# Properties

 Modulus $125$ Structure $$C_{100}$$ Order $100$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(125)

pari: g = idealstar(,125,2)

## 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 = $\chi_{125}(2,\cdot)$

## 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.

Character Orbit Order Primitive $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$7$$ $$8$$ $$9$$ $$11$$ $$12$$ $$13$$
$$\chi_{125}(1,\cdot)$$ 125.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{125}(2,\cdot)$$ 125.i 100 yes $$-1$$ $$1$$ $$e\left(\frac{1}{100}\right)$$ $$e\left(\frac{7}{100}\right)$$ $$e\left(\frac{1}{50}\right)$$ $$e\left(\frac{2}{25}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{3}{100}\right)$$ $$e\left(\frac{7}{50}\right)$$ $$e\left(\frac{19}{25}\right)$$ $$e\left(\frac{9}{100}\right)$$ $$e\left(\frac{39}{100}\right)$$
$$\chi_{125}(3,\cdot)$$ 125.i 100 yes $$-1$$ $$1$$ $$e\left(\frac{7}{100}\right)$$ $$e\left(\frac{49}{100}\right)$$ $$e\left(\frac{7}{50}\right)$$ $$e\left(\frac{14}{25}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{21}{100}\right)$$ $$e\left(\frac{49}{50}\right)$$ $$e\left(\frac{8}{25}\right)$$ $$e\left(\frac{63}{100}\right)$$ $$e\left(\frac{73}{100}\right)$$
$$\chi_{125}(4,\cdot)$$ 125.h 50 yes $$1$$ $$1$$ $$e\left(\frac{1}{50}\right)$$ $$e\left(\frac{7}{50}\right)$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{50}\right)$$ $$e\left(\frac{7}{25}\right)$$ $$e\left(\frac{13}{25}\right)$$ $$e\left(\frac{9}{50}\right)$$ $$e\left(\frac{39}{50}\right)$$
$$\chi_{125}(6,\cdot)$$ 125.g 25 yes $$1$$ $$1$$ $$e\left(\frac{2}{25}\right)$$ $$e\left(\frac{14}{25}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{6}{25}\right)$$ $$e\left(\frac{3}{25}\right)$$ $$e\left(\frac{2}{25}\right)$$ $$e\left(\frac{18}{25}\right)$$ $$e\left(\frac{3}{25}\right)$$
$$\chi_{125}(7,\cdot)$$ 125.f 20 no $$-1$$ $$1$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$i$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{3}{20}\right)$$
$$\chi_{125}(8,\cdot)$$ 125.i 100 yes $$-1$$ $$1$$ $$e\left(\frac{3}{100}\right)$$ $$e\left(\frac{21}{100}\right)$$ $$e\left(\frac{3}{50}\right)$$ $$e\left(\frac{6}{25}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{9}{100}\right)$$ $$e\left(\frac{21}{50}\right)$$ $$e\left(\frac{7}{25}\right)$$ $$e\left(\frac{27}{100}\right)$$ $$e\left(\frac{17}{100}\right)$$
$$\chi_{125}(9,\cdot)$$ 125.h 50 yes $$1$$ $$1$$ $$e\left(\frac{7}{50}\right)$$ $$e\left(\frac{49}{50}\right)$$ $$e\left(\frac{7}{25}\right)$$ $$e\left(\frac{3}{25}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{21}{50}\right)$$ $$e\left(\frac{24}{25}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{13}{50}\right)$$ $$e\left(\frac{23}{50}\right)$$
$$\chi_{125}(11,\cdot)$$ 125.g 25 yes $$1$$ $$1$$ $$e\left(\frac{19}{25}\right)$$ $$e\left(\frac{8}{25}\right)$$ $$e\left(\frac{13}{25}\right)$$ $$e\left(\frac{2}{25}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{7}{25}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{19}{25}\right)$$ $$e\left(\frac{21}{25}\right)$$ $$e\left(\frac{16}{25}\right)$$
$$\chi_{125}(12,\cdot)$$ 125.i 100 yes $$-1$$ $$1$$ $$e\left(\frac{9}{100}\right)$$ $$e\left(\frac{63}{100}\right)$$ $$e\left(\frac{9}{50}\right)$$ $$e\left(\frac{18}{25}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{27}{100}\right)$$ $$e\left(\frac{13}{50}\right)$$ $$e\left(\frac{21}{25}\right)$$ $$e\left(\frac{81}{100}\right)$$ $$e\left(\frac{51}{100}\right)$$
