# Properties

 Label 5185.331 Modulus $5185$ Conductor $1037$ Order $120$ Real no Primitive no Minimal yes Parity odd

# Related objects

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

sage: H = DirichletGroup(5185, base_ring=CyclotomicField(120))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,75,82]))

pari: [g,chi] = znchar(Mod(331,5185))

## Basic properties

 Modulus: $$5185$$ Conductor: $$1037$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$120$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{1037}(331,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 5185.jj

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: $\Q(\zeta_{120})$ Fixed field: Number field defined by a degree 120 polynomial (not computed)

## Values on generators

$$(3112,4576,2381)$$ → $$(1,e\left(\frac{5}{8}\right),e\left(\frac{41}{60}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$7$$ $$8$$ $$9$$ $$11$$ $$12$$ $$13$$ $$-1$$ $$1$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{29}{40}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{19}{120}\right)$$ $$e\left(\frac{43}{120}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{71}{120}\right)$$ $$e\left(\frac{5}{6}\right)$$
 value at e.g. 2