Properties

Label 1805.1084
Modulus $1805$
Conductor $5$
Order $2$
Real yes
Primitive no
Minimal yes
Parity even

Related objects

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1805, base_ring=CyclotomicField(2))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1,0]))
 
pari: [g,chi] = znchar(Mod(1084,1805))
 

Basic properties

Modulus: \(1805\)
Conductor: \(5\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: yes
Primitive: no, induced from \(\chi_{5}(4,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1805.b

\(\chi_{1805}(1084,\cdot)\)

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: \(\Q\)
Fixed field: \(\Q(\sqrt{5}) \)

Values on generators

\((362,1446)\) → \((-1,1)\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 1805 }(1084, a) \) \(1\)\(1\)\(-1\)\(-1\)\(1\)\(1\)\(-1\)\(-1\)\(1\)\(1\)\(-1\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1805 }(1084,a) \;\) at \(\;a = \) e.g. 2