# Properties

 Modulus $79$ Structure $$C_{78}$$ Order $78$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(79)

pari: g = idealstar(,79,2)

## Character group

 sage: G.order()  pari: g.no Order = 78 sage: H.invariants()  pari: g.cyc Structure = $$C_{78}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{79}(3,\cdot)$

## First 32 of 78 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$
$$\chi_{79}(1,\cdot)$$ 79.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{79}(2,\cdot)$$ 79.g 39 yes $$1$$ $$1$$ $$e\left(\frac{8}{39}\right)$$ $$e\left(\frac{2}{39}\right)$$ $$e\left(\frac{16}{39}\right)$$ $$e\left(\frac{7}{39}\right)$$ $$e\left(\frac{10}{39}\right)$$ $$e\left(\frac{28}{39}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{4}{39}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{19}{39}\right)$$
$$\chi_{79}(3,\cdot)$$ 79.h 78 yes $$-1$$ $$1$$ $$e\left(\frac{2}{39}\right)$$ $$e\left(\frac{1}{78}\right)$$ $$e\left(\frac{4}{39}\right)$$ $$e\left(\frac{31}{39}\right)$$ $$e\left(\frac{5}{78}\right)$$ $$e\left(\frac{53}{78}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{1}{39}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{34}{39}\right)$$
$$\chi_{79}(4,\cdot)$$ 79.g 39 yes $$1$$ $$1$$ $$e\left(\frac{16}{39}\right)$$ $$e\left(\frac{4}{39}\right)$$ $$e\left(\frac{32}{39}\right)$$ $$e\left(\frac{14}{39}\right)$$ $$e\left(\frac{20}{39}\right)$$ $$e\left(\frac{17}{39}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{8}{39}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{38}{39}\right)$$
$$\chi_{79}(5,\cdot)$$ 79.g 39 yes $$1$$ $$1$$ $$e\left(\frac{7}{39}\right)$$ $$e\left(\frac{31}{39}\right)$$ $$e\left(\frac{14}{39}\right)$$ $$e\left(\frac{11}{39}\right)$$ $$e\left(\frac{38}{39}\right)$$ $$e\left(\frac{5}{39}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{23}{39}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{2}{39}\right)$$
$$\chi_{79}(6,\cdot)$$ 79.h 78 yes $$-1$$ $$1$$ $$e\left(\frac{10}{39}\right)$$ $$e\left(\frac{5}{78}\right)$$ $$e\left(\frac{20}{39}\right)$$ $$e\left(\frac{38}{39}\right)$$ $$e\left(\frac{25}{78}\right)$$ $$e\left(\frac{31}{78}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{5}{39}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{14}{39}\right)$$
$$\chi_{79}(7,\cdot)$$ 79.h 78 yes $$-1$$ $$1$$ $$e\left(\frac{28}{39}\right)$$ $$e\left(\frac{53}{78}\right)$$ $$e\left(\frac{17}{39}\right)$$ $$e\left(\frac{5}{39}\right)$$ $$e\left(\frac{31}{78}\right)$$ $$e\left(\frac{1}{78}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{14}{39}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{8}{39}\right)$$
$$\chi_{79}(8,\cdot)$$ 79.e 13 yes $$1$$ $$1$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{6}{13}\right)$$
$$\chi_{79}(9,\cdot)$$ 79.g 39 yes $$1$$ $$1$$ $$e\left(\frac{4}{39}\right)$$ $$e\left(\frac{1}{39}\right)$$ $$e\left(\frac{8}{39}\right)$$ $$e\left(\frac{23}{39}\right)$$ $$e\left(\frac{5}{39}\right)$$ $$e\left(\frac{14}{39}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{2}{39}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{29}{39}\right)$$
$$\chi_{79}(10,\cdot)$$ 79.e 13 yes $$1$$ $$1$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{7}{13}\right)$$
$$\chi_{79}(11,\cdot)$$ 79.g 39 yes $$1$$ $$1$$ $$e\left(\frac{19}{39}\right)$$ $$e\left(\frac{34}{39}\right)$$ $$e\left(\frac{38}{39}\right)$$ $$e\left(\frac{2}{39}\right)$$ $$e\left(\frac{14}{39}\right)$$ $$e\left(\frac{8}{39}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{29}{39}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{11}{39}\right)$$
$$\chi_{79}(12,\cdot)$$ 79.f 26 yes $$-1$$ $$1$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{11}{13}\right)$$
$$\chi_{79}(13,\cdot)$$ 79.g 39 yes $$1$$ $$1$$ $$e\left(\frac{29}{39}\right)$$ $$e\left(\frac{17}{39}\right)$$ $$e\left(\frac{19}{39}\right)$$ $$e\left(\frac{1}{39}\right)$$ $$e\left(\frac{7}{39}\right)$$ $$e\left(\frac{4}{39}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{34}{39}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{25}{39}\right)$$
$$\chi_{79}(14,\cdot)$$ 79.f 26 yes $$-1$$ $$1$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{19}{26}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{17}{26}\right)$$ $$e\left(\frac{19}{26}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{9}{13}\right)$$
$$\chi_{79}(15,\cdot)$$ 79.f 26 yes $$-1$$ $$1$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{21}{26}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{1}{26}\right)$$ $$e\left(\frac{21}{26}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{12}{13}\right)$$
