# Properties

 Label 4729.3838 Modulus $4729$ Conductor $4729$ Order $1576$ 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(1576))

M = H._module

chi = DirichletCharacter(H, M([1417]))

pari: [g,chi] = znchar(Mod(3838,4729))

## Basic properties

 Modulus: $$4729$$ Conductor: $$4729$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$1576$$ 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.n

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: $\Q(\zeta_{1576})$ Fixed field: Number field defined by a degree 1576 polynomial (not computed)

## Values on generators

$$17$$ → $$e\left(\frac{1417}{1576}\right)$$

## First values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$ $$\chi_{ 4729 }(3838, a)$$ $$-1$$ $$1$$ $$e\left(\frac{723}{788}\right)$$ $$e\left(\frac{275}{788}\right)$$ $$e\left(\frac{329}{394}\right)$$ $$e\left(\frac{527}{788}\right)$$ $$e\left(\frac{105}{394}\right)$$ $$e\left(\frac{749}{788}\right)$$ $$e\left(\frac{593}{788}\right)$$ $$e\left(\frac{275}{394}\right)$$ $$e\left(\frac{231}{394}\right)$$ $$e\left(\frac{359}{1576}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 4729 }(3838,a) \;$$ at $$\;a =$$ e.g. 2