu^{2} + 8\nu - 8 ) / 8$ (-v^7 - v^6 + 7v^3 + 6v^2 + 8*v - 8) / 8
$\beta_{3}$	$=$	$( -\nu^{7} + \nu^{6} - 2\nu^{5} + 4\nu^{4} - \nu^{3} + 8\nu^{2} - 8\nu + 8 ) / 8$ (-v^7 + v^6 - 2v^5 + 4v^4 - v^3 + 8v^2 - 8v + 8) / 8
$\beta_{4}$	$=$	$( -\nu^{7} - 3\nu^{5} - \nu^{3} + 7\nu^{2} + 2\nu + 16 ) / 4$ (-v^7 - 3v^5 - v^3 + 7v^2 + 2*v + 16) / 4
$\beta_{5}$	$=$	$( \nu^{7} + \nu^{6} + 4\nu^{5} + \nu^{3} - 10\nu^{2} - 8\nu - 16 ) / 4$ (v^7 + v^6 + 4v^5 + v^3 - 10v^2 - 8*v - 16) / 4
$\beta_{6}$	$=$	$( 3\nu^{7} + \nu^{6} + 6\nu^{5} + 3\nu^{3} - 20\nu^{2} - 4\nu - 32 ) / 8$ (3v^7 + v^6 + 6v^5 + 3v^3 - 20v^2 - 4*v - 32) / 8
$\beta_{7}$	$=$	$( \nu^{7} + 2\nu^{5} - \nu^{4} + \nu^{3} - 5\nu^{2} + \nu - 8 ) / 2$ (v^7 + 2v^5 - v^4 + v^3 - 5v^2 + v - 8) / 2

Introduction

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Newspace parameters

Newform invariants

$q$ -expansion

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Inner twists

Twists

Hecke kernels

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	399.2.j.e

Level	$399$
Weight	$2$
Character orbit	399.j
Analytic conductor	$3.186$
Analytic rank	$0$
Dimension	$8$
Inner twists	$2$

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{7} - \beta_{6} + \beta_{3} - 1$ b7 - b6 + b3 - 1
$\nu^{3}$	$=$	$\beta_{6} + \beta_{4} + \beta_{2} - \beta _1 + 1$ b6 + b4 + b2 - b1 + 1
$\nu^{4}$	$=$	$-\beta_{7} + \beta_{6} - \beta_{5} - 2\beta_{4} + \beta_{3} + \beta _1 + 3$ -b7 + b6 - b5 - 2*b4 + b3 + b1 + 3
$\nu^{5}$	$=$	$\beta_{7} - 3\beta_{6} + \beta_{5} - 2\beta_{4} + \beta_{3} + 2\beta _1 + 3$ b7 - 3b6 + b5 - 2b4 + b3 + 2*b1 + 3
$\nu^{6}$	$=$	$2\beta_{7} + 3\beta_{5} + 6\beta_{4} + 2\beta_{3} + 4\beta _1 - 6$ 2b7 + 3b5 + 6b4 + 2b3 + 4*b1 - 6
$\nu^{7}$	$=$	$4\beta_{7} + \beta_{6} - 3\beta_{5} + \beta_{4} + 4\beta_{3} - \beta_{2} - 3\beta _1 - 1$ 4b7 + b6 - 3b5 + b4 + 4b3 - b2 - 3b1 - 1

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 58.4
Significant digits:
Format: