# Properties

 Modulus 101 Structure $$C_{100}$$ Order 100

Show commands for: SageMath / Pari/GP

sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(101)
pari: g = idealstar(,101,2)

## Character group

 sage: G.order() pari: g.no Order = 100 sage: H.invariants() pari: g.cyc Structure = $$C_{100}$$ sage: H.gens() pari: g.gen Generators = $\chi_{101}(2,\cdot)$

## First 32 of 100 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
$$\chi_{101}(1,\cdot)$$ 101.a 1 No $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{101}(2,\cdot)$$ 101.i 100 Yes $$-1$$ $$1$$ $$e\left(\frac{1}{100}\right)$$ $$e\left(\frac{69}{100}\right)$$ $$e\left(\frac{1}{50}\right)$$ $$e\left(\frac{6}{25}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{9}{100}\right)$$ $$e\left(\frac{3}{100}\right)$$ $$e\left(\frac{19}{50}\right)$$ $$i$$ $$e\left(\frac{13}{100}\right)$$
$$\chi_{101}(3,\cdot)$$ 101.i 100 Yes $$-1$$ $$1$$ $$e\left(\frac{69}{100}\right)$$ $$e\left(\frac{61}{100}\right)$$ $$e\left(\frac{19}{50}\right)$$ $$e\left(\frac{14}{25}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{21}{100}\right)$$ $$e\left(\frac{7}{100}\right)$$ $$e\left(\frac{11}{50}\right)$$ $$i$$ $$e\left(\frac{97}{100}\right)$$
$$\chi_{101}(4,\cdot)$$ 101.h 50 Yes $$1$$ $$1$$ $$e\left(\frac{1}{50}\right)$$ $$e\left(\frac{19}{50}\right)$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{12}{25}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{9}{50}\right)$$ $$e\left(\frac{3}{50}\right)$$ $$e\left(\frac{19}{25}\right)$$ $$-1$$ $$e\left(\frac{13}{50}\right)$$
$$\chi_{101}(5,\cdot)$$ 101.g 25 Yes $$1$$ $$1$$ $$e\left(\frac{6}{25}\right)$$ $$e\left(\frac{14}{25}\right)$$ $$e\left(\frac{12}{25}\right)$$ $$e\left(\frac{19}{25}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{18}{25}\right)$$ $$e\left(\frac{3}{25}\right)$$ $$1$$ $$e\left(\frac{3}{25}\right)$$
$$\chi_{101}(6,\cdot)$$ 101.e 10 Yes $$1$$ $$1$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$1$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$-1$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{101}(7,\cdot)$$ 101.i 100 Yes $$-1$$ $$1$$ $$e\left(\frac{9}{100}\right)$$ $$e\left(\frac{21}{100}\right)$$ $$e\left(\frac{9}{50}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{81}{100}\right)$$ $$e\left(\frac{27}{100}\right)$$ $$e\left(\frac{21}{50}\right)$$ $$i$$ $$e\left(\frac{17}{100}\right)$$
$$\chi_{101}(8,\cdot)$$ 101.i 100 Yes $$-1$$ $$1$$ $$e\left(\frac{3}{100}\right)$$ $$e\left(\frac{7}{100}\right)$$ $$e\left(\frac{3}{50}\right)$$ $$e\left(\frac{18}{25}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{27}{100}\right)$$ $$e\left(\frac{9}{100}\right)$$ $$e\left(\frac{7}{50}\right)$$ $$-i$$ $$e\left(\frac{39}{100}\right)$$
$$\chi_{101}(9,\cdot)$$ 101.h 50 Yes $$1$$ $$1$$ $$e\left(\frac{19}{50}\right)$$ $$e\left(\frac{11}{50}\right)$$ $$e\left(\frac{19}{25}\right)$$ $$e\left(\frac{3}{25}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{21}{50}\right)$$ $$e\left(\frac{7}{50}\right)$$ $$e\left(\frac{11}{25}\right)$$ $$-1$$ $$e\left(\frac{47}{50}\right)$$
$$\chi_{101}(10,\cdot)$$ 101.c 4 Yes $$-1$$ $$1$$ $$i$$ $$i$$ $$-1$$ $$1$$ $$-1$$ $$i$$ $$-i$$ $$-1$$ $$i$$ $$i$$
$$\chi_{101}(11,\cdot)$$ 101.i 100 Yes $$-1$$ $$1$$ $$e\left(\frac{13}{100}\right)$$ $$e\left(\frac{97}{100}\right)$$ $$e\left(\frac{13}{50}\right)$$ $$e\left(\frac{3}{25}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{17}{100}\right)$$ $$e\left(\frac{39}{100}\right)$$ $$e\left(\frac{47}{50}\right)$$ $$i$$ $$e\left(\frac{69}{100}\right)$$
