# Properties

 Modulus 113 Structure $$C_{112}$$ Order 112

Show commands for: SageMath / Pari/GP

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

sage: H = DirichletGroup_conrey(113)

pari: g = idealstar(,113,2)

## Character group

 sage: G.order()  pari: g.no Order = 112 sage: H.invariants()  pari: g.cyc Structure = $$C_{112}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{113}(3,\cdot)$

## First 32 of 112 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
$$\chi_{113}(1,\cdot)$$ 113.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{113}(2,\cdot)$$ 113.h 28 yes $$1$$ $$1$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{13}{14}\right)$$
$$\chi_{113}(3,\cdot)$$ 113.j 112 yes $$-1$$ $$1$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{1}{112}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{83}{112}\right)$$ $$e\left(\frac{13}{112}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{1}{56}\right)$$ $$e\left(\frac{95}{112}\right)$$ $$e\left(\frac{37}{56}\right)$$
$$\chi_{113}(4,\cdot)$$ 113.f 14 yes $$1$$ $$1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$
$$\chi_{113}(5,\cdot)$$ 113.j 112 yes $$-1$$ $$1$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{83}{112}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{57}{112}\right)$$ $$e\left(\frac{71}{112}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{27}{56}\right)$$ $$e\left(\frac{45}{112}\right)$$ $$e\left(\frac{47}{56}\right)$$
$$\chi_{113}(6,\cdot)$$ 113.j 112 yes $$-1$$ $$1$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{13}{112}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{71}{112}\right)$$ $$e\left(\frac{57}{112}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{13}{56}\right)$$ $$e\left(\frac{3}{112}\right)$$ $$e\left(\frac{33}{56}\right)$$
$$\chi_{113}(7,\cdot)$$ 113.f 14 yes $$1$$ $$1$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$
$$\chi_{113}(8,\cdot)$$ 113.h 28 yes $$1$$ $$1$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{11}{14}\right)$$
$$\chi_{113}(9,\cdot)$$ 113.i 56 yes $$1$$ $$1$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{1}{56}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{27}{56}\right)$$ $$e\left(\frac{13}{56}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{39}{56}\right)$$ $$e\left(\frac{9}{28}\right)$$
$$\chi_{113}(10,\cdot)$$ 113.j 112 yes $$-1$$ $$1$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{95}{112}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{45}{112}\right)$$ $$e\left(\frac{3}{112}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{39}{56}\right)$$ $$e\left(\frac{65}{112}\right)$$ $$e\left(\frac{43}{56}\right)$$
$$\chi_{113}(11,\cdot)$$ 113.i 56 yes $$1$$ $$1$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{37}{56}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{47}{56}\right)$$ $$e\left(\frac{33}{56}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{43}{56}\right)$$ $$e\left(\frac{25}{28}\right)$$
$$\chi_{113}(12,\cdot)$$ 113.j 112 yes $$-1$$ $$1$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{25}{112}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{59}{112}\right)$$ $$e\left(\frac{101}{112}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{25}{56}\right)$$ $$e\left(\frac{23}{112}\right)$$ $$e\left(\frac{29}{56}\right)$$
$$\chi_{113}(13,\cdot)$$ 113.i 56 yes $$1$$ $$1$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{11}{56}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{17}{56}\right)$$ $$e\left(\frac{31}{56}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{37}{56}\right)$$ $$e\left(\frac{15}{28}\right)$$
$$\chi_{113}(14,\cdot)$$ 113.h 28 yes $$1$$ $$1$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{3}{14}\right)$$
$$\chi_{113}(15,\cdot)$$ 113.c 4 yes $$1$$ $$1$$ $$1$$ $$-i$$ $$1$$ $$i$$ $$-i$$ $$1$$ $$1$$ $$-1$$ $$i$$ $$-1$$
$$\chi_{113}(16,\cdot)$$ 113.d 7 yes $$1$$ $$1$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{113}(17,\cdot)$$ 113.j 112 yes $$-1$$ $$1$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{5}{112}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{79}{112}\right)$$ $$e\left(\frac{65}{112}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{5}{56}\right)$$ $$e\left(\frac{27}{112}\right)$$ $$e\left(\frac{17}{56}\right)$$
