LMFDB - Newform orbit 841.2.d.e

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:	$\Z[a_1, a_2]$
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

1.62349

0.781831i

0.277479

−

0.347948i

0.777479

0.974928i

−0.321552

−

0.154851i

0.722521

−

0.347948i

−2.52446

3.16557i

−0.301938

−

1.32288i

0.623490

2.73169i

−0.400969

−

0.502799i

571.1

1.62349

−

0.781831i

0.277479

0.347948i

0.777479

−

0.974928i

−0.321552

0.154851i

0.722521

0.347948i

−2.52446

−

3.16557i

−0.301938

1.32288i

0.623490

−

2.73169i

−0.400969

0.502799i

574.1

0.777479

−

0.974928i

−0.400969

1.75676i

0.0990311

0.433884i

−2.52446

3.16557i

1.40097

1.75676i

−0.153989

0.674671i

2.74698

1.32288i

−0.222521

−

0.107160i

1.12349

4.92233i

605.1

0.0990311

−

0.433884i

1.12349

−

0.541044i

1.62349

0.781831i

−0.153989

0.674671i

−0.123490

−

0.541044i

−0.321552

0.154851i

1.05496

−

1.32288i

−0.900969

1.12978i

0.277479

0.133627i

645.1

0.0990311

0.433884i

1.12349

0.541044i

1.62349

−

0.781831i

−0.153989

−

0.674671i

−0.123490

0.541044i

−0.321552

−

0.154851i

1.05496

1.32288i

−0.900969

−

1.12978i

0.277479

−

0.133627i

778.1

0.777479

0.974928i

−0.400969

−

1.75676i

0.0990311

−

0.433884i

−2.52446

−

3.16557i

1.40097

−

1.75676i

−0.153989

−

0.674671i

2.74698

−

1.32288i

−0.222521

0.107160i

1.12349

−

4.92233i

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.e		6
29.b	even	2	1		841.2.d.a		6
29.c	odd	4	2		841.2.e.b		12
29.d	even	7	2		29.2.d.a	✓	6
29.d	even	7	1		841.2.a.e		3
29.d	even	7	2		841.2.d.b		6
29.d	even	7	1	inner	841.2.d.e		6
29.e	even	14	1		841.2.a.f		3
29.e	even	14	1		841.2.d.a		6
29.e	even	14	2		841.2.d.c		6
29.e	even	14	2		841.2.d.d		6
29.f	odd	28	2		841.2.b.c		6
29.f	odd	28	2		841.2.e.b		12
29.f	odd	28	4		841.2.e.c		12
29.f	odd	28	4		841.2.e.d		12
87.h	odd	14	1		7569.2.a.p		3
87.j	odd	14	2		261.2.k.a		6
87.j	odd	14	1		7569.2.a.r		3
116.j	odd	14	2		464.2.u.f		6
145.n	even	14	2		725.2.l.b		6
145.p	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.d	even	7	2
261.2.k.a		6	87.j	odd	14	2
464.2.u.f		6	116.j	odd	14	2
725.2.l.b		6	145.n	even	14	2
725.2.r.b		12	145.p	odd	28	4
841.2.a.e		3	29.d	even	7	1
841.2.a.f		3	29.e	even	14	1
841.2.b.c		6	29.f	odd	28	2
841.2.d.a		6	29.b	even	2	1
841.2.d.a		6	29.e	even	14	1
841.2.d.b		6	29.d	even	7	2
841.2.d.c		6	29.e	even	14	2
841.2.d.d		6	29.e	even	14	2
841.2.d.e		6	1.a	even	1	1	trivial
841.2.d.e		6	29.d	even	7	1	inner
841.2.e.b		12	29.c	odd	4	2
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	29.f	odd	28	4
7569.2.a.p		3	87.h	odd	14	1
7569.2.a.r		3	87.j	odd	14	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{6} - 5T_{2}^{5} + 11T_{2}^{4} - 13T_{2}^{3} + 9T_{2}^{2} - 3T_{2} + 1 $ T2^6 - 5*T2^5 + 11*T2^4 - 13*T2^3 + 9*T2^2 - 3*T2 + 1 acting on $S_{2}^{\mathrm{new}}(841, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} - 5 T^{5} + \cdots + 1 $ T^6 - 5T^5 + 11T^4 - 13T^3 + 9T^2 - 3*T + 1
