Properties

Conductor 10003
Order 2
Real Yes
Primitive Yes
Parity Odd

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(10003)
sage: chi = H[10002]
pari: [g,chi] = znchar(Mod(10002,10003))

Kronecker symbol representation

sage: kronecker_character(-10003)
pari: znchartokronecker(g,chi)

\(\displaystyle\left(\frac{-10003}{\bullet}\right)\)

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 10003
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 2
Real = Yes
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 = Odd

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 ]

\(\chi_{10003}(10002,\cdot)\)

Values on generators

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

\((2864,2859)\) → \((-1,-1)\)

First values

1234568910111213151617181920222324252627293031323334
1\(-1\)\(-1\)1\(-1\)1\(-1\)11\(-1\)\(-1\)\(-1\)11\(-1\)\(-1\)\(-1\)\(-1\)11111\(-1\)\(-1\)\(-1\)1\(-1\)11
value at  e.g. 2

Related number fields

Field of values \(\Q\)