# Properties

 Label 2304.5 Modulus $2304$ Conductor $2304$ Order $192$ Real no Primitive yes Minimal yes Parity odd

# Learn more

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

sage: H = DirichletGroup(2304, base_ring=CyclotomicField(192))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,3,160]))

pari: [g,chi] = znchar(Mod(5,2304))

## Basic properties

 Modulus: $$2304$$ Conductor: $$2304$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$192$$ 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 2304.cb

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: $\Q(\zeta_{192})$ Fixed field: Number field defined by a degree 192 polynomial (not computed)

## Values on generators

$$(1279,2053,1793)$$ → $$(1,e\left(\frac{1}{64}\right),e\left(\frac{5}{6}\right))$$

## Values

 $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$ $$-1$$ $$1$$ $$e\left(\frac{35}{192}\right)$$ $$e\left(\frac{47}{96}\right)$$ $$e\left(\frac{31}{192}\right)$$ $$e\left(\frac{77}{192}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{23}{64}\right)$$ $$e\left(\frac{37}{96}\right)$$ $$e\left(\frac{35}{96}\right)$$ $$e\left(\frac{145}{192}\right)$$ $$e\left(\frac{19}{24}\right)$$
 value at e.g. 2