# Properties

 Label 1200.257 Modulus $1200$ Conductor $15$ Order $4$ Real no Primitive no Minimal no Parity even

# Related objects

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

H = DirichletGroup(1200, base_ring=CyclotomicField(4))

M = H._module

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

pari: [g,chi] = znchar(Mod(257,1200))

## Basic properties

 Modulus: $$1200$$ Conductor: $$15$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$4$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{15}(2,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 1200.v

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: $$\mathbb{Q}(i)$$ Fixed field: $$\Q(\zeta_{15})^+$$

## Values on generators

$$(751,901,401,577)$$ → $$(1,1,-1,i)$$

## First values

 $$a$$ $$-1$$ $$1$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$29$$ $$31$$ $$37$$ $$41$$ $$\chi_{ 1200 }(257, a)$$ $$1$$ $$1$$ $$i$$ $$-1$$ $$-i$$ $$-i$$ $$-1$$ $$i$$ $$1$$ $$1$$ $$i$$ $$-1$$
sage: chi.jacobi_sum(n)

$$\chi_{ 1200 }(257,a) \;$$ at $$\;a =$$ e.g. 2