# Properties

 Modulus $663$ Structure $$C_{2}\times C_{4}\times C_{48}$$ Order $384$

# Learn more

Show commands: PariGP / SageMath

sage: H = DirichletGroup(663)

pari: g = idealstar(,663,2)

## Character group

 sage: G.order()  pari: g.no Order = 384 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{4}\times C_{48}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{663}(443,\cdot)$, $\chi_{663}(613,\cdot)$, $\chi_{663}(547,\cdot)$

## First 32 of 384 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 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$14$$ $$16$$ $$19$$
$$\chi_{663}(1,\cdot)$$ 663.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{663}(2,\cdot)$$ 663.ck 24 yes $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$-1$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{663}(4,\cdot)$$ 663.bk 12 no $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$i$$ $$e\left(\frac{1}{12}\right)$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$-i$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{663}(5,\cdot)$$ 663.bz 16 yes $$-1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$
$$\chi_{663}(7,\cdot)$$ 663.cm 48 no $$1$$ $$1$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{31}{48}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{24}\right)$$
$$\chi_{663}(8,\cdot)$$ 663.bd 8 yes $$1$$ $$1$$ $$-1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$-1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$1$$ $$1$$
$$\chi_{663}(10,\cdot)$$ 663.cn 48 no $$-1$$ $$1$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{43}{48}\right)$$ $$e\left(\frac{7}{48}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{19}{24}\right)$$
$$\chi_{663}(11,\cdot)$$ 663.ct 48 yes $$-1$$ $$1$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{48}\right)$$ $$e\left(\frac{31}{48}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{24}\right)$$
$$\chi_{663}(14,\cdot)$$ 663.by 16 no $$1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$
$$\chi_{663}(16,\cdot)$$ 663.w 6 no $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{663}(19,\cdot)$$ 663.ce 24 no $$-1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$1$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{663}(20,\cdot)$$ 663.cq 48 yes $$-1$$ $$1$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{37}{48}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{17}{48}\right)$$ $$e\left(\frac{17}{48}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{24}\right)$$
$$\chi_{663}(22,\cdot)$$ 663.co 48 no $$-1$$ $$1$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{37}{48}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{29}{48}\right)$$ $$e\left(\frac{41}{48}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{17}{24}\right)$$
$$\chi_{663}(23,\cdot)$$ 663.cs 48 yes $$1$$ $$1$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{23}{48}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{48}\right)$$ $$e\left(\frac{43}{48}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{24}\right)$$
$$\chi_{663}(25,\cdot)$$ 663.bh 8 no $$1$$ $$1$$ $$i$$ $$-1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$1$$ $$i$$
$$\chi_{663}(28,\cdot)$$ 663.cp 48 no $$1$$ $$1$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{35}{48}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{48}\right)$$ $$e\left(\frac{31}{48}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{13}{24}\right)$$
$$\chi_{663}(29,\cdot)$$ 663.cr 48 yes $$1$$ $$1$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{29}{48}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{37}{48}\right)$$ $$e\left(\frac{25}{48}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{24}\right)$$
$$\chi_{663}(31,\cdot)$$ 663.ca 16 no $$1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$-1$$ $$e\left(\frac{5}{8}\right)$$
$$\chi_{663}(32,\cdot)$$ 663.ck 24 yes $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$-1$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{663}(35,\cdot)$$ 663.y 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{663}(37,\cdot)$$ 663.cp 48 no $$1$$ $$1$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{5}{48}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{48}\right)$$ $$e\left(\frac{25}{48}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{19}{24}\right)$$
$$\chi_{663}(38,\cdot)$$ 663.k 4 yes $$-1$$ $$1$$ $$-1$$ $$1$$ $$-i$$ $$-i$$ $$-1$$ $$i$$ $$i$$ $$i$$ $$1$$ $$1$$
$$\chi_{663}(40,\cdot)$$ 663.cb 16 no $$-1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$i$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$
$$\chi_{663}(41,\cdot)$$ 663.cq 48 yes $$-1$$ $$1$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{23}{48}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{43}{48}\right)$$ $$e\left(\frac{43}{48}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{24}\right)$$
$$\chi_{663}(43,\cdot)$$ 663.ch 24 no $$1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$-i$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{7}{12}\right)$$
$$\chi_{663}(44,\cdot)$$ 663.bw 16 yes $$-1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$-1$$ $$e\left(\frac{3}{8}\right)$$
$$\chi_{663}(46,\cdot)$$ 663.cp 48 no $$1$$ $$1$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{1}{48}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{29}{48}\right)$$ $$e\left(\frac{5}{48}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{23}{24}\right)$$
$$\chi_{663}(47,\cdot)$$ 663.r 4 yes $$1$$ $$1$$ $$i$$ $$-1$$ $$1$$ $$-1$$ $$-i$$ $$i$$ $$1$$ $$-i$$ $$1$$ $$-i$$
$$\chi_{663}(49,\cdot)$$ 663.ch 24 no $$1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$i$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{12}\right)$$
$$\chi_{663}(50,\cdot)$$ 663.bp 12 yes $$1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$i$$ $$e\left(\frac{11}{12}\right)$$ $$i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$
$$\chi_{663}(53,\cdot)$$ 663.bf 8 no $$-1$$ $$1$$ $$-i$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$1$$ $$i$$
$$\chi_{663}(55,\cdot)$$ 663.bv 12 no $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-i$$ $$e\left(\frac{11}{12}\right)$$ $$-1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$-i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$
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