# Properties

 Modulus $129$ Structure $$C_{42}\times C_{2}$$ Order $84$

Show commands for: Pari/GP / SageMath

sage: H = DirichletGroup(129)

pari: g = idealstar(,129,2)

## Character group

 sage: G.order()  pari: g.no Order = 84 sage: H.invariants()  pari: g.cyc Structure = $$C_{42}\times C_{2}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{129}(44,\cdot)$, $\chi_{129}(46,\cdot)$

## First 32 of 84 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 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$
$$\chi_{129}(1,\cdot)$$ 129.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{129}(2,\cdot)$$ 129.j 14 yes $$1$$ $$1$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$-1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$
$$\chi_{129}(4,\cdot)$$ 129.i 7 no $$1$$ $$1$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$1$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$
$$\chi_{129}(5,\cdot)$$ 129.n 42 yes $$1$$ $$1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{2}{7}\right)$$
$$\chi_{129}(7,\cdot)$$ 129.g 6 no $$-1$$ $$1$$ $$-1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$
$$\chi_{129}(8,\cdot)$$ 129.j 14 yes $$1$$ $$1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$-1$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$
$$\chi_{129}(10,\cdot)$$ 129.m 21 no $$1$$ $$1$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{129}(11,\cdot)$$ 129.l 14 yes $$-1$$ $$1$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$1$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{129}(13,\cdot)$$ 129.m 21 no $$1$$ $$1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$
$$\chi_{129}(14,\cdot)$$ 129.o 42 yes $$-1$$ $$1$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{3}{7}\right)$$
$$\chi_{129}(16,\cdot)$$ 129.i 7 no $$1$$ $$1$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$1$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{129}(17,\cdot)$$ 129.o 42 yes $$-1$$ $$1$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{129}(19,\cdot)$$ 129.p 42 no $$-1$$ $$1$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$
$$\chi_{129}(20,\cdot)$$ 129.n 42 yes $$1$$ $$1$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{129}(22,\cdot)$$ 129.k 14 no $$-1$$ $$1$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$-1$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$
$$\chi_{129}(23,\cdot)$$ 129.o 42 yes $$-1$$ $$1$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{129}(25,\cdot)$$ 129.m 21 no $$1$$ $$1$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$
$$\chi_{129}(26,\cdot)$$ 129.n 42 yes $$1$$ $$1$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{129}(28,\cdot)$$ 129.p 42 no $$-1$$ $$1$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$
$$\chi_{129}(29,\cdot)$$ 129.n 42 yes $$1$$ $$1$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{3}{7}\right)$$
$$\chi_{129}(31,\cdot)$$ 129.m 21 no $$1$$ $$1$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$
$$\chi_{129}(32,\cdot)$$ 129.j 14 yes $$1$$ $$1$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$-1$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{129}(34,\cdot)$$ 129.p 42 no $$-1$$ $$1$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{129}(35,\cdot)$$ 129.l 14 yes $$-1$$ $$1$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$1$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$
$$\chi_{129}(37,\cdot)$$ 129.g 6 no $$-1$$ $$1$$ $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$
$$\chi_{129}(38,\cdot)$$ 129.o 42 yes $$-1$$ $$1$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{2}{7}\right)$$
$$\chi_{129}(40,\cdot)$$ 129.m 21 no $$1$$ $$1$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$
$$\chi_{129}(41,\cdot)$$ 129.l 14 yes $$-1$$ $$1$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$1$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$
$$\chi_{129}(44,\cdot)$$ 129.c 2 no $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$
$$\chi_{129}(46,\cdot)$$ 129.p 42 no $$-1$$ $$1$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$
$$\chi_{129}(47,\cdot)$$ 129.l 14 yes $$-1$$ $$1$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$1$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$
$$\chi_{129}(49,\cdot)$$ 129.e 3 no $$1$$ $$1$$ $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$