# Properties

 Modulus $47$ Structure $$C_{46}$$ Order $46$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(47)

pari: g = idealstar(,47,2)

## Character group

 sage: G.order()  pari: g.no Order = 46 sage: H.invariants()  pari: g.cyc Structure = $$C_{46}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{47}(5,\cdot)$

## First 32 of 46 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$
$$\chi_{47}(1,\cdot)$$ 47.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{47}(2,\cdot)$$ 47.c 23 yes $$1$$ $$1$$ $$e\left(\frac{1}{23}\right)$$ $$e\left(\frac{19}{23}\right)$$ $$e\left(\frac{2}{23}\right)$$ $$e\left(\frac{9}{23}\right)$$ $$e\left(\frac{20}{23}\right)$$ $$e\left(\frac{12}{23}\right)$$ $$e\left(\frac{3}{23}\right)$$ $$e\left(\frac{15}{23}\right)$$ $$e\left(\frac{10}{23}\right)$$ $$e\left(\frac{17}{23}\right)$$
$$\chi_{47}(3,\cdot)$$ 47.c 23 yes $$1$$ $$1$$ $$e\left(\frac{19}{23}\right)$$ $$e\left(\frac{16}{23}\right)$$ $$e\left(\frac{15}{23}\right)$$ $$e\left(\frac{10}{23}\right)$$ $$e\left(\frac{12}{23}\right)$$ $$e\left(\frac{21}{23}\right)$$ $$e\left(\frac{11}{23}\right)$$ $$e\left(\frac{9}{23}\right)$$ $$e\left(\frac{6}{23}\right)$$ $$e\left(\frac{1}{23}\right)$$
$$\chi_{47}(4,\cdot)$$ 47.c 23 yes $$1$$ $$1$$ $$e\left(\frac{2}{23}\right)$$ $$e\left(\frac{15}{23}\right)$$ $$e\left(\frac{4}{23}\right)$$ $$e\left(\frac{18}{23}\right)$$ $$e\left(\frac{17}{23}\right)$$ $$e\left(\frac{1}{23}\right)$$ $$e\left(\frac{6}{23}\right)$$ $$e\left(\frac{7}{23}\right)$$ $$e\left(\frac{20}{23}\right)$$ $$e\left(\frac{11}{23}\right)$$
$$\chi_{47}(5,\cdot)$$ 47.d 46 yes $$-1$$ $$1$$ $$e\left(\frac{9}{23}\right)$$ $$e\left(\frac{10}{23}\right)$$ $$e\left(\frac{18}{23}\right)$$ $$e\left(\frac{1}{46}\right)$$ $$e\left(\frac{19}{23}\right)$$ $$e\left(\frac{16}{23}\right)$$ $$e\left(\frac{4}{23}\right)$$ $$e\left(\frac{20}{23}\right)$$ $$e\left(\frac{19}{46}\right)$$ $$e\left(\frac{7}{46}\right)$$
$$\chi_{47}(6,\cdot)$$ 47.c 23 yes $$1$$ $$1$$ $$e\left(\frac{20}{23}\right)$$ $$e\left(\frac{12}{23}\right)$$ $$e\left(\frac{17}{23}\right)$$ $$e\left(\frac{19}{23}\right)$$ $$e\left(\frac{9}{23}\right)$$ $$e\left(\frac{10}{23}\right)$$ $$e\left(\frac{14}{23}\right)$$ $$e\left(\frac{1}{23}\right)$$ $$e\left(\frac{16}{23}\right)$$ $$e\left(\frac{18}{23}\right)$$
