sage: H = DirichletGroup(384)
pari: g = idealstar(,384,2)
Character group
sage: G.order()
pari: g.no
| ||
Order | = | 128 |
sage: H.invariants()
pari: g.cyc
| ||
Structure | = | \(C_{2}\times C_{2}\times C_{32}\) |
sage: H.gens()
pari: g.gen
| ||
Generators | = | $\chi_{384}(127,\cdot)$, $\chi_{384}(133,\cdot)$, $\chi_{384}(257,\cdot)$ |
First 32 of 128 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 | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{384}(1,\cdot)\) | 384.a | 1 | no | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) |
\(\chi_{384}(5,\cdot)\) | 384.x | 32 | yes | \(-1\) | \(1\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(i\) |
\(\chi_{384}(7,\cdot)\) | 384.t | 16 | no | \(-1\) | \(1\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(-i\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(1\) |
\(\chi_{384}(11,\cdot)\) | 384.w | 32 | yes | \(1\) | \(1\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(-i\) |
\(\chi_{384}(13,\cdot)\) | 384.v | 32 | no | \(1\) | \(1\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(-i\) |
\(\chi_{384}(17,\cdot)\) | 384.p | 8 | no | \(-1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(-i\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(-i\) | \(-i\) | \(e\left(\frac{1}{8}\right)\) | \(1\) |
\(\chi_{384}(19,\cdot)\) | 384.u | 32 | no | \(-1\) | \(1\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(i\) |
\(\chi_{384}(23,\cdot)\) | 384.s | 16 | no | \(1\) | \(1\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(-i\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(1\) |
\(\chi_{384}(25,\cdot)\) | 384.r | 16 | no | \(1\) | \(1\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(-i\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(-1\) |
\(\chi_{384}(29,\cdot)\) | 384.x | 32 | yes | \(-1\) | \(1\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(-i\) |
\(\chi_{384}(31,\cdot)\) | 384.l | 4 | no | \(-1\) | \(1\) | \(i\) | \(1\) | \(-i\) | \(-i\) | \(1\) | \(i\) | \(1\) | \(-1\) | \(-i\) | \(-1\) |
\(\chi_{384}(35,\cdot)\) | 384.w | 32 | yes | \(1\) | \(1\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(i\) |
\(\chi_{384}(37,\cdot)\) | 384.v | 32 | no | \(1\) | \(1\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(i\) |
\(\chi_{384}(41,\cdot)\) | 384.q | 16 | no | \(-1\) | \(1\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(-i\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(-1\) |
\(\chi_{384}(43,\cdot)\) | 384.u | 32 | no | \(-1\) | \(1\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(-i\) |
\(\chi_{384}(47,\cdot)\) | 384.o | 8 | no | \(1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(i\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(i\) | \(-i\) | \(e\left(\frac{5}{8}\right)\) | \(-1\) |
\(\chi_{384}(49,\cdot)\) | 384.n | 8 | no | \(1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(i\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(-1\) | \(e\left(\frac{3}{8}\right)\) | \(-i\) | \(i\) | \(e\left(\frac{7}{8}\right)\) | \(1\) |
\(\chi_{384}(53,\cdot)\) | 384.x | 32 | yes | \(-1\) | \(1\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(i\) |
\(\chi_{384}(55,\cdot)\) | 384.t | 16 | no | \(-1\) | \(1\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(i\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(1\) |
\(\chi_{384}(59,\cdot)\) | 384.w | 32 | yes | \(1\) | \(1\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(-i\) |
\(\chi_{384}(61,\cdot)\) | 384.v | 32 | no | \(1\) | \(1\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(-i\) |
\(\chi_{384}(65,\cdot)\) | 384.h | 2 | no | \(-1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(-1\) | \(-1\) | \(-1\) | \(-1\) | \(1\) | \(1\) | \(1\) |
\(\chi_{384}(67,\cdot)\) | 384.u | 32 | no | \(-1\) | \(1\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(i\) |
\(\chi_{384}(71,\cdot)\) | 384.s | 16 | no | \(1\) | \(1\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(i\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(1\) |
\(\chi_{384}(73,\cdot)\) | 384.r | 16 | no | \(1\) | \(1\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(i\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(-1\) |
\(\chi_{384}(77,\cdot)\) | 384.x | 32 | yes | \(-1\) | \(1\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(-i\) |
\(\chi_{384}(79,\cdot)\) | 384.m | 8 | no | \(-1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(-i\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(-1\) | \(e\left(\frac{7}{8}\right)\) | \(i\) | \(i\) | \(e\left(\frac{7}{8}\right)\) | \(-1\) |
\(\chi_{384}(83,\cdot)\) | 384.w | 32 | yes | \(1\) | \(1\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(i\) |
\(\chi_{384}(85,\cdot)\) | 384.v | 32 | no | \(1\) | \(1\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(i\) |
\(\chi_{384}(89,\cdot)\) | 384.q | 16 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(i\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(-1\) |
\(\chi_{384}(91,\cdot)\) | 384.u | 32 | no | \(-1\) | \(1\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(-i\) |
\(\chi_{384}(95,\cdot)\) | 384.k | 4 | no | \(1\) | \(1\) | \(i\) | \(1\) | \(-i\) | \(i\) | \(-1\) | \(-i\) | \(-1\) | \(-1\) | \(-i\) | \(-1\) |