$3$	$ T^{6} - 2 T^{5} + \cdots + 1 $ T^6 - 2T^5 + 4T^4 - 8T^3 + 9T^2 - 4*T + 1
$5$	$ T^{6} + 6 T^{5} + \cdots + 1 $ T^6 + 6T^5 + 22T^4 + 20T^3 + 15T^2 + 6*T + 1
$7$	$ T^{6} + 6 T^{5} + \cdots + 1 $ T^6 + 6T^5 + 22T^4 + 20T^3 + 15T^2 + 6*T + 1
$11$	$ T^{6} - 10 T^{5} + \cdots + 1681 $ T^6 - 10T^5 + 58T^4 - 188T^3 + 515T^2 - 1230*T + 1681
$13$	$ T^{6} + 12 T^{5} + \cdots + 1849 $ T^6 + 12T^5 + 102T^4 + 482T^3 + 1717T^2 + 2838*T + 1849
$17$	$ (T^{3} - 4 T^{2} - 4 T + 8)^{2} $ (T^3 - 4T^2 - 4T + 8)^2
$19$	$ T^{6} - 8 T^{5} + \cdots + 169 $ T^6 - 8T^5 + 36T^4 - 106T^3 + 211T^2 - 260*T + 169
$23$	$ T^{6} - 98 T^{3} + \cdots + 2401 $ T^6 - 98T^3 + 343T^2 + 2401
$29$	$ T^{6} $ T^6
$31$	$ T^{6} - 12 T^{5} + \cdots + 6889 $ T^6 - 12T^5 + 88T^4 - 398T^3 + 723T^2 + 1992*T + 6889
$37$	$ T^{6} + 10 T^{5} + \cdots + 1681 $ T^6 + 10T^5 + 58T^4 + 188T^3 + 515T^2 + 1230*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 + 22T^4 + 50T^3 + 337T^2 - 286*T + 169
$47$	$ T^{6} - 4 T^{5} + \cdots + 57121 $ T^6 - 4T^5 + 58T^4 - 190T^3 + 1257T^2 - 13862*T + 57121
$53$	$ T^{6} - 10 T^{5} + \cdots + 1 $ T^6 - 10T^5 + 58T^4 - 216T^3 + 431T^2 - 26*T + 1
$59$	$ (T^{3} + 28 T^{2} + \cdots + 728)^{2} $ (T^3 + 28T^2 + 252T + 728)^2
$61$	$ T^{6} + 4 T^{5} + \cdots + 169 $ T^6 + 4T^5 + 16T^4 - 90T^3 + 179T^2 - 208*T + 169
$67$	$ T^{6} + 30 T^{5} + \cdots + 169 $ T^6 + 30T^5 + 312T^4 + 848T^3 + 1381T^2 + 312*T + 169
$71$	$ T^{6} + 126 T^{4} + \cdots + 35721 $ T^6 + 126T^4 + 378T^3 + 3969T^2 - 23814T + 35721
$73$	$ T^{6} - 24 T^{5} + \cdots + 175561 $ T^6 - 24T^5 + 282T^4 - 1994T^3 + 10693T^2 - 47766*T + 175561
$79$	$ T^{6} - 12 T^{5} + \cdots + 729 $ T^6 - 12T^5 + 18T^4 + 162T^3 + 2025T^2 - 486*T + 729
$83$	$ T^{6} + 18 T^{5} + \cdots + 28561 $ T^6 + 18T^5 + 184T^4 + 904T^3 + 3357T^2 + 12168*T + 28561
$89$	$ T^{6} - 14 T^{5} + \cdots + 8281 $ T^6 - 14T^5 + 154T^4 - 308T^3 + 2499T^2 + 8918*T + 8281
$97$	$ T^{6} - 22 T^{5} + \cdots + 169 $ T^6 - 22T^5 + 316T^4 - 1016T^3 + 5489T^2 - 1716*T + 169
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Overview	Random
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Classical	Maass
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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:	\( N \)	\(=\)	\( 841 = 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	841.d (of order \(7\), degree \(6\), not minimal)

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

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Hecke kernels

Hecke characteristic polynomials

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Label	841.2.d.e

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:	\(\Z[a_1, a_2]\)
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:

