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Dirichlet character orbit 162.c
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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
Q
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F
q
\F_{q}
F
q
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p
p
p
-adic fields
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Properties
Label
162.c
Modulus
162
162
1
6
2
Conductor
9
9
9
Order
3
3
3
Real
no
Primitive
no
Minimal
no
Parity
even
Related objects
Character group
Value field
Primitive orbit 9.c
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Completeness of the data
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Dirichlet character labels
Dirichlet character orbit labels
Show commands:
Pari/GP
/
SageMath
sage:
from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(6)) M = H._module chi = DirichletCharacter(H, M([4])) chi.galois_orbit()
pari:
[g,chi] = znchar(Mod(55,162)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus
:
162
162
1
6
2
Conductor
:
9
9
9
sage:
chi.conductor()
pari:
znconreyconductor(g,chi)
Order
:
3
3
3
sage:
chi.multiplicative_order()
pari:
charorder(g,chi)
Real
:
no
Primitive
:
no, induced from
9.c
sage:
chi.is_primitive()
pari:
#znconreyconductor(g,chi)==1
Minimal
:
no
Parity
:
even
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
:
Q
(
ζ
9
)
+
\Q(\zeta_{9})^+
Q
(
ζ
9
)
+
Characters
in Galois orbit
Character
−
1
-1
−
1
1
1
1
5
5
5
7
7
7
11
11
1
1
13
13
1
3
17
17
1
7
19
19
1
9
23
23
2
3
25
25
2
5
29
29
2
9
31
31
3
1
χ
162
(
55
,
⋅
)
\chi_{162}(55,\cdot)
χ
1
6
2
(
5
5
,
⋅
)
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
)
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
(
1
3
)
e\left(\frac{1}{3}\right)
e
(
3
1
)
e
(
2
3
)
e\left(\frac{2}{3}\right)
e
(
3
2
)
e
(
2
3
)
e\left(\frac{2}{3}\right)
e
(
3
2
)
e
(
1
3
)
e\left(\frac{1}{3}\right)
e
(
3
1
)
χ
162
(
109
,
⋅
)
\chi_{162}(109,\cdot)
χ
1
6
2
(
1
0
9
,
⋅
)
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
)
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
(
2
3
)
e\left(\frac{2}{3}\right)
e
(
3
2
)
e
(
1
3
)
e\left(\frac{1}{3}\right)
e
(
3
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
)
