# Properties

 Label 344.z Modulus $344$ Conductor $344$ Order $42$ Real no Primitive yes Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(344, base_ring=CyclotomicField(42))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,21,32]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(13,344))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$344$$ Conductor: $$344$$ 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: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$21$$
$$\chi_{344}(13,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{13}{14}\right)$$
$$\chi_{344}(53,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{1}{14}\right)$$
$$\chi_{344}(101,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{11}{14}\right)$$
$$\chi_{344}(109,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{3}{14}\right)$$
$$\chi_{344}(117,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{9}{14}\right)$$
$$\chi_{344}(181,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{3}{14}\right)$$
$$\chi_{344}(189,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{1}{14}\right)$$
$$\chi_{344}(197,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{5}{14}\right)$$
$$\chi_{344}(229,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{9}{14}\right)$$
$$\chi_{344}(253,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{13}{14}\right)$$
$$\chi_{344}(325,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{11}{14}\right)$$
$$\chi_{344}(341,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{5}{14}\right)$$