Properties

Label 6027.806
Modulus $6027$
Conductor $6027$
Order $56$
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(56))
 
M = H._module
 
chi = DirichletCharacter(H, M([28,32,7]))
 
pari: [g,chi] = znchar(Mod(806,6027))
 

Basic properties

Modulus: \(6027\)
Conductor: \(6027\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(56\)
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.cz

\(\chi_{6027}(260,\cdot)\) \(\chi_{6027}(407,\cdot)\) \(\chi_{6027}(659,\cdot)\) \(\chi_{6027}(806,\cdot)\) \(\chi_{6027}(1121,\cdot)\) \(\chi_{6027}(1268,\cdot)\) \(\chi_{6027}(1982,\cdot)\) \(\chi_{6027}(2129,\cdot)\) \(\chi_{6027}(2381,\cdot)\) \(\chi_{6027}(2528,\cdot)\) \(\chi_{6027}(3242,\cdot)\) \(\chi_{6027}(3389,\cdot)\) \(\chi_{6027}(3704,\cdot)\) \(\chi_{6027}(3851,\cdot)\) \(\chi_{6027}(4103,\cdot)\) \(\chi_{6027}(4250,\cdot)\) \(\chi_{6027}(4565,\cdot)\) \(\chi_{6027}(4712,\cdot)\) \(\chi_{6027}(4964,\cdot)\) \(\chi_{6027}(5111,\cdot)\) \(\chi_{6027}(5426,\cdot)\) \(\chi_{6027}(5573,\cdot)\) \(\chi_{6027}(5825,\cdot)\) \(\chi_{6027}(5972,\cdot)\)

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: $\Q(\zeta_{56})$
Fixed field: Number field defined by a degree 56 polynomial

Values on generators

\((4019,493,2794)\) → \((-1,e\left(\frac{4}{7}\right),e\left(\frac{1}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 6027 }(806, a) \) \(1\)\(1\)\(e\left(\frac{17}{28}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{23}{28}\right)\)\(e\left(\frac{23}{28}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{41}{56}\right)\)\(e\left(\frac{41}{56}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{51}{56}\right)\)\(e\left(\frac{1}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6027 }(806,a) \;\) at \(\;a = \) e.g. 2