# Properties

 Label 1444.39 Modulus $1444$ Conductor $1444$ Order $38$ Real no Primitive yes Minimal yes Parity odd

# Related objects

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

H = DirichletGroup(1444, base_ring=CyclotomicField(38))

M = H._module

chi = DirichletCharacter(H, M([19,14]))

pari: [g,chi] = znchar(Mod(39,1444))

## Basic properties

 Modulus: $$1444$$ Conductor: $$1444$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$38$$ 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 1444.p

sage: chi.galois_orbit()

order = charorder(g,chi)

[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

## Values on generators

$$(723,1085)$$ → $$(-1,e\left(\frac{7}{19}\right))$$

## First values

 $$a$$ $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$21$$ $$23$$ $$\chi_{ 1444 }(39, a)$$ $$-1$$ $$1$$ $$e\left(\frac{27}{38}\right)$$ $$e\left(\frac{9}{19}\right)$$ $$e\left(\frac{29}{38}\right)$$ $$e\left(\frac{8}{19}\right)$$ $$e\left(\frac{3}{38}\right)$$ $$e\left(\frac{4}{19}\right)$$ $$e\left(\frac{7}{38}\right)$$ $$e\left(\frac{5}{19}\right)$$ $$e\left(\frac{9}{19}\right)$$ $$e\left(\frac{11}{38}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 1444 }(39,a) \;$$ at $$\;a =$$ e.g. 2