# Properties

 Modulus $177$ Structure $$C_{2}\times C_{58}$$ Order $116$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(177)

pari: g = idealstar(,177,2)

## Character group

 sage: G.order()  pari: g.no Order = 116 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{58}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{177}(119,\cdot)$, $\chi_{177}(61,\cdot)$

## First 32 of 116 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 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$
$$\chi_{177}(1,\cdot)$$ 177.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{177}(2,\cdot)$$ 177.f 58 yes $$1$$ $$1$$ $$e\left(\frac{15}{29}\right)$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{35}{58}\right)$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{7}{58}\right)$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{45}{58}\right)$$ $$e\left(\frac{24}{29}\right)$$ $$e\left(\frac{2}{29}\right)$$
$$\chi_{177}(4,\cdot)$$ 177.e 29 no $$1$$ $$1$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{18}{29}\right)$$ $$e\left(\frac{3}{29}\right)$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{25}{29}\right)$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{4}{29}\right)$$
$$\chi_{177}(5,\cdot)$$ 177.h 58 yes $$-1$$ $$1$$ $$e\left(\frac{35}{58}\right)$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{7}{58}\right)$$ $$e\left(\frac{25}{29}\right)$$ $$e\left(\frac{47}{58}\right)$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{5}{58}\right)$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{27}{58}\right)$$ $$e\left(\frac{12}{29}\right)$$
$$\chi_{177}(7,\cdot)$$ 177.e 29 no $$1$$ $$1$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{18}{29}\right)$$ $$e\left(\frac{25}{29}\right)$$ $$e\left(\frac{17}{29}\right)$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{22}{29}\right)$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{7}{29}\right)$$
$$\chi_{177}(8,\cdot)$$ 177.f 58 yes $$1$$ $$1$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{3}{29}\right)$$ $$e\left(\frac{47}{58}\right)$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{21}{58}\right)$$ $$e\left(\frac{23}{29}\right)$$ $$e\left(\frac{19}{58}\right)$$ $$e\left(\frac{14}{29}\right)$$ $$e\left(\frac{6}{29}\right)$$
$$\chi_{177}(10,\cdot)$$ 177.g 58 no $$-1$$ $$1$$ $$e\left(\frac{7}{58}\right)$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{21}{58}\right)$$ $$e\left(\frac{49}{58}\right)$$ $$e\left(\frac{1}{58}\right)$$ $$e\left(\frac{25}{58}\right)$$ $$e\left(\frac{17}{58}\right)$$ $$e\left(\frac{14}{29}\right)$$
$$\chi_{177}(11,\cdot)$$ 177.f 58 yes $$1$$ $$1$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{25}{29}\right)$$ $$e\left(\frac{5}{58}\right)$$ $$e\left(\frac{22}{29}\right)$$ $$e\left(\frac{23}{29}\right)$$ $$e\left(\frac{1}{58}\right)$$ $$e\left(\frac{8}{29}\right)$$ $$e\left(\frac{23}{58}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{21}{29}\right)$$
$$\chi_{177}(13,\cdot)$$ 177.g 58 no $$-1$$ $$1$$ $$e\left(\frac{45}{58}\right)$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{19}{58}\right)$$ $$e\left(\frac{25}{58}\right)$$ $$e\left(\frac{23}{58}\right)$$ $$e\left(\frac{53}{58}\right)$$ $$e\left(\frac{43}{58}\right)$$ $$e\left(\frac{3}{29}\right)$$
$$\chi_{177}(14,\cdot)$$ 177.f 58 yes $$1$$ $$1$$ $$e\left(\frac{24}{29}\right)$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{27}{58}\right)$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{14}{29}\right)$$ $$e\left(\frac{17}{58}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{43}{58}\right)$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{9}{29}\right)$$