$p$	$F_p(T)$
$2$	\( T^{6} - 5 T^{5} + \cdots + 1 \) T^6 - 5T^5 + 11T^4 - 13T^3 + 9T^2 - 3*T + 1
$3$	\( T^{6} - 2 T^{5} + \cdots + 1 \) T^6 - 2T^5 + 4T^4 - 8T^3 + 9T^2 - 4*T + 1
$5$	\( T^{6} + 6 T^{5} + \cdots + 1 \) T^6 + 6T^5 + 22T^4 + 20T^3 + 15T^2 + 6*T + 1
$7$	\( T^{6} + 6 T^{5} + \cdots + 1 \) T^6 + 6T^5 + 22T^4 + 20T^3 + 15T^2 + 6*T + 1
$11$	\( T^{6} - 10 T^{5} + \cdots + 1681 \) T^6 - 10T^5 + 58T^4 - 188T^3 + 515T^2 - 1230*T + 1681
$13$	\( T^{6} + 12 T^{5} + \cdots + 1849 \) T^6 + 12T^5 + 102T^4 + 482T^3 + 1717T^2 + 2838*T + 1849
$17$	\( (T^{3} - 4 T^{2} - 4 T + 8)^{2} \) (T^3 - 4T^2 - 4T + 8)^2
$19$	\( T^{6} - 8 T^{5} + \cdots + 169 \) T^6 - 8T^5 + 36T^4 - 106T^3 + 211T^2 - 260*T + 169
$23$	\( T^{6} - 98 T^{3} + \cdots + 2401 \) T^6 - 98T^3 + 343T^2 + 2401
$29$	\( T^{6} \) T^6
$31$	\( T^{6} - 12 T^{5} + \cdots + 6889 \) T^6 - 12T^5 + 88T^4 - 398T^3 + 723T^2 + 1992*T + 6889
$37$	\( T^{6} + 10 T^{5} + \cdots + 1681 \) T^6 + 10T^5 + 58T^4 + 188T^3 + 515T^2 + 1230*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 + 22T^4 + 50T^3 + 337T^2 - 286*T + 169
$47$	\( T^{6} - 4 T^{5} + \cdots + 57121 \) T^6 - 4T^5 + 58T^4 - 190T^3 + 1257T^2 - 13862*T + 57121
$53$	\( T^{6} - 10 T^{5} + \cdots + 1 \) T^6 - 10T^5 + 58T^4 - 216T^3 + 431T^2 - 26*T + 1
$59$	\( (T^{3} + 28 T^{2} + \cdots + 728)^{2} \) (T^3 + 28T^2 + 252T + 728)^2
$61$	\( T^{6} + 4 T^{5} + \cdots + 169 \) T^6 + 4T^5 + 16T^4 - 90T^3 + 179T^2 - 208*T + 169
$67$	\( T^{6} + 30 T^{5} + \cdots + 169 \) T^6 + 30T^5 + 312T^4 + 848T^3 + 1381T^2 + 312*T + 169
$71$	\( T^{6} + 126 T^{4} + \cdots + 35721 \) T^6 + 126T^4 + 378T^3 + 3969T^2 - 23814T + 35721
$73$	\( T^{6} - 24 T^{5} + \cdots + 175561 \) T^6 - 24T^5 + 282T^4 - 1994T^3 + 10693T^2 - 47766*T + 175561
$79$	\( T^{6} - 12 T^{5} + \cdots + 729 \) T^6 - 12T^5 + 18T^4 + 162T^3 + 2025T^2 - 486*T + 729
$83$	\( T^{6} + 18 T^{5} + \cdots + 28561 \) T^6 + 18T^5 + 184T^4 + 904T^3 + 3357T^2 + 12168*T + 28561
$89$	\( T^{6} - 14 T^{5} + \cdots + 8281 \) T^6 - 14T^5 + 154T^4 - 308T^3 + 2499T^2 + 8918*T + 8281
$97$	\( T^{6} - 22 T^{5} + \cdots + 169 \) T^6 - 22T^5 + 316T^4 - 1016T^3 + 5489T^2 - 1716*T + 169
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\(n\)	\(2\)
\(\chi(n)\)	\(-\zeta_{14}\)