Properties

Label 6027.2638
Modulus $6027$
Conductor $2009$
Order $56$
Real no
Primitive no
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([0,20,35]))
 
pari: [g,chi] = znchar(Mod(2638,6027))
 

Basic properties

Modulus: \(6027\)
Conductor: \(2009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(56\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2009}(629,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6027.dc

\(\chi_{6027}(55,\cdot)\) \(\chi_{6027}(202,\cdot)\) \(\chi_{6027}(454,\cdot)\) \(\chi_{6027}(601,\cdot)\) \(\chi_{6027}(916,\cdot)\) \(\chi_{6027}(1063,\cdot)\) \(\chi_{6027}(1315,\cdot)\) \(\chi_{6027}(1462,\cdot)\) \(\chi_{6027}(1777,\cdot)\) \(\chi_{6027}(1924,\cdot)\) \(\chi_{6027}(2176,\cdot)\) \(\chi_{6027}(2323,\cdot)\) \(\chi_{6027}(2638,\cdot)\) \(\chi_{6027}(2785,\cdot)\) \(\chi_{6027}(3499,\cdot)\) \(\chi_{6027}(3646,\cdot)\) \(\chi_{6027}(3898,\cdot)\) \(\chi_{6027}(4045,\cdot)\) \(\chi_{6027}(4759,\cdot)\) \(\chi_{6027}(4906,\cdot)\) \(\chi_{6027}(5221,\cdot)\) \(\chi_{6027}(5368,\cdot)\) \(\chi_{6027}(5620,\cdot)\) \(\chi_{6027}(5767,\cdot)\)

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

\((4019,493,2794)\) → \((1,e\left(\frac{5}{14}\right),e\left(\frac{5}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 6027 }(2638, a) \) \(1\)\(1\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{3}{28}\right)\)\(e\left(\frac{17}{28}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{9}{56}\right)\)\(e\left(\frac{9}{56}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{31}{56}\right)\)\(e\left(\frac{1}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6027 }(2638,a) \;\) at \(\;a = \) e.g. 2