$$\chi_{177}(16,\cdot)$$ 177.e 29 no $$1$$ $$1$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{14}{29}\right)$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{3}{29}\right)$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{8}{29}\right)$$
$$\chi_{177}(17,\cdot)$$ 177.h 58 yes $$-1$$ $$1$$ $$e\left(\frac{11}{58}\right)$$ $$e\left(\frac{11}{29}\right)$$ $$e\left(\frac{37}{58}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{33}{58}\right)$$ $$e\left(\frac{24}{29}\right)$$ $$e\left(\frac{43}{58}\right)$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{35}{58}\right)$$ $$e\left(\frac{22}{29}\right)$$
$$\chi_{177}(19,\cdot)$$ 177.e 29 no $$1$$ $$1$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{23}{29}\right)$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{17}{29}\right)$$ $$e\left(\frac{11}{29}\right)$$ $$e\left(\frac{14}{29}\right)$$ $$e\left(\frac{13}{29}\right)$$ $$e\left(\frac{18}{29}\right)$$
$$\chi_{177}(20,\cdot)$$ 177.h 58 yes $$-1$$ $$1$$ $$e\left(\frac{37}{58}\right)$$ $$e\left(\frac{8}{29}\right)$$ $$e\left(\frac{19}{58}\right)$$ $$e\left(\frac{14}{29}\right)$$ $$e\left(\frac{53}{58}\right)$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{55}{58}\right)$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{7}{58}\right)$$ $$e\left(\frac{16}{29}\right)$$
$$\chi_{177}(22,\cdot)$$ 177.e 29 no $$1$$ $$1$$ $$e\left(\frac{13}{29}\right)$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{10}{29}\right)$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{15}{29}\right)$$ $$e\left(\frac{23}{29}\right)$$
$$\chi_{177}(23,\cdot)$$ 177.f 58 yes $$1$$ $$1$$ $$e\left(\frac{22}{29}\right)$$ $$e\left(\frac{15}{29}\right)$$ $$e\left(\frac{3}{58}\right)$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{8}{29}\right)$$ $$e\left(\frac{47}{58}\right)$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{37}{58}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{1}{29}\right)$$
$$\chi_{177}(25,\cdot)$$ 177.e 29 no $$1$$ $$1$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{18}{29}\right)$$ $$e\left(\frac{13}{29}\right)$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{24}{29}\right)$$
$$\chi_{177}(26,\cdot)$$ 177.h 58 yes $$-1$$ $$1$$ $$e\left(\frac{17}{58}\right)$$ $$e\left(\frac{17}{29}\right)$$ $$e\left(\frac{15}{58}\right)$$ $$e\left(\frac{8}{29}\right)$$ $$e\left(\frac{51}{58}\right)$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{19}{58}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{33}{58}\right)$$ $$e\left(\frac{5}{29}\right)$$
$$\chi_{177}(28,\cdot)$$ 177.e 29 no $$1$$ $$1$$ $$e\left(\frac{10}{29}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{18}{29}\right)$$ $$e\left(\frac{15}{29}\right)$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{11}{29}\right)$$
$$\chi_{177}(29,\cdot)$$ 177.h 58 yes $$-1$$ $$1$$ $$e\left(\frac{57}{58}\right)$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{23}{58}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{55}{58}\right)$$ $$e\left(\frac{11}{29}\right)$$ $$e\left(\frac{33}{58}\right)$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{39}{58}\right)$$ $$e\left(\frac{27}{29}\right)$$
$$\chi_{177}(31,\cdot)$$ 177.g 58 no $$-1$$ $$1$$ $$e\left(\frac{49}{58}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{31}{58}\right)$$ $$e\left(\frac{53}{58}\right)$$ $$e\left(\frac{7}{58}\right)$$ $$e\left(\frac{1}{58}\right)$$ $$e\left(\frac{3}{58}\right)$$ $$e\left(\frac{11}{29}\right)$$
