# Properties

 Modulus 403 Structure $$C_{60}\times C_{6}$$ Order 360

Show commands for: SageMath / Pari/GP

sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed

sage: H = DirichletGroup_conrey(403)

pari: g = idealstar(,403,2)

## Character group

 sage: G.order()  pari: g.no Order = 360 sage: H.invariants()  pari: g.cyc Structure = $$C_{60}\times C_{6}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{403}(189,\cdot)$, $\chi_{403}(243,\cdot)$

## First 32 of 360 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.

orbit label order primitive -1 1 2 3 4 5 6 7 8 9 10 11
$$\chi_{403}(1,\cdot)$$ 403.a 1 No $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{403}(2,\cdot)$$ 403.ce 60 Yes $$-1$$ $$1$$ $$e\left(\frac{17}{60}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$-i$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{19}{60}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{59}{60}\right)$$
$$\chi_{403}(3,\cdot)$$ 403.bv 30 Yes $$-1$$ $$1$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{403}(4,\cdot)$$ 403.bs 30 Yes $$1$$ $$1$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{29}{30}\right)$$
$$\chi_{403}(5,\cdot)$$ 403.bb 12 Yes $$-1$$ $$1$$ $$-i$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$
$$\chi_{403}(6,\cdot)$$ 403.ba 12 Yes $$1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$i$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$
$$\chi_{403}(7,\cdot)$$ 403.ca 60 Yes $$-1$$ $$1$$ $$e\left(\frac{19}{60}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{13}{60}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{53}{60}\right)$$
$$\chi_{403}(8,\cdot)$$ 403.bm 20 Yes $$-1$$ $$1$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$i$$ $$i$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{19}{20}\right)$$
$$\chi_{403}(9,\cdot)$$ 403.bk 15 Yes $$1$$ $$1$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{403}(10,\cdot)$$ 403.bp 30 Yes $$1$$ $$1$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{17}{30}\right)$$
$$\chi_{403}(11,\cdot)$$ 403.ch 60 Yes $$1$$ $$1$$ $$e\left(\frac{59}{60}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{53}{60}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{43}{60}\right)$$
$$\chi_{403}(12,\cdot)$$ 403.by 30 Yes $$-1$$ $$1$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$
$$\chi_{403}(14,\cdot)$$ 403.bi 15 No $$1$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$
$$\chi_{403}(15,\cdot)$$ 403.cb 60 Yes $$1$$ $$1$$ $$e\left(\frac{53}{60}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$-i$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{31}{60}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{41}{60}\right)$$
$$\chi_{403}(16,\cdot)$$ 403.bl 15 Yes $$1$$ $$1$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$
$$\chi_{403}(17,\cdot)$$ 403.bu 30 Yes $$-1$$ $$1$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$
$$\chi_{403}(18,\cdot)$$ 403.cg 60 Yes $$-1$$ $$1$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{31}{60}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{11}{60}\right)$$
$$\chi_{403}(19,\cdot)$$ 403.ca 60 Yes $$-1$$ $$1$$ $$e\left(\frac{37}{60}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{19}{60}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{59}{60}\right)$$
$$\chi_{403}(20,\cdot)$$ 403.cf 60 Yes $$-1$$ $$1$$ $$e\left(\frac{19}{60}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$i$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{11}{20}\right)$$
$$\chi_{403}(21,\cdot)$$ 403.cd 60 Yes $$1$$ $$1$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{49}{60}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{59}{60}\right)$$
$$\chi_{403}(22,\cdot)$$ 403.bv 30 Yes $$-1$$ $$1$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{403}(23,\cdot)$$ 403.bq 30 Yes $$-1$$ $$1$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$
$$\chi_{403}(24,\cdot)$$ 403.cc 60 Yes $$1$$ $$1$$ $$e\left(\frac{59}{60}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$-i$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{20}\right)$$
$$\chi_{403}(25,\cdot)$$ 403.l 6 Yes $$1$$ $$1$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{403}(27,\cdot)$$ 403.x 10 No $$-1$$ $$1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{403}(28,\cdot)$$ 403.cf 60 Yes $$-1$$ $$1$$ $$e\left(\frac{53}{60}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$-i$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{17}{20}\right)$$
$$\chi_{403}(29,\cdot)$$ 403.bw 30 Yes $$-1$$ $$1$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{7}{30}\right)$$
$$\chi_{403}(30,\cdot)$$ 403.t 6 Yes $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{403}(32,\cdot)$$ 403.bd 12 No $$-1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-i$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$
$$\chi_{403}(33,\cdot)$$ 403.ce 60 Yes $$-1$$ $$1$$ $$e\left(\frac{7}{60}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$i$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{29}{60}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{49}{60}\right)$$
$$\chi_{403}(34,\cdot)$$ 403.cd 60 Yes $$1$$ $$1$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{41}{60}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{31}{60}\right)$$
$$\chi_{403}(35,\cdot)$$ 403.bl 15 Yes $$1$$ $$1$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$