Properties

Label 8015.2608
Modulus $8015$
Conductor $8015$
Order $12$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8015, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([9,8,7]))
 
pari: [g,chi] = znchar(Mod(2608,8015))
 

Basic properties

Modulus: \(8015\)
Conductor: \(8015\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
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 8015.cc

\(\chi_{8015}(2608,\cdot)\) \(\chi_{8015}(3453,\cdot)\) \(\chi_{8015}(5707,\cdot)\) \(\chi_{8015}(7697,\cdot)\)

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: \(\Q(\zeta_{12})\)
Fixed field: Number field defined by a degree 12 polynomial

Values on generators

\((3207,4581,4586)\) → \((-i,e\left(\frac{2}{3}\right),e\left(\frac{7}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 8015 }(2608, a) \) \(1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(i\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{7}{12}\right)\)\(1\)\(-1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{11}{12}\right)\)\(-1\)\(e\left(\frac{1}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8015 }(2608,a) \;\) at \(\;a = \) e.g. 2