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Show commands for: Pari/GP / SageMath
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(22050)
 
sage: chi = H[1]
 
pari: [g,chi] = znchar(Mod(1,22050))
 

Basic properties

Modulus: \(22050\)
Conductor: \(1\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(1\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: yes
Primitive: no, induced from \(\chi_{1}(1,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 22050.None

\(\chi_{22050}(1,\cdot)\)

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

Values on generators

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

\((14213,4801,9899)\) → \((1,1,1)\)

First values

\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)\(47\)\(53\)\(59\)\(61\)\(67\)\(71\)\(73\)\(79\)\(83\)\(89\)\(97\)\(101\)\(103\)\(107\)\(109\)\(113\)\(121\)\(127\)\(131\)
111111111111111111111111111111
value at e.g. 2

Related number fields

Field of values: \(\Q\)