# Properties

 Modulus $667$ Structure $$C_{308}\times C_{2}$$ Order $616$

Show commands for: Pari/GP / SageMath

sage: H = DirichletGroup(667)

pari: g = idealstar(,667,2)

## Character group

 sage: G.order()  pari: g.no Order = 616 sage: H.invariants()  pari: g.cyc Structure = $$C_{308}\times C_{2}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{667}(465,\cdot)$, $\chi_{667}(553,\cdot)$

## First 32 of 616 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$
$$\chi_{667}(1,\cdot)$$ 667.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{667}(2,\cdot)$$ 667.w 308 yes $$-1$$ $$1$$ $$e\left(\frac{67}{308}\right)$$ $$e\left(\frac{195}{308}\right)$$ $$e\left(\frac{67}{154}\right)$$ $$e\left(\frac{135}{154}\right)$$ $$e\left(\frac{131}{154}\right)$$ $$e\left(\frac{12}{77}\right)$$ $$e\left(\frac{201}{308}\right)$$ $$e\left(\frac{41}{154}\right)$$ $$e\left(\frac{29}{308}\right)$$ $$e\left(\frac{219}{308}\right)$$
$$\chi_{667}(3,\cdot)$$ 667.w 308 yes $$-1$$ $$1$$ $$e\left(\frac{195}{308}\right)$$ $$e\left(\frac{163}{308}\right)$$ $$e\left(\frac{41}{154}\right)$$ $$e\left(\frac{101}{154}\right)$$ $$e\left(\frac{25}{154}\right)$$ $$e\left(\frac{74}{77}\right)$$ $$e\left(\frac{277}{308}\right)$$ $$e\left(\frac{9}{154}\right)$$ $$e\left(\frac{89}{308}\right)$$ $$e\left(\frac{3}{308}\right)$$
$$\chi_{667}(4,\cdot)$$ 667.u 154 yes $$1$$ $$1$$ $$e\left(\frac{67}{154}\right)$$ $$e\left(\frac{41}{154}\right)$$ $$e\left(\frac{67}{77}\right)$$ $$e\left(\frac{58}{77}\right)$$ $$e\left(\frac{54}{77}\right)$$ $$e\left(\frac{24}{77}\right)$$ $$e\left(\frac{47}{154}\right)$$ $$e\left(\frac{41}{77}\right)$$ $$e\left(\frac{29}{154}\right)$$ $$e\left(\frac{65}{154}\right)$$
$$\chi_{667}(5,\cdot)$$ 667.t 154 yes $$-1$$ $$1$$ $$e\left(\frac{135}{154}\right)$$ $$e\left(\frac{101}{154}\right)$$ $$e\left(\frac{58}{77}\right)$$ $$e\left(\frac{51}{154}\right)$$ $$e\left(\frac{41}{77}\right)$$ $$e\left(\frac{45}{154}\right)$$ $$e\left(\frac{97}{154}\right)$$ $$e\left(\frac{24}{77}\right)$$ $$e\left(\frac{16}{77}\right)$$ $$e\left(\frac{4}{77}\right)$$
$$\chi_{667}(6,\cdot)$$ 667.u 154 yes $$1$$ $$1$$ $$e\left(\frac{131}{154}\right)$$ $$e\left(\frac{25}{154}\right)$$ $$e\left(\frac{54}{77}\right)$$ $$e\left(\frac{41}{77}\right)$$ $$e\left(\frac{1}{77}\right)$$ $$e\left(\frac{9}{77}\right)$$ $$e\left(\frac{85}{154}\right)$$ $$e\left(\frac{25}{77}\right)$$ $$e\left(\frac{59}{154}\right)$$ $$e\left(\frac{111}{154}\right)$$
$$\chi_{667}(7,\cdot)$$ 667.v 154 yes $$-1$$ $$1$$ $$e\left(\frac{12}{77}\right)$$ $$e\left(\frac{74}{77}\right)$$ $$e\left(\frac{24}{77}\right)$$ $$e\left(\frac{45}{154}\right)$$ $$e\left(\frac{9}{77}\right)$$ $$e\left(\frac{85}{154}\right)$$ $$e\left(\frac{36}{77}\right)$$ $$e\left(\frac{71}{77}\right)$$ $$e\left(\frac{69}{154}\right)$$ $$e\left(\frac{75}{154}\right)$$
