# Properties

 Modulus $59$ Structure $$C_{58}$$ Order $58$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(59)

pari: g = idealstar(,59,2)

## Character group

 sage: G.order()  pari: g.no Order = 58 sage: H.invariants()  pari: g.cyc Structure = $$C_{58}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{59}(2,\cdot)$

## First 32 of 58 characters

Each row describes a character. When available, the columns show the orbit label, order of the character, whether the character is primitive, and several values of the character.

Character Orbit Order Primitive $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$
$$\chi_{59}(1,\cdot)$$ 59.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{59}(2,\cdot)$$ 59.d 58 yes $$-1$$ $$1$$ $$e\left(\frac{1}{58}\right)$$ $$e\left(\frac{25}{29}\right)$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{3}{29}\right)$$ $$e\left(\frac{51}{58}\right)$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{3}{58}\right)$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{7}{58}\right)$$ $$e\left(\frac{25}{58}\right)$$
$$\chi_{59}(3,\cdot)$$ 59.c 29 yes $$1$$ $$1$$ $$e\left(\frac{25}{29}\right)$$ $$e\left(\frac{3}{29}\right)$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{15}{29}\right)$$ $$e\left(\frac{17}{29}\right)$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{16}{29}\right)$$
$$\chi_{59}(4,\cdot)$$ 59.c 29 yes $$1$$ $$1$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{22}{29}\right)$$ $$e\left(\frac{18}{29}\right)$$ $$e\left(\frac{3}{29}\right)$$ $$e\left(\frac{13}{29}\right)$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{25}{29}\right)$$
$$\chi_{59}(5,\cdot)$$ 59.c 29 yes $$1$$ $$1$$ $$e\left(\frac{3}{29}\right)$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{18}{29}\right)$$ $$e\left(\frac{8}{29}\right)$$ $$e\left(\frac{25}{29}\right)$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{10}{29}\right)$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{17}{29}\right)$$
$$\chi_{59}(6,\cdot)$$ 59.d 58 yes $$-1$$ $$1$$ $$e\left(\frac{51}{58}\right)$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{22}{29}\right)$$ $$e\left(\frac{8}{29}\right)$$ $$e\left(\frac{49}{58}\right)$$ $$e\left(\frac{24}{29}\right)$$ $$e\left(\frac{37}{58}\right)$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{9}{58}\right)$$ $$e\left(\frac{57}{58}\right)$$
$$\chi_{59}(7,\cdot)$$ 59.c 29 yes $$1$$ $$1$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{15}{29}\right)$$ $$e\left(\frac{18}{29}\right)$$ $$e\left(\frac{25}{29}\right)$$ $$e\left(\frac{24}{29}\right)$$ $$e\left(\frac{17}{29}\right)$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{22}{29}\right)$$
$$\chi_{59}(8,\cdot)$$ 59.d 58 yes $$-1$$ $$1$$ $$e\left(\frac{3}{58}\right)$$ $$e\left(\frac{17}{29}\right)$$ $$e\left(\frac{3}{29}\right)$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{37}{58}\right)$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{9}{58}\right)$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{21}{58}\right)$$ $$e\left(\frac{17}{58}\right)$$
$$\chi_{59}(9,\cdot)$$ 59.c 29 yes $$1$$ $$1$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{13}{29}\right)$$ $$e\left(\frac{10}{29}\right)$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{3}{29}\right)$$
$$\chi_{59}(10,\cdot)$$ 59.d 58 yes $$-1$$ $$1$$ $$e\left(\frac{7}{58}\right)$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{9}{58}\right)$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{21}{58}\right)$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{49}{58}\right)$$ $$e\left(\frac{1}{58}\right)$$
$$\chi_{59}(11,\cdot)$$ 59.d 58 yes $$-1$$ $$1$$ $$e\left(\frac{25}{58}\right)$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{25}{29}\right)$$ $$e\left(\frac{17}{29}\right)$$ $$e\left(\frac{57}{58}\right)$$ $$e\left(\frac{22}{29}\right)$$ $$e\left(\frac{17}{58}\right)$$ $$e\left(\frac{3}{29}\right)$$ $$e\left(\frac{1}{58}\right)$$ $$e\left(\frac{45}{58}\right)$$
$$\chi_{59}(12,\cdot)$$ 59.c 29 yes $$1$$ $$1$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{24}{29}\right)$$ $$e\left(\frac{23}{29}\right)$$ $$e\left(\frac{11}{29}\right)$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{8}{29}\right)$$ $$e\left(\frac{12}{29}\right)$$
$$\chi_{59}(13,\cdot)$$ 59.d 58 yes $$-1$$ $$1$$ $$e\left(\frac{45}{58}\right)$$ $$e\left(\frac{23}{29}\right)$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{33}{58}\right)$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{19}{58}\right)$$ $$e\left(\frac{17}{29}\right)$$ $$e\left(\frac{25}{58}\right)$$ $$e\left(\frac{23}{58}\right)$$
$$\chi_{59}(14,\cdot)$$ 59.d 58 yes $$-1$$ $$1$$ $$e\left(\frac{19}{58}\right)$$ $$e\left(\frac{11}{29}\right)$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{41}{58}\right)$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{57}{58}\right)$$ $$e\left(\frac{22}{29}\right)$$ $$e\left(\frac{17}{58}\right)$$ $$e\left(\frac{11}{58}\right)$$
$$\chi_{59}(15,\cdot)$$ 59.c 29 yes $$1$$ $$1$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{8}{29}\right)$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{23}{29}\right)$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{11}{29}\right)$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{22}{29}\right)$$ $$e\left(\frac{4}{29}\right)$$
