# Properties

 Modulus $735$ Structure $$C_{2}\times C_{2}\times C_{84}$$ Order $336$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(735)

pari: g = idealstar(,735,2)

## Character group

 sage: G.order()  pari: g.no Order = 336 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{2}\times C_{84}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{735}(491,\cdot)$, $\chi_{735}(442,\cdot)$, $\chi_{735}(346,\cdot)$

## First 32 of 336 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 $$-1$$ $$1$$ $$2$$ $$4$$ $$8$$ $$11$$ $$13$$ $$16$$ $$17$$ $$19$$ $$22$$ $$23$$
$$\chi_{735}(1,\cdot)$$ 735.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{735}(2,\cdot)$$ 735.bt 84 yes $$1$$ $$1$$ $$e\left(\frac{71}{84}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{19}{84}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{65}{84}\right)$$
$$\chi_{735}(4,\cdot)$$ 735.bo 42 no $$1$$ $$1$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{23}{42}\right)$$
$$\chi_{735}(8,\cdot)$$ 735.bj 28 yes $$1$$ $$1$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$-1$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{9}{28}\right)$$
$$\chi_{735}(11,\cdot)$$ 735.bl 42 no $$-1$$ $$1$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{29}{42}\right)$$
$$\chi_{735}(13,\cdot)$$ 735.bi 28 no $$1$$ $$1$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$1$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{3}{28}\right)$$
$$\chi_{735}(16,\cdot)$$ 735.bg 21 no $$1$$ $$1$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{2}{21}\right)$$
$$\chi_{735}(17,\cdot)$$ 735.bs 84 yes $$-1$$ $$1$$ $$e\left(\frac{19}{84}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{53}{84}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{73}{84}\right)$$
$$\chi_{735}(19,\cdot)$$ 735.r 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{735}(22,\cdot)$$ 735.bh 28 no $$-1$$ $$1$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$-1$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{13}{28}\right)$$
$$\chi_{735}(23,\cdot)$$ 735.bt 84 yes $$1$$ $$1$$ $$e\left(\frac{65}{84}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{73}{84}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{11}{84}\right)$$
$$\chi_{735}(26,\cdot)$$ 735.bm 42 no $$1$$ $$1$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{37}{42}\right)$$
$$\chi_{735}(29,\cdot)$$ 735.bb 14 yes $$-1$$ $$1$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$1$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$
$$\chi_{735}(31,\cdot)$$ 735.n 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{735}(32,\cdot)$$ 735.bt 84 yes $$1$$ $$1$$ $$e\left(\frac{19}{84}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{11}{84}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{73}{84}\right)$$
$$\chi_{735}(34,\cdot)$$ 735.bc 14 no $$-1$$ $$1$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$-1$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{9}{14}\right)$$
$$\chi_{735}(37,\cdot)$$ 735.bv 84 no $$-1$$ $$1$$ $$e\left(\frac{5}{84}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{25}{84}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{59}{84}\right)$$
$$\chi_{735}(38,\cdot)$$ 735.bs 84 yes $$-1$$ $$1$$ $$e\left(\frac{1}{84}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{47}{84}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{79}{84}\right)$$
$$\chi_{735}(41,\cdot)$$ 735.bf 14 no $$1$$ $$1$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$-1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$
$$\chi_{735}(43,\cdot)$$ 735.bh 28 no $$-1$$ $$1$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$-1$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{19}{28}\right)$$
$$\chi_{735}(44,\cdot)$$ 735.bq 42 yes $$-1$$ $$1$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{5}{21}\right)$$
$$\chi_{735}(46,\cdot)$$ 735.bg 21 no $$1$$ $$1$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{19}{21}\right)$$
$$\chi_{735}(47,\cdot)$$ 735.bs 84 yes $$-1$$ $$1$$ $$e\left(\frac{71}{84}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{61}{84}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{65}{84}\right)$$
$$\chi_{735}(52,\cdot)$$ 735.bu 84 no $$1$$ $$1$$ $$e\left(\frac{73}{84}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{71}{84}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{55}{84}\right)$$
$$\chi_{735}(53,\cdot)$$ 735.bt 84 yes $$1$$ $$1$$ $$e\left(\frac{37}{84}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{17}{84}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{67}{84}\right)$$
$$\chi_{735}(58,\cdot)$$ 735.bv 84 no $$-1$$ $$1$$ $$e\left(\frac{83}{84}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{79}{84}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{5}{84}\right)$$
$$\chi_{735}(59,\cdot)$$ 735.bp 42 yes $$1$$ $$1$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{16}{21}\right)$$
$$\chi_{735}(61,\cdot)$$ 735.br 42 no $$-1$$ $$1$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{20}{21}\right)$$
$$\chi_{735}(62,\cdot)$$ 735.bk 28 yes $$-1$$ $$1$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$1$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{3}{28}\right)$$
$$\chi_{735}(64,\cdot)$$ 735.bd 14 no $$1$$ $$1$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$1$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{9}{14}\right)$$
$$\chi_{735}(67,\cdot)$$ 735.w 12 no $$-1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-i$$ $$e\left(\frac{2}{3}\right)$$ $$-i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$i$$ $$e\left(\frac{1}{12}\right)$$
$$\chi_{735}(68,\cdot)$$ 735.x 12 no $$-1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-i$$ $$e\left(\frac{5}{6}\right)$$ $$-i$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-i$$ $$e\left(\frac{5}{12}\right)$$