# Properties

 Modulus $784$ Structure $$C_{2}\times C_{2}\times C_{84}$$ Order $336$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(784)

pari: g = idealstar(,784,2)

## 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 = $\chi_{784}(687,\cdot)$, $\chi_{784}(197,\cdot)$, $\chi_{784}(689,\cdot)$

## 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 $$-1$$ $$1$$ $$3$$ $$5$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$23$$ $$25$$
$$\chi_{784}(1,\cdot)$$ 784.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{784}(3,\cdot)$$ 784.bu 84 yes $$1$$ $$1$$ $$e\left(\frac{65}{84}\right)$$ $$e\left(\frac{37}{84}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{17}{84}\right)$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{37}{42}\right)$$
$$\chi_{784}(5,\cdot)$$ 784.bs 84 yes $$-1$$ $$1$$ $$e\left(\frac{37}{84}\right)$$ $$e\left(\frac{23}{84}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{73}{84}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{23}{42}\right)$$
$$\chi_{784}(9,\cdot)$$ 784.bl 42 no $$1$$ $$1$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$
$$\chi_{784}(11,\cdot)$$ 784.bv 84 yes $$-1$$ $$1$$ $$e\left(\frac{17}{84}\right)$$ $$e\left(\frac{73}{84}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{71}{84}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{31}{42}\right)$$
$$\chi_{784}(13,\cdot)$$ 784.bi 28 yes $$-1$$ $$1$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$-i$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{1}{14}\right)$$
$$\chi_{784}(15,\cdot)$$ 784.bd 14 no $$-1$$ $$1$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$-1$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$
$$\chi_{784}(17,\cdot)$$ 784.bm 42 no $$-1$$ $$1$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$
$$\chi_{784}(19,\cdot)$$ 784.w 12 no $$1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$-i$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{784}(23,\cdot)$$ 784.bq 42 no $$-1$$ $$1$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{10}{21}\right)$$
$$\chi_{784}(25,\cdot)$$ 784.bl 42 no $$1$$ $$1$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$
$$\chi_{784}(27,\cdot)$$ 784.bk 28 yes $$1$$ $$1$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$-i$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$
$$\chi_{784}(29,\cdot)$$ 784.bh 28 yes $$1$$ $$1$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$i$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{5}{14}\right)$$
$$\chi_{784}(31,\cdot)$$ 784.p 6 no $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{784}(33,\cdot)$$ 784.bm 42 no $$-1$$ $$1$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$
$$\chi_{784}(37,\cdot)$$ 784.bt 84 yes $$1$$ $$1$$ $$e\left(\frac{43}{84}\right)$$ $$e\left(\frac{29}{84}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{61}{84}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{29}{42}\right)$$
$$\chi_{784}(39,\cdot)$$ 784.bq 42 no $$-1$$ $$1$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{20}{21}\right)$$
$$\chi_{784}(41,\cdot)$$ 784.z 14 no $$-1$$ $$1$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{784}(43,\cdot)$$ 784.bj 28 yes $$-1$$ $$1$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$i$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$
$$\chi_{784}(45,\cdot)$$ 784.bs 84 yes $$-1$$ $$1$$ $$e\left(\frac{83}{84}\right)$$ $$e\left(\frac{13}{84}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{23}{84}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{13}{42}\right)$$
$$\chi_{784}(47,\cdot)$$ 784.bp 42 no $$1$$ $$1$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{19}{21}\right)$$
$$\chi_{784}(51,\cdot)$$ 784.bv 84 yes $$-1$$ $$1$$ $$e\left(\frac{31}{84}\right)$$ $$e\left(\frac{59}{84}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{1}{84}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{17}{42}\right)$$
$$\chi_{784}(53,\cdot)$$ 784.bt 84 yes $$1$$ $$1$$ $$e\left(\frac{83}{84}\right)$$ $$e\left(\frac{13}{84}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{65}{84}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{13}{42}\right)$$
$$\chi_{784}(55,\cdot)$$ 784.bc 14 no $$1$$ $$1$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$-1$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$
$$\chi_{784}(57,\cdot)$$ 784.bf 14 no $$1$$ $$1$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$-1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{784}(59,\cdot)$$ 784.bu 84 yes $$1$$ $$1$$ $$e\left(\frac{47}{84}\right)$$ $$e\left(\frac{19}{84}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{11}{84}\right)$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{19}{42}\right)$$
$$\chi_{784}(61,\cdot)$$ 784.bs 84 yes $$-1$$ $$1$$ $$e\left(\frac{43}{84}\right)$$ $$e\left(\frac{29}{84}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{19}{84}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{29}{42}\right)$$
$$\chi_{784}(65,\cdot)$$ 784.bg 21 no $$1$$ $$1$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$
$$\chi_{784}(67,\cdot)$$ 784.v 12 no $$-1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$i$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{784}(69,\cdot)$$ 784.bi 28 yes $$-1$$ $$1$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$i$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{5}{14}\right)$$
$$\chi_{784}(71,\cdot)$$ 784.ba 14 no $$-1$$ $$1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$1$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{784}(73,\cdot)$$ 784.br 42 no $$-1$$ $$1$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$