$$\chi_{667}(8,\cdot)$$ 667.w 308 yes $$-1$$ $$1$$ $$e\left(\frac{201}{308}\right)$$ $$e\left(\frac{277}{308}\right)$$ $$e\left(\frac{47}{154}\right)$$ $$e\left(\frac{97}{154}\right)$$ $$e\left(\frac{85}{154}\right)$$ $$e\left(\frac{36}{77}\right)$$ $$e\left(\frac{295}{308}\right)$$ $$e\left(\frac{123}{154}\right)$$ $$e\left(\frac{87}{308}\right)$$ $$e\left(\frac{41}{308}\right)$$
$$\chi_{667}(9,\cdot)$$ 667.u 154 yes $$1$$ $$1$$ $$e\left(\frac{41}{154}\right)$$ $$e\left(\frac{9}{154}\right)$$ $$e\left(\frac{41}{77}\right)$$ $$e\left(\frac{24}{77}\right)$$ $$e\left(\frac{25}{77}\right)$$ $$e\left(\frac{71}{77}\right)$$ $$e\left(\frac{123}{154}\right)$$ $$e\left(\frac{9}{77}\right)$$ $$e\left(\frac{89}{154}\right)$$ $$e\left(\frac{3}{154}\right)$$
$$\chi_{667}(10,\cdot)$$ 667.x 308 yes $$1$$ $$1$$ $$e\left(\frac{29}{308}\right)$$ $$e\left(\frac{89}{308}\right)$$ $$e\left(\frac{29}{154}\right)$$ $$e\left(\frac{16}{77}\right)$$ $$e\left(\frac{59}{154}\right)$$ $$e\left(\frac{69}{154}\right)$$ $$e\left(\frac{87}{308}\right)$$ $$e\left(\frac{89}{154}\right)$$ $$e\left(\frac{93}{308}\right)$$ $$e\left(\frac{235}{308}\right)$$
$$\chi_{667}(11,\cdot)$$ 667.x 308 yes $$1$$ $$1$$ $$e\left(\frac{219}{308}\right)$$ $$e\left(\frac{3}{308}\right)$$ $$e\left(\frac{65}{154}\right)$$ $$e\left(\frac{4}{77}\right)$$ $$e\left(\frac{111}{154}\right)$$ $$e\left(\frac{75}{154}\right)$$ $$e\left(\frac{41}{308}\right)$$ $$e\left(\frac{3}{154}\right)$$ $$e\left(\frac{235}{308}\right)$$ $$e\left(\frac{1}{308}\right)$$
$$\chi_{667}(12,\cdot)$$ 667.r 44 yes $$-1$$ $$1$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{19}{44}\right)$$
$$\chi_{667}(13,\cdot)$$ 667.u 154 yes $$1$$ $$1$$ $$e\left(\frac{141}{154}\right)$$ $$e\left(\frac{61}{154}\right)$$ $$e\left(\frac{64}{77}\right)$$ $$e\left(\frac{60}{77}\right)$$ $$e\left(\frac{24}{77}\right)$$ $$e\left(\frac{62}{77}\right)$$ $$e\left(\frac{115}{154}\right)$$ $$e\left(\frac{61}{77}\right)$$ $$e\left(\frac{107}{154}\right)$$ $$e\left(\frac{123}{154}\right)$$
$$\chi_{667}(14,\cdot)$$ 667.x 308 yes $$1$$ $$1$$ $$e\left(\frac{115}{308}\right)$$ $$e\left(\frac{183}{308}\right)$$ $$e\left(\frac{115}{154}\right)$$ $$e\left(\frac{13}{77}\right)$$ $$e\left(\frac{149}{154}\right)$$ $$e\left(\frac{109}{154}\right)$$ $$e\left(\frac{37}{308}\right)$$ $$e\left(\frac{29}{154}\right)$$ $$e\left(\frac{167}{308}\right)$$ $$e\left(\frac{61}{308}\right)$$
