Properties

Label 39.d
Modulus $39$
Conductor $39$
Order $2$
Real yes
Primitive yes
Minimal yes
Parity odd

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(2))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1,1]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(38,39))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Kronecker symbol representation

sage: kronecker_character(-39)
 
pari: znchartokronecker(g,chi)
 

\(\displaystyle\left(\frac{-39}{\bullet}\right)\)

Basic properties

Modulus: \(39\)
Conductor: \(39\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: yes
Primitive: yes
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: \(\Q\)
Fixed field: \(\Q(\sqrt{-39}) \)

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(4\) \(5\) \(7\) \(8\) \(10\) \(11\) \(14\) \(16\) \(17\)
\(\chi_{39}(38,\cdot)\) \(-1\) \(1\) \(1\) \(1\) \(1\) \(-1\) \(1\) \(1\) \(1\) \(-1\) \(1\) \(-1\)