Properties

Conductor 720
Order 12
Real No
Primitive No
Parity Odd
Orbit Label 3600.dh

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(3600)
 
sage: chi = H[493]
 
pari: [g,chi] = znchar(Mod(493,3600))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 720
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 12
Real = No
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = No
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = Odd
Orbit label = 3600.dh
Orbit index = 86

Galois orbit

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

\(\chi_{3600}(493,\cdot)\) \(\chi_{3600}(1957,\cdot)\) \(\chi_{3600}(2893,\cdot)\) \(\chi_{3600}(3157,\cdot)\)

Inducing primitive character

\(\chi_{720}(493,\cdot)\)

Values on generators

\((3151,901,2801,577)\) → \((1,-i,e\left(\frac{2}{3}\right),-i)\)

Values

-117111317192329313741
\(-1\)\(1\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(-i\)\(-i\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{3}\right)\)\(-1\)\(e\left(\frac{5}{6}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{12})\)