Properties

Conductor 360
Order 12
Real No
Primitive No
Parity Odd
Orbit Label 3600.dl

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[407]
pari: [g,chi] = znchar(Mod(407,3600))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 360
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.dl
Orbit index = 90

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 ]

\(\chi_{3600}(407,\cdot)\) \(\chi_{3600}(743,\cdot)\) \(\chi_{3600}(1607,\cdot)\) \(\chi_{3600}(3143,\cdot)\)

Inducing primitive character

\(\chi_{360}(227,\cdot)\)

Values on generators

\((3151,901,2801,577)\) → \((-1,-1,e\left(\frac{1}{6}\right),i)\)

Values

-117111317192329313741
\(-1\)\(1\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{7}{12}\right)\)\(-i\)\(-1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(-i\)\(e\left(\frac{5}{6}\right)\)
value at  e.g. 2

Related number fields

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