$$\chi_{79}(16,\cdot)$$ 79.g 39 yes $$1$$ $$1$$ $$e\left(\frac{32}{39}\right)$$ $$e\left(\frac{8}{39}\right)$$ $$e\left(\frac{25}{39}\right)$$ $$e\left(\frac{28}{39}\right)$$ $$e\left(\frac{1}{39}\right)$$ $$e\left(\frac{34}{39}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{16}{39}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{37}{39}\right)$$
$$\chi_{79}(17,\cdot)$$ 79.f 26 yes $$-1$$ $$1$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{7}{26}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{9}{26}\right)$$ $$e\left(\frac{7}{26}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{4}{13}\right)$$
$$\chi_{79}(18,\cdot)$$ 79.e 13 yes $$1$$ $$1$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{3}{13}\right)$$
$$\chi_{79}(19,\cdot)$$ 79.g 39 yes $$1$$ $$1$$ $$e\left(\frac{25}{39}\right)$$ $$e\left(\frac{16}{39}\right)$$ $$e\left(\frac{11}{39}\right)$$ $$e\left(\frac{17}{39}\right)$$ $$e\left(\frac{2}{39}\right)$$ $$e\left(\frac{29}{39}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{32}{39}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{35}{39}\right)$$
$$\chi_{79}(20,\cdot)$$ 79.g 39 yes $$1$$ $$1$$ $$e\left(\frac{23}{39}\right)$$ $$e\left(\frac{35}{39}\right)$$ $$e\left(\frac{7}{39}\right)$$ $$e\left(\frac{25}{39}\right)$$ $$e\left(\frac{19}{39}\right)$$ $$e\left(\frac{22}{39}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{31}{39}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{1}{39}\right)$$
$$\chi_{79}(21,\cdot)$$ 79.e 13 yes $$1$$ $$1$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{1}{13}\right)$$
$$\chi_{79}(22,\cdot)$$ 79.e 13 yes $$1$$ $$1$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{10}{13}\right)$$
$$\chi_{79}(23,\cdot)$$ 79.c 3 yes $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{79}(24,\cdot)$$ 79.d 6 yes $$-1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{79}(25,\cdot)$$ 79.g 39 yes $$1$$ $$1$$ $$e\left(\frac{14}{39}\right)$$ $$e\left(\frac{23}{39}\right)$$ $$e\left(\frac{28}{39}\right)$$ $$e\left(\frac{22}{39}\right)$$ $$e\left(\frac{37}{39}\right)$$ $$e\left(\frac{10}{39}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{7}{39}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{4}{39}\right)$$
$$\chi_{79}(26,\cdot)$$ 79.g 39 yes $$1$$ $$1$$ $$e\left(\frac{37}{39}\right)$$ $$e\left(\frac{19}{39}\right)$$ $$e\left(\frac{35}{39}\right)$$ $$e\left(\frac{8}{39}\right)$$ $$e\left(\frac{17}{39}\right)$$ $$e\left(\frac{32}{39}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{38}{39}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{5}{39}\right)$$
$$\chi_{79}(27,\cdot)$$ 79.f 26 yes $$-1$$ $$1$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{1}{26}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{5}{26}\right)$$ $$e\left(\frac{1}{26}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{8}{13}\right)$$
$$\chi_{79}(28,\cdot)$$ 79.h 78 yes $$-1$$ $$1$$ $$e\left(\frac{5}{39}\right)$$ $$e\left(\frac{61}{78}\right)$$ $$e\left(\frac{10}{39}\right)$$ $$e\left(\frac{19}{39}\right)$$ $$e\left(\frac{71}{78}\right)$$ $$e\left(\frac{35}{78}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{22}{39}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{7}{39}\right)$$
$$\chi_{79}(29,\cdot)$$ 79.h 78 yes $$-1$$ $$1$$ $$e\left(\frac{22}{39}\right)$$ $$e\left(\frac{11}{78}\right)$$ $$e\left(\frac{5}{39}\right)$$ $$e\left(\frac{29}{39}\right)$$ $$e\left(\frac{55}{78}\right)$$ $$e\left(\frac{37}{78}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{11}{39}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{23}{39}\right)$$
$$\chi_{79}(30,\cdot)$$ 79.h 78 yes $$-1$$ $$1$$ $$e\left(\frac{17}{39}\right)$$ $$e\left(\frac{67}{78}\right)$$ $$e\left(\frac{34}{39}\right)$$ $$e\left(\frac{10}{39}\right)$$ $$e\left(\frac{23}{78}\right)$$ $$e\left(\frac{41}{78}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{28}{39}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{16}{39}\right)$$
$$\chi_{79}(31,\cdot)$$ 79.g 39 yes $$1$$ $$1$$ $$e\left(\frac{34}{39}\right)$$ $$e\left(\frac{28}{39}\right)$$ $$e\left(\frac{29}{39}\right)$$ $$e\left(\frac{20}{39}\right)$$ $$e\left(\frac{23}{39}\right)$$ $$e\left(\frac{2}{39}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{17}{39}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{32}{39}\right)$$
$$\chi_{79}(32,\cdot)$$ 79.g 39 yes $$1$$ $$1$$ $$e\left(\frac{1}{39}\right)$$ $$e\left(\frac{10}{39}\right)$$ $$e\left(\frac{2}{39}\right)$$ $$e\left(\frac{35}{39}\right)$$ $$e\left(\frac{11}{39}\right)$$ $$e\left(\frac{23}{39}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{20}{39}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{17}{39}\right)$$
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