# Properties

 Modulus $89$ Structure $$C_{88}$$ Order $88$

Show commands for: Pari/GP / SageMath

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

sage: H = DirichletGroup_conrey(89)

pari: g = idealstar(,89,2)

## Character group

 sage: G.order()  pari: g.no Order = 88 sage: H.invariants()  pari: g.cyc Structure = $$C_{88}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{89}(3,\cdot)$

## First 32 of 88 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$
$$\chi_{89}(1,\cdot)$$ 89.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{89}(2,\cdot)$$ 89.e 11 yes $$1$$ $$1$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$
$$\chi_{89}(3,\cdot)$$ 89.h 88 yes $$-1$$ $$1$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{1}{88}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{17}{88}\right)$$ $$e\left(\frac{81}{88}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{21}{22}\right)$$
$$\chi_{89}(4,\cdot)$$ 89.e 11 yes $$1$$ $$1$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$
$$\chi_{89}(5,\cdot)$$ 89.g 44 yes $$1$$ $$1$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$
$$\chi_{89}(6,\cdot)$$ 89.h 88 yes $$-1$$ $$1$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{17}{88}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{25}{88}\right)$$ $$e\left(\frac{57}{88}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$
$$\chi_{89}(7,\cdot)$$ 89.h 88 yes $$-1$$ $$1$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{81}{88}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{57}{88}\right)$$ $$e\left(\frac{49}{88}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{7}{22}\right)$$
$$\chi_{89}(8,\cdot)$$ 89.e 11 yes $$1$$ $$1$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$
$$\chi_{89}(9,\cdot)$$ 89.g 44 yes $$1$$ $$1$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$
$$\chi_{89}(10,\cdot)$$ 89.g 44 yes $$1$$ $$1$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$
$$\chi_{89}(11,\cdot)$$ 89.f 22 yes $$1$$ $$1$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$
$$\chi_{89}(12,\cdot)$$ 89.d 8 yes $$-1$$ $$1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$1$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$1$$ $$-i$$ $$i$$ $$-1$$
$$\chi_{89}(13,\cdot)$$ 89.h 88 yes $$-1$$ $$1$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{23}{88}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{39}{88}\right)$$ $$e\left(\frac{15}{88}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{21}{22}\right)$$
$$\chi_{89}(14,\cdot)$$ 89.h 88 yes $$-1$$ $$1$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{9}{88}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{65}{88}\right)$$ $$e\left(\frac{25}{88}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$
$$\chi_{89}(15,\cdot)$$ 89.h 88 yes $$-1$$ $$1$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{71}{88}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{63}{88}\right)$$ $$e\left(\frac{31}{88}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$
$$\chi_{89}(16,\cdot)$$ 89.e 11 yes $$1$$ $$1$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$
$$\chi_{89}(17,\cdot)$$ 89.g 44 yes $$1$$ $$1$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$
$$\chi_{89}(18,\cdot)$$ 89.g 44 yes $$1$$ $$1$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{25}{44}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$
$$\chi_{89}(19,\cdot)$$ 89.h 88 yes $$-1$$ $$1$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{35}{88}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{67}{88}\right)$$ $$e\left(\frac{19}{88}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{9}{22}\right)$$
$$\chi_{89}(20,\cdot)$$ 89.g 44 yes $$1$$ $$1$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$
$$\chi_{89}(21,\cdot)$$ 89.g 44 yes $$1$$ $$1$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$
$$\chi_{89}(22,\cdot)$$ 89.f 22 yes $$1$$ $$1$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$
$$\chi_{89}(23,\cdot)$$ 89.h 88 yes $$-1$$ $$1$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{57}{88}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{1}{88}\right)$$ $$e\left(\frac{41}{88}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{9}{22}\right)$$
$$\chi_{89}(24,\cdot)$$ 89.h 88 yes $$-1$$ $$1$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{49}{88}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{41}{88}\right)$$ $$e\left(\frac{9}{88}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$
$$\chi_{89}(25,\cdot)$$ 89.f 22 yes $$1$$ $$1$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$
$$\chi_{89}(26,\cdot)$$ 89.h 88 yes $$-1$$ $$1$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{39}{88}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{47}{88}\right)$$ $$e\left(\frac{79}{88}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$
$$\chi_{89}(27,\cdot)$$ 89.h 88 yes $$-1$$ $$1$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{3}{88}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{51}{88}\right)$$ $$e\left(\frac{67}{88}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{19}{22}\right)$$
$$\chi_{89}(28,\cdot)$$ 89.h 88 yes $$-1$$ $$1$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{25}{88}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{73}{88}\right)$$ $$e\left(\frac{1}{88}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{25}{44}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{19}{22}\right)$$
$$\chi_{89}(29,\cdot)$$ 89.h 88 yes $$-1$$ $$1$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{59}{88}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{35}{88}\right)$$ $$e\left(\frac{27}{88}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{7}{22}\right)$$
$$\chi_{89}(30,\cdot)$$ 89.h 88 yes $$-1$$ $$1$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{87}{88}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{71}{88}\right)$$ $$e\left(\frac{7}{88}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$
$$\chi_{89}(31,\cdot)$$ 89.h 88 yes $$-1$$ $$1$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{31}{88}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{87}{88}\right)$$ $$e\left(\frac{47}{88}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$
$$\chi_{89}(32,\cdot)$$ 89.e 11 yes $$1$$ $$1$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$