Properties

Label 6027.1718
Modulus $6027$
Conductor $6027$
Order $210$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6027, base_ring=CyclotomicField(210))
 
M = H._module
 
chi = DirichletCharacter(H, M([105,5,168]))
 
pari: [g,chi] = znchar(Mod(1718,6027))
 

Basic properties

Modulus: \(6027\)
Conductor: \(6027\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(210\)
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 6027.ej

\(\chi_{6027}(59,\cdot)\) \(\chi_{6027}(467,\cdot)\) \(\chi_{6027}(488,\cdot)\) \(\chi_{6027}(551,\cdot)\) \(\chi_{6027}(584,\cdot)\) \(\chi_{6027}(836,\cdot)\) \(\chi_{6027}(857,\cdot)\) \(\chi_{6027}(920,\cdot)\) \(\chi_{6027}(1076,\cdot)\) \(\chi_{6027}(1328,\cdot)\) \(\chi_{6027}(1349,\cdot)\) \(\chi_{6027}(1412,\cdot)\) \(\chi_{6027}(1445,\cdot)\) \(\chi_{6027}(1718,\cdot)\) \(\chi_{6027}(1781,\cdot)\) \(\chi_{6027}(1937,\cdot)\) \(\chi_{6027}(2189,\cdot)\) \(\chi_{6027}(2210,\cdot)\) \(\chi_{6027}(2306,\cdot)\) \(\chi_{6027}(2558,\cdot)\) \(\chi_{6027}(2642,\cdot)\) \(\chi_{6027}(2798,\cdot)\) \(\chi_{6027}(3050,\cdot)\) \(\chi_{6027}(3071,\cdot)\) \(\chi_{6027}(3134,\cdot)\) \(\chi_{6027}(3419,\cdot)\) \(\chi_{6027}(3440,\cdot)\) \(\chi_{6027}(3503,\cdot)\) \(\chi_{6027}(3659,\cdot)\) \(\chi_{6027}(3911,\cdot)\) ...

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: $\Q(\zeta_{105})$
Fixed field: Number field defined by a degree 210 polynomial (not computed)

Values on generators

\((4019,493,2794)\) → \((-1,e\left(\frac{1}{42}\right),e\left(\frac{4}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 6027 }(1718, a) \) \(1\)\(1\)\(e\left(\frac{193}{210}\right)\)\(e\left(\frac{88}{105}\right)\)\(e\left(\frac{83}{105}\right)\)\(e\left(\frac{53}{70}\right)\)\(e\left(\frac{149}{210}\right)\)\(e\left(\frac{179}{210}\right)\)\(e\left(\frac{41}{70}\right)\)\(e\left(\frac{71}{105}\right)\)\(e\left(\frac{52}{105}\right)\)\(e\left(\frac{1}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6027 }(1718,a) \;\) at \(\;a = \) e.g. 2