$$\chi_{47}(7,\cdot)$$ 47.c 23 yes $$1$$ $$1$$ $$e\left(\frac{12}{23}\right)$$ $$e\left(\frac{21}{23}\right)$$ $$e\left(\frac{1}{23}\right)$$ $$e\left(\frac{16}{23}\right)$$ $$e\left(\frac{10}{23}\right)$$ $$e\left(\frac{6}{23}\right)$$ $$e\left(\frac{13}{23}\right)$$ $$e\left(\frac{19}{23}\right)$$ $$e\left(\frac{5}{23}\right)$$ $$e\left(\frac{20}{23}\right)$$
$$\chi_{47}(8,\cdot)$$ 47.c 23 yes $$1$$ $$1$$ $$e\left(\frac{3}{23}\right)$$ $$e\left(\frac{11}{23}\right)$$ $$e\left(\frac{6}{23}\right)$$ $$e\left(\frac{4}{23}\right)$$ $$e\left(\frac{14}{23}\right)$$ $$e\left(\frac{13}{23}\right)$$ $$e\left(\frac{9}{23}\right)$$ $$e\left(\frac{22}{23}\right)$$ $$e\left(\frac{7}{23}\right)$$ $$e\left(\frac{5}{23}\right)$$
$$\chi_{47}(9,\cdot)$$ 47.c 23 yes $$1$$ $$1$$ $$e\left(\frac{15}{23}\right)$$ $$e\left(\frac{9}{23}\right)$$ $$e\left(\frac{7}{23}\right)$$ $$e\left(\frac{20}{23}\right)$$ $$e\left(\frac{1}{23}\right)$$ $$e\left(\frac{19}{23}\right)$$ $$e\left(\frac{22}{23}\right)$$ $$e\left(\frac{18}{23}\right)$$ $$e\left(\frac{12}{23}\right)$$ $$e\left(\frac{2}{23}\right)$$
$$\chi_{47}(10,\cdot)$$ 47.d 46 yes $$-1$$ $$1$$ $$e\left(\frac{10}{23}\right)$$ $$e\left(\frac{6}{23}\right)$$ $$e\left(\frac{20}{23}\right)$$ $$e\left(\frac{19}{46}\right)$$ $$e\left(\frac{16}{23}\right)$$ $$e\left(\frac{5}{23}\right)$$ $$e\left(\frac{7}{23}\right)$$ $$e\left(\frac{12}{23}\right)$$ $$e\left(\frac{39}{46}\right)$$ $$e\left(\frac{41}{46}\right)$$
$$\chi_{47}(11,\cdot)$$ 47.d 46 yes $$-1$$ $$1$$ $$e\left(\frac{17}{23}\right)$$ $$e\left(\frac{1}{23}\right)$$ $$e\left(\frac{11}{23}\right)$$ $$e\left(\frac{7}{46}\right)$$ $$e\left(\frac{18}{23}\right)$$ $$e\left(\frac{20}{23}\right)$$ $$e\left(\frac{5}{23}\right)$$ $$e\left(\frac{2}{23}\right)$$ $$e\left(\frac{41}{46}\right)$$ $$e\left(\frac{3}{46}\right)$$
$$\chi_{47}(12,\cdot)$$ 47.c 23 yes $$1$$ $$1$$ $$e\left(\frac{21}{23}\right)$$ $$e\left(\frac{8}{23}\right)$$ $$e\left(\frac{19}{23}\right)$$ $$e\left(\frac{5}{23}\right)$$ $$e\left(\frac{6}{23}\right)$$ $$e\left(\frac{22}{23}\right)$$ $$e\left(\frac{17}{23}\right)$$ $$e\left(\frac{16}{23}\right)$$ $$e\left(\frac{3}{23}\right)$$ $$e\left(\frac{12}{23}\right)$$
$$\chi_{47}(13,\cdot)$$ 47.d 46 yes $$-1$$ $$1$$ $$e\left(\frac{7}{23}\right)$$ $$e\left(\frac{18}{23}\right)$$ $$e\left(\frac{14}{23}\right)$$ $$e\left(\frac{11}{46}\right)$$ $$e\left(\frac{2}{23}\right)$$ $$e\left(\frac{15}{23}\right)$$ $$e\left(\frac{21}{23}\right)$$ $$e\left(\frac{13}{23}\right)$$ $$e\left(\frac{25}{46}\right)$$ $$e\left(\frac{31}{46}\right)$$
