Properties

Label 1805.8
Modulus $1805$
Conductor $1805$
Order $228$
Real no
Primitive yes
Minimal yes
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(228))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([171,2]))
 
pari: [g,chi] = znchar(Mod(8,1805))
 

Basic properties

Modulus: \(1805\)
Conductor: \(1805\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(228\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1805.be

\(\chi_{1805}(8,\cdot)\) \(\chi_{1805}(12,\cdot)\) \(\chi_{1805}(27,\cdot)\) \(\chi_{1805}(88,\cdot)\) \(\chi_{1805}(103,\cdot)\) \(\chi_{1805}(107,\cdot)\) \(\chi_{1805}(122,\cdot)\) \(\chi_{1805}(183,\cdot)\) \(\chi_{1805}(198,\cdot)\) \(\chi_{1805}(202,\cdot)\) \(\chi_{1805}(217,\cdot)\) \(\chi_{1805}(278,\cdot)\) \(\chi_{1805}(297,\cdot)\) \(\chi_{1805}(312,\cdot)\) \(\chi_{1805}(373,\cdot)\) \(\chi_{1805}(388,\cdot)\) \(\chi_{1805}(392,\cdot)\) \(\chi_{1805}(407,\cdot)\) \(\chi_{1805}(468,\cdot)\) \(\chi_{1805}(483,\cdot)\) \(\chi_{1805}(487,\cdot)\) \(\chi_{1805}(502,\cdot)\) \(\chi_{1805}(563,\cdot)\) \(\chi_{1805}(578,\cdot)\) \(\chi_{1805}(582,\cdot)\) \(\chi_{1805}(597,\cdot)\) \(\chi_{1805}(658,\cdot)\) \(\chi_{1805}(673,\cdot)\) \(\chi_{1805}(677,\cdot)\) \(\chi_{1805}(692,\cdot)\) ...

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: $\Q(\zeta_{228})$
Fixed field: Number field defined by a degree 228 polynomial (not computed)

Values on generators

\((362,1446)\) → \((-i,e\left(\frac{1}{114}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 1805 }(8, a) \) \(1\)\(1\)\(e\left(\frac{173}{228}\right)\)\(e\left(\frac{107}{228}\right)\)\(e\left(\frac{59}{114}\right)\)\(e\left(\frac{13}{57}\right)\)\(e\left(\frac{5}{76}\right)\)\(e\left(\frac{21}{76}\right)\)\(e\left(\frac{107}{114}\right)\)\(e\left(\frac{17}{19}\right)\)\(e\left(\frac{75}{76}\right)\)\(e\left(\frac{31}{228}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1805 }(8,a) \;\) at \(\;a = \) e.g. 2