# Properties

 Label 1480.573 Modulus $1480$ Conductor $1480$ Order $36$ Real no Primitive yes Minimal yes Parity even

# Related objects

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

H = DirichletGroup(1480, base_ring=CyclotomicField(36))

M = H._module

chi = DirichletCharacter(H, M([0,18,27,17]))

pari: [g,chi] = znchar(Mod(573,1480))

## Basic properties

 Modulus: $$1480$$ Conductor: $$1480$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$36$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 1480.ds

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: $$\Q(\zeta_{36})$$ Fixed field: Number field defined by a degree 36 polynomial

## Values on generators

$$(1111,741,297,1001)$$ → $$(1,-1,-i,e\left(\frac{17}{36}\right))$$

## First values

 $$a$$ $$-1$$ $$1$$ $$3$$ $$7$$ $$9$$ $$11$$ $$13$$ $$17$$ $$19$$ $$21$$ $$23$$ $$27$$ $$\chi_{ 1480 }(573, a)$$ $$1$$ $$1$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 1480 }(573,a) \;$$ at $$\;a =$$ e.g. 2