# Properties

 Modulus $507$ Structure $$C_{2}\times C_{156}$$ Order $312$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(507)

pari: g = idealstar(,507,2)

## Character group

 sage: G.order()  pari: g.no Order = 312 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{156}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{507}(170,\cdot)$, $\chi_{507}(340,\cdot)$

## First 32 of 312 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 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$14$$ $$16$$ $$17$$
$$\chi_{507}(1,\cdot)$$ 507.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{507}(2,\cdot)$$ 507.x 156 yes $$1$$ $$1$$ $$e\left(\frac{79}{156}\right)$$ $$e\left(\frac{1}{78}\right)$$ $$e\left(\frac{29}{52}\right)$$ $$e\left(\frac{107}{156}\right)$$ $$e\left(\frac{27}{52}\right)$$ $$e\left(\frac{5}{78}\right)$$ $$e\left(\frac{25}{156}\right)$$ $$e\left(\frac{5}{26}\right)$$ $$e\left(\frac{1}{39}\right)$$ $$e\left(\frac{17}{39}\right)$$
$$\chi_{507}(4,\cdot)$$ 507.t 78 no $$1$$ $$1$$ $$e\left(\frac{1}{78}\right)$$ $$e\left(\frac{1}{39}\right)$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{29}{78}\right)$$ $$e\left(\frac{1}{26}\right)$$ $$e\left(\frac{5}{39}\right)$$ $$e\left(\frac{25}{78}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{2}{39}\right)$$ $$e\left(\frac{34}{39}\right)$$
$$\chi_{507}(5,\cdot)$$ 507.s 52 yes $$1$$ $$1$$ $$e\left(\frac{29}{52}\right)$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{1}{52}\right)$$ $$e\left(\frac{9}{52}\right)$$ $$e\left(\frac{35}{52}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$e\left(\frac{23}{52}\right)$$ $$e\left(\frac{19}{26}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{12}{13}\right)$$
$$\chi_{507}(7,\cdot)$$ 507.w 156 no $$-1$$ $$1$$ $$e\left(\frac{107}{156}\right)$$ $$e\left(\frac{29}{78}\right)$$ $$e\left(\frac{9}{52}\right)$$ $$e\left(\frac{61}{156}\right)$$ $$e\left(\frac{3}{52}\right)$$ $$e\left(\frac{67}{78}\right)$$ $$e\left(\frac{101}{156}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{29}{39}\right)$$ $$e\left(\frac{11}{78}\right)$$
$$\chi_{507}(8,\cdot)$$ 507.s 52 yes $$1$$ $$1$$ $$e\left(\frac{27}{52}\right)$$ $$e\left(\frac{1}{26}\right)$$ $$e\left(\frac{35}{52}\right)$$ $$e\left(\frac{3}{52}\right)$$ $$e\left(\frac{29}{52}\right)$$ $$e\left(\frac{5}{26}\right)$$ $$e\left(\frac{25}{52}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{4}{13}\right)$$
$$\chi_{507}(10,\cdot)$$ 507.t 78 no $$1$$ $$1$$ $$e\left(\frac{5}{78}\right)$$ $$e\left(\frac{5}{39}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$e\left(\frac{67}{78}\right)$$ $$e\left(\frac{5}{26}\right)$$ $$e\left(\frac{25}{39}\right)$$ $$e\left(\frac{47}{78}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{10}{39}\right)$$ $$e\left(\frac{14}{39}\right)$$
$$\chi_{507}(11,\cdot)$$ 507.x 156 yes $$1$$ $$1$$ $$e\left(\frac{25}{156}\right)$$ $$e\left(\frac{25}{78}\right)$$ $$e\left(\frac{23}{52}\right)$$ $$e\left(\frac{101}{156}\right)$$ $$e\left(\frac{25}{52}\right)$$ $$e\left(\frac{47}{78}\right)$$ $$e\left(\frac{79}{156}\right)$$ $$e\left(\frac{21}{26}\right)$$ $$e\left(\frac{25}{39}\right)$$ $$e\left(\frac{35}{39}\right)$$
$$\chi_{507}(14,\cdot)$$ 507.o 26 yes $$-1$$ $$1$$ $$e\left(\frac{5}{26}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{19}{26}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{21}{26}\right)$$ $$e\left(\frac{7}{26}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{15}{26}\right)$$
