Properties

Label 5269.3
Modulus $5269$
Conductor $5269$
Order $1195$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5269, base_ring=CyclotomicField(2390))
 
M = H._module
 
chi = DirichletCharacter(H, M([1912,2230]))
 
pari: [g,chi] = znchar(Mod(3,5269))
 

Basic properties

Modulus: \(5269\)
Conductor: \(5269\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(1195\)
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 5269.m

\(\chi_{5269}(3,\cdot)\) \(\chi_{5269}(4,\cdot)\) \(\chi_{5269}(5,\cdot)\) \(\chi_{5269}(9,\cdot)\) \(\chi_{5269}(14,\cdot)\) \(\chi_{5269}(15,\cdot)\) \(\chi_{5269}(16,\cdot)\) \(\chi_{5269}(20,\cdot)\) \(\chi_{5269}(25,\cdot)\) \(\chi_{5269}(27,\cdot)\) \(\chi_{5269}(36,\cdot)\) \(\chi_{5269}(42,\cdot)\) \(\chi_{5269}(48,\cdot)\) \(\chi_{5269}(49,\cdot)\) \(\chi_{5269}(60,\cdot)\) \(\chi_{5269}(64,\cdot)\) \(\chi_{5269}(69,\cdot)\) \(\chi_{5269}(70,\cdot)\) \(\chi_{5269}(71,\cdot)\) \(\chi_{5269}(75,\cdot)\) \(\chi_{5269}(80,\cdot)\) \(\chi_{5269}(81,\cdot)\) \(\chi_{5269}(92,\cdot)\) \(\chi_{5269}(97,\cdot)\) \(\chi_{5269}(103,\cdot)\) \(\chi_{5269}(108,\cdot)\) \(\chi_{5269}(115,\cdot)\) \(\chi_{5269}(125,\cdot)\) \(\chi_{5269}(126,\cdot)\) \(\chi_{5269}(135,\cdot)\) ...

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: $\Q(\zeta_{1195})$
Fixed field: Number field defined by a degree 1195 polynomial (not computed)

Values on generators

\((959,4324)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{223}{239}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 5269 }(3, a) \) \(1\)\(1\)\(e\left(\frac{676}{1195}\right)\)\(e\left(\frac{648}{1195}\right)\)\(e\left(\frac{157}{1195}\right)\)\(e\left(\frac{274}{1195}\right)\)\(e\left(\frac{129}{1195}\right)\)\(e\left(\frac{92}{1195}\right)\)\(e\left(\frac{833}{1195}\right)\)\(e\left(\frac{101}{1195}\right)\)\(e\left(\frac{190}{239}\right)\)\(e\left(\frac{161}{239}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5269 }(3,a) \;\) at \(\;a = \) e.g. 2