# Properties

 Modulus $576$ Structure $$C_{2}\times C_{2}\times C_{48}$$ Order $192$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(576)

pari: g = idealstar(,576,2)

## Character group

 sage: G.order()  pari: g.no Order = 192 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{2}\times C_{48}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{576}(127,\cdot)$, $\chi_{576}(325,\cdot)$, $\chi_{576}(65,\cdot)$

## First 32 of 192 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 $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$
$$\chi_{576}(1,\cdot)$$ 576.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{576}(5,\cdot)$$ 576.bn 48 yes $$-1$$ $$1$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{7}{48}\right)$$ $$e\left(\frac{29}{48}\right)$$ $$i$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{25}{48}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{576}(7,\cdot)$$ 576.bh 24 no $$-1$$ $$1$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{576}(11,\cdot)$$ 576.bl 48 yes $$1$$ $$1$$ $$e\left(\frac{7}{48}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{1}{48}\right)$$ $$i$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{29}{48}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{576}(13,\cdot)$$ 576.bm 48 yes $$1$$ $$1$$ $$e\left(\frac{29}{48}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{1}{48}\right)$$ $$e\left(\frac{35}{48}\right)$$ $$i$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{31}{48}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{576}(17,\cdot)$$ 576.j 4 no $$-1$$ $$1$$ $$i$$ $$-1$$ $$i$$ $$i$$ $$-1$$ $$i$$ $$1$$ $$-1$$ $$-i$$ $$1$$
$$\chi_{576}(19,\cdot)$$ 576.bf 16 no $$-1$$ $$1$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$i$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$1$$
$$\chi_{576}(23,\cdot)$$ 576.bj 24 no $$1$$ $$1$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{576}(25,\cdot)$$ 576.bg 24 no $$1$$ $$1$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{576}(29,\cdot)$$ 576.bn 48 yes $$-1$$ $$1$$ $$e\left(\frac{25}{48}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{29}{48}\right)$$ $$e\left(\frac{31}{48}\right)$$ $$-i$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{35}{48}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{576}(31,\cdot)$$ 576.t 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{576}(35,\cdot)$$ 576.be 16 no $$1$$ $$1$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$-i$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$1$$
$$\chi_{576}(37,\cdot)$$ 576.bd 16 no $$1$$ $$1$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$-i$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$-1$$
$$\chi_{576}(41,\cdot)$$ 576.bi 24 no $$-1$$ $$1$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{576}(43,\cdot)$$ 576.bk 48 yes $$-1$$ $$1$$ $$e\left(\frac{7}{48}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{25}{48}\right)$$ $$-i$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{29}{48}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{576}(47,\cdot)$$ 576.y 12 no $$1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$-1$$ $$-i$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{576}(49,\cdot)$$ 576.bb 12 no $$1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$1$$ $$-i$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{576}(53,\cdot)$$ 576.bc 16 no $$-1$$ $$1$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$i$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$-1$$
$$\chi_{576}(55,\cdot)$$ 576.u 8 no $$-1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$-1$$
$$\chi_{576}(59,\cdot)$$ 576.bl 48 yes $$1$$ $$1$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{31}{48}\right)$$ $$e\left(\frac{29}{48}\right)$$ $$i$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{25}{48}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{576}(61,\cdot)$$ 576.bm 48 yes $$1$$ $$1$$ $$e\left(\frac{25}{48}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{29}{48}\right)$$ $$e\left(\frac{7}{48}\right)$$ $$i$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{35}{48}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{576}(65,\cdot)$$ 576.q 6 no $$-1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{576}(67,\cdot)$$ 576.bk 48 yes $$-1$$ $$1$$ $$e\left(\frac{41}{48}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{37}{48}\right)$$ $$e\left(\frac{23}{48}\right)$$ $$i$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{19}{48}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{576}(71,\cdot)$$ 576.w 8 no $$1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$-1$$
$$\chi_{576}(73,\cdot)$$ 576.v 8 no $$1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$-1$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$1$$
$$\chi_{576}(77,\cdot)$$ 576.bn 48 yes $$-1$$ $$1$$ $$e\left(\frac{5}{48}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{25}{48}\right)$$ $$e\left(\frac{35}{48}\right)$$ $$-i$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{7}{48}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{576}(79,\cdot)$$ 576.z 12 no $$-1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$1$$ $$i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{576}(83,\cdot)$$ 576.bl 48 yes $$1$$ $$1$$ $$e\left(\frac{13}{48}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{41}{48}\right)$$ $$e\left(\frac{43}{48}\right)$$ $$-i$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{47}{48}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{576}(85,\cdot)$$ 576.bm 48 yes $$1$$ $$1$$ $$e\left(\frac{23}{48}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{19}{48}\right)$$ $$e\left(\frac{41}{48}\right)$$ $$-i$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{13}{48}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{576}(89,\cdot)$$ 576.x 8 no $$-1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$i$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$1$$
$$\chi_{576}(91,\cdot)$$ 576.bf 16 no $$-1$$ $$1$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$-i$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$1$$
$$\chi_{576}(95,\cdot)$$ 576.p 6 no $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$