$$\chi_{177}(32,\cdot)$$ 177.f 58 yes $$1$$ $$1$$ $$e\left(\frac{17}{29}\right)$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{1}{58}\right)$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{22}{29}\right)$$ $$e\left(\frac{35}{58}\right)$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{51}{58}\right)$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{10}{29}\right)$$
$$\chi_{177}(34,\cdot)$$ 177.g 58 no $$-1$$ $$1$$ $$e\left(\frac{41}{58}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{7}{58}\right)$$ $$e\left(\frac{55}{58}\right)$$ $$e\left(\frac{39}{58}\right)$$ $$e\left(\frac{47}{58}\right)$$ $$e\left(\frac{25}{58}\right)$$ $$e\left(\frac{24}{29}\right)$$
$$\chi_{177}(35,\cdot)$$ 177.h 58 yes $$-1$$ $$1$$ $$e\left(\frac{53}{58}\right)$$ $$e\left(\frac{24}{29}\right)$$ $$e\left(\frac{57}{58}\right)$$ $$e\left(\frac{13}{29}\right)$$ $$e\left(\frac{43}{58}\right)$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{49}{58}\right)$$ $$e\left(\frac{18}{29}\right)$$ $$e\left(\frac{21}{58}\right)$$ $$e\left(\frac{19}{29}\right)$$
$$\chi_{177}(37,\cdot)$$ 177.g 58 no $$-1$$ $$1$$ $$e\left(\frac{55}{58}\right)$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{49}{58}\right)$$ $$e\left(\frac{37}{58}\right)$$ $$e\left(\frac{41}{58}\right)$$ $$e\left(\frac{39}{58}\right)$$ $$e\left(\frac{1}{58}\right)$$ $$e\left(\frac{23}{29}\right)$$
$$\chi_{177}(38,\cdot)$$ 177.f 58 yes $$1$$ $$1$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{10}{29}\right)$$ $$e\left(\frac{31}{58}\right)$$ $$e\left(\frac{3}{29}\right)$$ $$e\left(\frac{15}{29}\right)$$ $$e\left(\frac{41}{58}\right)$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{15}{58}\right)$$ $$e\left(\frac{8}{29}\right)$$ $$e\left(\frac{20}{29}\right)$$
$$\chi_{177}(40,\cdot)$$ 177.g 58 no $$-1$$ $$1$$ $$e\left(\frac{9}{58}\right)$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{23}{29}\right)$$ $$e\left(\frac{27}{58}\right)$$ $$e\left(\frac{5}{58}\right)$$ $$e\left(\frac{51}{58}\right)$$ $$e\left(\frac{57}{58}\right)$$ $$e\left(\frac{55}{58}\right)$$ $$e\left(\frac{18}{29}\right)$$
$$\chi_{177}(41,\cdot)$$ 177.h 58 yes $$-1$$ $$1$$ $$e\left(\frac{43}{58}\right)$$ $$e\left(\frac{14}{29}\right)$$ $$e\left(\frac{55}{58}\right)$$ $$e\left(\frac{10}{29}\right)$$ $$e\left(\frac{13}{58}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{31}{58}\right)$$ $$e\left(\frac{25}{29}\right)$$ $$e\left(\frac{5}{58}\right)$$ $$e\left(\frac{28}{29}\right)$$
$$\chi_{177}(43,\cdot)$$ 177.g 58 no $$-1$$ $$1$$ $$e\left(\frac{33}{58}\right)$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{41}{58}\right)$$ $$e\left(\frac{57}{58}\right)$$ $$e\left(\frac{13}{58}\right)$$ $$e\left(\frac{35}{58}\right)$$ $$e\left(\frac{47}{58}\right)$$ $$e\left(\frac{8}{29}\right)$$
$$\chi_{177}(44,\cdot)$$ 177.f 58 yes $$1$$ $$1$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{17}{58}\right)$$ $$e\left(\frac{11}{29}\right)$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{15}{58}\right)$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{55}{58}\right)$$ $$e\left(\frac{10}{29}\right)$$ $$e\left(\frac{25}{29}\right)$$
$$\chi_{177}(46,\cdot)$$ 177.e 29 no $$1$$ $$1$$ $$e\left(\frac{8}{29}\right)$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{24}{29}\right)$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{3}{29}\right)$$
$$\chi_{177}(47,\cdot)$$ 177.f 58 yes $$1$$ $$1$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{23}{29}\right)$$ $$e\left(\frac{51}{58}\right)$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{45}{58}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{49}{58}\right)$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{17}{29}\right)$$