Properties

Label 6001.999
Modulus $6001$
Conductor $6001$
Order $16$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6001, base_ring=CyclotomicField(16))
 
M = H._module
 
chi = DirichletCharacter(H, M([4,5]))
 
pari: [g,chi] = znchar(Mod(999,6001))
 

Basic properties

Modulus: \(6001\)
Conductor: \(6001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
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 6001.ba

\(\chi_{6001}(999,\cdot)\) \(\chi_{6001}(2435,\cdot)\) \(\chi_{6001}(2571,\cdot)\) \(\chi_{6001}(2724,\cdot)\) \(\chi_{6001}(2860,\cdot)\) \(\chi_{6001}(3226,\cdot)\) \(\chi_{6001}(3481,\cdot)\) \(\chi_{6001}(5708,\cdot)\)

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: \(\Q(\zeta_{16})\)
Fixed field: Number field defined by a degree 16 polynomial

Values on generators

\((2825,3180)\) → \((i,e\left(\frac{5}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 6001 }(999, a) \) \(1\)\(1\)\(-i\)\(e\left(\frac{9}{16}\right)\)\(-1\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{11}{16}\right)\)\(i\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{9}{16}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6001 }(999,a) \;\) at \(\;a = \) e.g. 2