# Properties

 Modulus 6048 Structure $$C_{72}\times C_{6}\times C_{2}\times C_{2}$$ Order 1728

Show commands for: SageMath / Pari/GP

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

sage: H = DirichletGroup_conrey(6048)

pari: g = idealstar(,6048,2)

## Character group

 sage: G.order()  pari: g.no Order = 1728 sage: H.invariants()  pari: g.cyc Structure = $$C_{72}\times C_{6}\times C_{2}\times C_{2}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{6048}(5645,\cdot)$, $\chi_{6048}(1745,\cdot)$, $\chi_{6048}(433,\cdot)$, $\chi_{6048}(4159,\cdot)$

## First 32 of 1728 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 5 11 13 17 19 23 25 29 31 37
$$\chi_{6048}(1,\cdot)$$ 6048.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{6048}(5,\cdot)$$ 6048.iy 72 yes $$1$$ $$1$$ $$e\left(\frac{49}{72}\right)$$ $$e\left(\frac{41}{72}\right)$$ $$e\left(\frac{43}{72}\right)$$ $$-1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{47}{72}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{11}{24}\right)$$
$$\chi_{6048}(11,\cdot)$$ 6048.jv 72 yes $$1$$ $$1$$ $$e\left(\frac{41}{72}\right)$$ $$e\left(\frac{49}{72}\right)$$ $$e\left(\frac{11}{72}\right)$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{43}{72}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{7}{24}\right)$$
$$\chi_{6048}(13,\cdot)$$ 6048.jc 72 yes $$-1$$ $$1$$ $$e\left(\frac{43}{72}\right)$$ $$e\left(\frac{11}{72}\right)$$ $$e\left(\frac{13}{72}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{5}{72}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{13}{24}\right)$$
$$\chi_{6048}(17,\cdot)$$ 6048.bg 6 no $$1$$ $$1$$ $$-1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{6048}(19,\cdot)$$ 6048.hd 24 no $$1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{13}{24}\right)$$
$$\chi_{6048}(23,\cdot)$$ 6048.id 36 no $$1$$ $$1$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$-1$$ $$-i$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{12}\right)$$
$$\chi_{6048}(25,\cdot)$$ 6048.iv 36 no $$1$$ $$1$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$1$$ $$-i$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{11}{12}\right)$$
$$\chi_{6048}(29,\cdot)$$ 6048.jd 72 no $$-1$$ $$1$$ $$e\left(\frac{47}{72}\right)$$ $$e\left(\frac{43}{72}\right)$$ $$e\left(\frac{5}{72}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{13}{72}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{17}{24}\right)$$
$$\chi_{6048}(31,\cdot)$$ 6048.gd 18 no $$1$$ $$1$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$1$$
$$\chi_{6048}(37,\cdot)$$ 6048.hu 24 no $$1$$ $$1$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$1$$ $$e\left(\frac{19}{24}\right)$$
$$\chi_{6048}(41,\cdot)$$ 6048.ij 36 no $$1$$ $$1$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{5}{12}\right)$$
$$\chi_{6048}(43,\cdot)$$ 6048.jp 72 no $$-1$$ $$1$$ $$e\left(\frac{53}{72}\right)$$ $$e\left(\frac{37}{72}\right)$$ $$e\left(\frac{11}{72}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{7}{72}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{23}{24}\right)$$
$$\chi_{6048}(47,\cdot)$$ 6048.fp 18 no $$-1$$ $$1$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$-1$$
$$\chi_{6048}(53,\cdot)$$ 6048.hb 24 no $$-1$$ $$1$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{23}{24}\right)$$
$$\chi_{6048}(55,\cdot)$$ 6048.x 4 no $$1$$ $$1$$ $$i$$ $$i$$ $$-i$$ $$-1$$ $$i$$ $$1$$ $$-1$$ $$i$$ $$1$$ $$-i$$
$$\chi_{6048}(59,\cdot)$$ 6048.jr 72 yes $$-1$$ $$1$$ $$e\left(\frac{25}{72}\right)$$ $$e\left(\frac{29}{72}\right)$$ $$e\left(\frac{43}{72}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{47}{72}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{8}\right)$$
$$\chi_{6048}(61,\cdot)$$ 6048.jf 72 yes $$-1$$ $$1$$ $$e\left(\frac{71}{72}\right)$$ $$e\left(\frac{55}{72}\right)$$ $$e\left(\frac{17}{72}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{1}{72}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{3}{8}\right)$$
$$\chi_{6048}(65,\cdot)$$ 6048.ge 18 no $$-1$$ $$1$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$1$$
$$\chi_{6048}(67,\cdot)$$ 6048.jq 72 yes $$-1$$ $$1$$ $$e\left(\frac{67}{72}\right)$$ $$e\left(\frac{59}{72}\right)$$ $$e\left(\frac{13}{72}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{41}{72}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{3}{8}\right)$$
$$\chi_{6048}(71,\cdot)$$ 6048.dx 12 no $$1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$-1$$ $$i$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$i$$
$$\chi_{6048}(73,\cdot)$$ 6048.er 12 no $$-1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$-1$$ $$e\left(\frac{1}{12}\right)$$
$$\chi_{6048}(79,\cdot)$$ 6048.fk 18 no $$-1$$ $$1$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$-1$$
$$\chi_{6048}(83,\cdot)$$ 6048.jo 72 yes $$-1$$ $$1$$ $$e\left(\frac{47}{72}\right)$$ $$e\left(\frac{43}{72}\right)$$ $$e\left(\frac{5}{72}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{49}{72}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{24}\right)$$
$$\chi_{6048}(85,\cdot)$$ 6048.jn 72 no $$1$$ $$1$$ $$e\left(\frac{13}{72}\right)$$ $$e\left(\frac{41}{72}\right)$$ $$e\left(\frac{19}{72}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{71}{72}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{24}\right)$$
$$\chi_{6048}(89,\cdot)$$ 6048.ej 12 no $$1$$ $$1$$ $$i$$ $$-i$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$
$$\chi_{6048}(95,\cdot)$$ 6048.gm 18 no $$1$$ $$1$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$1$$
$$\chi_{6048}(97,\cdot)$$ 6048.gi 18 no $$-1$$ $$1$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{6048}(101,\cdot)$$ 6048.iy 72 yes $$1$$ $$1$$ $$e\left(\frac{65}{72}\right)$$ $$e\left(\frac{25}{72}\right)$$ $$e\left(\frac{35}{72}\right)$$ $$-1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{55}{72}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{19}{24}\right)$$
$$\chi_{6048}(103,\cdot)$$ 6048.ia 36 no $$1$$ $$1$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{17}{36}\right)$$ $$-1$$ $$-i$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{7}{12}\right)$$
$$\chi_{6048}(107,\cdot)$$ 6048.hh 24 no $$1$$ $$1$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{24}\right)$$
$$\chi_{6048}(109,\cdot)$$ 6048.hp 24 no $$1$$ $$1$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{24}\right)$$