# Properties

 Label 43.g Modulus $43$ Conductor $43$ Order $21$ Real no Primitive yes Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(43, base_ring=CyclotomicField(42))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([2]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(9,43))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$43$$ Conductor: $$43$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$21$$ 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)

## Related number fields

 Field of values: $$\Q(\zeta_{21})$$ Fixed field: $$\Q(\zeta_{43})^+$$

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$
$$\chi_{43}(9,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$
$$\chi_{43}(10,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{43}(13,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$
$$\chi_{43}(14,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$
$$\chi_{43}(15,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$
$$\chi_{43}(17,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{43}(23,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$
$$\chi_{43}(24,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$
$$\chi_{43}(25,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{43}(31,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$
$$\chi_{43}(38,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$
$$\chi_{43}(40,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$