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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(10020)
 
sage: chi = H[10019]
 
pari: [g,chi] = znchar(Mod(10019,10020))
 

Kronecker symbol representation

sage: kronecker_character(-10020)
 
pari: znchartokronecker(g,chi)
 

\(\displaystyle\left(\frac{-10020}{\bullet}\right)\)

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 10020
Conductor = 10020
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.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

\(\chi_{10020}(10019,\cdot)\)

Values on generators

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

\((673,1669,3341,5011)\) → \((1,-1,-1,-1)\)

First values

17111317192329313741434749535961677173777983899197101103107109
1111\(-1\)\(-1\)1\(-1\)\(-1\)11\(-1\)\(-1\)1\(-1\)\(-1\)1\(-1\)\(-1\)1111\(-1\)1\(-1\)1\(-1\)\(-1\)\(-1\)
value at  e.g. 2

Related number fields

Field of values \(\Q\)