Properties

Label 8015.1496
Modulus $8015$
Conductor $1603$
Order $12$
Real no
Primitive no
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([0,10,3]))
 
pari: [g,chi] = znchar(Mod(1496,8015))
 

Basic properties

Modulus: \(8015\)
Conductor: \(1603\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1603}(1496,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8015.dj

\(\chi_{8015}(1496,\cdot)\) \(\chi_{8015}(3771,\cdot)\) \(\chi_{8015}(4931,\cdot)\) \(\chi_{8015}(7206,\cdot)\)

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

\((3207,4581,4586)\) → \((1,e\left(\frac{5}{6}\right),i)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 8015 }(1496, a) \) \(1\)\(1\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(-i\)\(-i\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(-i\)\(e\left(\frac{2}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8015 }(1496,a) \;\) at \(\;a = \) e.g. 2