Properties

 Modulus 55 Structure $$C_{20}\times C_{2}$$ Order 40

Show commands for: SageMath / Pari/GP

sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(55)
pari: g = idealstar(,55,2)

Character group

 sage: G.order() pari: g.no Order = 40 sage: H.invariants() pari: g.cyc Structure = $$C_{20}\times C_{2}$$ sage: H.gens() pari: g.gen Generators = $\chi_{55}(13,\cdot)$, $\chi_{55}(21,\cdot)$

First 32 of 40 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.

orbit label order primitive -1 1 2 3 4 6 7 8 9 12 13 14
$$\chi_{55}(1,\cdot)$$ 55.a 1 No $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{55}(2,\cdot)$$ 55.l 20 Yes $$1$$ $$1$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$i$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{55}(3,\cdot)$$ 55.k 20 Yes $$-1$$ $$1$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$-i$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{55}(4,\cdot)$$ 55.j 10 Yes $$1$$ $$1$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$-1$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$
$$\chi_{55}(6,\cdot)$$ 55.i 10 No $$-1$$ $$1$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$1$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{55}(7,\cdot)$$ 55.l 20 Yes $$1$$ $$1$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$i$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{55}(8,\cdot)$$ 55.l 20 Yes $$1$$ $$1$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$-i$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{55}(9,\cdot)$$ 55.j 10 Yes $$1$$ $$1$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$-1$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{55}(12,\cdot)$$ 55.f 4 No $$-1$$ $$1$$ $$i$$ $$-i$$ $$-1$$ $$1$$ $$i$$ $$-i$$ $$-1$$ $$i$$ $$-i$$ $$-1$$
$$\chi_{55}(13,\cdot)$$ 55.l 20 Yes $$1$$ $$1$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$-i$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{55}(14,\cdot)$$ 55.j 10 Yes $$1$$ $$1$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$-1$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$
$$\chi_{55}(16,\cdot)$$ 55.g 5 No $$1$$ $$1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{55}(17,\cdot)$$ 55.l 20 Yes $$1$$ $$1$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$i$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{55}(18,\cdot)$$ 55.l 20 Yes $$1$$ $$1$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$-i$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{55}(19,\cdot)$$ 55.h 10 Yes $$-1$$ $$1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$-1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$
$$\chi_{55}(21,\cdot)$$ 55.c 2 No $$-1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$1$$
$$\chi_{55}(23,\cdot)$$ 55.f 4 No $$-1$$ $$1$$ $$-i$$ $$i$$ $$-1$$ $$1$$ $$-i$$ $$i$$ $$-1$$ $$-i$$ $$i$$ $$-1$$
$$\chi_{55}(24,\cdot)$$ 55.h 10 Yes $$-1$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$-1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{55}(26,\cdot)$$ 55.g 5 No $$1$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$
$$\chi_{55}(27,\cdot)$$ 55.k 20 Yes $$-1$$ $$1$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$i$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{55}(28,\cdot)$$ 55.l 20 Yes $$1$$ $$1$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$-i$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{55}(29,\cdot)$$ 55.h 10 Yes $$-1$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$-1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$
$$\chi_{55}(31,\cdot)$$ 55.g 5 No $$1$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{55}(32,\cdot)$$ 55.e 4 Yes $$1$$ $$1$$ $$-i$$ $$-i$$ $$-1$$ $$-1$$ $$-i$$ $$i$$ $$-1$$ $$i$$ $$i$$ $$-1$$
$$\chi_{55}(34,\cdot)$$ 55.b 2 No $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$ $$1$$
$$\chi_{55}(36,\cdot)$$ 55.g 5 No $$1$$ $$1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$
$$\chi_{55}(37,\cdot)$$ 55.k 20 Yes $$-1$$ $$1$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$i$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{55}(38,\cdot)$$ 55.k 20 Yes $$-1$$ $$1$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$-i$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{55}(39,\cdot)$$ 55.h 10 Yes $$-1$$ $$1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$-1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{55}(41,\cdot)$$ 55.i 10 No $$-1$$ $$1$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$1$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$
$$\chi_{55}(42,\cdot)$$ 55.k 20 Yes $$-1$$ $$1$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$i$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{55}(43,\cdot)$$ 55.e 4 Yes $$1$$ $$1$$ $$i$$ $$i$$ $$-1$$ $$-1$$ $$i$$ $$-i$$ $$-1$$ $$-i$$ $$-i$$ $$-1$$