Properties

Label 1045.208
Modulus $1045$
Conductor $1045$
Order $4$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1045, base_ring=CyclotomicField(4))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([3,2,2]))
 
pari: [g,chi] = znchar(Mod(208,1045))
 

Basic properties

Modulus: \(1045\)
Conductor: \(1045\)
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)
 

Galois orbit 1045.k

\(\chi_{1045}(208,\cdot)\) \(\chi_{1045}(417,\cdot)\)

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: 4.0.5460125.4

Values on generators

\((837,761,496)\) → \((-i,-1,-1)\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 1045 }(208, a) \) \(-1\)\(1\)\(-i\)\(-i\)\(-1\)\(-1\)\(i\)\(i\)\(-1\)\(i\)\(i\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1045 }(208,a) \;\) at \(\;a = \) e.g. 2