# Properties

 Label 392.bf Modulus $392$ Conductor $392$ Order $42$ Real no Primitive yes Minimal yes Parity odd

# Related objects

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

sage: H = DirichletGroup(392, base_ring=CyclotomicField(42))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,21,29]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(5,392))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$392$$ Conductor: $$392$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$42$$ 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)

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$3$$ $$5$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$23$$ $$25$$
$$\chi_{392}(5,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$
$$\chi_{392}(45,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{17}{21}\right)$$
$$\chi_{392}(61,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$
$$\chi_{392}(101,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$
$$\chi_{392}(157,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$
$$\chi_{392}(173,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$
$$\chi_{392}(213,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$
$$\chi_{392}(229,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$
$$\chi_{392}(269,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$
$$\chi_{392}(285,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$
$$\chi_{392}(341,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$
$$\chi_{392}(381,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$