# Properties

 Modulus $67$ Structure $$C_{66}$$ Order $66$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(67)

pari: g = idealstar(,67,2)

## Character group

 sage: G.order()  pari: g.no Order = 66 sage: H.invariants()  pari: g.cyc Structure = $$C_{66}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{67}(2,\cdot)$

## First 32 of 66 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$
$$\chi_{67}(1,\cdot)$$ 67.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{67}(2,\cdot)$$ 67.h 66 yes $$-1$$ $$1$$ $$e\left(\frac{1}{66}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{1}{33}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{23}{66}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{59}{66}\right)$$
$$\chi_{67}(3,\cdot)$$ 67.f 22 yes $$-1$$ $$1$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{19}{22}\right)$$
$$\chi_{67}(4,\cdot)$$ 67.g 33 yes $$1$$ $$1$$ $$e\left(\frac{1}{33}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{2}{33}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{23}{33}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{26}{33}\right)$$
$$\chi_{67}(5,\cdot)$$ 67.f 22 yes $$-1$$ $$1$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$
$$\chi_{67}(6,\cdot)$$ 67.g 33 yes $$1$$ $$1$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{31}{33}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{23}{33}\right)$$ $$e\left(\frac{25}{33}\right)$$
$$\chi_{67}(7,\cdot)$$ 67.h 66 yes $$-1$$ $$1$$ $$e\left(\frac{23}{66}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{23}{33}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{31}{33}\right)$$ $$e\left(\frac{1}{66}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{37}{66}\right)$$
$$\chi_{67}(8,\cdot)$$ 67.f 22 yes $$-1$$ $$1$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{15}{22}\right)$$
$$\chi_{67}(9,\cdot)$$ 67.e 11 yes $$1$$ $$1$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$
$$\chi_{67}(10,\cdot)$$ 67.g 33 yes $$1$$ $$1$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{23}{33}\right)$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{10}{33}\right)$$
$$\chi_{67}(11,\cdot)$$ 67.h 66 yes $$-1$$ $$1$$ $$e\left(\frac{59}{66}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{25}{33}\right)$$ $$e\left(\frac{37}{66}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{10}{33}\right)$$ $$e\left(\frac{49}{66}\right)$$
$$\chi_{67}(12,\cdot)$$ 67.h 66 yes $$-1$$ $$1$$ $$e\left(\frac{41}{66}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{28}{33}\right)$$ $$e\left(\frac{19}{66}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{31}{33}\right)$$ $$e\left(\frac{43}{66}\right)$$
$$\chi_{67}(13,\cdot)$$ 67.h 66 yes $$-1$$ $$1$$ $$e\left(\frac{19}{66}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{17}{33}\right)$$ $$e\left(\frac{41}{66}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{65}{66}\right)$$
$$\chi_{67}(14,\cdot)$$ 67.e 11 yes $$1$$ $$1$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$
$$\chi_{67}(15,\cdot)$$ 67.e 11 yes $$1$$ $$1$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$
$$\chi_{67}(16,\cdot)$$ 67.g 33 yes $$1$$ $$1$$ $$e\left(\frac{2}{33}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{4}{33}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{13}{33}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{19}{33}\right)$$
$$\chi_{67}(17,\cdot)$$ 67.g 33 yes $$1$$ $$1$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{31}{33}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{10}{33}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{17}{33}\right)$$ $$e\left(\frac{7}{33}\right)$$
$$\chi_{67}(18,\cdot)$$ 67.h 66 yes $$-1$$ $$1$$ $$e\left(\frac{13}{66}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{13}{33}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{35}{66}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{5}{33}\right)$$ $$e\left(\frac{41}{66}\right)$$
$$\chi_{67}(19,\cdot)$$ 67.g 33 yes $$1$$ $$1$$ $$e\left(\frac{5}{33}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{10}{33}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{2}{33}\right)$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{31}{33}\right)$$
$$\chi_{67}(20,\cdot)$$ 67.h 66 yes $$-1$$ $$1$$ $$e\left(\frac{17}{66}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{17}{33}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{10}{33}\right)$$ $$e\left(\frac{61}{66}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{4}{33}\right)$$ $$e\left(\frac{13}{66}\right)$$
$$\chi_{67}(21,\cdot)$$ 67.g 33 yes $$1$$ $$1$$ $$e\left(\frac{31}{33}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{1}{33}\right)$$ $$e\left(\frac{14}{33}\right)$$
$$\chi_{67}(22,\cdot)$$ 67.e 11 yes $$1$$ $$1$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$
$$\chi_{67}(23,\cdot)$$ 67.g 33 yes $$1$$ $$1$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{28}{33}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{25}{33}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{1}{33}\right)$$
$$\chi_{67}(24,\cdot)$$ 67.e 11 yes $$1$$ $$1$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$
$$\chi_{67}(25,\cdot)$$ 67.e 11 yes $$1$$ $$1$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$
$$\chi_{67}(26,\cdot)$$ 67.g 33 yes $$1$$ $$1$$ $$e\left(\frac{10}{33}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{4}{33}\right)$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{28}{33}\right)$$ $$e\left(\frac{29}{33}\right)$$
$$\chi_{67}(27,\cdot)$$ 67.f 22 yes $$-1$$ $$1$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{13}{22}\right)$$
$$\chi_{67}(28,\cdot)$$ 67.h 66 yes $$-1$$ $$1$$ $$e\left(\frac{25}{66}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{25}{33}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{5}{33}\right)$$ $$e\left(\frac{47}{66}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{2}{33}\right)$$ $$e\left(\frac{23}{66}\right)$$
$$\chi_{67}(29,\cdot)$$ 67.c 3 yes $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{67}(30,\cdot)$$ 67.d 6 yes $$-1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{67}(31,\cdot)$$ 67.h 66 yes $$-1$$ $$1$$ $$e\left(\frac{47}{66}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{25}{66}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{13}{33}\right)$$ $$e\left(\frac{1}{66}\right)$$
$$\chi_{67}(32,\cdot)$$ 67.h 66 yes $$-1$$ $$1$$ $$e\left(\frac{5}{66}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{5}{33}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{1}{33}\right)$$ $$e\left(\frac{49}{66}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{31}{66}\right)$$