$$\chi_{125}(13,\cdot)$$ 125.i 100 yes $$-1$$ $$1$$ $$e\left(\frac{39}{100}\right)$$ $$e\left(\frac{73}{100}\right)$$ $$e\left(\frac{39}{50}\right)$$ $$e\left(\frac{3}{25}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{17}{100}\right)$$ $$e\left(\frac{23}{50}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{51}{100}\right)$$ $$e\left(\frac{21}{100}\right)$$
$$\chi_{125}(14,\cdot)$$ 125.h 50 yes $$1$$ $$1$$ $$e\left(\frac{43}{50}\right)$$ $$e\left(\frac{1}{50}\right)$$ $$e\left(\frac{18}{25}\right)$$ $$e\left(\frac{22}{25}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{29}{50}\right)$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{9}{25}\right)$$ $$e\left(\frac{37}{50}\right)$$ $$e\left(\frac{27}{50}\right)$$
$$\chi_{125}(16,\cdot)$$ 125.g 25 yes $$1$$ $$1$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{7}{25}\right)$$ $$e\left(\frac{2}{25}\right)$$ $$e\left(\frac{8}{25}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{25}\right)$$ $$e\left(\frac{14}{25}\right)$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{9}{25}\right)$$ $$e\left(\frac{14}{25}\right)$$
$$\chi_{125}(17,\cdot)$$ 125.i 100 yes $$-1$$ $$1$$ $$e\left(\frac{73}{100}\right)$$ $$e\left(\frac{11}{100}\right)$$ $$e\left(\frac{23}{50}\right)$$ $$e\left(\frac{21}{25}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{19}{100}\right)$$ $$e\left(\frac{11}{50}\right)$$ $$e\left(\frac{12}{25}\right)$$ $$e\left(\frac{57}{100}\right)$$ $$e\left(\frac{47}{100}\right)$$
$$\chi_{125}(18,\cdot)$$ 125.f 20 no $$-1$$ $$1$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$-i$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{17}{20}\right)$$
$$\chi_{125}(19,\cdot)$$ 125.h 50 yes $$1$$ $$1$$ $$e\left(\frac{9}{50}\right)$$ $$e\left(\frac{13}{50}\right)$$ $$e\left(\frac{9}{25}\right)$$ $$e\left(\frac{11}{25}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{27}{50}\right)$$ $$e\left(\frac{13}{25}\right)$$ $$e\left(\frac{17}{25}\right)$$ $$e\left(\frac{31}{50}\right)$$ $$e\left(\frac{1}{50}\right)$$
$$\chi_{125}(21,\cdot)$$ 125.g 25 yes $$1$$ $$1$$ $$e\left(\frac{23}{25}\right)$$ $$e\left(\frac{11}{25}\right)$$ $$e\left(\frac{21}{25}\right)$$ $$e\left(\frac{9}{25}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{19}{25}\right)$$ $$e\left(\frac{22}{25}\right)$$ $$e\left(\frac{23}{25}\right)$$ $$e\left(\frac{7}{25}\right)$$ $$e\left(\frac{22}{25}\right)$$
$$\chi_{125}(22,\cdot)$$ 125.i 100 yes $$-1$$ $$1$$ $$e\left(\frac{77}{100}\right)$$ $$e\left(\frac{39}{100}\right)$$ $$e\left(\frac{27}{50}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{31}{100}\right)$$ $$e\left(\frac{39}{50}\right)$$ $$e\left(\frac{13}{25}\right)$$ $$e\left(\frac{93}{100}\right)$$ $$e\left(\frac{3}{100}\right)$$
$$\chi_{125}(23,\cdot)$$ 125.i 100 yes $$-1$$ $$1$$ $$e\left(\frac{31}{100}\right)$$ $$e\left(\frac{17}{100}\right)$$ $$e\left(\frac{31}{50}\right)$$ $$e\left(\frac{12}{25}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{93}{100}\right)$$ $$e\left(\frac{17}{50}\right)$$ $$e\left(\frac{14}{25}\right)$$ $$e\left(\frac{79}{100}\right)$$ $$e\left(\frac{9}{100}\right)$$
$$\chi_{125}(24,\cdot)$$ 125.e 10 no $$1$$ $$1$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$-1$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{125}(26,\cdot)$$ 125.d 5 no $$1$$ $$1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$
$$\chi_{125}(27,\cdot)$$ 125.i 100 yes $$-1$$ $$1$$ $$e\left(\frac{21}{100}\right)$$ $$e\left(\frac{47}{100}\right)$$ $$e\left(\frac{21}{50}\right)$$ $$e\left(\frac{17}{25}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{63}{100}\right)$$ $$e\left(\frac{47}{50}\right)$$ $$e\left(\frac{24}{25}\right)$$ $$e\left(\frac{89}{100}\right)$$ $$e\left(\frac{19}{100}\right)$$
