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Properties

Label 15.e
Modulus $15$
Conductor $15$
Order $4$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(15, base_ring=CyclotomicField(4)) M = H._module chi = DirichletCharacter(H, M([2,1])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(2,15)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(15\)
Conductor: \(15\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(4\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: yes
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Related number fields

Field of values: \(\mathbb{Q}(i)\)
Fixed field: \(\Q(\zeta_{15})^+\)

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(4\) \(7\) \(8\) \(11\) \(13\)
\(\chi_{15}(2,\cdot)\) \(1\) \(1\) \(-i\) \(-1\) \(i\) \(i\) \(-1\) \(-i\)
\(\chi_{15}(8,\cdot)\) \(1\) \(1\) \(i\) \(-1\) \(-i\) \(-i\) \(-1\) \(i\)