# Properties

 Modulus $476$ Structure $$C_{2}\times C_{2}\times C_{48}$$ Order $192$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(476)

pari: g = idealstar(,476,2)

## 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 = $\chi_{476}(239,\cdot)$, $\chi_{476}(409,\cdot)$, $\chi_{476}(309,\cdot)$

## 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 $$-1$$ $$1$$ $$3$$ $$5$$ $$9$$ $$11$$ $$13$$ $$15$$ $$19$$ $$23$$ $$25$$ $$27$$
$$\chi_{476}(1,\cdot)$$ 476.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{476}(3,\cdot)$$ 476.bk 48 yes $$-1$$ $$1$$ $$e\left(\frac{35}{48}\right)$$ $$e\left(\frac{7}{48}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{29}{48}\right)$$ $$-i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{37}{48}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{3}{16}\right)$$
$$\chi_{476}(5,\cdot)$$ 476.bl 48 no $$1$$ $$1$$ $$e\left(\frac{7}{48}\right)$$ $$e\left(\frac{35}{48}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{25}{48}\right)$$ $$-i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{17}{48}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{7}{16}\right)$$
$$\chi_{476}(9,\cdot)$$ 476.bh 24 no $$1$$ $$1$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$-1$$ $$-i$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{3}{8}\right)$$
$$\chi_{476}(11,\cdot)$$ 476.bm 48 yes $$1$$ $$1$$ $$e\left(\frac{29}{48}\right)$$ $$e\left(\frac{25}{48}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{11}{48}\right)$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{19}{48}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{13}{16}\right)$$
$$\chi_{476}(13,\cdot)$$ 476.j 4 no $$-1$$ $$1$$ $$-i$$ $$-i$$ $$-1$$ $$-i$$ $$-1$$ $$-1$$ $$1$$ $$-i$$ $$-1$$ $$i$$
$$\chi_{476}(15,\cdot)$$ 476.v 8 no $$-1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$-1$$ $$-i$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$
$$\chi_{476}(19,\cdot)$$ 476.bj 24 yes $$1$$ $$1$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$1$$ $$-i$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{8}\right)$$
$$\chi_{476}(23,\cdot)$$ 476.bm 48 yes $$1$$ $$1$$ $$e\left(\frac{37}{48}\right)$$ $$e\left(\frac{17}{48}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{19}{48}\right)$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{5}{16}\right)$$
$$\chi_{476}(25,\cdot)$$ 476.bh 24 no $$1$$ $$1$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$-1$$ $$-i$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{7}{8}\right)$$
$$\chi_{476}(27,\cdot)$$ 476.bf 16 yes $$-1$$ $$1$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$i$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{9}{16}\right)$$
$$\chi_{476}(29,\cdot)$$ 476.bc 16 no $$-1$$ $$1$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{16}\right)$$
$$\chi_{476}(31,\cdot)$$ 476.bk 48 yes $$-1$$ $$1$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{31}{48}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{5}{48}\right)$$ $$-i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{13}{48}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{11}{16}\right)$$
$$\chi_{476}(33,\cdot)$$ 476.n 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$
$$\chi_{476}(37,\cdot)$$ 476.bn 48 no $$-1$$ $$1$$ $$e\left(\frac{19}{48}\right)$$ $$e\left(\frac{47}{48}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{37}{48}\right)$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{29}{48}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{3}{16}\right)$$
$$\chi_{476}(39,\cdot)$$ 476.bm 48 yes $$1$$ $$1$$ $$e\left(\frac{23}{48}\right)$$ $$e\left(\frac{43}{48}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{17}{48}\right)$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{25}{48}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{7}{16}\right)$$
$$\chi_{476}(41,\cdot)$$ 476.be 16 no $$1$$ $$1$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{9}{16}\right)$$
$$\chi_{476}(43,\cdot)$$ 476.v 8 no $$-1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$-1$$ $$i$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$i$$ $$e\left(\frac{7}{8}\right)$$
$$\chi_{476}(45,\cdot)$$ 476.bl 48 no $$1$$ $$1$$ $$e\left(\frac{29}{48}\right)$$ $$e\left(\frac{1}{48}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{35}{48}\right)$$ $$i$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{43}{48}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{13}{16}\right)$$
$$\chi_{476}(47,\cdot)$$ 476.z 12 yes $$1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-i$$
$$\chi_{476}(53,\cdot)$$ 476.bh 24 no $$1$$ $$1$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$-1$$ $$i$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{8}\right)$$
$$\chi_{476}(55,\cdot)$$ 476.k 4 yes $$1$$ $$1$$ $$-i$$ $$i$$ $$-1$$ $$-i$$ $$-1$$ $$1$$ $$-1$$ $$-i$$ $$-1$$ $$i$$
$$\chi_{476}(57,\cdot)$$ 476.bc 16 no $$-1$$ $$1$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$
$$\chi_{476}(59,\cdot)$$ 476.bj 24 yes $$1$$ $$1$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$1$$ $$i$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{7}{8}\right)$$
$$\chi_{476}(61,\cdot)$$ 476.bl 48 no $$1$$ $$1$$ $$e\left(\frac{1}{48}\right)$$ $$e\left(\frac{5}{48}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{31}{48}\right)$$ $$i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{23}{48}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{1}{16}\right)$$
$$\chi_{476}(65,\cdot)$$ 476.bn 48 no $$-1$$ $$1$$ $$e\left(\frac{43}{48}\right)$$ $$e\left(\frac{23}{48}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{13}{48}\right)$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{5}{48}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{11}{16}\right)$$
$$\chi_{476}(67,\cdot)$$ 476.o 6 yes $$-1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$
$$\chi_{476}(69,\cdot)$$ 476.c 2 no $$-1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$-1$$
$$\chi_{476}(71,\cdot)$$ 476.bd 16 no $$1$$ $$1$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$
$$\chi_{476}(73,\cdot)$$ 476.bl 48 no $$1$$ $$1$$ $$e\left(\frac{23}{48}\right)$$ $$e\left(\frac{19}{48}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{41}{48}\right)$$ $$-i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{1}{48}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{7}{16}\right)$$
$$\chi_{476}(75,\cdot)$$ 476.bk 48 yes $$-1$$ $$1$$ $$e\left(\frac{1}{48}\right)$$ $$e\left(\frac{29}{48}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{31}{48}\right)$$ $$i$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{23}{48}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{1}{16}\right)$$
$$\chi_{476}(79,\cdot)$$ 476.bm 48 yes $$1$$ $$1$$ $$e\left(\frac{13}{48}\right)$$ $$e\left(\frac{41}{48}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{43}{48}\right)$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{35}{48}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{13}{16}\right)$$
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