# Properties

 Label 1440.1297 Modulus $1440$ Conductor $40$ Order $4$ Real no Primitive no Minimal no Parity odd

# Related objects

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

sage: H = DirichletGroup(1440, base_ring=CyclotomicField(4))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,2,0,1]))

pari: [g,chi] = znchar(Mod(1297,1440))

## Basic properties

 Modulus: $$1440$$ Conductor: $$40$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$4$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{40}(37,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 1440.y

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: $$\Q(\sqrt{-1})$$ Fixed field: 4.0.8000.2

## Values on generators

$$(991,901,641,577)$$ → $$(1,-1,1,i)$$

## Values

 $$a$$ $$-1$$ $$1$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$29$$ $$31$$ $$37$$ $$41$$ $$\chi_{ 1440 }(1297, a)$$ $$-1$$ $$1$$ $$i$$ $$-1$$ $$i$$ $$i$$ $$1$$ $$-i$$ $$1$$ $$1$$ $$-i$$ $$1$$
sage: chi.jacobi_sum(n)

$$\chi_{ 1440 }(1297,a) \;$$ at $$\;a =$$ e.g. 2