LMFDB - Newform orbit 841.2.d.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [841,2,Mod(190,841)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(841, base_ring=CyclotomicField(14))

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

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

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

chi := DirichletCharacter("841.190");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 841 = 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	841.d (of order $7$, degree $6$, not 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:	$6.71541880999$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{14})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 29)
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/841\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-\zeta_{14}$

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} $

190.1

−0.623490	−	0.781831i
−0.623490	+	0.781831i
0.222521	+	0.974928i
0.900969	+	0.433884i
0.900969	−	0.433884i
0.222521	−	0.974928i

1.12349

0.541044i

−1.12349

1.40881i

−0.277479

−

0.347948i

3.64795

1.75676i

−2.02446

0.974928i

0.431468

−

0.541044i

−0.678448

−

2.97247i

−0.0549581

−

0.240787i

3.14795

3.94740i

571.1

1.12349

−

0.541044i

−1.12349

−

1.40881i

−0.277479

0.347948i

3.64795

−

1.75676i

−2.02446

−

0.974928i

0.431468

0.541044i

−0.678448

2.97247i

−0.0549581

0.240787i

3.14795

−

3.94740i

574.1

0.277479

−

0.347948i

−0.277479

1.21572i

0.400969

1.75676i

0.431468

−

0.541044i

0.346011

0.433884i

−0.0794168

0.347948i

1.52446

0.734141i

1.30194

0.626980i

−0.0685317

−

0.300257i

605.1

−0.400969

1.75676i

0.400969

−

0.193096i

−1.12349

−

0.541044i

−0.0794168

0.347948i

0.178448

0.781831i

3.64795

−

1.75676i

−0.846011

1.06086i

−1.74698

2.19064i

−0.579417

−

0.279032i

645.1

−0.400969

−

1.75676i

0.400969

0.193096i

−1.12349

0.541044i

−0.0794168

−

0.347948i

0.178448

−

0.781831i

3.64795

1.75676i

−0.846011

−

1.06086i

−1.74698

−

2.19064i

−0.579417

0.279032i

778.1

0.277479

0.347948i

−0.277479

−

1.21572i

0.400969

−

1.75676i

0.431468

0.541044i

0.346011

−

0.433884i

−0.0794168

−

0.347948i

1.52446

−

0.734141i

1.30194

−

0.626980i

−0.0685317

0.300257i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
29.d	even	7	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	841.2.d.c		6
29.b	even	2	1		841.2.d.b		6
29.c	odd	4	2		841.2.e.c		12
29.d	even	7	1		841.2.a.f		3
29.d	even	7	2		841.2.d.a		6
29.d	even	7	1	inner	841.2.d.c		6
29.d	even	7	2		841.2.d.d		6
29.e	even	14	2		29.2.d.a	✓	6
29.e	even	14	1		841.2.a.e		3
29.e	even	14	1		841.2.d.b		6
29.e	even	14	2		841.2.d.e		6
29.f	odd	28	2		841.2.b.c		6
29.f	odd	28	4		841.2.e.b		12
29.f	odd	28	2		841.2.e.c		12
29.f	odd	28	4		841.2.e.d		12
87.h	odd	14	2		261.2.k.a		6
87.h	odd	14	1		7569.2.a.r		3
87.j	odd	14	1		7569.2.a.p		3
116.h	odd	14	2		464.2.u.f		6
145.l	even	14	2		725.2.l.b		6
145.q	odd	28	4		725.2.r.b		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
29.2.d.a	✓	6	29.e	even	14	2
261.2.k.a		6	87.h	odd	14	2
464.2.u.f		6	116.h	odd	14	2
