Properties

Conductor 10009
Order 3
Real No
Primitive Yes
Parity Even

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(10009)
sage: chi = H[1044]
pari: [g,chi] = znchar(Mod(1044,10009))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 10009
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 3
Real = No
sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
Primitive = Yes
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Even

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 ]

\(\chi_{10009}(1044,\cdot)\) \(\chi_{10009}(8964,\cdot)\)

Values on generators

sage: chi(k) for k in H.gens()
pari: [ chareval(g,chi,x) | x <- g.gen ] \\ value in Q/Z

\(11\) → \(e\left(\frac{2}{3}\right)\)

First values

123456789101112131415161718192021222324252627282930
11\(e\left(\frac{2}{3}\right)\)1\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)11\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{2}{3}\right)\)1111\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{3}\right)\)1\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{2}{3}\right)\)111\(e\left(\frac{2}{3}\right)\)1
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{3})\)