Properties

Label 63.h
Modulus $63$
Conductor $63$
Order $3$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(63, base_ring=CyclotomicField(6))
 
M = H._module
 
chi = DirichletCharacter(H, M([4,4]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(25,63))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(63\)
Conductor: \(63\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(3\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: \(\mathbb{Q}(\zeta_3)\)
Fixed field: 3.3.3969.1

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(4\) \(5\) \(8\) \(10\) \(11\) \(13\) \(16\) \(17\) \(19\)
\(\chi_{63}(25,\cdot)\) \(1\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{63}(58,\cdot)\) \(1\) \(1\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\)