$$\chi_{47}(14,\cdot)$$ 47.c 23 yes $$1$$ $$1$$ $$e\left(\frac{13}{23}\right)$$ $$e\left(\frac{17}{23}\right)$$ $$e\left(\frac{3}{23}\right)$$ $$e\left(\frac{2}{23}\right)$$ $$e\left(\frac{7}{23}\right)$$ $$e\left(\frac{18}{23}\right)$$ $$e\left(\frac{16}{23}\right)$$ $$e\left(\frac{11}{23}\right)$$ $$e\left(\frac{15}{23}\right)$$ $$e\left(\frac{14}{23}\right)$$
$$\chi_{47}(15,\cdot)$$ 47.d 46 yes $$-1$$ $$1$$ $$e\left(\frac{5}{23}\right)$$ $$e\left(\frac{3}{23}\right)$$ $$e\left(\frac{10}{23}\right)$$ $$e\left(\frac{21}{46}\right)$$ $$e\left(\frac{8}{23}\right)$$ $$e\left(\frac{14}{23}\right)$$ $$e\left(\frac{15}{23}\right)$$ $$e\left(\frac{6}{23}\right)$$ $$e\left(\frac{31}{46}\right)$$ $$e\left(\frac{9}{46}\right)$$
$$\chi_{47}(16,\cdot)$$ 47.c 23 yes $$1$$ $$1$$ $$e\left(\frac{4}{23}\right)$$ $$e\left(\frac{7}{23}\right)$$ $$e\left(\frac{8}{23}\right)$$ $$e\left(\frac{13}{23}\right)$$ $$e\left(\frac{11}{23}\right)$$ $$e\left(\frac{2}{23}\right)$$ $$e\left(\frac{12}{23}\right)$$ $$e\left(\frac{14}{23}\right)$$ $$e\left(\frac{17}{23}\right)$$ $$e\left(\frac{22}{23}\right)$$
$$\chi_{47}(17,\cdot)$$ 47.c 23 yes $$1$$ $$1$$ $$e\left(\frac{6}{23}\right)$$ $$e\left(\frac{22}{23}\right)$$ $$e\left(\frac{12}{23}\right)$$ $$e\left(\frac{8}{23}\right)$$ $$e\left(\frac{5}{23}\right)$$ $$e\left(\frac{3}{23}\right)$$ $$e\left(\frac{18}{23}\right)$$ $$e\left(\frac{21}{23}\right)$$ $$e\left(\frac{14}{23}\right)$$ $$e\left(\frac{10}{23}\right)$$
$$\chi_{47}(18,\cdot)$$ 47.c 23 yes $$1$$ $$1$$ $$e\left(\frac{16}{23}\right)$$ $$e\left(\frac{5}{23}\right)$$ $$e\left(\frac{9}{23}\right)$$ $$e\left(\frac{6}{23}\right)$$ $$e\left(\frac{21}{23}\right)$$ $$e\left(\frac{8}{23}\right)$$ $$e\left(\frac{2}{23}\right)$$ $$e\left(\frac{10}{23}\right)$$ $$e\left(\frac{22}{23}\right)$$ $$e\left(\frac{19}{23}\right)$$
$$\chi_{47}(19,\cdot)$$ 47.d 46 yes $$-1$$ $$1$$ $$e\left(\frac{14}{23}\right)$$ $$e\left(\frac{13}{23}\right)$$ $$e\left(\frac{5}{23}\right)$$ $$e\left(\frac{45}{46}\right)$$ $$e\left(\frac{4}{23}\right)$$ $$e\left(\frac{7}{23}\right)$$ $$e\left(\frac{19}{23}\right)$$ $$e\left(\frac{3}{23}\right)$$ $$e\left(\frac{27}{46}\right)$$ $$e\left(\frac{39}{46}\right)$$