$$\chi_{667}(15,\cdot)$$ 667.x 308 yes $$1$$ $$1$$ $$e\left(\frac{157}{308}\right)$$ $$e\left(\frac{57}{308}\right)$$ $$e\left(\frac{3}{154}\right)$$ $$e\left(\frac{76}{77}\right)$$ $$e\left(\frac{107}{154}\right)$$ $$e\left(\frac{39}{154}\right)$$ $$e\left(\frac{163}{308}\right)$$ $$e\left(\frac{57}{154}\right)$$ $$e\left(\frac{153}{308}\right)$$ $$e\left(\frac{19}{308}\right)$$
$$\chi_{667}(16,\cdot)$$ 667.s 77 yes $$1$$ $$1$$ $$e\left(\frac{67}{77}\right)$$ $$e\left(\frac{41}{77}\right)$$ $$e\left(\frac{57}{77}\right)$$ $$e\left(\frac{39}{77}\right)$$ $$e\left(\frac{31}{77}\right)$$ $$e\left(\frac{48}{77}\right)$$ $$e\left(\frac{47}{77}\right)$$ $$e\left(\frac{5}{77}\right)$$ $$e\left(\frac{29}{77}\right)$$ $$e\left(\frac{65}{77}\right)$$
$$\chi_{667}(17,\cdot)$$ 667.q 44 yes $$1$$ $$1$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{27}{44}\right)$$
$$\chi_{667}(18,\cdot)$$ 667.w 308 yes $$-1$$ $$1$$ $$e\left(\frac{149}{308}\right)$$ $$e\left(\frac{213}{308}\right)$$ $$e\left(\frac{149}{154}\right)$$ $$e\left(\frac{29}{154}\right)$$ $$e\left(\frac{27}{154}\right)$$ $$e\left(\frac{6}{77}\right)$$ $$e\left(\frac{139}{308}\right)$$ $$e\left(\frac{59}{154}\right)$$ $$e\left(\frac{207}{308}\right)$$ $$e\left(\frac{225}{308}\right)$$
$$\chi_{667}(19,\cdot)$$ 667.x 308 yes $$1$$ $$1$$ $$e\left(\frac{211}{308}\right)$$ $$e\left(\frac{159}{308}\right)$$ $$e\left(\frac{57}{154}\right)$$ $$e\left(\frac{58}{77}\right)$$ $$e\left(\frac{31}{154}\right)$$ $$e\left(\frac{125}{154}\right)$$ $$e\left(\frac{17}{308}\right)$$ $$e\left(\frac{5}{154}\right)$$ $$e\left(\frac{135}{308}\right)$$ $$e\left(\frac{53}{308}\right)$$
$$\chi_{667}(20,\cdot)$$ 667.v 154 yes $$-1$$ $$1$$ $$e\left(\frac{24}{77}\right)$$ $$e\left(\frac{71}{77}\right)$$ $$e\left(\frac{48}{77}\right)$$ $$e\left(\frac{13}{154}\right)$$ $$e\left(\frac{18}{77}\right)$$ $$e\left(\frac{93}{154}\right)$$ $$e\left(\frac{72}{77}\right)$$ $$e\left(\frac{65}{77}\right)$$ $$e\left(\frac{61}{154}\right)$$ $$e\left(\frac{73}{154}\right)$$
$$\chi_{667}(21,\cdot)$$ 667.x 308 yes $$1$$ $$1$$ $$e\left(\frac{243}{308}\right)$$ $$e\left(\frac{151}{308}\right)$$ $$e\left(\frac{89}{154}\right)$$ $$e\left(\frac{73}{77}\right)$$ $$e\left(\frac{43}{154}\right)$$ $$e\left(\frac{79}{154}\right)$$ $$e\left(\frac{113}{308}\right)$$ $$e\left(\frac{151}{154}\right)$$ $$e\left(\frac{227}{308}\right)$$ $$e\left(\frac{153}{308}\right)$$
$$\chi_{667}(22,\cdot)$$ 667.k 14 yes $$-1$$ $$1$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{667}(24,\cdot)$$ 667.g 7 no $$1$$ $$1$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{667}(25,\cdot)$$ 667.s 77 yes $$1$$ $$1$$ $$e\left(\frac{58}{77}\right)$$ $$e\left(\frac{24}{77}\right)$$ $$e\left(\frac{39}{77}\right)$$ $$e\left(\frac{51}{77}\right)$$ $$e\left(\frac{5}{77}\right)$$ $$e\left(\frac{45}{77}\right)$$ $$e\left(\frac{20}{77}\right)$$ $$e\left(\frac{48}{77}\right)$$ $$e\left(\frac{32}{77}\right)$$ $$e\left(\frac{8}{77}\right)$$