$$\chi_{101}(12,\cdot)$$ 101.i 100 Yes $$-1$$ $$1$$ $$e\left(\frac{71}{100}\right)$$ $$e\left(\frac{99}{100}\right)$$ $$e\left(\frac{21}{50}\right)$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{39}{100}\right)$$ $$e\left(\frac{13}{100}\right)$$ $$e\left(\frac{49}{50}\right)$$ $$-i$$ $$e\left(\frac{23}{100}\right)$$
$$\chi_{101}(13,\cdot)$$ 101.h 50 Yes $$1$$ $$1$$ $$e\left(\frac{33}{50}\right)$$ $$e\left(\frac{27}{50}\right)$$ $$e\left(\frac{8}{25}\right)$$ $$e\left(\frac{21}{25}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{47}{50}\right)$$ $$e\left(\frac{49}{50}\right)$$ $$e\left(\frac{2}{25}\right)$$ $$-1$$ $$e\left(\frac{29}{50}\right)$$
$$\chi_{101}(14,\cdot)$$ 101.e 10 Yes $$1$$ $$1$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$1$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$-1$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{101}(15,\cdot)$$ 101.i 100 Yes $$-1$$ $$1$$ $$e\left(\frac{93}{100}\right)$$ $$e\left(\frac{17}{100}\right)$$ $$e\left(\frac{43}{50}\right)$$ $$e\left(\frac{8}{25}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{37}{100}\right)$$ $$e\left(\frac{79}{100}\right)$$ $$e\left(\frac{17}{50}\right)$$ $$i$$ $$e\left(\frac{9}{100}\right)$$
$$\chi_{101}(16,\cdot)$$ 101.g 25 Yes $$1$$ $$1$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{19}{25}\right)$$ $$e\left(\frac{2}{25}\right)$$ $$e\left(\frac{24}{25}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{9}{25}\right)$$ $$e\left(\frac{3}{25}\right)$$ $$e\left(\frac{13}{25}\right)$$ $$1$$ $$e\left(\frac{13}{25}\right)$$
$$\chi_{101}(17,\cdot)$$ 101.e 10 Yes $$1$$ $$1$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$1$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$-1$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{101}(18,\cdot)$$ 101.i 100 Yes $$-1$$ $$1$$ $$e\left(\frac{39}{100}\right)$$ $$e\left(\frac{91}{100}\right)$$ $$e\left(\frac{39}{50}\right)$$ $$e\left(\frac{9}{25}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{51}{100}\right)$$ $$e\left(\frac{17}{100}\right)$$ $$e\left(\frac{41}{50}\right)$$ $$-i$$ $$e\left(\frac{7}{100}\right)$$
$$\chi_{101}(19,\cdot)$$ 101.g 25 Yes $$1$$ $$1$$ $$e\left(\frac{24}{25}\right)$$ $$e\left(\frac{6}{25}\right)$$ $$e\left(\frac{23}{25}\right)$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{22}{25}\right)$$ $$e\left(\frac{12}{25}\right)$$ $$1$$ $$e\left(\frac{12}{25}\right)$$
$$\chi_{101}(20,\cdot)$$ 101.h 50 Yes $$1$$ $$1$$ $$e\left(\frac{13}{50}\right)$$ $$e\left(\frac{47}{50}\right)$$ $$e\left(\frac{13}{25}\right)$$ $$e\left(\frac{6}{25}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{17}{50}\right)$$ $$e\left(\frac{39}{50}\right)$$ $$e\left(\frac{22}{25}\right)$$ $$-1$$ $$e\left(\frac{19}{50}\right)$$
$$\chi_{101}(21,\cdot)$$ 101.h 50 Yes $$1$$ $$1$$ $$e\left(\frac{39}{50}\right)$$ $$e\left(\frac{41}{50}\right)$$ $$e\left(\frac{14}{25}\right)$$ $$e\left(\frac{18}{25}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{50}\right)$$ $$e\left(\frac{17}{50}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$-1$$ $$e\left(\frac{7}{50}\right)$$
