# Properties

 Label 4410.1733 Modulus $4410$ Conductor $315$ Order $12$ Real no Primitive no Minimal no Parity even

# Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter

sage: H = DirichletGroup(4410)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([10,9,8]))

pari: [g,chi] = znchar(Mod(1733,4410))

## Basic properties

 Modulus: $$4410$$ Conductor: $$315$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$12$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{315}(158,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 4410.bu

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

$$(3431,2647,1081)$$ → $$(e\left(\frac{5}{6}\right),-i,e\left(\frac{2}{3}\right))$$

## Values

 $$-1$$ $$1$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$29$$ $$31$$ $$37$$ $$41$$ $$43$$ $$1$$ $$1$$ $$-1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $$\Q(\zeta_{12})$$ Fixed field: 12.12.4362113325015017578125.2