# Properties

 Modulus $207$ Structure $$C_{2}\times C_{66}$$ Order $132$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(207)

pari: g = idealstar(,207,2)

## Character group

 sage: G.order()  pari: g.no Order = 132 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{66}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{207}(47,\cdot)$, $\chi_{207}(28,\cdot)$

## First 32 of 132 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 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$
$$\chi_{207}(1,\cdot)$$ 207.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{207}(2,\cdot)$$ 207.n 66 yes $$-1$$ $$1$$ $$e\left(\frac{23}{66}\right)$$ $$e\left(\frac{23}{33}\right)$$ $$e\left(\frac{61}{66}\right)$$ $$e\left(\frac{13}{33}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{65}{66}\right)$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{49}{66}\right)$$ $$e\left(\frac{13}{33}\right)$$
$$\chi_{207}(4,\cdot)$$ 207.m 33 yes $$1$$ $$1$$ $$e\left(\frac{23}{33}\right)$$ $$e\left(\frac{13}{33}\right)$$ $$e\left(\frac{28}{33}\right)$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{26}{33}\right)$$
$$\chi_{207}(5,\cdot)$$ 207.o 66 yes $$1$$ $$1$$ $$e\left(\frac{61}{66}\right)$$ $$e\left(\frac{28}{33}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{13}{66}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{10}{33}\right)$$ $$e\left(\frac{4}{33}\right)$$ $$e\left(\frac{23}{33}\right)$$
$$\chi_{207}(7,\cdot)$$ 207.p 66 yes $$-1$$ $$1$$ $$e\left(\frac{13}{33}\right)$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{13}{66}\right)$$ $$e\left(\frac{5}{66}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{29}{66}\right)$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{31}{66}\right)$$ $$e\left(\frac{19}{33}\right)$$
$$\chi_{207}(8,\cdot)$$ 207.l 22 no $$-1$$ $$1$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$
$$\chi_{207}(10,\cdot)$$ 207.j 22 no $$-1$$ $$1$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$
$$\chi_{207}(11,\cdot)$$ 207.o 66 yes $$1$$ $$1$$ $$e\left(\frac{65}{66}\right)$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{29}{66}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{28}{33}\right)$$ $$e\left(\frac{2}{33}\right)$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{31}{33}\right)$$
$$\chi_{207}(13,\cdot)$$ 207.m 33 yes $$1$$ $$1$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{10}{33}\right)$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{2}{33}\right)$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{1}{33}\right)$$ $$e\left(\frac{14}{33}\right)$$
$$\chi_{207}(14,\cdot)$$ 207.o 66 yes $$1$$ $$1$$ $$e\left(\frac{49}{66}\right)$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{4}{33}\right)$$ $$e\left(\frac{31}{66}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{1}{33}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{32}{33}\right)$$
$$\chi_{207}(16,\cdot)$$ 207.m 33 yes $$1$$ $$1$$ $$e\left(\frac{13}{33}\right)$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{23}{33}\right)$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{31}{33}\right)$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{19}{33}\right)$$
$$\chi_{207}(17,\cdot)$$ 207.k 22 no $$1$$ $$1$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$
$$\chi_{207}(19,\cdot)$$ 207.j 22 no $$-1$$ $$1$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$
$$\chi_{207}(20,\cdot)$$ 207.o 66 yes $$1$$ $$1$$ $$e\left(\frac{41}{66}\right)$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{2}{33}\right)$$ $$e\left(\frac{65}{66}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{17}{33}\right)$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{16}{33}\right)$$
$$\chi_{207}(22,\cdot)$$ 207.f 6 yes $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{207}(25,\cdot)$$ 207.m 33 yes $$1$$ $$1$$ $$e\left(\frac{28}{33}\right)$$ $$e\left(\frac{23}{33}\right)$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{13}{33}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{13}{33}\right)$$
