Properties

Label 1339.102
Modulus $1339$
Conductor $1339$
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(1339, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([7,6]))
 
pari: [g,chi] = znchar(Mod(102,1339))
 

Basic properties

Modulus: \(1339\)
Conductor: \(1339\)
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 1339.bc

\(\chi_{1339}(102,\cdot)\) \(\chi_{1339}(514,\cdot)\) \(\chi_{1339}(617,\cdot)\) \(\chi_{1339}(1029,\cdot)\)

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: 12.12.2139933234248197407397573.1

Values on generators

\((1237,417)\) → \((e\left(\frac{7}{12}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1339 }(102, a) \) \(1\)\(1\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{6}\right)\)\(-i\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(-i\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{7}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1339 }(102,a) \;\) at \(\;a = \) e.g. 2