Properties

Label 1859.7
Modulus $1859$
Conductor $1859$
Order $780$
Real no
Primitive yes
Minimal yes
Parity even

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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([546,535]))
 
pari: [g,chi] = znchar(Mod(7,1859))
 

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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1859.bv

\(\chi_{1859}(2,\cdot)\) \(\chi_{1859}(6,\cdot)\) \(\chi_{1859}(7,\cdot)\) \(\chi_{1859}(24,\cdot)\) \(\chi_{1859}(28,\cdot)\) \(\chi_{1859}(41,\cdot)\) \(\chi_{1859}(46,\cdot)\) \(\chi_{1859}(50,\cdot)\) \(\chi_{1859}(63,\cdot)\) \(\chi_{1859}(72,\cdot)\) \(\chi_{1859}(84,\cdot)\) \(\chi_{1859}(85,\cdot)\) \(\chi_{1859}(106,\cdot)\) \(\chi_{1859}(123,\cdot)\) \(\chi_{1859}(128,\cdot)\) \(\chi_{1859}(145,\cdot)\) \(\chi_{1859}(149,\cdot)\) \(\chi_{1859}(162,\cdot)\) \(\chi_{1859}(167,\cdot)\) \(\chi_{1859}(171,\cdot)\) \(\chi_{1859}(184,\cdot)\) \(\chi_{1859}(189,\cdot)\) \(\chi_{1859}(193,\cdot)\) \(\chi_{1859}(206,\cdot)\) \(\chi_{1859}(215,\cdot)\) \(\chi_{1859}(227,\cdot)\) \(\chi_{1859}(228,\cdot)\) \(\chi_{1859}(266,\cdot)\) \(\chi_{1859}(271,\cdot)\) \(\chi_{1859}(288,\cdot)\) ...

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

\((508,1354)\) → \((e\left(\frac{7}{10}\right),e\left(\frac{107}{156}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 1859 }(7, a) \) \(1\)\(1\)\(e\left(\frac{301}{780}\right)\)\(e\left(\frac{127}{195}\right)\)\(e\left(\frac{301}{390}\right)\)\(e\left(\frac{253}{260}\right)\)\(e\left(\frac{29}{780}\right)\)\(e\left(\frac{227}{780}\right)\)\(e\left(\frac{41}{260}\right)\)\(e\left(\frac{59}{195}\right)\)\(e\left(\frac{14}{39}\right)\)\(e\left(\frac{11}{26}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1859 }(7,a) \;\) at \(\;a = \) e.g. 2