# Properties

 Conductor 8041 Order 336 Real No Primitive Yes Parity Even Orbit Label 8041.fl

# 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(8041)

sage: chi = H[10]

pari: [g,chi] = znchar(Mod(10,8041))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Conductor = 8041 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 336 Real = No 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 = Even Orbit label = 8041.fl Orbit index = 142

## 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

$$(6580,2366,562)$$ → $$(-1,e\left(\frac{3}{16}\right),e\left(\frac{5}{21}\right))$$

## Values

 -1 1 2 3 4 5 6 7 8 9 10 12 $$1$$ $$1$$ $$e\left(\frac{31}{56}\right)$$ $$e\left(\frac{143}{336}\right)$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{299}{336}\right)$$ $$e\left(\frac{47}{48}\right)$$ $$e\left(\frac{43}{48}\right)$$ $$e\left(\frac{37}{56}\right)$$ $$e\left(\frac{143}{168}\right)$$ $$e\left(\frac{149}{336}\right)$$ $$e\left(\frac{179}{336}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{336})$$