Properties

Label 1805.776
Modulus $1805$
Conductor $19$
Order $9$
Real no
Primitive no
Minimal no
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1805, base_ring=CyclotomicField(18))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,4]))
 
pari: [g,chi] = znchar(Mod(776,1805))
 

Basic properties

Modulus: \(1805\)
Conductor: \(19\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(9\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{19}(16,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1805.k

\(\chi_{1805}(606,\cdot)\) \(\chi_{1805}(776,\cdot)\) \(\chi_{1805}(821,\cdot)\) \(\chi_{1805}(956,\cdot)\) \(\chi_{1805}(1111,\cdot)\) \(\chi_{1805}(1506,\cdot)\)

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: \(\Q(\zeta_{9})\)
Fixed field: \(\Q(\zeta_{19})^+\)

Values on generators

\((362,1446)\) → \((1,e\left(\frac{2}{9}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 1805 }(776, a) \) \(1\)\(1\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1805 }(776,a) \;\) at \(\;a = \) e.g. 2