725.2.l.b		6	145.l	even	14	2
725.2.r.b		12	145.q	odd	28	4
841.2.a.e		3	29.e	even	14	1
841.2.a.f		3	29.d	even	7	1
841.2.b.c		6	29.f	odd	28	2
841.2.d.a		6	29.d	even	7	2
841.2.d.b		6	29.b	even	2	1
841.2.d.b		6	29.e	even	14	1
841.2.d.c		6	1.a	even	1	1	trivial
841.2.d.c		6	29.d	even	7	1	inner
841.2.d.d		6	29.d	even	7	2
841.2.d.e		6	29.e	even	14	2
841.2.e.b		12	29.f	odd	28	4
841.2.e.c		12	29.c	odd	4	2
841.2.e.c		12	29.f	odd	28	2
841.2.e.d		12	29.f	odd	28	4
7569.2.a.p		3	87.j	odd	14	1
7569.2.a.r		3	87.h	odd	14	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{6} - 2T_{2}^{5} + 4T_{2}^{4} - 8T_{2}^{3} + 9T_{2}^{2} - 4T_{2} + 1 $ T2^6 - 2*T2^5 + 4*T2^4 - 8*T2^3 + 9*T2^2 - 4*T2 + 1 acting on $S_{2}^{\mathrm{new}}(841, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} - 2 T^{5} + \cdots + 1 $ T^6 - 2T^5 + 4T^4 - 8T^3 + 9T^2 - 4*T + 1
$3$	$ T^{6} + 2 T^{5} + \cdots + 1 $ T^6 + 2T^5 + 4T^4 + T^3 + 2T^2 - 3T + 1
$5$	$ T^{6} - 8 T^{5} + \cdots + 1 $ T^6 - 8T^5 + 22T^4 - 15T^3 + 8T^2 - T + 1
$7$	$ T^{6} - 8 T^{5} + \cdots + 1 $ T^6 - 8T^5 + 22T^4 - 15T^3 + 8T^2 - T + 1
$11$	$ T^{6} - 4 T^{5} + \cdots + 1681 $ T^6 - 4T^5 + 2T^4 - 29T^3 + 228T^2 + 369*T + 1681
$13$	$ T^{6} - 16 T^{5} + \cdots + 1849 $ T^6 - 16T^5 + 116T^4 - 435T^3 + 814T^2 - 473*T + 1849
$17$	$ (T^{3} + 4 T^{2} - 4 T - 8)^{2} $ (T^3 + 4T^2 - 4T - 8)^2
$19$	$ T^{6} - 6 T^{5} + \cdots + 169 $ T^6 - 6T^5 + 22T^4 - 55T^3 + 120T^2 - 195*T + 169
$23$	$ T^{6} - 14 T^{5} + \cdots + 2401 $ T^6 - 14T^5 + 98T^4 - 441T^3 + 1372T^2 - 2401*T + 2401
$29$	$ T^{6} $ T^6
$31$	$ T^{6} - 2 T^{5} + \cdots + 6889 $ T^6 - 2T^5 + 46T^4 - 15T^3 + 1304T^2 + 5561*T + 6889
$37$	$ T^{6} + 4 T^{5} + \cdots + 1681 $ T^6 + 4T^5 + 2T^4 + 29T^3 + 228T^2 - 369*T + 1681
$41$	$ (T^{3} + 10 T^{2} + 24 T + 8)^{2} $ (T^3 + 10T^2 + 24T + 8)^2
$43$	$ T^{6} - 8 T^{5} + \cdots + 169 $ T^6 - 8T^5 + 64T^4 - 239T^3 + 610T^2 - 533*T + 169
$47$	$ T^{6} + 4 T^{5} + \cdots + 57121 $ T^6 + 4T^5 + 16T^4 + 323T^3 + 2930T^2 + 12189*T + 57121
$53$	$ T^{6} + 4 T^{5} + \cdots + 1 $ T^6 + 4T^5 + 30T^4 + 71T^3 + 424T^2 + 37*T + 1
$59$	$ (T^{3} + 28 T^{2} + \cdots + 728)^{2} $ (T^3 + 28T^2 + 252T + 728)^2
$61$	$ T^{6} + 10 T^{5} + \cdots + 169 $ T^6 + 10T^5 + 30T^4 - T^3 + 88T^2 - 65T + 169
$67$	$ T^{6} + 2 T^{5} + \cdots + 169 $ T^6 + 2T^5 + 200T^4 - 993T^3 + 1654T^2 - 689*T + 169
$71$	$ T^{6} - 945 T^{3} + \cdots + 35721 $ T^6 - 945T^3 + 7938T^2 - 11907*T + 35721
$73$	$ T^{6} - 4 T^{5} + \cdots + 175561 $ T^6 - 4T^5 + 44T^4 - 645T^3 + 1894T^2 + 9637*T + 175561
$79$	$ T^{6} + 12 T^{5} + \cdots + 729 $ T^6 + 12T^5 + 144T^4 + 783T^3 + 1458T^2 - 1215*T + 729
$83$	$ T^{6} - 10 T^{5} + \cdots + 28561 $ T^6 - 10T^5 + 16T^4 - 27T^3 + 2174T^2 + 6253*T + 28561
$89$	$ T^{6} - 28 T^{5} + \cdots + 8281 $ T^6 - 28T^5 + 322T^4 - 1799T^3 + 5684T^2 - 5733*T + 8281
$97$	$ T^{6} - 34 T^{5} + \cdots + 169 $ T^6 - 34T^5 + 428T^4 - 1973T^3 + 5034T^2 + 351*T + 169
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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:	\( N \)	\(=\)	\( 841 = 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	841.d (of order \(7\), degree \(6\), not minimal)

