sage: H = DirichletGroup(1728)
pari: g = idealstar(,1728,2)
Character group
sage: G.order()
pari: g.no
| ||
Order | = | 576 |
sage: H.invariants()
pari: g.cyc
| ||
Structure | = | \(C_{2}\times C_{2}\times C_{144}\) |
sage: H.gens()
pari: g.gen
| ||
Generators | = | $\chi_{1728}(703,\cdot)$, $\chi_{1728}(325,\cdot)$, $\chi_{1728}(1217,\cdot)$ |
First 32 of 576 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 | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{1728}(1,\cdot)\) | 1728.a | 1 | no | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) |
\(\chi_{1728}(5,\cdot)\) | 1728.cf | 144 | yes | \(-1\) | \(1\) | \(e\left(\frac{65}{144}\right)\) | \(e\left(\frac{5}{72}\right)\) | \(e\left(\frac{133}{144}\right)\) | \(e\left(\frac{23}{144}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{67}{72}\right)\) | \(e\left(\frac{65}{72}\right)\) | \(e\left(\frac{139}{144}\right)\) | \(e\left(\frac{1}{18}\right)\) |
\(\chi_{1728}(7,\cdot)\) | 1728.cd | 72 | no | \(-1\) | \(1\) | \(e\left(\frac{5}{72}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{13}{72}\right)\) | \(e\left(\frac{35}{72}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{55}{72}\right)\) | \(e\left(\frac{5}{18}\right)\) |
\(\chi_{1728}(11,\cdot)\) | 1728.ch | 144 | yes | \(1\) | \(1\) | \(e\left(\frac{133}{144}\right)\) | \(e\left(\frac{13}{72}\right)\) | \(e\left(\frac{65}{144}\right)\) | \(e\left(\frac{67}{144}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{59}{72}\right)\) | \(e\left(\frac{61}{72}\right)\) | \(e\left(\frac{23}{144}\right)\) | \(e\left(\frac{4}{9}\right)\) |
\(\chi_{1728}(13,\cdot)\) | 1728.ce | 144 | yes | \(1\) | \(1\) | \(e\left(\frac{23}{144}\right)\) | \(e\left(\frac{35}{72}\right)\) | \(e\left(\frac{67}{144}\right)\) | \(e\left(\frac{89}{144}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{1}{72}\right)\) | \(e\left(\frac{23}{72}\right)\) | \(e\left(\frac{109}{144}\right)\) | \(e\left(\frac{7}{18}\right)\) |
\(\chi_{1728}(17,\cdot)\) | 1728.bb | 12 | no | \(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(-1\) | \(i\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{1728}(19,\cdot)\) | 1728.bw | 48 | no | \(-1\) | \(1\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(i\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{1728}(23,\cdot)\) | 1728.cb | 72 | no | \(1\) | \(1\) | \(e\left(\frac{67}{72}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{59}{72}\right)\) | \(e\left(\frac{1}{72}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{17}{72}\right)\) | \(e\left(\frac{13}{18}\right)\) |
\(\chi_{1728}(25,\cdot)\) | 1728.cc | 72 | no | \(1\) | \(1\) | \(e\left(\frac{65}{72}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{61}{72}\right)\) | \(e\left(\frac{23}{72}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{67}{72}\right)\) | \(e\left(\frac{1}{9}\right)\) |
\(\chi_{1728}(29,\cdot)\) | 1728.cf | 144 | yes | \(-1\) | \(1\) | \(e\left(\frac{139}{144}\right)\) | \(e\left(\frac{55}{72}\right)\) | \(e\left(\frac{23}{144}\right)\) | \(e\left(\frac{109}{144}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{17}{72}\right)\) | \(e\left(\frac{67}{72}\right)\) | \(e\left(\frac{89}{144}\right)\) | \(e\left(\frac{11}{18}\right)\) |
\(\chi_{1728}(31,\cdot)\) | 1728.bh | 18 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) |
\(\chi_{1728}(35,\cdot)\) | 1728.bx | 48 | no | \(1\) | \(1\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(-i\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{1728}(37,\cdot)\) | 1728.by | 48 | no | \(1\) | \(1\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(-i\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{1728}(41,\cdot)\) | 1728.ca | 72 | no | \(-1\) | \(1\) | \(e\left(\frac{43}{72}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{47}{72}\right)\) | \(e\left(\frac{49}{72}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{41}{72}\right)\) | \(e\left(\frac{8}{9}\right)\) |
