Properties

Label 6336.119
Modulus $6336$
Conductor $3168$
Order $120$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6336, base_ring=CyclotomicField(120))
 
M = H._module
 
chi = DirichletCharacter(H, M([60,45,20,72]))
 
pari: [g,chi] = znchar(Mod(119,6336))
 

Basic properties

Modulus: \(6336\)
Conductor: \(3168\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(120\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{3168}(515,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6336.fo

\(\chi_{6336}(119,\cdot)\) \(\chi_{6336}(311,\cdot)\) \(\chi_{6336}(455,\cdot)\) \(\chi_{6336}(599,\cdot)\) \(\chi_{6336}(839,\cdot)\) \(\chi_{6336}(983,\cdot)\) \(\chi_{6336}(1127,\cdot)\) \(\chi_{6336}(1175,\cdot)\) \(\chi_{6336}(1703,\cdot)\) \(\chi_{6336}(1895,\cdot)\) \(\chi_{6336}(2039,\cdot)\) \(\chi_{6336}(2183,\cdot)\) \(\chi_{6336}(2423,\cdot)\) \(\chi_{6336}(2567,\cdot)\) \(\chi_{6336}(2711,\cdot)\) \(\chi_{6336}(2759,\cdot)\) \(\chi_{6336}(3287,\cdot)\) \(\chi_{6336}(3479,\cdot)\) \(\chi_{6336}(3623,\cdot)\) \(\chi_{6336}(3767,\cdot)\) \(\chi_{6336}(4007,\cdot)\) \(\chi_{6336}(4151,\cdot)\) \(\chi_{6336}(4295,\cdot)\) \(\chi_{6336}(4343,\cdot)\) \(\chi_{6336}(4871,\cdot)\) \(\chi_{6336}(5063,\cdot)\) \(\chi_{6336}(5207,\cdot)\) \(\chi_{6336}(5351,\cdot)\) \(\chi_{6336}(5591,\cdot)\) \(\chi_{6336}(5735,\cdot)\) ...

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: $\Q(\zeta_{120})$
Fixed field: Number field defined by a degree 120 polynomial (not computed)

Values on generators

\((4159,4357,3521,1729)\) → \((-1,e\left(\frac{3}{8}\right),e\left(\frac{1}{6}\right),e\left(\frac{3}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 6336 }(119, a) \) \(1\)\(1\)\(e\left(\frac{73}{120}\right)\)\(e\left(\frac{7}{60}\right)\)\(e\left(\frac{67}{120}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{37}{40}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{59}{120}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{29}{40}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6336 }(119,a) \;\) at \(\;a = \) e.g. 2