Properties

Label 1805.2
Modulus $1805$
Conductor $1805$
Order $684$
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(684))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([171,2]))
 
pari: [g,chi] = znchar(Mod(2,1805))
 

Basic properties

Modulus: \(1805\)
Conductor: \(1805\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(684\)
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.bi

\(\chi_{1805}(2,\cdot)\) \(\chi_{1805}(3,\cdot)\) \(\chi_{1805}(13,\cdot)\) \(\chi_{1805}(22,\cdot)\) \(\chi_{1805}(32,\cdot)\) \(\chi_{1805}(33,\cdot)\) \(\chi_{1805}(48,\cdot)\) \(\chi_{1805}(52,\cdot)\) \(\chi_{1805}(53,\cdot)\) \(\chi_{1805}(67,\cdot)\) \(\chi_{1805}(72,\cdot)\) \(\chi_{1805}(78,\cdot)\) \(\chi_{1805}(97,\cdot)\) \(\chi_{1805}(98,\cdot)\) \(\chi_{1805}(108,\cdot)\) \(\chi_{1805}(117,\cdot)\) \(\chi_{1805}(128,\cdot)\) \(\chi_{1805}(143,\cdot)\) \(\chi_{1805}(147,\cdot)\) \(\chi_{1805}(148,\cdot)\) \(\chi_{1805}(162,\cdot)\) \(\chi_{1805}(167,\cdot)\) \(\chi_{1805}(173,\cdot)\) \(\chi_{1805}(192,\cdot)\) \(\chi_{1805}(193,\cdot)\) \(\chi_{1805}(203,\cdot)\) \(\chi_{1805}(212,\cdot)\) \(\chi_{1805}(222,\cdot)\) \(\chi_{1805}(223,\cdot)\) \(\chi_{1805}(238,\cdot)\) ...

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: $\Q(\zeta_{684})$
Fixed field: Number field defined by a degree 684 polynomial (not computed)

Values on generators

\((362,1446)\) → \((i,e\left(\frac{1}{342}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 1805 }(2, a) \) \(1\)\(1\)\(e\left(\frac{173}{684}\right)\)\(e\left(\frac{107}{684}\right)\)\(e\left(\frac{173}{342}\right)\)\(e\left(\frac{70}{171}\right)\)\(e\left(\frac{157}{228}\right)\)\(e\left(\frac{173}{228}\right)\)\(e\left(\frac{107}{342}\right)\)\(e\left(\frac{17}{57}\right)\)\(e\left(\frac{151}{228}\right)\)\(e\left(\frac{487}{684}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1805 }(2,a) \;\) at \(\;a = \) e.g. 2