$$\chi_{47}(20,\cdot)$$ 47.d 46 yes $$-1$$ $$1$$ $$e\left(\frac{11}{23}\right)$$ $$e\left(\frac{2}{23}\right)$$ $$e\left(\frac{22}{23}\right)$$ $$e\left(\frac{37}{46}\right)$$ $$e\left(\frac{13}{23}\right)$$ $$e\left(\frac{17}{23}\right)$$ $$e\left(\frac{10}{23}\right)$$ $$e\left(\frac{4}{23}\right)$$ $$e\left(\frac{13}{46}\right)$$ $$e\left(\frac{29}{46}\right)$$
$$\chi_{47}(21,\cdot)$$ 47.c 23 yes $$1$$ $$1$$ $$e\left(\frac{8}{23}\right)$$ $$e\left(\frac{14}{23}\right)$$ $$e\left(\frac{16}{23}\right)$$ $$e\left(\frac{3}{23}\right)$$ $$e\left(\frac{22}{23}\right)$$ $$e\left(\frac{4}{23}\right)$$ $$e\left(\frac{1}{23}\right)$$ $$e\left(\frac{5}{23}\right)$$ $$e\left(\frac{11}{23}\right)$$ $$e\left(\frac{21}{23}\right)$$
$$\chi_{47}(22,\cdot)$$ 47.d 46 yes $$-1$$ $$1$$ $$e\left(\frac{18}{23}\right)$$ $$e\left(\frac{20}{23}\right)$$ $$e\left(\frac{13}{23}\right)$$ $$e\left(\frac{25}{46}\right)$$ $$e\left(\frac{15}{23}\right)$$ $$e\left(\frac{9}{23}\right)$$ $$e\left(\frac{8}{23}\right)$$ $$e\left(\frac{17}{23}\right)$$ $$e\left(\frac{15}{46}\right)$$ $$e\left(\frac{37}{46}\right)$$
$$\chi_{47}(23,\cdot)$$ 47.d 46 yes $$-1$$ $$1$$ $$e\left(\frac{22}{23}\right)$$ $$e\left(\frac{4}{23}\right)$$ $$e\left(\frac{21}{23}\right)$$ $$e\left(\frac{5}{46}\right)$$ $$e\left(\frac{3}{23}\right)$$ $$e\left(\frac{11}{23}\right)$$ $$e\left(\frac{20}{23}\right)$$ $$e\left(\frac{8}{23}\right)$$ $$e\left(\frac{3}{46}\right)$$ $$e\left(\frac{35}{46}\right)$$
$$\chi_{47}(24,\cdot)$$ 47.c 23 yes $$1$$ $$1$$ $$e\left(\frac{22}{23}\right)$$ $$e\left(\frac{4}{23}\right)$$ $$e\left(\frac{21}{23}\right)$$ $$e\left(\frac{14}{23}\right)$$ $$e\left(\frac{3}{23}\right)$$ $$e\left(\frac{11}{23}\right)$$ $$e\left(\frac{20}{23}\right)$$ $$e\left(\frac{8}{23}\right)$$ $$e\left(\frac{13}{23}\right)$$ $$e\left(\frac{6}{23}\right)$$
$$\chi_{47}(25,\cdot)$$ 47.c 23 yes $$1$$ $$1$$ $$e\left(\frac{18}{23}\right)$$ $$e\left(\frac{20}{23}\right)$$ $$e\left(\frac{13}{23}\right)$$ $$e\left(\frac{1}{23}\right)$$ $$e\left(\frac{15}{23}\right)$$ $$e\left(\frac{9}{23}\right)$$ $$e\left(\frac{8}{23}\right)$$ $$e\left(\frac{17}{23}\right)$$ $$e\left(\frac{19}{23}\right)$$ $$e\left(\frac{7}{23}\right)$$
$$\chi_{47}(26,\cdot)$$ 47.d 46 yes $$-1$$ $$1$$ $$e\left(\frac{8}{23}\right)$$ $$e\left(\frac{14}{23}\right)$$ $$e\left(\frac{16}{23}\right)$$ $$e\left(\frac{29}{46}\right)$$ $$e\left(\frac{22}{23}\right)$$ $$e\left(\frac{4}{23}\right)$$ $$e\left(\frac{1}{23}\right)$$ $$e\left(\frac{5}{23}\right)$$ $$e\left(\frac{45}{46}\right)$$ $$e\left(\frac{19}{46}\right)$$
