# Properties

 Modulus 73 Structure $$C_{72}$$ Order 72

Show commands for: SageMath / Pari/GP

sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(73)
pari: g = idealstar(,73,2)

## Character group

 sage: G.order() pari: g.no Order = 72 sage: H.invariants() pari: g.cyc Structure = $$C_{72}$$ sage: H.gens() pari: g.gen Generators = $\chi_{73}(5,\cdot)$

## First 32 of 72 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.

orbit label order primitive -1 1 2 3 4 5 6 7 8 9 10 11
$$\chi_{73}(1,\cdot)$$ 73.a 1 No $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{73}(2,\cdot)$$ 73.g 9 Yes $$1$$ $$1$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{1}{9}\right)$$
$$\chi_{73}(3,\cdot)$$ 73.h 12 Yes $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-i$$ $$1$$ $$1$$ $$-i$$ $$e\left(\frac{7}{12}\right)$$
$$\chi_{73}(4,\cdot)$$ 73.g 9 Yes $$1$$ $$1$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{2}{9}\right)$$
$$\chi_{73}(5,\cdot)$$ 73.l 72 Yes $$-1$$ $$1$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{72}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{55}{72}\right)$$
$$\chi_{73}(6,\cdot)$$ 73.k 36 Yes $$1$$ $$1$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-i$$ $$e\left(\frac{25}{36}\right)$$
$$\chi_{73}(7,\cdot)$$ 73.j 24 Yes $$-1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$-i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{5}{24}\right)$$
$$\chi_{73}(8,\cdot)$$ 73.c 3 Yes $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$1$$ $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{73}(9,\cdot)$$ 73.e 6 Yes $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{73}(10,\cdot)$$ 73.f 8 Yes $$-1$$ $$1$$ $$1$$ $$-i$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$
$$\chi_{73}(11,\cdot)$$ 73.l 72 Yes $$-1$$ $$1$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{55}{72}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{72}\right)$$
$$\chi_{73}(12,\cdot)$$ 73.k 36 Yes $$1$$ $$1$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-i$$ $$e\left(\frac{29}{36}\right)$$
$$\chi_{73}(13,\cdot)$$ 73.l 72 Yes $$-1$$ $$1$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{59}{72}\right)$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{72}\right)$$
$$\chi_{73}(14,\cdot)$$ 73.l 72 Yes $$-1$$ $$1$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{41}{72}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{23}{72}\right)$$
$$\chi_{73}(15,\cdot)$$ 73.l 72 Yes $$-1$$ $$1$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{72}\right)$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{25}{72}\right)$$
$$\chi_{73}(16,\cdot)$$ 73.g 9 Yes $$1$$ $$1$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{4}{9}\right)$$
$$\chi_{73}(17,\cdot)$$ 73.j 24 Yes $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$-i$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{24}\right)$$
$$\chi_{73}(18,\cdot)$$ 73.i 18 Yes $$1$$ $$1$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{5}{18}\right)$$
$$\chi_{73}(19,\cdot)$$ 73.k 36 Yes $$1$$ $$1$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-i$$ $$e\left(\frac{13}{36}\right)$$
$$\chi_{73}(20,\cdot)$$ 73.l 72 Yes $$-1$$ $$1$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{17}{72}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{71}{72}\right)$$
$$\chi_{73}(21,\cdot)$$ 73.j 24 Yes $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$i$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{19}{24}\right)$$
$$\chi_{73}(22,\cdot)$$ 73.f 8 Yes $$-1$$ $$1$$ $$1$$ $$i$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$
$$\chi_{73}(23,\cdot)$$ 73.k 36 Yes $$1$$ $$1$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-i$$ $$e\left(\frac{5}{36}\right)$$
$$\chi_{73}(24,\cdot)$$ 73.h 12 Yes $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-i$$ $$1$$ $$1$$ $$-i$$ $$e\left(\frac{11}{12}\right)$$
$$\chi_{73}(25,\cdot)$$ 73.k 36 Yes $$1$$ $$1$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$i$$ $$e\left(\frac{19}{36}\right)$$
$$\chi_{73}(26,\cdot)$$ 73.l 72 Yes $$-1$$ $$1$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{67}{72}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{13}{72}\right)$$
$$\chi_{73}(27,\cdot)$$ 73.d 4 Yes $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$i$$ $$-1$$ $$i$$ $$1$$ $$1$$ $$i$$ $$-i$$
$$\chi_{73}(28,\cdot)$$ 73.l 72 Yes $$-1$$ $$1$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{49}{72}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{31}{72}\right)$$
$$\chi_{73}(29,\cdot)$$ 73.l 72 Yes $$-1$$ $$1$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{35}{72}\right)$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{53}{72}\right)$$
$$\chi_{73}(30,\cdot)$$ 73.j 24 Yes $$-1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{11}{24}\right)$$
$$\chi_{73}(31,\cdot)$$ 73.l 72 Yes $$-1$$ $$1$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{11}{72}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{29}{72}\right)$$
$$\chi_{73}(32,\cdot)$$ 73.g 9 Yes $$1$$ $$1$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{5}{9}\right)$$