Properties

Label 1859.15
Modulus $1859$
Conductor $1859$
Order $780$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1859, base_ring=CyclotomicField(780))
 
M = H._module
 
chi = DirichletCharacter(H, M([156,665]))
 
pari: [g,chi] = znchar(Mod(15,1859))
 

Basic properties

Modulus: \(1859\)
Conductor: \(1859\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(780\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1859.bu

\(\chi_{1859}(15,\cdot)\) \(\chi_{1859}(20,\cdot)\) \(\chi_{1859}(37,\cdot)\) \(\chi_{1859}(58,\cdot)\) \(\chi_{1859}(59,\cdot)\) \(\chi_{1859}(71,\cdot)\) \(\chi_{1859}(93,\cdot)\) \(\chi_{1859}(97,\cdot)\) \(\chi_{1859}(102,\cdot)\) \(\chi_{1859}(115,\cdot)\) \(\chi_{1859}(119,\cdot)\) \(\chi_{1859}(124,\cdot)\) \(\chi_{1859}(136,\cdot)\) \(\chi_{1859}(137,\cdot)\) \(\chi_{1859}(141,\cdot)\) \(\chi_{1859}(158,\cdot)\) \(\chi_{1859}(163,\cdot)\) \(\chi_{1859}(180,\cdot)\) \(\chi_{1859}(201,\cdot)\) \(\chi_{1859}(202,\cdot)\) \(\chi_{1859}(214,\cdot)\) \(\chi_{1859}(223,\cdot)\) \(\chi_{1859}(236,\cdot)\) \(\chi_{1859}(240,\cdot)\) \(\chi_{1859}(245,\cdot)\) \(\chi_{1859}(262,\cdot)\) \(\chi_{1859}(267,\cdot)\) \(\chi_{1859}(279,\cdot)\) \(\chi_{1859}(280,\cdot)\) \(\chi_{1859}(284,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{780})$
Fixed field: Number field defined by a degree 780 polynomial (not computed)

Values on generators

\((508,1354)\) → \((e\left(\frac{1}{5}\right),e\left(\frac{133}{156}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 1859 }(15, a) \) \(-1\)\(1\)\(e\left(\frac{41}{780}\right)\)\(e\left(\frac{62}{195}\right)\)\(e\left(\frac{41}{390}\right)\)\(e\left(\frac{123}{260}\right)\)\(e\left(\frac{289}{780}\right)\)\(e\left(\frac{487}{780}\right)\)\(e\left(\frac{41}{260}\right)\)\(e\left(\frac{124}{195}\right)\)\(e\left(\frac{41}{78}\right)\)\(e\left(\frac{11}{26}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1859 }(15,a) \;\) at \(\;a = \) e.g. 2