Properties

 Modulus $448$ Structure $$C_{2}\times C_{2}\times C_{48}$$ Order $192$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(448)

pari: g = idealstar(,448,2)

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 = $\chi_{448}(127,\cdot)$, $\chi_{448}(197,\cdot)$, $\chi_{448}(129,\cdot)$

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 $$-1$$ $$1$$ $$3$$ $$5$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$23$$ $$25$$
$$\chi_{448}(1,\cdot)$$ 448.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{448}(3,\cdot)$$ 448.bm 48 yes $$1$$ $$1$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{1}{48}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{5}{48}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$i$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{31}{48}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{1}{24}\right)$$
$$\chi_{448}(5,\cdot)$$ 448.bk 48 yes $$-1$$ $$1$$ $$e\left(\frac{1}{48}\right)$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{31}{48}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$i$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{29}{48}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{11}{24}\right)$$
$$\chi_{448}(9,\cdot)$$ 448.bh 24 no $$1$$ $$1$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$
$$\chi_{448}(11,\cdot)$$ 448.bl 48 yes $$-1$$ $$1$$ $$e\left(\frac{5}{48}\right)$$ $$e\left(\frac{31}{48}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{35}{48}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$-i$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{48}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{7}{24}\right)$$
$$\chi_{448}(13,\cdot)$$ 448.bf 16 yes $$-1$$ $$1$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$
$$\chi_{448}(15,\cdot)$$ 448.k 4 no $$-1$$ $$1$$ $$i$$ $$i$$ $$-1$$ $$-i$$ $$-i$$ $$-1$$ $$1$$ $$i$$ $$1$$ $$-1$$
$$\chi_{448}(17,\cdot)$$ 448.bb 12 no $$-1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$-i$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{448}(19,\cdot)$$ 448.bm 48 yes $$1$$ $$1$$ $$e\left(\frac{31}{48}\right)$$ $$e\left(\frac{29}{48}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{1}{48}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$i$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{35}{48}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{5}{24}\right)$$
$$\chi_{448}(23,\cdot)$$ 448.bj 24 no $$-1$$ $$1$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$
$$\chi_{448}(25,\cdot)$$ 448.bh 24 no $$1$$ $$1$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$
$$\chi_{448}(27,\cdot)$$ 448.bd 16 yes $$1$$ $$1$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$-i$$ $$i$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$
$$\chi_{448}(29,\cdot)$$ 448.bc 16 no $$1$$ $$1$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$-i$$ $$i$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$
$$\chi_{448}(31,\cdot)$$ 448.q 6 no $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{448}(33,\cdot)$$ 448.n 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{448}(37,\cdot)$$ 448.bn 48 yes $$1$$ $$1$$ $$e\left(\frac{1}{48}\right)$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{7}{48}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$i$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{29}{48}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{11}{24}\right)$$
$$\chi_{448}(39,\cdot)$$ 448.bj 24 no $$-1$$ $$1$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$
$$\chi_{448}(41,\cdot)$$ 448.v 8 no $$-1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$-i$$
$$\chi_{448}(43,\cdot)$$ 448.be 16 no $$-1$$ $$1$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$
$$\chi_{448}(45,\cdot)$$ 448.bk 48 yes $$-1$$ $$1$$ $$e\left(\frac{23}{48}\right)$$ $$e\left(\frac{13}{48}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{41}{48}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$-i$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{43}{48}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{13}{24}\right)$$
$$\chi_{448}(47,\cdot)$$ 448.z 12 no $$1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$-i$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{448}(51,\cdot)$$ 448.bl 48 yes $$-1$$ $$1$$ $$e\left(\frac{31}{48}\right)$$ $$e\left(\frac{29}{48}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{25}{48}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$i$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{35}{48}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{5}{24}\right)$$
$$\chi_{448}(53,\cdot)$$ 448.bn 48 yes $$1$$ $$1$$ $$e\left(\frac{29}{48}\right)$$ $$e\left(\frac{31}{48}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$i$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{25}{48}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{7}{24}\right)$$
$$\chi_{448}(55,\cdot)$$ 448.x 8 no $$1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$-i$$ $$-i$$
$$\chi_{448}(57,\cdot)$$ 448.u 8 no $$1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$-1$$ $$-1$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$i$$
$$\chi_{448}(59,\cdot)$$ 448.bm 48 yes $$1$$ $$1$$ $$e\left(\frac{41}{48}\right)$$ $$e\left(\frac{43}{48}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{23}{48}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$-i$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{37}{48}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{19}{24}\right)$$
$$\chi_{448}(61,\cdot)$$ 448.bk 48 yes $$-1$$ $$1$$ $$e\left(\frac{19}{48}\right)$$ $$e\left(\frac{17}{48}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{13}{48}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$-i$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{23}{48}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{17}{24}\right)$$
$$\chi_{448}(65,\cdot)$$ 448.i 3 no $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{448}(67,\cdot)$$ 448.bl 48 yes $$-1$$ $$1$$ $$e\left(\frac{35}{48}\right)$$ $$e\left(\frac{25}{48}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{5}{48}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$i$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{7}{48}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{1}{24}\right)$$
$$\chi_{448}(69,\cdot)$$ 448.bf 16 yes $$-1$$ $$1$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$i$$ $$i$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$
$$\chi_{448}(71,\cdot)$$ 448.w 8 no $$-1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$ $$i$$ $$i$$
$$\chi_{448}(73,\cdot)$$ 448.bg 24 no $$-1$$ $$1$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$
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