# Properties

 Label 8033.407 Modulus $8033$ Conductor $277$ Order $138$ Real no Primitive no Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(8033)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,47]))

pari: [g,chi] = znchar(Mod(407,8033))

## Basic properties

 Modulus: $$8033$$ Conductor: $$277$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$138$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{277}(130,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 8033.cb

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

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

## Values on generators

$$(5541,1944)$$ → $$(1,e\left(\frac{47}{138}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$ $$1$$ $$1$$ $$e\left(\frac{3}{46}\right)$$ $$e\left(\frac{2}{69}\right)$$ $$e\left(\frac{3}{23}\right)$$ $$e\left(\frac{47}{138}\right)$$ $$e\left(\frac{13}{138}\right)$$ $$e\left(\frac{34}{69}\right)$$ $$e\left(\frac{9}{46}\right)$$ $$e\left(\frac{4}{69}\right)$$ $$e\left(\frac{28}{69}\right)$$ $$e\left(\frac{53}{138}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $\Q(\zeta_{69})$ Fixed field: Number field defined by a degree 138 polynomial