Properties

Label 1805.39
Modulus $1805$
Conductor $1805$
Order $38$
Real no
Primitive yes
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(38))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([19,14]))
 
pari: [g,chi] = znchar(Mod(39,1805))
 

Basic properties

Modulus: \(1805\)
Conductor: \(1805\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(38\)
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.v

\(\chi_{1805}(39,\cdot)\) \(\chi_{1805}(134,\cdot)\) \(\chi_{1805}(229,\cdot)\) \(\chi_{1805}(324,\cdot)\) \(\chi_{1805}(419,\cdot)\) \(\chi_{1805}(514,\cdot)\) \(\chi_{1805}(609,\cdot)\) \(\chi_{1805}(704,\cdot)\) \(\chi_{1805}(799,\cdot)\) \(\chi_{1805}(894,\cdot)\) \(\chi_{1805}(989,\cdot)\) \(\chi_{1805}(1179,\cdot)\) \(\chi_{1805}(1274,\cdot)\) \(\chi_{1805}(1369,\cdot)\) \(\chi_{1805}(1464,\cdot)\) \(\chi_{1805}(1559,\cdot)\) \(\chi_{1805}(1654,\cdot)\) \(\chi_{1805}(1749,\cdot)\)

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: \(\Q(\zeta_{19})\)
Fixed field: 38.38.2242277458404766702002557226129081814951173635373007362690887769154380596585299665301923770160675048828125.1

Values on generators

\((362,1446)\) → \((-1,e\left(\frac{7}{19}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 1805 }(39, a) \) \(1\)\(1\)\(e\left(\frac{33}{38}\right)\)\(e\left(\frac{27}{38}\right)\)\(e\left(\frac{14}{19}\right)\)\(e\left(\frac{11}{19}\right)\)\(e\left(\frac{29}{38}\right)\)\(e\left(\frac{23}{38}\right)\)\(e\left(\frac{8}{19}\right)\)\(e\left(\frac{11}{19}\right)\)\(e\left(\frac{17}{38}\right)\)\(e\left(\frac{27}{38}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1805 }(39,a) \;\) at \(\;a = \) e.g. 2