Properties

Label 6039.2809
Modulus $6039$
Conductor $671$
Order $10$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6039, base_ring=CyclotomicField(10))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,2,1]))
 
pari: [g,chi] = znchar(Mod(2809,6039))
 

Basic properties

Modulus: \(6039\)
Conductor: \(671\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(10\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{671}(125,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6039.dj

\(\chi_{6039}(2809,\cdot)\) \(\chi_{6039}(3457,\cdot)\) \(\chi_{6039}(5212,\cdot)\) \(\chi_{6039}(5725,\cdot)\)

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: \(\Q(\zeta_{5})\)
Fixed field: 10.10.2506744070710448583356221.4

Values on generators

\((1343,5491,5248)\) → \((1,e\left(\frac{1}{5}\right),e\left(\frac{1}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(17\)
\( \chi_{ 6039 }(2809, a) \) \(1\)\(1\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(1\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6039 }(2809,a) \;\) at \(\;a = \) e.g. 2