$$\chi_{667}(26,\cdot)$$ 667.w 308 yes $$-1$$ $$1$$ $$e\left(\frac{41}{308}\right)$$ $$e\left(\frac{9}{308}\right)$$ $$e\left(\frac{41}{154}\right)$$ $$e\left(\frac{101}{154}\right)$$ $$e\left(\frac{25}{154}\right)$$ $$e\left(\frac{74}{77}\right)$$ $$e\left(\frac{123}{308}\right)$$ $$e\left(\frac{9}{154}\right)$$ $$e\left(\frac{243}{308}\right)$$ $$e\left(\frac{157}{308}\right)$$
$$\chi_{667}(27,\cdot)$$ 667.w 308 yes $$-1$$ $$1$$ $$e\left(\frac{277}{308}\right)$$ $$e\left(\frac{181}{308}\right)$$ $$e\left(\frac{123}{154}\right)$$ $$e\left(\frac{149}{154}\right)$$ $$e\left(\frac{75}{154}\right)$$ $$e\left(\frac{68}{77}\right)$$ $$e\left(\frac{215}{308}\right)$$ $$e\left(\frac{27}{154}\right)$$ $$e\left(\frac{267}{308}\right)$$ $$e\left(\frac{9}{308}\right)$$
$$\chi_{667}(28,\cdot)$$ 667.n 22 yes $$-1$$ $$1$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$
$$\chi_{667}(30,\cdot)$$ 667.l 22 no $$-1$$ $$1$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{17}{22}\right)$$
$$\chi_{667}(31,\cdot)$$ 667.w 308 yes $$-1$$ $$1$$ $$e\left(\frac{179}{308}\right)$$ $$e\left(\frac{167}{308}\right)$$ $$e\left(\frac{25}{154}\right)$$ $$e\left(\frac{9}{154}\right)$$ $$e\left(\frac{19}{154}\right)$$ $$e\left(\frac{47}{77}\right)$$ $$e\left(\frac{229}{308}\right)$$ $$e\left(\frac{13}{154}\right)$$ $$e\left(\frac{197}{308}\right)$$ $$e\left(\frac{107}{308}\right)$$
$$\chi_{667}(32,\cdot)$$ 667.w 308 yes $$-1$$ $$1$$ $$e\left(\frac{27}{308}\right)$$ $$e\left(\frac{51}{308}\right)$$ $$e\left(\frac{27}{154}\right)$$ $$e\left(\frac{59}{154}\right)$$ $$e\left(\frac{39}{154}\right)$$ $$e\left(\frac{60}{77}\right)$$ $$e\left(\frac{81}{308}\right)$$ $$e\left(\frac{51}{154}\right)$$ $$e\left(\frac{145}{308}\right)$$ $$e\left(\frac{171}{308}\right)$$
$$\chi_{667}(33,\cdot)$$ 667.t 154 yes $$-1$$ $$1$$ $$e\left(\frac{53}{154}\right)$$ $$e\left(\frac{83}{154}\right)$$ $$e\left(\frac{53}{77}\right)$$ $$e\left(\frac{109}{154}\right)$$ $$e\left(\frac{68}{77}\right)$$ $$e\left(\frac{69}{154}\right)$$ $$e\left(\frac{5}{154}\right)$$ $$e\left(\frac{6}{77}\right)$$ $$e\left(\frac{4}{77}\right)$$ $$e\left(\frac{1}{77}\right)$$
$$\chi_{667}(34,\cdot)$$ 667.t 154 yes $$-1$$ $$1$$ $$e\left(\frac{93}{154}\right)$$ $$e\left(\frac{73}{154}\right)$$ $$e\left(\frac{16}{77}\right)$$ $$e\left(\frac{107}{154}\right)$$ $$e\left(\frac{6}{77}\right)$$ $$e\left(\frac{31}{154}\right)$$ $$e\left(\frac{125}{154}\right)$$ $$e\left(\frac{73}{77}\right)$$ $$e\left(\frac{23}{77}\right)$$ $$e\left(\frac{25}{77}\right)$$