Properties

Label 304.j
Modulus $304$
Conductor $304$
Order $4$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(304, base_ring=CyclotomicField(4))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,1,2]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(37,304))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(304\)
Conductor: \(304\)
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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\sqrt{-1}) \)
Fixed field: 4.0.739328.2

Characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(5\) \(7\) \(9\) \(11\) \(13\) \(15\) \(17\) \(21\) \(23\)
\(\chi_{304}(37,\cdot)\) \(-1\) \(1\) \(i\) \(i\) \(-1\) \(-1\) \(i\) \(i\) \(-1\) \(1\) \(-i\) \(-1\)
\(\chi_{304}(189,\cdot)\) \(-1\) \(1\) \(-i\) \(-i\) \(-1\) \(-1\) \(-i\) \(-i\) \(-1\) \(1\) \(i\) \(-1\)