$$\chi_{125}(28,\cdot)$$ 125.i 100 yes $$-1$$ $$1$$ $$e\left(\frac{87}{100}\right)$$ $$e\left(\frac{9}{100}\right)$$ $$e\left(\frac{37}{50}\right)$$ $$e\left(\frac{24}{25}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{61}{100}\right)$$ $$e\left(\frac{9}{50}\right)$$ $$e\left(\frac{3}{25}\right)$$ $$e\left(\frac{83}{100}\right)$$ $$e\left(\frac{93}{100}\right)$$
$$\chi_{125}(29,\cdot)$$ 125.h 50 yes $$1$$ $$1$$ $$e\left(\frac{31}{50}\right)$$ $$e\left(\frac{17}{50}\right)$$ $$e\left(\frac{6}{25}\right)$$ $$e\left(\frac{24}{25}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{43}{50}\right)$$ $$e\left(\frac{17}{25}\right)$$ $$e\left(\frac{3}{25}\right)$$ $$e\left(\frac{29}{50}\right)$$ $$e\left(\frac{9}{50}\right)$$
$$\chi_{125}(31,\cdot)$$ 125.g 25 yes $$1$$ $$1$$ $$e\left(\frac{12}{25}\right)$$ $$e\left(\frac{9}{25}\right)$$ $$e\left(\frac{24}{25}\right)$$ $$e\left(\frac{21}{25}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{11}{25}\right)$$ $$e\left(\frac{18}{25}\right)$$ $$e\left(\frac{12}{25}\right)$$ $$e\left(\frac{8}{25}\right)$$ $$e\left(\frac{18}{25}\right)$$
$$\chi_{125}(32,\cdot)$$ 125.f 20 no $$-1$$ $$1$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$i$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{19}{20}\right)$$
$$\chi_{125}(33,\cdot)$$ 125.i 100 yes $$-1$$ $$1$$ $$e\left(\frac{83}{100}\right)$$ $$e\left(\frac{81}{100}\right)$$ $$e\left(\frac{33}{50}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{49}{100}\right)$$ $$e\left(\frac{31}{50}\right)$$ $$e\left(\frac{2}{25}\right)$$ $$e\left(\frac{47}{100}\right)$$ $$e\left(\frac{37}{100}\right)$$
$$\chi_{125}(34,\cdot)$$ 125.h 50 yes $$1$$ $$1$$ $$e\left(\frac{37}{50}\right)$$ $$e\left(\frac{9}{50}\right)$$ $$e\left(\frac{12}{25}\right)$$ $$e\left(\frac{23}{25}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{11}{50}\right)$$ $$e\left(\frac{9}{25}\right)$$ $$e\left(\frac{6}{25}\right)$$ $$e\left(\frac{33}{50}\right)$$ $$e\left(\frac{43}{50}\right)$$
$$\chi_{125}(36,\cdot)$$ 125.g 25 yes $$1$$ $$1$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{3}{25}\right)$$ $$e\left(\frac{8}{25}\right)$$ $$e\left(\frac{7}{25}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{12}{25}\right)$$ $$e\left(\frac{6}{25}\right)$$ $$e\left(\frac{4}{25}\right)$$ $$e\left(\frac{11}{25}\right)$$ $$e\left(\frac{6}{25}\right)$$
$$\chi_{125}(37,\cdot)$$ 125.i 100 yes $$-1$$ $$1$$ $$e\left(\frac{29}{100}\right)$$ $$e\left(\frac{3}{100}\right)$$ $$e\left(\frac{29}{50}\right)$$ $$e\left(\frac{8}{25}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{87}{100}\right)$$ $$e\left(\frac{3}{50}\right)$$ $$e\left(\frac{1}{25}\right)$$ $$e\left(\frac{61}{100}\right)$$ $$e\left(\frac{31}{100}\right)$$
$$\chi_{125}(38,\cdot)$$ 125.i 100 yes $$-1$$ $$1$$ $$e\left(\frac{19}{100}\right)$$ $$e\left(\frac{33}{100}\right)$$ $$e\left(\frac{19}{50}\right)$$ $$e\left(\frac{13}{25}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{57}{100}\right)$$ $$e\left(\frac{33}{50}\right)$$ $$e\left(\frac{11}{25}\right)$$ $$e\left(\frac{71}{100}\right)$$ $$e\left(\frac{41}{100}\right)$$
$$\chi_{125}(39,\cdot)$$ 125.h 50 yes $$1$$ $$1$$ $$e\left(\frac{23}{50}\right)$$ $$e\left(\frac{11}{50}\right)$$ $$e\left(\frac{23}{25}\right)$$ $$e\left(\frac{17}{25}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{19}{50}\right)$$ $$e\left(\frac{11}{25}\right)$$ $$e\left(\frac{24}{25}\right)$$ $$e\left(\frac{7}{50}\right)$$ $$e\left(\frac{47}{50}\right)$$
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