# Properties

 Label 2009.900 Modulus $2009$ Conductor $287$ Order $60$ 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(2009, base_ring=CyclotomicField(60))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([40,9]))

pari: [g,chi] = znchar(Mod(900,2009))

## Basic properties

 Modulus: $$2009$$ Conductor: $$287$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$60$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{287}(39,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 2009.bp

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

## Related number fields

 Field of values: $$\Q(\zeta_{60})$$ Fixed field: Number field defined by a degree 60 polynomial

## Values on generators

$$(493,785)$$ → $$(e\left(\frac{2}{3}\right),e\left(\frac{3}{20}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$8$$ $$9$$ $$10$$ $$11$$ $$12$$ $$1$$ $$1$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{7}{60}\right)$$ $$e\left(\frac{23}{60}\right)$$
 value at e.g. 2