Properties

Label 1849.36
Modulus $1849$
Conductor $1849$
Order $129$
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(258))
 
M = H._module
 
chi = DirichletCharacter(H, M([182]))
 
pari: [g,chi] = znchar(Mod(36,1849))
 

Basic properties

Modulus: \(1849\)
Conductor: \(1849\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(129\)
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.k

\(\chi_{1849}(6,\cdot)\) \(\chi_{1849}(36,\cdot)\) \(\chi_{1849}(49,\cdot)\) \(\chi_{1849}(79,\cdot)\) \(\chi_{1849}(92,\cdot)\) \(\chi_{1849}(122,\cdot)\) \(\chi_{1849}(135,\cdot)\) \(\chi_{1849}(165,\cdot)\) \(\chi_{1849}(178,\cdot)\) \(\chi_{1849}(208,\cdot)\) \(\chi_{1849}(221,\cdot)\) \(\chi_{1849}(251,\cdot)\) \(\chi_{1849}(264,\cdot)\) \(\chi_{1849}(294,\cdot)\) \(\chi_{1849}(307,\cdot)\) \(\chi_{1849}(337,\cdot)\) \(\chi_{1849}(350,\cdot)\) \(\chi_{1849}(380,\cdot)\) \(\chi_{1849}(393,\cdot)\) \(\chi_{1849}(436,\cdot)\) \(\chi_{1849}(466,\cdot)\) \(\chi_{1849}(479,\cdot)\) \(\chi_{1849}(509,\cdot)\) \(\chi_{1849}(522,\cdot)\) \(\chi_{1849}(552,\cdot)\) \(\chi_{1849}(565,\cdot)\) \(\chi_{1849}(595,\cdot)\) \(\chi_{1849}(608,\cdot)\) \(\chi_{1849}(638,\cdot)\) \(\chi_{1849}(651,\cdot)\) ...

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: $\Q(\zeta_{129})$
Fixed field: Number field defined by a degree 129 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{91}{129}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1849 }(36, a) \) \(1\)\(1\)\(e\left(\frac{28}{43}\right)\)\(e\left(\frac{91}{129}\right)\)\(e\left(\frac{13}{43}\right)\)\(e\left(\frac{19}{129}\right)\)\(e\left(\frac{46}{129}\right)\)\(e\left(\frac{107}{129}\right)\)\(e\left(\frac{41}{43}\right)\)\(e\left(\frac{53}{129}\right)\)\(e\left(\frac{103}{129}\right)\)\(e\left(\frac{22}{43}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1849 }(36,a) \;\) at \(\;a = \) e.g. 2