$$\chi_{113}(18,\cdot)$$ 113.e 8 yes $$1$$ $$1$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$1$$ $$-1$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$i$$
$$\chi_{113}(19,\cdot)$$ 113.j 112 yes $$-1$$ $$1$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{99}{112}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{41}{112}\right)$$ $$e\left(\frac{55}{112}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{43}{56}\right)$$ $$e\left(\frac{109}{112}\right)$$ $$e\left(\frac{23}{56}\right)$$
$$\chi_{113}(20,\cdot)$$ 113.j 112 yes $$-1$$ $$1$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{107}{112}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{33}{112}\right)$$ $$e\left(\frac{47}{112}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{51}{56}\right)$$ $$e\left(\frac{85}{112}\right)$$ $$e\left(\frac{39}{56}\right)$$
$$\chi_{113}(21,\cdot)$$ 113.j 112 yes $$-1$$ $$1$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{9}{112}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{75}{112}\right)$$ $$e\left(\frac{5}{112}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{9}{56}\right)$$ $$e\left(\frac{71}{112}\right)$$ $$e\left(\frac{53}{56}\right)$$
$$\chi_{113}(22,\cdot)$$ 113.i 56 yes $$1$$ $$1$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{43}{56}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{41}{56}\right)$$ $$e\left(\frac{55}{56}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{53}{56}\right)$$ $$e\left(\frac{23}{28}\right)$$
$$\chi_{113}(23,\cdot)$$ 113.j 112 yes $$-1$$ $$1$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{41}{112}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{43}{112}\right)$$ $$e\left(\frac{85}{112}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{41}{56}\right)$$ $$e\left(\frac{87}{112}\right)$$ $$e\left(\frac{5}{56}\right)$$
$$\chi_{113}(24,\cdot)$$ 113.j 112 yes $$-1$$ $$1$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{37}{112}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{47}{112}\right)$$ $$e\left(\frac{33}{112}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{37}{56}\right)$$ $$e\left(\frac{43}{112}\right)$$ $$e\left(\frac{25}{56}\right)$$
$$\chi_{113}(25,\cdot)$$ 113.i 56 yes $$1$$ $$1$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{27}{56}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{56}\right)$$ $$e\left(\frac{15}{56}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{45}{56}\right)$$ $$e\left(\frac{19}{28}\right)$$
$$\chi_{113}(26,\cdot)$$ 113.i 56 yes $$1$$ $$1$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{17}{56}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{11}{56}\right)$$ $$e\left(\frac{53}{56}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{47}{56}\right)$$ $$e\left(\frac{13}{28}\right)$$
$$\chi_{113}(27,\cdot)$$ 113.j 112 yes $$-1$$ $$1$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{3}{112}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{25}{112}\right)$$ $$e\left(\frac{39}{112}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{3}{56}\right)$$ $$e\left(\frac{61}{112}\right)$$ $$e\left(\frac{55}{56}\right)$$
$$\chi_{113}(28,\cdot)$$ 113.d 7 yes $$1$$ $$1$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{113}(29,\cdot)$$ 113.j 112 yes $$-1$$ $$1$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{89}{112}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{107}{112}\right)$$ $$e\left(\frac{37}{112}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{33}{56}\right)$$ $$e\left(\frac{55}{112}\right)$$ $$e\left(\frac{45}{56}\right)$$
$$\chi_{113}(30,\cdot)$$ 113.d 7 yes $$1$$ $$1$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$
$$\chi_{113}(31,\cdot)$$ 113.i 56 yes $$1$$ $$1$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{25}{56}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{56}\right)$$ $$e\left(\frac{45}{56}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{23}{56}\right)$$ $$e\left(\frac{1}{28}\right)$$
$$\chi_{113}(32,\cdot)$$ 113.h 28 yes $$1$$ $$1$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{9}{14}\right)$$