Properties

 Conductor 4000 Order 200 Real no Primitive yes Minimal yes Parity odd Orbit label 4000.cz

Related objects

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed

sage: H = DirichletGroup_conrey(4000)

sage: chi = H[773]

pari: [g,chi] = znchar(Mod(773,4000))

Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Conductor = 4000 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 200 Real = no sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = yes Minimal = yes sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = odd Orbit label = 4000.cz Orbit index = 78

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 ]

Values on generators

$$(2751,2501,1377)$$ → $$(1,e\left(\frac{1}{8}\right),e\left(\frac{31}{100}\right))$$

Values

 -1 1 3 7 9 11 13 17 19 21 23 27 $$-1$$ $$1$$ $$e\left(\frac{109}{200}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{9}{100}\right)$$ $$e\left(\frac{37}{200}\right)$$ $$e\left(\frac{193}{200}\right)$$ $$e\left(\frac{13}{100}\right)$$ $$e\left(\frac{91}{200}\right)$$ $$e\left(\frac{29}{200}\right)$$ $$e\left(\frac{9}{25}\right)$$ $$e\left(\frac{127}{200}\right)$$
value at  e.g. 2

Related number fields

 Field of values $$\Q(\zeta_{200})$$