Properties

Label 175.g
Modulus $175$
Conductor $5$
Order $4$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(175, base_ring=CyclotomicField(4))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([3,0]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(43,175))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(175\)
Conductor: \(5\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from 5.c
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: \(\Q(\sqrt{-1}) \)
Fixed field: \(\Q(\zeta_{5})\)

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(6\) \(8\) \(9\) \(11\) \(12\) \(13\) \(16\)
\(\chi_{175}(43,\cdot)\) \(-1\) \(1\) \(-i\) \(i\) \(-1\) \(1\) \(i\) \(-1\) \(1\) \(-i\) \(i\) \(1\)
\(\chi_{175}(57,\cdot)\) \(-1\) \(1\) \(i\) \(-i\) \(-1\) \(1\) \(-i\) \(-1\) \(1\) \(i\) \(-i\) \(1\)