# Properties

 Label 59.c Modulus $59$ Conductor $59$ Order $29$ Real no Primitive yes Minimal yes Parity even

# Learn more

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

sage: H = DirichletGroup(59, base_ring=CyclotomicField(58))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([50]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(3,59))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$59$$ Conductor: $$59$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$29$$ 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: $\Q(\zeta_{29})$ Fixed field: $$\Q(\zeta_{59})^+$$

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$
$$\chi_{59}(3,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{25}{29}\right)$$ $$e\left(\frac{3}{29}\right)$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{15}{29}\right)$$ $$e\left(\frac{17}{29}\right)$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{16}{29}\right)$$
$$\chi_{59}(4,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{22}{29}\right)$$ $$e\left(\frac{18}{29}\right)$$ $$e\left(\frac{3}{29}\right)$$ $$e\left(\frac{13}{29}\right)$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{25}{29}\right)$$
$$\chi_{59}(5,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{3}{29}\right)$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{18}{29}\right)$$ $$e\left(\frac{8}{29}\right)$$ $$e\left(\frac{25}{29}\right)$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{10}{29}\right)$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{17}{29}\right)$$
$$\chi_{59}(7,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{15}{29}\right)$$ $$e\left(\frac{18}{29}\right)$$ $$e\left(\frac{25}{29}\right)$$ $$e\left(\frac{24}{29}\right)$$ $$e\left(\frac{17}{29}\right)$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{22}{29}\right)$$
$$\chi_{59}(9,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{13}{29}\right)$$ $$e\left(\frac{10}{29}\right)$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{3}{29}\right)$$
$$\chi_{59}(12,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{24}{29}\right)$$ $$e\left(\frac{23}{29}\right)$$ $$e\left(\frac{11}{29}\right)$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{8}{29}\right)$$ $$e\left(\frac{12}{29}\right)$$
$$\chi_{59}(15,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{8}{29}\right)$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{23}{29}\right)$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{11}{29}\right)$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{22}{29}\right)$$ $$e\left(\frac{4}{29}\right)$$
$$\chi_{59}(16,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{13}{29}\right)$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{15}{29}\right)$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{14}{29}\right)$$ $$e\left(\frac{21}{29}\right)$$
$$\chi_{59}(17,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{14}{29}\right)$$ $$e\left(\frac{11}{29}\right)$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{24}{29}\right)$$ $$e\left(\frac{7}{29}\right)$$
$$\chi_{59}(19,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{22}{29}\right)$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{23}{29}\right)$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{15}{29}\right)$$ $$e\left(\frac{17}{29}\right)$$ $$e\left(\frac{11}{29}\right)$$
$$\chi_{59}(20,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{8}{29}\right)$$ $$e\left(\frac{24}{29}\right)$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{14}{29}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{23}{29}\right)$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{13}{29}\right)$$
$$\chi_{59}(21,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{18}{29}\right)$$ $$e\left(\frac{10}{29}\right)$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{23}{29}\right)$$ $$e\left(\frac{3}{29}\right)$$ $$e\left(\frac{15}{29}\right)$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{9}{29}\right)$$
$$\chi_{59}(22,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{13}{29}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{25}{29}\right)$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{10}{29}\right)$$ $$e\left(\frac{24}{29}\right)$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{6}{29}\right)$$
$$\chi_{59}(25,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{10}{29}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{18}{29}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{13}{29}\right)$$ $$e\left(\frac{5}{29}\right)$$
$$\chi_{59}(26,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{23}{29}\right)$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{17}{29}\right)$$ $$e\left(\frac{22}{29}\right)$$ $$e\left(\frac{13}{29}\right)$$ $$e\left(\frac{8}{29}\right)$$ $$e\left(\frac{11}{29}\right)$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{24}{29}\right)$$
$$\chi_{59}(27,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{29}\right)$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{15}{29}\right)$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{22}{29}\right)$$ $$e\left(\frac{18}{29}\right)$$ $$e\left(\frac{3}{29}\right)$$ $$e\left(\frac{19}{29}\right)$$
$$\chi_{59}(28,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{10}{29}\right)$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{17}{29}\right)$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{14}{29}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{18}{29}\right)$$
$$\chi_{59}(29,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{14}{29}\right)$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{18}{29}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{13}{29}\right)$$ $$e\left(\frac{8}{29}\right)$$ $$e\left(\frac{11}{29}\right)$$ $$e\left(\frac{2}{29}\right)$$
$$\chi_{59}(35,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{24}{29}\right)$$ $$e\left(\frac{14}{29}\right)$$ $$e\left(\frac{3}{29}\right)$$ $$e\left(\frac{13}{29}\right)$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{11}{29}\right)$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{10}{29}\right)$$
$$\chi_{59}(36,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{22}{29}\right)$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{15}{29}\right)$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{8}{29}\right)$$ $$e\left(\frac{25}{29}\right)$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{28}{29}\right)$$
$$\chi_{59}(41,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{14}{29}\right)$$ $$e\left(\frac{13}{29}\right)$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{10}{29}\right)$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{1}{29}\right)$$
$$\chi_{59}(45,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{24}{29}\right)$$ $$e\left(\frac{11}{29}\right)$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{14}{29}\right)$$ $$e\left(\frac{22}{29}\right)$$ $$e\left(\frac{23}{29}\right)$$ $$e\left(\frac{20}{29}\right)$$
$$\chi_{59}(46,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{8}{29}\right)$$ $$e\left(\frac{23}{29}\right)$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{24}{29}\right)$$ $$e\left(\frac{17}{29}\right)$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{26}{29}\right)$$
$$\chi_{59}(48,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{25}{29}\right)$$ $$e\left(\frac{17}{29}\right)$$ $$e\left(\frac{14}{29}\right)$$ $$e\left(\frac{22}{29}\right)$$ $$e\left(\frac{23}{29}\right)$$ $$e\left(\frac{3}{29}\right)$$ $$e\left(\frac{15}{29}\right)$$ $$e\left(\frac{8}{29}\right)$$
$$\chi_{59}(49,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{18}{29}\right)$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{25}{29}\right)$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{10}{29}\right)$$ $$e\left(\frac{15}{29}\right)$$
$$\chi_{59}(51,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{17}{29}\right)$$ $$e\left(\frac{3}{29}\right)$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{25}{29}\right)$$ $$e\left(\frac{23}{29}\right)$$
$$\chi_{59}(53,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{11}{29}\right)$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{22}{29}\right)$$ $$e\left(\frac{8}{29}\right)$$ $$e\left(\frac{10}{29}\right)$$ $$e\left(\frac{24}{29}\right)$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{14}{29}\right)$$
$$\chi_{59}(57,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{15}{29}\right)$$ $$e\left(\frac{25}{29}\right)$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{3}{29}\right)$$ $$e\left(\frac{11}{29}\right)$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{18}{29}\right)$$ $$e\left(\frac{27}{29}\right)$$