Properties

Label 1849.4
Modulus $1849$
Conductor $1849$
Order $301$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1849, base_ring=CyclotomicField(602))
 
M = H._module
 
chi = DirichletCharacter(H, M([424]))
 
pari: [g,chi] = znchar(Mod(4,1849))
 

Basic properties

Modulus: \(1849\)
Conductor: \(1849\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(301\)
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 1849.m

\(\chi_{1849}(4,\cdot)\) \(\chi_{1849}(11,\cdot)\) \(\chi_{1849}(16,\cdot)\) \(\chi_{1849}(21,\cdot)\) \(\chi_{1849}(35,\cdot)\) \(\chi_{1849}(41,\cdot)\) \(\chi_{1849}(47,\cdot)\) \(\chi_{1849}(54,\cdot)\) \(\chi_{1849}(59,\cdot)\) \(\chi_{1849}(64,\cdot)\) \(\chi_{1849}(84,\cdot)\) \(\chi_{1849}(90,\cdot)\) \(\chi_{1849}(97,\cdot)\) \(\chi_{1849}(102,\cdot)\) \(\chi_{1849}(107,\cdot)\) \(\chi_{1849}(121,\cdot)\) \(\chi_{1849}(127,\cdot)\) \(\chi_{1849}(133,\cdot)\) \(\chi_{1849}(140,\cdot)\) \(\chi_{1849}(145,\cdot)\) \(\chi_{1849}(150,\cdot)\) \(\chi_{1849}(164,\cdot)\) \(\chi_{1849}(170,\cdot)\) \(\chi_{1849}(176,\cdot)\) \(\chi_{1849}(183,\cdot)\) \(\chi_{1849}(188,\cdot)\) \(\chi_{1849}(193,\cdot)\) \(\chi_{1849}(207,\cdot)\) \(\chi_{1849}(213,\cdot)\) \(\chi_{1849}(219,\cdot)\) ...

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: $\Q(\zeta_{301})$
Fixed field: Number field defined by a degree 301 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{212}{301}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1849 }(4, a) \) \(1\)\(1\)\(e\left(\frac{285}{301}\right)\)\(e\left(\frac{212}{301}\right)\)\(e\left(\frac{269}{301}\right)\)\(e\left(\frac{281}{301}\right)\)\(e\left(\frac{28}{43}\right)\)\(e\left(\frac{24}{43}\right)\)\(e\left(\frac{253}{301}\right)\)\(e\left(\frac{123}{301}\right)\)\(e\left(\frac{265}{301}\right)\)\(e\left(\frac{270}{301}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1849 }(4,a) \;\) at \(\;a = \) e.g. 2