# Properties

 Label 3645.487 Modulus $3645$ Conductor $45$ Order $12$ Real no Primitive no Minimal no Parity odd

# Related objects

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3645, base_ring=CyclotomicField(12))

M = H._module

chi = DirichletCharacter(H, M([8,3]))

pari: [g,chi] = znchar(Mod(487,3645))

## Basic properties

 Modulus: $$3645$$ Conductor: $$45$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$12$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{45}(7,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 3645.l

sage: chi.galois_orbit()

order = charorder(g,chi)

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

## Related number fields

 Field of values: $$\Q(\zeta_{12})$$ Fixed field: 12.0.84075626953125.1

## Values on generators

$$(731,2917)$$ → $$(e\left(\frac{2}{3}\right),i)$$

## First values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$4$$ $$7$$ $$8$$ $$11$$ $$13$$ $$14$$ $$16$$ $$17$$ $$19$$ $$\chi_{ 3645 }(487, a)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$-i$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$i$$ $$-1$$
sage: chi.jacobi_sum(n)

$$\chi_{ 3645 }(487,a) \;$$ at $$\;a =$$ e.g. 2