# Properties

 Modulus 87 Structure $$C_{28}\times C_{2}$$ Order 56

Show commands for: SageMath / Pari/GP

sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(87)
pari: g = idealstar(,87,2)

## Character group

 sage: G.order() pari: g.no Order = 56 sage: H.invariants() pari: g.cyc Structure = $$C_{28}\times C_{2}$$ sage: H.gens() pari: g.gen Generators = $\chi_{87}(31,\cdot)$, $\chi_{87}(59,\cdot)$

## First 32 of 56 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 4 5 7 8 10 11 13 14 16
$$\chi_{87}(1,\cdot)$$ 87.a 1 No $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{87}(2,\cdot)$$ 87.k 28 Yes $$1$$ $$1$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{87}(4,\cdot)$$ 87.i 14 No $$1$$ $$1$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$
$$\chi_{87}(5,\cdot)$$ 87.h 14 Yes $$-1$$ $$1$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{87}(7,\cdot)$$ 87.g 7 No $$1$$ $$1$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{87}(8,\cdot)$$ 87.k 28 Yes $$1$$ $$1$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{3}{7}\right)$$
$$\chi_{87}(10,\cdot)$$ 87.l 28 No $$-1$$ $$1$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{2}{7}\right)$$
$$\chi_{87}(11,\cdot)$$ 87.k 28 Yes $$1$$ $$1$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{4}{7}\right)$$
$$\chi_{87}(13,\cdot)$$ 87.i 14 No $$1$$ $$1$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$
$$\chi_{87}(14,\cdot)$$ 87.k 28 Yes $$1$$ $$1$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{6}{7}\right)$$
$$\chi_{87}(16,\cdot)$$ 87.g 7 No $$1$$ $$1$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$
$$\chi_{87}(17,\cdot)$$ 87.f 4 Yes $$1$$ $$1$$ $$i$$ $$-1$$ $$1$$ $$1$$ $$-i$$ $$i$$ $$i$$ $$-1$$ $$i$$ $$1$$
$$\chi_{87}(19,\cdot)$$ 87.l 28 No $$-1$$ $$1$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{2}{7}\right)$$
$$\chi_{87}(20,\cdot)$$ 87.j 14 Yes $$-1$$ $$1$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$
$$\chi_{87}(22,\cdot)$$ 87.i 14 No $$1$$ $$1$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{87}(23,\cdot)$$ 87.j 14 Yes $$-1$$ $$1$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$
$$\chi_{87}(25,\cdot)$$ 87.g 7 No $$1$$ $$1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$
$$\chi_{87}(26,\cdot)$$ 87.k 28 Yes $$1$$ $$1$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{87}(28,\cdot)$$ 87.c 2 No $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$1$$
$$\chi_{87}(31,\cdot)$$ 87.l 28 No $$-1$$ $$1$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{87}(32,\cdot)$$ 87.k 28 Yes $$1$$ $$1$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{87}(34,\cdot)$$ 87.i 14 No $$1$$ $$1$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{87}(35,\cdot)$$ 87.h 14 Yes $$-1$$ $$1$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$
$$\chi_{87}(37,\cdot)$$ 87.l 28 No $$-1$$ $$1$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{3}{7}\right)$$
$$\chi_{87}(38,\cdot)$$ 87.h 14 Yes $$-1$$ $$1$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$
$$\chi_{87}(40,\cdot)$$ 87.l 28 No $$-1$$ $$1$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{4}{7}\right)$$
$$\chi_{87}(41,\cdot)$$ 87.f 4 Yes $$1$$ $$1$$ $$-i$$ $$-1$$ $$1$$ $$1$$ $$i$$ $$-i$$ $$-i$$ $$-1$$ $$-i$$ $$1$$
$$\chi_{87}(43,\cdot)$$ 87.l 28 No $$-1$$ $$1$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{6}{7}\right)$$
$$\chi_{87}(44,\cdot)$$ 87.k 28 Yes $$1$$ $$1$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{6}{7}\right)$$
$$\chi_{87}(46,\cdot)$$ 87.e 4 No $$-1$$ $$1$$ $$-i$$ $$-1$$ $$-1$$ $$1$$ $$i$$ $$i$$ $$-i$$ $$-1$$ $$-i$$ $$1$$
$$\chi_{87}(47,\cdot)$$ 87.k 28 Yes $$1$$ $$1$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{4}{7}\right)$$
$$\chi_{87}(49,\cdot)$$ 87.g 7 No $$1$$ $$1$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$