Properties

Label 6001.3061
Modulus $6001$
Conductor $353$
Order $8$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6001, base_ring=CyclotomicField(8))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,7]))
 
pari: [g,chi] = znchar(Mod(3061,6001))
 

Basic properties

Modulus: \(6001\)
Conductor: \(353\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{353}(237,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6001.k

\(\chi_{6001}(3061,\cdot)\) \(\chi_{6001}(4166,\cdot)\) \(\chi_{6001}(4659,\cdot)\) \(\chi_{6001}(5764,\cdot)\)

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: \(\Q(\zeta_{8})\)
Fixed field: 8.8.683003513396280737.1

Values on generators

\((2825,3180)\) → \((1,e\left(\frac{7}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 6001 }(3061, a) \) \(1\)\(1\)\(-1\)\(e\left(\frac{7}{8}\right)\)\(1\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{5}{8}\right)\)\(-1\)\(-i\)\(e\left(\frac{7}{8}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6001 }(3061,a) \;\) at \(\;a = \) e.g. 2