Properties

Label 4029.614
Modulus $4029$
Conductor $4029$
Order $104$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4029, base_ring=CyclotomicField(104))
 
M = H._module
 
chi = DirichletCharacter(H, M([52,91,60]))
 
pari: [g,chi] = znchar(Mod(614,4029))
 

Basic properties

Modulus: \(4029\)
Conductor: \(4029\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(104\)
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 4029.ci

\(\chi_{4029}(185,\cdot)\) \(\chi_{4029}(308,\cdot)\) \(\chi_{4029}(410,\cdot)\) \(\chi_{4029}(491,\cdot)\) \(\chi_{4029}(614,\cdot)\) \(\chi_{4029}(644,\cdot)\) \(\chi_{4029}(665,\cdot)\) \(\chi_{4029}(689,\cdot)\) \(\chi_{4029}(848,\cdot)\) \(\chi_{4029}(926,\cdot)\) \(\chi_{4029}(1175,\cdot)\) \(\chi_{4029}(1199,\cdot)\) \(\chi_{4029}(1226,\cdot)\) \(\chi_{4029}(1256,\cdot)\) \(\chi_{4029}(1358,\cdot)\) \(\chi_{4029}(1436,\cdot)\) \(\chi_{4029}(1562,\cdot)\) \(\chi_{4029}(1607,\cdot)\) \(\chi_{4029}(1613,\cdot)\) \(\chi_{4029}(1844,\cdot)\) \(\chi_{4029}(1913,\cdot)\) \(\chi_{4029}(2066,\cdot)\) \(\chi_{4029}(2123,\cdot)\) \(\chi_{4029}(2150,\cdot)\) \(\chi_{4029}(2174,\cdot)\) \(\chi_{4029}(2270,\cdot)\) \(\chi_{4029}(2303,\cdot)\) \(\chi_{4029}(2348,\cdot)\) \(\chi_{4029}(2507,\cdot)\) \(\chi_{4029}(2678,\cdot)\) ...

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: $\Q(\zeta_{104})$
Fixed field: Number field defined by a degree 104 polynomial (not computed)

Values on generators

\((2687,3556,3163)\) → \((-1,e\left(\frac{7}{8}\right),e\left(\frac{15}{26}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 4029 }(614, a) \) \(1\)\(1\)\(e\left(\frac{3}{52}\right)\)\(e\left(\frac{3}{26}\right)\)\(e\left(\frac{67}{104}\right)\)\(e\left(\frac{21}{104}\right)\)\(e\left(\frac{9}{52}\right)\)\(e\left(\frac{73}{104}\right)\)\(e\left(\frac{89}{104}\right)\)\(e\left(\frac{3}{26}\right)\)\(e\left(\frac{27}{104}\right)\)\(e\left(\frac{3}{13}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4029 }(614,a) \;\) at \(\;a = \) e.g. 2