Properties

Label 39.f
Modulus $39$
Conductor $39$
Order $4$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(39, base_ring=CyclotomicField(4))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([2,3]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(5,39))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(39\)
Conductor: \(39\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
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: \(\Q(\sqrt{-1}) \)
Fixed field: 4.4.19773.1

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(4\) \(5\) \(7\) \(8\) \(10\) \(11\) \(14\) \(16\) \(17\)
\(\chi_{39}(5,\cdot)\) \(1\) \(1\) \(i\) \(-1\) \(i\) \(i\) \(-i\) \(-1\) \(-i\) \(-1\) \(1\) \(1\)
\(\chi_{39}(8,\cdot)\) \(1\) \(1\) \(-i\) \(-1\) \(-i\) \(-i\) \(i\) \(-1\) \(i\) \(-1\) \(1\) \(1\)