$$\chi_{101}(22,\cdot)$$ 101.h 50 Yes $$1$$ $$1$$ $$e\left(\frac{7}{50}\right)$$ $$e\left(\frac{33}{50}\right)$$ $$e\left(\frac{7}{25}\right)$$ $$e\left(\frac{9}{25}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{13}{50}\right)$$ $$e\left(\frac{21}{50}\right)$$ $$e\left(\frac{8}{25}\right)$$ $$-1$$ $$e\left(\frac{41}{50}\right)$$
$$\chi_{101}(23,\cdot)$$ 101.h 50 Yes $$1$$ $$1$$ $$e\left(\frac{43}{50}\right)$$ $$e\left(\frac{17}{50}\right)$$ $$e\left(\frac{18}{25}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{37}{50}\right)$$ $$e\left(\frac{29}{50}\right)$$ $$e\left(\frac{17}{25}\right)$$ $$-1$$ $$e\left(\frac{9}{50}\right)$$
$$\chi_{101}(24,\cdot)$$ 101.g 25 Yes $$1$$ $$1$$ $$e\left(\frac{18}{25}\right)$$ $$e\left(\frac{17}{25}\right)$$ $$e\left(\frac{11}{25}\right)$$ $$e\left(\frac{7}{25}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{12}{25}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{9}{25}\right)$$ $$1$$ $$e\left(\frac{9}{25}\right)$$
$$\chi_{101}(25,\cdot)$$ 101.g 25 Yes $$1$$ $$1$$ $$e\left(\frac{12}{25}\right)$$ $$e\left(\frac{3}{25}\right)$$ $$e\left(\frac{24}{25}\right)$$ $$e\left(\frac{13}{25}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{8}{25}\right)$$ $$e\left(\frac{11}{25}\right)$$ $$e\left(\frac{6}{25}\right)$$ $$1$$ $$e\left(\frac{6}{25}\right)$$
$$\chi_{101}(26,\cdot)$$ 101.i 100 Yes $$-1$$ $$1$$ $$e\left(\frac{67}{100}\right)$$ $$e\left(\frac{23}{100}\right)$$ $$e\left(\frac{17}{50}\right)$$ $$e\left(\frac{2}{25}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{100}\right)$$ $$e\left(\frac{1}{100}\right)$$ $$e\left(\frac{23}{50}\right)$$ $$-i$$ $$e\left(\frac{71}{100}\right)$$
$$\chi_{101}(27,\cdot)$$ 101.i 100 Yes $$-1$$ $$1$$ $$e\left(\frac{7}{100}\right)$$ $$e\left(\frac{83}{100}\right)$$ $$e\left(\frac{7}{50}\right)$$ $$e\left(\frac{17}{25}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{63}{100}\right)$$ $$e\left(\frac{21}{100}\right)$$ $$e\left(\frac{33}{50}\right)$$ $$-i$$ $$e\left(\frac{91}{100}\right)$$
$$\chi_{101}(28,\cdot)$$ 101.i 100 Yes $$-1$$ $$1$$ $$e\left(\frac{11}{100}\right)$$ $$e\left(\frac{59}{100}\right)$$ $$e\left(\frac{11}{50}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{99}{100}\right)$$ $$e\left(\frac{33}{100}\right)$$ $$e\left(\frac{9}{50}\right)$$ $$-i$$ $$e\left(\frac{43}{100}\right)$$
$$\chi_{101}(29,\cdot)$$ 101.i 100 Yes $$-1$$ $$1$$ $$e\left(\frac{91}{100}\right)$$ $$e\left(\frac{79}{100}\right)$$ $$e\left(\frac{41}{50}\right)$$ $$e\left(\frac{21}{25}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{19}{100}\right)$$ $$e\left(\frac{73}{100}\right)$$ $$e\left(\frac{29}{50}\right)$$ $$-i$$ $$e\left(\frac{83}{100}\right)$$
$$\chi_{101}(30,\cdot)$$ 101.h 50 Yes $$1$$ $$1$$ $$e\left(\frac{47}{50}\right)$$ $$e\left(\frac{43}{50}\right)$$ $$e\left(\frac{22}{25}\right)$$ $$e\left(\frac{14}{25}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{23}{50}\right)$$ $$e\left(\frac{41}{50}\right)$$ $$e\left(\frac{18}{25}\right)$$ $$-1$$ $$e\left(\frac{11}{50}\right)$$
$$\chi_{101}(31,\cdot)$$ 101.g 25 Yes $$1$$ $$1$$ $$e\left(\frac{21}{25}\right)$$ $$e\left(\frac{24}{25}\right)$$ $$e\left(\frac{17}{25}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{14}{25}\right)$$ $$e\left(\frac{13}{25}\right)$$ $$e\left(\frac{23}{25}\right)$$ $$1$$ $$e\left(\frac{23}{25}\right)$$
$$\chi_{101}(32,\cdot)$$ 101.f 20 Yes $$-1$$ $$1$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$-1$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$i$$ $$e\left(\frac{13}{20}\right)$$