Properties

Label 1148.729
Modulus $1148$
Conductor $41$
Order $4$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1148, base_ring=CyclotomicField(4))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,0,1]))
 
pari: [g,chi] = znchar(Mod(729,1148))
 

Basic properties

Modulus: \(1148\)
Conductor: \(41\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{41}(32,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1148.k

\(\chi_{1148}(337,\cdot)\) \(\chi_{1148}(729,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\sqrt{-1}) \)
Fixed field: 4.4.68921.1

Values on generators

\((575,493,785)\) → \((1,1,i)\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(1\)\(1\)\(-i\)\(-1\)\(-1\)\(-i\)\(-i\)\(i\)\(i\)\(i\)\(1\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1148 }(729,a) \;\) at \(\;a = \) e.g. 2