$$\chi_{507}(16,\cdot)$$ 507.q 39 no $$1$$ $$1$$ $$e\left(\frac{1}{39}\right)$$ $$e\left(\frac{2}{39}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{29}{39}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{10}{39}\right)$$ $$e\left(\frac{25}{39}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{4}{39}\right)$$ $$e\left(\frac{29}{39}\right)$$
$$\chi_{507}(17,\cdot)$$ 507.v 78 yes $$-1$$ $$1$$ $$e\left(\frac{17}{39}\right)$$ $$e\left(\frac{34}{39}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{11}{78}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{14}{39}\right)$$ $$e\left(\frac{35}{39}\right)$$ $$e\left(\frac{15}{26}\right)$$ $$e\left(\frac{29}{39}\right)$$ $$e\left(\frac{11}{78}\right)$$
$$\chi_{507}(19,\cdot)$$ 507.l 12 no $$-1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-i$$ $$e\left(\frac{7}{12}\right)$$ $$i$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{507}(20,\cdot)$$ 507.x 156 yes $$1$$ $$1$$ $$e\left(\frac{89}{156}\right)$$ $$e\left(\frac{11}{78}\right)$$ $$e\left(\frac{7}{52}\right)$$ $$e\left(\frac{85}{156}\right)$$ $$e\left(\frac{37}{52}\right)$$ $$e\left(\frac{55}{78}\right)$$ $$e\left(\frac{119}{156}\right)$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{11}{39}\right)$$ $$e\left(\frac{31}{39}\right)$$
$$\chi_{507}(22,\cdot)$$ 507.e 3 no $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{507}(23,\cdot)$$ 507.h 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{507}(25,\cdot)$$ 507.p 26 no $$1$$ $$1$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{1}{26}\right)$$ $$e\left(\frac{9}{26}\right)$$ $$e\left(\frac{9}{26}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{23}{26}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{11}{13}\right)$$
$$\chi_{507}(28,\cdot)$$ 507.w 156 no $$-1$$ $$1$$ $$e\left(\frac{109}{156}\right)$$ $$e\left(\frac{31}{78}\right)$$ $$e\left(\frac{15}{52}\right)$$ $$e\left(\frac{119}{156}\right)$$ $$e\left(\frac{5}{52}\right)$$ $$e\left(\frac{77}{78}\right)$$ $$e\left(\frac{151}{156}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{31}{39}\right)$$ $$e\left(\frac{1}{78}\right)$$
$$\chi_{507}(29,\cdot)$$ 507.u 78 yes $$-1$$ $$1$$ $$e\left(\frac{59}{78}\right)$$ $$e\left(\frac{20}{39}\right)$$ $$e\left(\frac{21}{26}\right)$$ $$e\left(\frac{17}{39}\right)$$ $$e\left(\frac{7}{26}\right)$$ $$e\left(\frac{22}{39}\right)$$ $$e\left(\frac{71}{78}\right)$$ $$e\left(\frac{5}{26}\right)$$ $$e\left(\frac{1}{39}\right)$$ $$e\left(\frac{73}{78}\right)$$
$$\chi_{507}(31,\cdot)$$ 507.r 52 no $$-1$$ $$1$$ $$e\left(\frac{7}{52}\right)$$ $$e\left(\frac{7}{26}\right)$$ $$e\left(\frac{11}{52}\right)$$ $$e\left(\frac{21}{52}\right)$$ $$e\left(\frac{21}{52}\right)$$ $$e\left(\frac{9}{26}\right)$$ $$e\left(\frac{45}{52}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{17}{26}\right)$$
$$\chi_{507}(32,\cdot)$$ 507.x 156 yes $$1$$ $$1$$ $$e\left(\frac{83}{156}\right)$$ $$e\left(\frac{5}{78}\right)$$ $$e\left(\frac{41}{52}\right)$$ $$e\left(\frac{67}{156}\right)$$ $$e\left(\frac{31}{52}\right)$$ $$e\left(\frac{25}{78}\right)$$ $$e\left(\frac{125}{156}\right)$$ $$e\left(\frac{25}{26}\right)$$ $$e\left(\frac{5}{39}\right)$$ $$e\left(\frac{7}{39}\right)$$
$$\chi_{507}(34,\cdot)$$ 507.r 52 no $$-1$$ $$1$$ $$e\left(\frac{49}{52}\right)$$ $$e\left(\frac{23}{26}\right)$$ $$e\left(\frac{25}{52}\right)$$ $$e\left(\frac{43}{52}\right)$$ $$e\left(\frac{43}{52}\right)$$ $$e\left(\frac{11}{26}\right)$$ $$e\left(\frac{3}{52}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{15}{26}\right)$$
$$\chi_{507}(35,\cdot)$$ 507.u 78 yes $$-1$$ $$1$$ $$e\left(\frac{19}{78}\right)$$ $$e\left(\frac{19}{39}\right)$$ $$e\left(\frac{5}{26}\right)$$ $$e\left(\frac{22}{39}\right)$$ $$e\left(\frac{19}{26}\right)$$ $$e\left(\frac{17}{39}\right)$$ $$e\left(\frac{7}{78}\right)$$ $$e\left(\frac{21}{26}\right)$$ $$e\left(\frac{38}{39}\right)$$ $$e\left(\frac{5}{78}\right)$$
