# Properties

 Modulus $637$ Structure $$C_{6}\times C_{84}$$ Order $504$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(637)

pari: g = idealstar(,637,2)

## Character group

 sage: G.order()  pari: g.no Order = 504 sage: H.invariants()  pari: g.cyc Structure = $$C_{6}\times C_{84}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{637}(248,\cdot)$, $\chi_{637}(197,\cdot)$

## First 32 of 504 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$8$$ $$9$$ $$10$$ $$11$$ $$12$$
$$\chi_{637}(1,\cdot)$$ 637.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{637}(2,\cdot)$$ 637.cg 84 yes $$-1$$ $$1$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{59}{84}\right)$$ $$e\left(\frac{11}{84}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{29}{84}\right)$$ $$e\left(\frac{13}{42}\right)$$
$$\chi_{637}(3,\cdot)$$ 637.bx 42 yes $$-1$$ $$1$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{11}{42}\right)$$
$$\chi_{637}(4,\cdot)$$ 637.bz 42 yes $$1$$ $$1$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{13}{21}\right)$$
$$\chi_{637}(5,\cdot)$$ 637.cc 84 yes $$1$$ $$1$$ $$e\left(\frac{59}{84}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{65}{84}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{73}{84}\right)$$ $$e\left(\frac{2}{21}\right)$$
$$\chi_{637}(6,\cdot)$$ 637.cb 84 yes $$1$$ $$1$$ $$e\left(\frac{11}{84}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{37}{84}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{53}{84}\right)$$ $$e\left(\frac{4}{7}\right)$$
$$\chi_{637}(8,\cdot)$$ 637.bm 28 yes $$-1$$ $$1$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{13}{14}\right)$$
$$\chi_{637}(9,\cdot)$$ 637.bj 21 yes $$1$$ $$1$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{11}{21}\right)$$
$$\chi_{637}(10,\cdot)$$ 637.bu 42 yes $$-1$$ $$1$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{17}{42}\right)$$
$$\chi_{637}(11,\cdot)$$ 637.ca 84 yes $$-1$$ $$1$$ $$e\left(\frac{29}{84}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{73}{84}\right)$$ $$e\left(\frac{53}{84}\right)$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{41}{42}\right)$$
$$\chi_{637}(12,\cdot)$$ 637.br 42 yes $$-1$$ $$1$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{37}{42}\right)$$
$$\chi_{637}(15,\cdot)$$ 637.cf 84 yes $$-1$$ $$1$$ $$e\left(\frac{55}{84}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{59}{84}\right)$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{13}{84}\right)$$ $$e\left(\frac{5}{14}\right)$$
$$\chi_{637}(16,\cdot)$$ 637.bi 21 yes $$1$$ $$1$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$
$$\chi_{637}(17,\cdot)$$ 637.by 42 yes $$-1$$ $$1$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{23}{42}\right)$$
$$\chi_{637}(18,\cdot)$$ 637.ba 12 no $$-1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$-i$$ $$i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{637}(19,\cdot)$$ 637.x 12 no $$1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$i$$ $$1$$ $$1$$ $$i$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{637}(20,\cdot)$$ 637.cb 84 yes $$1$$ $$1$$ $$e\left(\frac{5}{84}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{55}{84}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{47}{84}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{637}(22,\cdot)$$ 637.bk 21 yes $$1$$ $$1$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$
$$\chi_{637}(23,\cdot)$$ 637.bz 42 yes $$1$$ $$1$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{20}{21}\right)$$
$$\chi_{637}(24,\cdot)$$ 637.ch 84 yes $$1$$ $$1$$ $$e\left(\frac{41}{84}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{67}{84}\right)$$ $$e\left(\frac{59}{84}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{4}{21}\right)$$
$$\chi_{637}(25,\cdot)$$ 637.bs 42 yes $$1$$ $$1$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{4}{21}\right)$$
$$\chi_{637}(27,\cdot)$$ 637.bf 14 no $$-1$$ $$1$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$
$$\chi_{637}(29,\cdot)$$ 637.bk 21 yes $$1$$ $$1$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{637}(30,\cdot)$$ 637.u 6 no $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{637}(31,\cdot)$$ 637.bc 12 no $$1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$i$$ $$i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{637}(32,\cdot)$$ 637.cg 84 yes $$-1$$ $$1$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{43}{84}\right)$$ $$e\left(\frac{55}{84}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{61}{84}\right)$$ $$e\left(\frac{23}{42}\right)$$
$$\chi_{637}(33,\cdot)$$ 637.ch 84 yes $$1$$ $$1$$ $$e\left(\frac{25}{84}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{47}{84}\right)$$ $$e\left(\frac{79}{84}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{5}{21}\right)$$
$$\chi_{637}(34,\cdot)$$ 637.bn 28 yes $$1$$ $$1$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{6}{7}\right)$$
$$\chi_{637}(36,\cdot)$$ 637.bt 42 yes $$1$$ $$1$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{637}(37,\cdot)$$ 637.cg 84 yes $$-1$$ $$1$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{29}{84}\right)$$ $$e\left(\frac{41}{84}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{47}{84}\right)$$ $$e\left(\frac{37}{42}\right)$$
$$\chi_{637}(38,\cdot)$$ 637.br 42 yes $$-1$$ $$1$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{41}{42}\right)$$
$$\chi_{637}(40,\cdot)$$ 637.bv 42 no $$-1$$ $$1$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{1}{42}\right)$$