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Dirichlet character orbit 585.bc
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Hilbert
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Q
\Q
Q
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Q
(
α
)
\Q(\alpha)
Q
(
α
)
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\Q
Q
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q
\F_{q}
F
q
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p
p
p
-adic fields
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Properties
Label
585.bc
Modulus
585
585
5
8
5
Conductor
117
117
1
1
7
Order
6
6
6
Real
no
Primitive
no
Minimal
yes
Parity
odd
Related objects
Character group
Value field
Primitive orbit 117.m
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Completeness of the data
Reliability of the data
Dirichlet character labels
Dirichlet character orbit labels
Show commands:
Pari/GP
/
SageMath
sage:
from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(585, base_ring=CyclotomicField(6)) M = H._module chi = DirichletCharacter(H, M([1,0,1])) chi.galois_orbit()
pari:
[g,chi] = znchar(Mod(56,585)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus
:
585
585
5
8
5
Conductor
:
117
117
1
1
7
sage:
chi.conductor()
pari:
znconreyconductor(g,chi)
Order
:
6
6
6
sage:
chi.multiplicative_order()
pari:
charorder(g,chi)
Real
:
no
Primitive
:
no, induced from
117.m
sage:
chi.is_primitive()
pari:
#znconreyconductor(g,chi)==1
Minimal
:
yes
Parity
:
odd
sage:
chi.is_odd()
pari:
zncharisodd(g,chi)
Related number fields
Field of values
:
Q
(
ζ
3
)
\mathbb{Q}(\zeta_3)
Q
(
ζ
3
)
Fixed field
:
6.0.7308160119.2
Characters
in Galois orbit
Character
−
1
-1
−
1
1
1
1
2
2
2
4
4
4
7
7
7
8
8
8
11
11
1
1
14
14
1
4
16
16
1
6
17
17
1
7
19
19
1
9
22
22
2
2
χ
585
(
56
,
⋅
)
\chi_{585}(56,\cdot)
χ
5
8
5
(
5
6
,
⋅
)
−
1
-1
−
1
1
1
1
e
(
1
3
)
e\left(\frac{1}{3}\right)
e
(
3
1
)
e
(
2
3
)
e\left(\frac{2}{3}\right)
e
(
3
2
)
−
1
-1
−
1
1
1
1
e
(
1
3
)
e\left(\frac{1}{3}\right)
e
(
3
1
)
e
(
5
6
)
e\left(\frac{5}{6}\right)
e
(
6
5
)
e
(
1
3
)
e\left(\frac{1}{3}\right)
e
(
3
1
)
e
(
5
6
)
e\left(\frac{5}{6}\right)
e
(
6
5
)
e
(
5
6
)
e\left(\frac{5}{6}\right)
e
(
6
5
)
e
(
2
3
)
e\left(\frac{2}{3}\right)
e
(
3
2
)
χ
585
(
491
,
⋅
)
\chi_{585}(491,\cdot)
χ
5
8
5
(
4
9
1
,
⋅
)
−
1
-1
−
1
1
1
1
e
(
2
3
)
e\left(\frac{2}{3}\right)
e
(
3
2
)
e
(
1
3
)
e\left(\frac{1}{3}\right)
e
(
3
1
)
−
1
-1
−
1
1
1
1
e
(
2
3
)
e\left(\frac{2}{3}\right)
e
(
3
2
)
e
(
1
6
)
e\left(\frac{1}{6}\right)
e
(
6
1
)
e
(
2
3
)
e\left(\frac{2}{3}\right)
e
(
3
2
)
e
(
1
6
)
e\left(\frac{1}{6}\right)
e
(
6
1
)
e
(
1
6
)
e\left(\frac{1}{6}\right)
e
(
6
1
)
e
(
1
3
)
e\left(\frac{1}{3}\right)
e
(
3
1
)
