Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(10003)
 
sage: chi = H[7809]
 
pari: [g,chi] = znchar(Mod(7809,10003))
 

Basic properties

Modulus: \(10003\)
Conductor: \(10003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(3\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
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 10003.None

\(\chi_{10003}(7809,\cdot)\) \(\chi_{10003}(2193,\cdot)\)

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
 

\((2864,2859)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{2}{3}\right))\)

First values

\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)\(13\)\(15\)\(16\)\(17\)\(18\)\(19\)\(20\)\(22\)\(23\)\(24\)\(25\)\(26\)\(27\)\(29\)\(30\)\(31\)\(32\)\(33\)\(34\)
11\(e\left(\frac{1}{3}\right)\)1\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{3}\right)\)1\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{2}{3}\right)\)11\(e\left(\frac{2}{3}\right)\)1\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{2}{3}\right)\)1\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)1\(e\left(\frac{2}{3}\right)\)1
value at e.g. 2

Related number fields

Field of values: \(\Q(\sqrt{-3}) \)