# Properties

 Label 4729.17 Modulus $4729$ Conductor $4729$ Order $4728$ Real no Primitive yes Minimal yes Parity odd

# Related objects

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

H = DirichletGroup(4729, base_ring=CyclotomicField(4728))

M = H._module

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

pari: [g,chi] = znchar(Mod(17,4729))

## Basic properties

 Modulus: $$4729$$ Conductor: $$4729$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$4728$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 4729.p

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: $\Q(\zeta_{4728})$ Fixed field: Number field defined by a degree 4728 polynomial (not computed)

## Values on generators

$$17$$ → $$e\left(\frac{1}{4728}\right)$$

## First values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$ $$\chi_{ 4729 }(17, a)$$ $$-1$$ $$1$$ $$e\left(\frac{53}{788}\right)$$ $$e\left(\frac{685}{788}\right)$$ $$e\left(\frac{53}{394}\right)$$ $$e\left(\frac{299}{2364}\right)$$ $$e\left(\frac{369}{394}\right)$$ $$e\left(\frac{1829}{2364}\right)$$ $$e\left(\frac{159}{788}\right)$$ $$e\left(\frac{291}{394}\right)$$ $$e\left(\frac{229}{1182}\right)$$ $$e\left(\frac{1453}{1576}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 4729 }(17,a) \;$$ at $$\;a =$$ e.g. 2