Properties

Label 1260.419
Modulus $1260$
Conductor $1260$
Order $6$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1260, base_ring=CyclotomicField(6))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([3,5,3,3]))
 
pari: [g,chi] = znchar(Mod(419,1260))
 

Basic properties

Modulus: \(1260\)
Conductor: \(1260\)
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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1260.bp

\(\chi_{1260}(419,\cdot)\) \(\chi_{1260}(839,\cdot)\)

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

Related number fields

Field of values: \(\Q(\sqrt{-3}) \)
Fixed field: 6.0.54010152000.17

Values on generators

\((631,281,757,1081)\) → \((-1,e\left(\frac{5}{6}\right),-1,-1)\)

Values

\(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(-1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(-1\)\(1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(-1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1260 }(419,a) \;\) at \(\;a = \) e.g. 2