Properties

 Modulus $768$ Structure $$C_{2}\times C_{2}\times C_{64}$$ Order $256$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(768)

pari: g = idealstar(,768,2)

Character group

 sage: G.order()  pari: g.no Order = 256 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{2}\times C_{64}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{768}(511,\cdot)$, $\chi_{768}(517,\cdot)$, $\chi_{768}(257,\cdot)$

First 32 of 256 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 $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$
$$\chi_{768}(1,\cdot)$$ 768.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{768}(5,\cdot)$$ 768.y 64 yes $$-1$$ $$1$$ $$e\left(\frac{33}{64}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{53}{64}\right)$$ $$e\left(\frac{47}{64}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{23}{64}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{27}{64}\right)$$ $$e\left(\frac{1}{8}\right)$$
$$\chi_{768}(7,\cdot)$$ 768.u 32 no $$-1$$ $$1$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$-i$$
$$\chi_{768}(11,\cdot)$$ 768.ba 64 yes $$1$$ $$1$$ $$e\left(\frac{53}{64}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{57}{64}\right)$$ $$e\left(\frac{27}{64}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{3}{64}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{55}{64}\right)$$ $$e\left(\frac{1}{8}\right)$$
$$\chi_{768}(13,\cdot)$$ 768.z 64 no $$1$$ $$1$$ $$e\left(\frac{47}{64}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{27}{64}\right)$$ $$e\left(\frac{33}{64}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{57}{64}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{21}{64}\right)$$ $$e\left(\frac{7}{8}\right)$$
$$\chi_{768}(17,\cdot)$$ 768.q 16 no $$-1$$ $$1$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$-i$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$-1$$
$$\chi_{768}(19,\cdot)$$ 768.bb 64 no $$-1$$ $$1$$ $$e\left(\frac{23}{64}\right)$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{3}{64}\right)$$ $$e\left(\frac{57}{64}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{49}{64}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{13}{64}\right)$$ $$e\left(\frac{3}{8}\right)$$
$$\chi_{768}(23,\cdot)$$ 768.w 32 no $$1$$ $$1$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$i$$
$$\chi_{768}(25,\cdot)$$ 768.v 32 no $$1$$ $$1$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$i$$
$$\chi_{768}(29,\cdot)$$ 768.y 64 yes $$-1$$ $$1$$ $$e\left(\frac{27}{64}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{55}{64}\right)$$ $$e\left(\frac{21}{64}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{13}{64}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{57}{64}\right)$$ $$e\left(\frac{3}{8}\right)$$
$$\chi_{768}(31,\cdot)$$ 768.m 8 no $$-1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-1$$ $$e\left(\frac{3}{8}\right)$$ $$i$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$-1$$
$$\chi_{768}(35,\cdot)$$ 768.ba 64 yes $$1$$ $$1$$ $$e\left(\frac{43}{64}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{39}{64}\right)$$ $$e\left(\frac{5}{64}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{29}{64}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{41}{64}\right)$$ $$e\left(\frac{7}{8}\right)$$
$$\chi_{768}(37,\cdot)$$ 768.z 64 no $$1$$ $$1$$ $$e\left(\frac{25}{64}\right)$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{13}{64}\right)$$ $$e\left(\frac{23}{64}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{63}{64}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{3}{64}\right)$$ $$e\left(\frac{1}{8}\right)$$
$$\chi_{768}(41,\cdot)$$ 768.x 32 no $$-1$$ $$1$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$-i$$
$$\chi_{768}(43,\cdot)$$ 768.bb 64 no $$-1$$ $$1$$ $$e\left(\frac{61}{64}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{33}{64}\right)$$ $$e\left(\frac{51}{64}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{27}{64}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{15}{64}\right)$$ $$e\left(\frac{1}{8}\right)$$
$$\chi_{768}(47,\cdot)$$ 768.s 16 no $$1$$ $$1$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$-i$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$1$$
$$\chi_{768}(49,\cdot)$$ 768.r 16 no $$1$$ $$1$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$-i$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$-1$$
$$\chi_{768}(53,\cdot)$$ 768.y 64 yes $$-1$$ $$1$$ $$e\left(\frac{37}{64}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{9}{64}\right)$$ $$e\left(\frac{43}{64}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{51}{64}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{7}{64}\right)$$ $$e\left(\frac{5}{8}\right)$$
$$\chi_{768}(55,\cdot)$$ 768.u 32 no $$-1$$ $$1$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$i$$
$$\chi_{768}(59,\cdot)$$ 768.ba 64 yes $$1$$ $$1$$ $$e\left(\frac{49}{64}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{37}{64}\right)$$ $$e\left(\frac{31}{64}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{39}{64}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{11}{64}\right)$$ $$e\left(\frac{5}{8}\right)$$
$$\chi_{768}(61,\cdot)$$ 768.z 64 no $$1$$ $$1$$ $$e\left(\frac{19}{64}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{15}{64}\right)$$ $$e\left(\frac{61}{64}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{53}{64}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{33}{64}\right)$$ $$e\left(\frac{3}{8}\right)$$
$$\chi_{768}(65,\cdot)$$ 768.i 4 no $$-1$$ $$1$$ $$i$$ $$-1$$ $$i$$ $$i$$ $$-1$$ $$i$$ $$1$$ $$-1$$ $$-i$$ $$1$$
$$\chi_{768}(67,\cdot)$$ 768.bb 64 no $$-1$$ $$1$$ $$e\left(\frac{51}{64}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{15}{64}\right)$$ $$e\left(\frac{29}{64}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{53}{64}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{1}{64}\right)$$ $$e\left(\frac{7}{8}\right)$$
$$\chi_{768}(71,\cdot)$$ 768.w 32 no $$1$$ $$1$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$-i$$
$$\chi_{768}(73,\cdot)$$ 768.v 32 no $$1$$ $$1$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$-i$$
$$\chi_{768}(77,\cdot)$$ 768.y 64 yes $$-1$$ $$1$$ $$e\left(\frac{63}{64}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{43}{64}\right)$$ $$e\left(\frac{49}{64}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{9}{64}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{5}{64}\right)$$ $$e\left(\frac{7}{8}\right)$$
$$\chi_{768}(79,\cdot)$$ 768.t 16 no $$-1$$ $$1$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$-i$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$1$$
$$\chi_{768}(83,\cdot)$$ 768.ba 64 yes $$1$$ $$1$$ $$e\left(\frac{7}{64}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{51}{64}\right)$$ $$e\left(\frac{41}{64}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{33}{64}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{29}{64}\right)$$ $$e\left(\frac{3}{8}\right)$$
$$\chi_{768}(85,\cdot)$$ 768.z 64 no $$1$$ $$1$$ $$e\left(\frac{29}{64}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{33}{64}\right)$$ $$e\left(\frac{19}{64}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{27}{64}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{47}{64}\right)$$ $$e\left(\frac{5}{8}\right)$$
$$\chi_{768}(89,\cdot)$$ 768.x 32 no $$-1$$ $$1$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$i$$
$$\chi_{768}(91,\cdot)$$ 768.bb 64 no $$-1$$ $$1$$ $$e\left(\frac{57}{64}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{13}{64}\right)$$ $$e\left(\frac{55}{64}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{63}{64}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{35}{64}\right)$$ $$e\left(\frac{5}{8}\right)$$
$$\chi_{768}(95,\cdot)$$ 768.o 8 no $$1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$i$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$-1$$