$$\chi_{507}(37,\cdot)$$ 507.w 156 no $$-1$$ $$1$$ $$e\left(\frac{151}{156}\right)$$ $$e\left(\frac{73}{78}\right)$$ $$e\left(\frac{37}{52}\right)$$ $$e\left(\frac{89}{156}\right)$$ $$e\left(\frac{47}{52}\right)$$ $$e\left(\frac{53}{78}\right)$$ $$e\left(\frac{109}{156}\right)$$ $$e\left(\frac{7}{13}\right)$$ $$e\left(\frac{34}{39}\right)$$ $$e\left(\frac{25}{78}\right)$$
$$\chi_{507}(38,\cdot)$$ 507.n 26 yes $$-1$$ $$1$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{11}{13}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{7}{26}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{5}{26}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{7}{26}\right)$$
$$\chi_{507}(40,\cdot)$$ 507.m 13 no $$1$$ $$1$$ $$e\left(\frac{1}{13}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{3}{13}\right)$$ $$e\left(\frac{10}{13}\right)$$ $$e\left(\frac{12}{13}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{4}{13}\right)$$ $$e\left(\frac{3}{13}\right)$$
$$\chi_{507}(41,\cdot)$$ 507.x 156 yes $$1$$ $$1$$ $$e\left(\frac{7}{156}\right)$$ $$e\left(\frac{7}{78}\right)$$ $$e\left(\frac{21}{52}\right)$$ $$e\left(\frac{47}{156}\right)$$ $$e\left(\frac{7}{52}\right)$$ $$e\left(\frac{35}{78}\right)$$ $$e\left(\frac{97}{156}\right)$$ $$e\left(\frac{9}{26}\right)$$ $$e\left(\frac{7}{39}\right)$$ $$e\left(\frac{2}{39}\right)$$
$$\chi_{507}(43,\cdot)$$ 507.t 78 no $$1$$ $$1$$ $$e\left(\frac{61}{78}\right)$$ $$e\left(\frac{22}{39}\right)$$ $$e\left(\frac{1}{26}\right)$$ $$e\left(\frac{53}{78}\right)$$ $$e\left(\frac{9}{26}\right)$$ $$e\left(\frac{32}{39}\right)$$ $$e\left(\frac{43}{78}\right)$$ $$e\left(\frac{6}{13}\right)$$ $$e\left(\frac{5}{39}\right)$$ $$e\left(\frac{7}{39}\right)$$
$$\chi_{507}(44,\cdot)$$ 507.s 52 yes $$1$$ $$1$$ $$e\left(\frac{9}{52}\right)$$ $$e\left(\frac{9}{26}\right)$$ $$e\left(\frac{29}{52}\right)$$ $$e\left(\frac{1}{52}\right)$$ $$e\left(\frac{27}{52}\right)$$ $$e\left(\frac{19}{26}\right)$$ $$e\left(\frac{43}{52}\right)$$ $$e\left(\frac{5}{26}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{10}{13}\right)$$
$$\chi_{507}(46,\cdot)$$ 507.w 156 no $$-1$$ $$1$$ $$e\left(\frac{131}{156}\right)$$ $$e\left(\frac{53}{78}\right)$$ $$e\left(\frac{29}{52}\right)$$ $$e\left(\frac{133}{156}\right)$$ $$e\left(\frac{27}{52}\right)$$ $$e\left(\frac{31}{78}\right)$$ $$e\left(\frac{77}{156}\right)$$ $$e\left(\frac{9}{13}\right)$$ $$e\left(\frac{14}{39}\right)$$ $$e\left(\frac{47}{78}\right)$$
$$\chi_{507}(47,\cdot)$$ 507.s 52 yes $$1$$ $$1$$ $$e\left(\frac{47}{52}\right)$$ $$e\left(\frac{21}{26}\right)$$ $$e\left(\frac{7}{52}\right)$$ $$e\left(\frac{11}{52}\right)$$ $$e\left(\frac{37}{52}\right)$$ $$e\left(\frac{1}{26}\right)$$ $$e\left(\frac{5}{52}\right)$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{8}{13}\right)$$ $$e\left(\frac{6}{13}\right)$$
$$\chi_{507}(49,\cdot)$$ 507.t 78 no $$1$$ $$1$$ $$e\left(\frac{29}{78}\right)$$ $$e\left(\frac{29}{39}\right)$$ $$e\left(\frac{9}{26}\right)$$ $$e\left(\frac{61}{78}\right)$$ $$e\left(\frac{3}{26}\right)$$ $$e\left(\frac{28}{39}\right)$$ $$e\left(\frac{23}{78}\right)$$ $$e\left(\frac{2}{13}\right)$$ $$e\left(\frac{19}{39}\right)$$ $$e\left(\frac{11}{39}\right)$$
$$\chi_{507}(50,\cdot)$$ 507.x 156 yes $$1$$ $$1$$ $$e\left(\frac{97}{156}\right)$$ $$e\left(\frac{19}{78}\right)$$ $$e\left(\frac{31}{52}\right)$$ $$e\left(\frac{5}{156}\right)$$ $$e\left(\frac{45}{52}\right)$$ $$e\left(\frac{17}{78}\right)$$ $$e\left(\frac{7}{156}\right)$$ $$e\left(\frac{17}{26}\right)$$ $$e\left(\frac{19}{39}\right)$$ $$e\left(\frac{11}{39}\right)$$