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

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Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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Label	841.2.d.c

Level	$841$
Weight	$2$
Character orbit	841.d
Analytic conductor	$6.715$
Analytic rank	$0$
Dimension	$6$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(6.71541880999\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	\(\Q(\zeta_{14})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 29)
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 190.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{6} - 2 T^{5} + \cdots + 1 \) T^6 - 2T^5 + 4T^4 - 8T^3 + 9T^2 - 4*T + 1
$3$	\( T^{6} + 2 T^{5} + \cdots + 1 \) T^6 + 2T^5 + 4T^4 + T^3 + 2T^2 - 3T + 1
$5$	\( T^{6} - 8 T^{5} + \cdots + 1 \) T^6 - 8T^5 + 22T^4 - 15T^3 + 8T^2 - T + 1
$7$	\( T^{6} - 8 T^{5} + \cdots + 1 \) T^6 - 8T^5 + 22T^4 - 15T^3 + 8T^2 - T + 1
$11$	\( T^{6} - 4 T^{5} + \cdots + 1681 \) T^6 - 4T^5 + 2T^4 - 29T^3 + 228T^2 + 369*T + 1681
$13$	\( T^{6} - 16 T^{5} + \cdots + 1849 \) T^6 - 16T^5 + 116T^4 - 435T^3 + 814T^2 - 473*T + 1849
$17$	\( (T^{3} + 4 T^{2} - 4 T - 8)^{2} \) (T^3 + 4T^2 - 4T - 8)^2
$19$	\( T^{6} - 6 T^{5} + \cdots + 169 \) T^6 - 6T^5 + 22T^4 - 55T^3 + 120T^2 - 195*T + 169
$23$	\( T^{6} - 14 T^{5} + \cdots + 2401 \) T^6 - 14T^5 + 98T^4 - 441T^3 + 1372T^2 - 2401*T + 2401
$29$	\( T^{6} \) T^6
$31$	\( T^{6} - 2 T^{5} + \cdots + 6889 \) T^6 - 2T^5 + 46T^4 - 15T^3 + 1304T^2 + 5561*T + 6889
$37$	\( T^{6} + 4 T^{5} + \cdots + 1681 \) T^6 + 4T^5 + 2T^4 + 29T^3 + 228T^2 - 369*T + 1681
$41$	\( (T^{3} + 10 T^{2} + 24 T + 8)^{2} \) (T^3 + 10T^2 + 24T + 8)^2
$43$	\( T^{6} - 8 T^{5} + \cdots + 169 \) T^6 - 8T^5 + 64T^4 - 239T^3 + 610T^2 - 533*T + 169
$47$	\( T^{6} + 4 T^{5} + \cdots + 57121 \) T^6 + 4T^5 + 16T^4 + 323T^3 + 2930T^2 + 12189*T + 57121
$53$	\( T^{6} + 4 T^{5} + \cdots + 1 \) T^6 + 4T^5 + 30T^4 + 71T^3 + 424T^2 + 37*T + 1
$59$	\( (T^{3} + 28 T^{2} + \cdots + 728)^{2} \) (T^3 + 28T^2 + 252T + 728)^2
$61$	\( T^{6} + 10 T^{5} + \cdots + 169 \) T^6 + 10T^5 + 30T^4 - T^3 + 88T^2 - 65T + 169
$67$	\( T^{6} + 2 T^{5} + \cdots + 169 \) T^6 + 2T^5 + 200T^4 - 993T^3 + 1654T^2 - 689*T + 169
$71$	\( T^{6} - 945 T^{3} + \cdots + 35721 \) T^6 - 945T^3 + 7938T^2 - 11907*T + 35721
$73$	\( T^{6} - 4 T^{5} + \cdots + 175561 \) T^6 - 4T^5 + 44T^4 - 645T^3 + 1894T^2 + 9637*T + 175561
$79$	\( T^{6} + 12 T^{5} + \cdots + 729 \) T^6 + 12T^5 + 144T^4 + 783T^3 + 1458T^2 - 1215*T + 729
$83$	\( T^{6} - 10 T^{5} + \cdots + 28561 \) T^6 - 10T^5 + 16T^4 - 27T^3 + 2174T^2 + 6253*T + 28561
$89$	\( T^{6} - 28 T^{5} + \cdots + 8281 \) T^6 - 28T^5 + 322T^4 - 1799T^3 + 5684T^2 - 5733*T + 8281
$97$	\( T^{6} - 34 T^{5} + \cdots + 169 \) T^6 - 34T^5 + 428T^4 - 1973T^3 + 5034T^2 + 351*T + 169
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\(n\)	\(2\)
\(\chi(n)\)	\(-\zeta_{14}\)