$$\chi_{47}(27,\cdot)$$ 47.c 23 yes $$1$$ $$1$$ $$e\left(\frac{11}{23}\right)$$ $$e\left(\frac{2}{23}\right)$$ $$e\left(\frac{22}{23}\right)$$ $$e\left(\frac{7}{23}\right)$$ $$e\left(\frac{13}{23}\right)$$ $$e\left(\frac{17}{23}\right)$$ $$e\left(\frac{10}{23}\right)$$ $$e\left(\frac{4}{23}\right)$$ $$e\left(\frac{18}{23}\right)$$ $$e\left(\frac{3}{23}\right)$$
$$\chi_{47}(28,\cdot)$$ 47.c 23 yes $$1$$ $$1$$ $$e\left(\frac{14}{23}\right)$$ $$e\left(\frac{13}{23}\right)$$ $$e\left(\frac{5}{23}\right)$$ $$e\left(\frac{11}{23}\right)$$ $$e\left(\frac{4}{23}\right)$$ $$e\left(\frac{7}{23}\right)$$ $$e\left(\frac{19}{23}\right)$$ $$e\left(\frac{3}{23}\right)$$ $$e\left(\frac{2}{23}\right)$$ $$e\left(\frac{8}{23}\right)$$
$$\chi_{47}(29,\cdot)$$ 47.d 46 yes $$-1$$ $$1$$ $$e\left(\frac{16}{23}\right)$$ $$e\left(\frac{5}{23}\right)$$ $$e\left(\frac{9}{23}\right)$$ $$e\left(\frac{35}{46}\right)$$ $$e\left(\frac{21}{23}\right)$$ $$e\left(\frac{8}{23}\right)$$ $$e\left(\frac{2}{23}\right)$$ $$e\left(\frac{10}{23}\right)$$ $$e\left(\frac{21}{46}\right)$$ $$e\left(\frac{15}{46}\right)$$
$$\chi_{47}(30,\cdot)$$ 47.d 46 yes $$-1$$ $$1$$ $$e\left(\frac{6}{23}\right)$$ $$e\left(\frac{22}{23}\right)$$ $$e\left(\frac{12}{23}\right)$$ $$e\left(\frac{39}{46}\right)$$ $$e\left(\frac{5}{23}\right)$$ $$e\left(\frac{3}{23}\right)$$ $$e\left(\frac{18}{23}\right)$$ $$e\left(\frac{21}{23}\right)$$ $$e\left(\frac{5}{46}\right)$$ $$e\left(\frac{43}{46}\right)$$
$$\chi_{47}(31,\cdot)$$ 47.d 46 yes $$-1$$ $$1$$ $$e\left(\frac{4}{23}\right)$$ $$e\left(\frac{7}{23}\right)$$ $$e\left(\frac{8}{23}\right)$$ $$e\left(\frac{3}{46}\right)$$ $$e\left(\frac{11}{23}\right)$$ $$e\left(\frac{2}{23}\right)$$ $$e\left(\frac{12}{23}\right)$$ $$e\left(\frac{14}{23}\right)$$ $$e\left(\frac{11}{46}\right)$$ $$e\left(\frac{21}{46}\right)$$
$$\chi_{47}(32,\cdot)$$ 47.c 23 yes $$1$$ $$1$$ $$e\left(\frac{5}{23}\right)$$ $$e\left(\frac{3}{23}\right)$$ $$e\left(\frac{10}{23}\right)$$ $$e\left(\frac{22}{23}\right)$$ $$e\left(\frac{8}{23}\right)$$ $$e\left(\frac{14}{23}\right)$$ $$e\left(\frac{15}{23}\right)$$ $$e\left(\frac{6}{23}\right)$$ $$e\left(\frac{4}{23}\right)$$ $$e\left(\frac{16}{23}\right)$$