\(\chi_{1728}(43,\cdot)\) | 1728.cg | 144 | yes | \(-1\) | \(1\) | \(e\left(\frac{133}{144}\right)\) | \(e\left(\frac{13}{72}\right)\) | \(e\left(\frac{65}{144}\right)\) | \(e\left(\frac{139}{144}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{23}{72}\right)\) | \(e\left(\frac{61}{72}\right)\) | \(e\left(\frac{23}{144}\right)\) | \(e\left(\frac{4}{9}\right)\) |
\(\chi_{1728}(47,\cdot)\) | 1728.bv | 36 | no | \(1\) | \(1\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) |
\(\chi_{1728}(49,\cdot)\) | 1728.bs | 36 | no | \(1\) | \(1\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) |
\(\chi_{1728}(53,\cdot)\) | 1728.bd | 16 | no | \(-1\) | \(1\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(i\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(-1\) |
\(\chi_{1728}(55,\cdot)\) | 1728.u | 8 | no | \(-1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(-1\) | \(e\left(\frac{1}{8}\right)\) | \(-i\) | \(-i\) | \(e\left(\frac{1}{8}\right)\) | \(-1\) |
\(\chi_{1728}(59,\cdot)\) | 1728.ch | 144 | yes | \(1\) | \(1\) | \(e\left(\frac{65}{144}\right)\) | \(e\left(\frac{41}{72}\right)\) | \(e\left(\frac{61}{144}\right)\) | \(e\left(\frac{23}{144}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{31}{72}\right)\) | \(e\left(\frac{65}{72}\right)\) | \(e\left(\frac{139}{144}\right)\) | \(e\left(\frac{5}{9}\right)\) |
\(\chi_{1728}(61,\cdot)\) | 1728.ce | 144 | yes | \(1\) | \(1\) | \(e\left(\frac{91}{144}\right)\) | \(e\left(\frac{7}{72}\right)\) | \(e\left(\frac{71}{144}\right)\) | \(e\left(\frac{133}{144}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{29}{72}\right)\) | \(e\left(\frac{19}{72}\right)\) | \(e\left(\frac{137}{144}\right)\) | \(e\left(\frac{5}{18}\right)\) |
\(\chi_{1728}(65,\cdot)\) | 1728.bk | 18 | no | \(-1\) | \(1\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) |
\(\chi_{1728}(67,\cdot)\) | 1728.cg | 144 | yes | \(-1\) | \(1\) | \(e\left(\frac{59}{144}\right)\) | \(e\left(\frac{35}{72}\right)\) | \(e\left(\frac{31}{144}\right)\) | \(e\left(\frac{53}{144}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{1}{72}\right)\) | \(e\left(\frac{59}{72}\right)\) | \(e\left(\frac{73}{144}\right)\) | \(e\left(\frac{8}{9}\right)\) |
\(\chi_{1728}(71,\cdot)\) | 1728.br | 24 | no | \(1\) | \(1\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{1728}(73,\cdot)\) | 1728.bo | 24 | no | \(1\) | \(1\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(-1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{1728}(77,\cdot)\) | 1728.cf | 144 | yes | \(-1\) | \(1\) | \(e\left(\frac{143}{144}\right)\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{91}{144}\right)\) | \(e\left(\frac{137}{144}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{61}{72}\right)\) | \(e\left(\frac{71}{72}\right)\) | \(e\left(\frac{133}{144}\right)\) | \(e\left(\frac{13}{18}\right)\) |
\(\chi_{1728}(79,\cdot)\) | 1728.bt | 36 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) |
\(\chi_{1728}(83,\cdot)\) | 1728.ch | 144 | yes | \(1\) | \(1\) | \(e\left(\frac{103}{144}\right)\) | \(e\left(\frac{55}{72}\right)\) | \(e\left(\frac{59}{144}\right)\) | \(e\left(\frac{1}{144}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{17}{72}\right)\) | \(e\left(\frac{31}{72}\right)\) | \(e\left(\frac{125}{144}\right)\) | \(e\left(\frac{1}{9}\right)\) |
\(\chi_{1728}(85,\cdot)\) | 1728.ce | 144 | yes | \(1\) | \(1\) | \(e\left(\frac{53}{144}\right)\) | \(e\left(\frac{65}{72}\right)\) | \(e\left(\frac{73}{144}\right)\) | \(e\left(\frac{11}{144}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{43}{72}\right)\) | \(e\left(\frac{53}{72}\right)\) | \(e\left(\frac{7}{144}\right)\) | \(e\left(\frac{13}{18}\right)\) |
\(\chi_{1728}(89,\cdot)\) | 1728.bq | 24 | no | \(-1\) | \(1\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{1728}(91,\cdot)\) | 1728.bw | 48 | no | \(-1\) | \(1\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(-i\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{1728}(95,\cdot)\) | 1728.bl | 18 | no | \(1\) | \(1\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) |