# Properties

 Modulus 4004 Structure $$C_{60}\times C_{6}\times C_{2}\times C_{2}$$ Order 1440

# Learn more about

Show commands for: SageMath / Pari/GP

sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(4004)
pari: g = idealstar(,4004,2)

## Character group

 sage: G.order() pari: g.no Order = 1440 sage: H.invariants() pari: g.cyc Structure = $$C_{60}\times C_{6}\times C_{2}\times C_{2}$$ sage: H.gens() pari: g.gen Generators = $\chi_{4004}(2173,\cdot)$, $\chi_{4004}(2089,\cdot)$, $\chi_{4004}(3277,\cdot)$, $\chi_{4004}(2003,\cdot)$

## First 32 of 1440 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 3 5 9 15 17 19 23 25 27 29
$$\chi_{4004}(1,\cdot)$$ 4004.a 1 No $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{4004}(3,\cdot)$$ 4004.ft 30 Yes $$1$$ $$1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{14}{15}\right)$$
$$\chi_{4004}(5,\cdot)$$ 4004.im 60 No $$1$$ $$1$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{31}{60}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{7}{60}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{4004}(9,\cdot)$$ 4004.fg 15 No $$1$$ $$1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{13}{15}\right)$$
$$\chi_{4004}(15,\cdot)$$ 4004.ih 60 No $$1$$ $$1$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{59}{60}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{31}{60}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{11}{15}\right)$$
$$\chi_{4004}(17,\cdot)$$ 4004.gm 30 No $$1$$ $$1$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$1$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{29}{30}\right)$$
$$\chi_{4004}(19,\cdot)$$ 4004.ia 60 Yes $$1$$ $$1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{7}{60}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{31}{60}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{23}{30}\right)$$
$$\chi_{4004}(23,\cdot)$$ 4004.cu 6 No $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{4004}(25,\cdot)$$ 4004.hg 30 No $$1$$ $$1$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$
$$\chi_{4004}(27,\cdot)$$ 4004.dq 10 No $$1$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$-1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{4004}(29,\cdot)$$ 4004.gx 30 No $$-1$$ $$1$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{7}{30}\right)$$
$$\chi_{4004}(31,\cdot)$$ 4004.iw 60 Yes $$-1$$ $$1$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{59}{60}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{53}{60}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{4004}(37,\cdot)$$ 4004.ja 60 No $$-1$$ $$1$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{43}{60}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{59}{60}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{11}{60}\right)$$ $$-1$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{11}{15}\right)$$
$$\chi_{4004}(41,\cdot)$$ 4004.ij 60 No $$-1$$ $$1$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{41}{60}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{49}{60}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{13}{30}\right)$$
$$\chi_{4004}(43,\cdot)$$ 4004.cj 6 No $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{4004}(45,\cdot)$$ 4004.fa 12 No $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{4004}(47,\cdot)$$ 4004.iw 60 Yes $$-1$$ $$1$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{37}{60}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{19}{60}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$
$$\chi_{4004}(51,\cdot)$$ 4004.ge 30 Yes $$1$$ $$1$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{4004}(53,\cdot)$$ 4004.fi 15 No $$1$$ $$1$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{4004}(57,\cdot)$$ 4004.fk 20 No $$1$$ $$1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$-1$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{4004}(59,\cdot)$$ 4004.ir 60 Yes $$-1$$ $$1$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{53}{60}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{49}{60}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{31}{60}\right)$$ $$1$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{15}\right)$$
$$\chi_{4004}(61,\cdot)$$ 4004.hu 30 No $$1$$ $$1$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{29}{30}\right)$$
$$\chi_{4004}(67,\cdot)$$ 4004.ec 12 No $$1$$ $$1$$ $$-1$$ $$e\left(\frac{1}{12}\right)$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$i$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{4004}(69,\cdot)$$ 4004.fz 30 No $$-1$$ $$1$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{4}{15}\right)$$
$$\chi_{4004}(71,\cdot)$$ 4004.ih 60 No $$1$$ $$1$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{43}{60}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{47}{60}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{7}{15}\right)$$
$$\chi_{4004}(73,\cdot)$$ 4004.ii 60 No $$-1$$ $$1$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{53}{60}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{11}{60}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{4004}(75,\cdot)$$ 4004.hm 30 Yes $$1$$ $$1$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$-1$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{8}{15}\right)$$
$$\chi_{4004}(79,\cdot)$$ 4004.gq 30 No $$1$$ $$1$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{4004}(81,\cdot)$$ 4004.fg 15 No $$1$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{11}{15}\right)$$
$$\chi_{4004}(83,\cdot)$$ 4004.fn 20 Yes $$1$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$1$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{4004}(85,\cdot)$$ 4004.iz 60 No $$1$$ $$1$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{31}{60}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{29}{60}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{23}{30}\right)$$
$$\chi_{4004}(87,\cdot)$$ 4004.cb 6 Yes $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$