# Properties

 Label 1600.109 Modulus $1600$ Conductor $1600$ Order $80$ Real no Primitive yes Minimal yes Parity even

# Learn more

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

sage: H = DirichletGroup(1600, base_ring=CyclotomicField(80))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,35,56]))

pari: [g,chi] = znchar(Mod(109,1600))

## Basic properties

 Modulus: $$1600$$ Conductor: $$1600$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$80$$ 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 1600.cq

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

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

## Related number fields

 Field of values: $\Q(\zeta_{80})$ Fixed field: Number field defined by a degree 80 polynomial

## Values on generators

$$(1151,901,577)$$ → $$(1,e\left(\frac{7}{16}\right),e\left(\frac{7}{10}\right))$$

## Values

 $$-1$$ $$1$$ $$3$$ $$7$$ $$9$$ $$11$$ $$13$$ $$17$$ $$19$$ $$21$$ $$23$$ $$27$$ $$1$$ $$1$$ $$e\left(\frac{17}{80}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{17}{40}\right)$$ $$e\left(\frac{31}{80}\right)$$ $$e\left(\frac{69}{80}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{53}{80}\right)$$ $$e\left(\frac{7}{80}\right)$$ $$e\left(\frac{33}{40}\right)$$ $$e\left(\frac{51}{80}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 1600 }(109,a) \;$$ at $$\;a =$$ e.g. 2