Properties

Label 1339.355
Modulus $1339$
Conductor $1339$
Order $6$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1339, base_ring=CyclotomicField(6))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1,4]))
 
pari: [g,chi] = znchar(Mod(355,1339))
 

Basic properties

Modulus: \(1339\)
Conductor: \(1339\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(6\)
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.k

\(\chi_{1339}(355,\cdot)\) \(\chi_{1339}(777,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\((1237,417)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{2}{3}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(-1\)\(e\left(\frac{2}{3}\right)\)\(1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{6}\right)\)\(-1\)\(-1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{6}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\sqrt{-3}) \)
Fixed field: 6.6.41789354259133.1