Properties

Label 55.e
Modulus $55$
Conductor $55$
Order $4$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(55, base_ring=CyclotomicField(4))
 
M = H._module
 
chi = DirichletCharacter(H, M([1,2]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(32,55))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(55\)
Conductor: \(55\)
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: \(\mathbb{Q}(i)\)
Fixed field: 4.4.15125.1

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(6\) \(7\) \(8\) \(9\) \(12\) \(13\) \(14\)
\(\chi_{55}(32,\cdot)\) \(1\) \(1\) \(-i\) \(-i\) \(-1\) \(-1\) \(-i\) \(i\) \(-1\) \(i\) \(i\) \(-1\)
\(\chi_{55}(43,\cdot)\) \(1\) \(1\) \(i\) \(i\) \(-1\) \(-1\) \(i\) \(-i\) \(-1\) \(-i\) \(-i\) \(-1\)