# Properties

 Modulus $86$ Structure $$C_{42}$$ Order $42$

Show commands for: Pari/GP / SageMath

sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed

sage: H = DirichletGroup_conrey(86)

pari: g = idealstar(,86,2)

## Character group

 sage: G.order()  pari: g.no Order = 42 sage: H.invariants()  pari: g.cyc Structure = $$C_{42}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{86}(3,\cdot)$

## First 32 of 42 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 $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$21$$
$$\chi_{86}(1,\cdot)$$ 86.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{86}(3,\cdot)$$ 86.h 42 no $$-1$$ $$1$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{6}{7}\right)$$
$$\chi_{86}(5,\cdot)$$ 86.h 42 no $$-1$$ $$1$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{3}{7}\right)$$
$$\chi_{86}(7,\cdot)$$ 86.d 6 no $$-1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$1$$
$$\chi_{86}(9,\cdot)$$ 86.g 21 no $$1$$ $$1$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{86}(11,\cdot)$$ 86.e 7 no $$1$$ $$1$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$1$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{86}(13,\cdot)$$ 86.g 21 no $$1$$ $$1$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$
$$\chi_{86}(15,\cdot)$$ 86.g 21 no $$1$$ $$1$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$
$$\chi_{86}(17,\cdot)$$ 86.g 21 no $$1$$ $$1$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$
$$\chi_{86}(19,\cdot)$$ 86.h 42 no $$-1$$ $$1$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{2}{7}\right)$$
$$\chi_{86}(21,\cdot)$$ 86.e 7 no $$1$$ $$1$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$1$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$
$$\chi_{86}(23,\cdot)$$ 86.g 21 no $$1$$ $$1$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{86}(25,\cdot)$$ 86.g 21 no $$1$$ $$1$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$
$$\chi_{86}(27,\cdot)$$ 86.f 14 no $$-1$$ $$1$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$-1$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$
$$\chi_{86}(29,\cdot)$$ 86.h 42 no $$-1$$ $$1$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{86}(31,\cdot)$$ 86.g 21 no $$1$$ $$1$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{86}(33,\cdot)$$ 86.h 42 no $$-1$$ $$1$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{4}{7}\right)$$
$$\chi_{86}(35,\cdot)$$ 86.e 7 no $$1$$ $$1$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$1$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$
$$\chi_{86}(37,\cdot)$$ 86.d 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$
$$\chi_{86}(39,\cdot)$$ 86.f 14 no $$-1$$ $$1$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$-1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$
$$\chi_{86}(41,\cdot)$$ 86.e 7 no $$1$$ $$1$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$1$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{86}(45,\cdot)$$ 86.f 14 no $$-1$$ $$1$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$-1$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{86}(47,\cdot)$$ 86.e 7 no $$1$$ $$1$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$
$$\chi_{86}(49,\cdot)$$ 86.c 3 no $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$
$$\chi_{86}(51,\cdot)$$ 86.f 14 no $$-1$$ $$1$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$-1$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$
$$\chi_{86}(53,\cdot)$$ 86.g 21 no $$1$$ $$1$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$
$$\chi_{86}(55,\cdot)$$ 86.h 42 no $$-1$$ $$1$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{86}(57,\cdot)$$ 86.g 21 no $$1$$ $$1$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{86}(59,\cdot)$$ 86.e 7 no $$1$$ $$1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$1$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$
$$\chi_{86}(61,\cdot)$$ 86.h 42 no $$-1$$ $$1$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{6}{7}\right)$$
$$\chi_{86}(63,\cdot)$$ 86.h 42 no $$-1$$ $$1$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{86}(65,\cdot)$$ 86.f 14 no $$-1$$ $$1$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$-1$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$