# Properties

 Label 1148.65 Modulus $1148$ Conductor $287$ 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(1148, base_ring=CyclotomicField(120))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,40,39]))

pari: [g,chi] = znchar(Mod(65,1148))

## Basic properties

 Modulus: $$1148$$ Conductor: $$287$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$120$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{287}(65,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 1148.ci

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

$$(575,493,785)$$ → $$(1,e\left(\frac{1}{3}\right),e\left(\frac{13}{40}\right))$$

## Values

 $$-1$$ $$1$$ $$3$$ $$5$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$23$$ $$25$$ $$-1$$ $$1$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{49}{60}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{37}{120}\right)$$ $$e\left(\frac{3}{40}\right)$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{7}{120}\right)$$ $$e\left(\frac{71}{120}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{19}{30}\right)$$
 value at e.g. 2