$$\chi_{207}(26,\cdot)$$ 207.l 22 no $$-1$$ $$1$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$
$$\chi_{207}(28,\cdot)$$ 207.j 22 no $$-1$$ $$1$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$
$$\chi_{207}(29,\cdot)$$ 207.n 66 yes $$-1$$ $$1$$ $$e\left(\frac{53}{66}\right)$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{43}{66}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{35}{66}\right)$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{1}{66}\right)$$ $$e\left(\frac{7}{33}\right)$$
$$\chi_{207}(31,\cdot)$$ 207.m 33 yes $$1$$ $$1$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{25}{33}\right)$$ $$e\left(\frac{31}{33}\right)$$ $$e\left(\frac{17}{33}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{13}{33}\right)$$ $$e\left(\frac{17}{33}\right)$$
$$\chi_{207}(32,\cdot)$$ 207.n 66 yes $$-1$$ $$1$$ $$e\left(\frac{49}{66}\right)$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{41}{66}\right)$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{61}{66}\right)$$ $$e\left(\frac{1}{33}\right)$$ $$e\left(\frac{47}{66}\right)$$ $$e\left(\frac{32}{33}\right)$$
$$\chi_{207}(34,\cdot)$$ 207.p 66 yes $$-1$$ $$1$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{49}{66}\right)$$ $$e\left(\frac{29}{66}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{23}{66}\right)$$ $$e\left(\frac{2}{33}\right)$$ $$e\left(\frac{61}{66}\right)$$ $$e\left(\frac{31}{33}\right)$$
$$\chi_{207}(35,\cdot)$$ 207.l 22 no $$-1$$ $$1$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$
$$\chi_{207}(37,\cdot)$$ 207.j 22 no $$-1$$ $$1$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$
$$\chi_{207}(38,\cdot)$$ 207.o 66 yes $$1$$ $$1$$ $$e\left(\frac{47}{66}\right)$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{23}{66}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{4}{33}\right)$$ $$e\left(\frac{5}{33}\right)$$ $$e\left(\frac{2}{33}\right)$$ $$e\left(\frac{28}{33}\right)$$
$$\chi_{207}(40,\cdot)$$ 207.p 66 yes $$-1$$ $$1$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{31}{33}\right)$$ $$e\left(\frac{65}{66}\right)$$ $$e\left(\frac{25}{66}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{13}{66}\right)$$ $$e\left(\frac{4}{33}\right)$$ $$e\left(\frac{23}{66}\right)$$ $$e\left(\frac{29}{33}\right)$$
$$\chi_{207}(41,\cdot)$$ 207.n 66 yes $$-1$$ $$1$$ $$e\left(\frac{61}{66}\right)$$ $$e\left(\frac{28}{33}\right)$$ $$e\left(\frac{47}{66}\right)$$ $$e\left(\frac{23}{33}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{49}{66}\right)$$ $$e\left(\frac{10}{33}\right)$$ $$e\left(\frac{41}{66}\right)$$ $$e\left(\frac{23}{33}\right)$$
$$\chi_{207}(43,\cdot)$$ 207.p 66 yes $$-1$$ $$1$$ $$e\left(\frac{4}{33}\right)$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{37}{66}\right)$$ $$e\left(\frac{65}{66}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{47}{66}\right)$$ $$e\left(\frac{17}{33}\right)$$ $$e\left(\frac{7}{66}\right)$$ $$e\left(\frac{16}{33}\right)$$
$$\chi_{207}(44,\cdot)$$ 207.k 22 no $$1$$ $$1$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$
$$\chi_{207}(47,\cdot)$$ 207.h 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{207}(49,\cdot)$$ 207.m 33 yes $$1$$ $$1$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{13}{33}\right)$$ $$e\left(\frac{5}{33}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{28}{33}\right)$$ $$e\left(\frac{31}{33}\right)$$ $$e\left(\frac{5}{33}\right)$$
$$\chi_{207}(50,\cdot)$$ 207.n 66 yes $$-1$$ $$1$$ $$e\left(\frac{13}{66}\right)$$ $$e\left(\frac{13}{33}\right)$$ $$e\left(\frac{23}{66}\right)$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{31}{66}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{65}{66}\right)$$ $$e\left(\frac{26}{33}\right)$$