$$\chi_{59}(16,\cdot)$$ 59.c 29 yes $$1$$ $$1$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{13}{29}\right)$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{15}{29}\right)$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{14}{29}\right)$$ $$e\left(\frac{21}{29}\right)$$
$$\chi_{59}(17,\cdot)$$ 59.c 29 yes $$1$$ $$1$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{14}{29}\right)$$ $$e\left(\frac{11}{29}\right)$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{24}{29}\right)$$ $$e\left(\frac{7}{29}\right)$$
$$\chi_{59}(18,\cdot)$$ 59.d 58 yes $$-1$$ $$1$$ $$e\left(\frac{43}{58}\right)$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{14}{29}\right)$$ $$e\left(\frac{13}{29}\right)$$ $$e\left(\frac{47}{58}\right)$$ $$e\left(\frac{10}{29}\right)$$ $$e\left(\frac{13}{58}\right)$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{11}{58}\right)$$ $$e\left(\frac{31}{58}\right)$$
$$\chi_{59}(19,\cdot)$$ 59.c 29 yes $$1$$ $$1$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{22}{29}\right)$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{23}{29}\right)$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{15}{29}\right)$$ $$e\left(\frac{17}{29}\right)$$ $$e\left(\frac{11}{29}\right)$$
$$\chi_{59}(20,\cdot)$$ 59.c 29 yes $$1$$ $$1$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{8}{29}\right)$$ $$e\left(\frac{24}{29}\right)$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{14}{29}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{23}{29}\right)$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{13}{29}\right)$$
$$\chi_{59}(21,\cdot)$$ 59.c 29 yes $$1$$ $$1$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{18}{29}\right)$$ $$e\left(\frac{10}{29}\right)$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{23}{29}\right)$$ $$e\left(\frac{3}{29}\right)$$ $$e\left(\frac{15}{29}\right)$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{9}{29}\right)$$
$$\chi_{59}(22,\cdot)$$ 59.c 29 yes $$1$$ $$1$$ $$e\left(\frac{13}{29}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{25}{29}\right)$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{10}{29}\right)$$ $$e\left(\frac{24}{29}\right)$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{6}{29}\right)$$
$$\chi_{59}(23,\cdot)$$ 59.d 58 yes $$-1$$ $$1$$ $$e\left(\frac{15}{58}\right)$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{15}{29}\right)$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{11}{58}\right)$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{45}{58}\right)$$ $$e\left(\frac{25}{29}\right)$$ $$e\left(\frac{47}{58}\right)$$ $$e\left(\frac{27}{58}\right)$$
$$\chi_{59}(24,\cdot)$$ 59.d 58 yes $$-1$$ $$1$$ $$e\left(\frac{53}{58}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{24}{29}\right)$$ $$e\left(\frac{14}{29}\right)$$ $$e\left(\frac{35}{58}\right)$$ $$e\left(\frac{13}{29}\right)$$ $$e\left(\frac{43}{58}\right)$$ $$e\left(\frac{11}{29}\right)$$ $$e\left(\frac{23}{58}\right)$$ $$e\left(\frac{49}{58}\right)$$
$$\chi_{59}(25,\cdot)$$ 59.c 29 yes $$1$$ $$1$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{10}{29}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{18}{29}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{13}{29}\right)$$ $$e\left(\frac{5}{29}\right)$$
$$\chi_{59}(26,\cdot)$$ 59.c 29 yes $$1$$ $$1$$ $$e\left(\frac{23}{29}\right)$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{17}{29}\right)$$ $$e\left(\frac{22}{29}\right)$$ $$e\left(\frac{13}{29}\right)$$ $$e\left(\frac{8}{29}\right)$$ $$e\left(\frac{11}{29}\right)$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{24}{29}\right)$$
$$\chi_{59}(27,\cdot)$$ 59.c 29 yes $$1$$ $$1$$ $$e\left(\frac{17}{29}\right)$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{15}{29}\right)$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{22}{29}\right)$$ $$e\left(\frac{18}{29}\right)$$ $$e\left(\frac{3}{29}\right)$$ $$e\left(\frac{19}{29}\right)$$
$$\chi_{59}(28,\cdot)$$ 59.c 29 yes $$1$$ $$1$$ $$e\left(\frac{10}{29}\right)$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{17}{29}\right)$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{14}{29}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{18}{29}\right)$$
$$\chi_{59}(29,\cdot)$$ 59.c 29 yes $$1$$ $$1$$ $$e\left(\frac{14}{29}\right)$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{18}{29}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{13}{29}\right)$$ $$e\left(\frac{8}{29}\right)$$ $$e\left(\frac{11}{29}\right)$$ $$e\left(\frac{2}{29}\right)$$
$$\chi_{59}(30,\cdot)$$ 59.d 58 yes $$-1$$ $$1$$ $$e\left(\frac{57}{58}\right)$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{7}{58}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{55}{58}\right)$$ $$e\left(\frac{8}{29}\right)$$ $$e\left(\frac{51}{58}\right)$$ $$e\left(\frac{33}{58}\right)$$
$$\chi_{59}(31,\cdot)$$ 59.d 58 yes $$-1$$ $$1$$ $$e\left(\frac{49}{58}\right)$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{5}{58}\right)$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{31}{58}\right)$$ $$e\left(\frac{14}{29}\right)$$ $$e\left(\frac{53}{58}\right)$$ $$e\left(\frac{7}{58}\right)$$
$$\chi_{59}(32,\cdot)$$ 59.d 58 yes $$-1$$ $$1$$ $$e\left(\frac{5}{58}\right)$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{15}{29}\right)$$ $$e\left(\frac{23}{58}\right)$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{15}{58}\right)$$ $$e\left(\frac{18}{29}\right)$$ $$e\left(\frac{35}{58}\right)$$ $$e\left(\frac{9}{58}\right)$$