Properties

Label 6039.236
Modulus $6039$
Conductor $6039$
Order $60$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6039, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([10,24,33]))
 
pari: [g,chi] = znchar(Mod(236,6039))
 

Basic properties

Modulus: \(6039\)
Conductor: \(6039\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
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 6039.oc

\(\chi_{6039}(236,\cdot)\) \(\chi_{6039}(1004,\cdot)\) \(\chi_{6039}(1379,\cdot)\) \(\chi_{6039}(1670,\cdot)\) \(\chi_{6039}(1868,\cdot)\) \(\chi_{6039}(3017,\cdot)\) \(\chi_{6039}(3083,\cdot)\) \(\chi_{6039}(3623,\cdot)\) \(\chi_{6039}(3668,\cdot)\) \(\chi_{6039}(3683,\cdot)\) \(\chi_{6039}(3881,\cdot)\) \(\chi_{6039}(4262,\cdot)\) \(\chi_{6039}(5096,\cdot)\) \(\chi_{6039}(5405,\cdot)\) \(\chi_{6039}(5636,\cdot)\) \(\chi_{6039}(5681,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((1343,5491,5248)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{2}{5}\right),e\left(\frac{11}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(17\)
\( \chi_{ 6039 }(236, a) \) \(1\)\(1\)\(e\left(\frac{7}{60}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{19}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6039 }(236,a) \;\) at \(\;a = \) e.g. 2