Properties

 Modulus 85 Structure $$C_{16}\times C_{4}$$ Order 64

Show commands for: SageMath / Pari/GP

sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed

sage: H = DirichletGroup_conrey(85)

pari: g = idealstar(,85,2)

Character group

 sage: G.order()  pari: g.no Order = 64 sage: H.invariants()  pari: g.cyc Structure = $$C_{16}\times C_{4}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{85}(71,\cdot)$, $\chi_{85}(52,\cdot)$

First 32 of 64 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 11 12 13
$$\chi_{85}(1,\cdot)$$ 85.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{85}(2,\cdot)$$ 85.k 8 yes $$-1$$ $$1$$ $$-1$$ $$e\left(\frac{5}{8}\right)$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-1$$ $$i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$i$$
$$\chi_{85}(3,\cdot)$$ 85.o 16 yes $$1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$i$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$-1$$
$$\chi_{85}(4,\cdot)$$ 85.j 4 yes $$1$$ $$1$$ $$1$$ $$i$$ $$1$$ $$i$$ $$-i$$ $$1$$ $$-1$$ $$i$$ $$i$$ $$-1$$
$$\chi_{85}(6,\cdot)$$ 85.q 16 no $$-1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$i$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$-i$$
$$\chi_{85}(7,\cdot)$$ 85.o 16 yes $$1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$-i$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$-1$$
$$\chi_{85}(8,\cdot)$$ 85.k 8 yes $$-1$$ $$1$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$-1$$ $$-i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$
$$\chi_{85}(9,\cdot)$$ 85.m 8 yes $$1$$ $$1$$ $$i$$ $$e\left(\frac{5}{8}\right)$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$1$$
$$\chi_{85}(11,\cdot)$$ 85.q 16 no $$-1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$i$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$-i$$
$$\chi_{85}(12,\cdot)$$ 85.r 16 yes $$1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$i$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$1$$
$$\chi_{85}(13,\cdot)$$ 85.f 4 yes $$-1$$ $$1$$ $$i$$ $$-1$$ $$-1$$ $$-i$$ $$-1$$ $$-i$$ $$1$$ $$-i$$ $$1$$ $$i$$
$$\chi_{85}(14,\cdot)$$ 85.p 16 yes $$-1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$-i$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$-i$$
$$\chi_{85}(16,\cdot)$$ 85.d 2 no $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$
$$\chi_{85}(18,\cdot)$$ 85.h 4 no $$-1$$ $$1$$ $$-i$$ $$i$$ $$-1$$ $$1$$ $$-i$$ $$i$$ $$-1$$ $$1$$ $$-i$$ $$i$$
$$\chi_{85}(19,\cdot)$$ 85.m 8 yes $$1$$ $$1$$ $$-i$$ $$e\left(\frac{3}{8}\right)$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$i$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$1$$
$$\chi_{85}(21,\cdot)$$ 85.e 4 no $$1$$ $$1$$ $$-1$$ $$-i$$ $$1$$ $$i$$ $$i$$ $$-1$$ $$-1$$ $$i$$ $$-i$$ $$1$$
$$\chi_{85}(22,\cdot)$$ 85.r 16 yes $$1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$i$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$1$$
$$\chi_{85}(23,\cdot)$$ 85.r 16 yes $$1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$-i$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$1$$
$$\chi_{85}(24,\cdot)$$ 85.p 16 yes $$-1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$i$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$i$$
$$\chi_{85}(26,\cdot)$$ 85.l 8 no $$1$$ $$1$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$i$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$-1$$
$$\chi_{85}(27,\cdot)$$ 85.o 16 yes $$1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$-i$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$-1$$
$$\chi_{85}(28,\cdot)$$ 85.r 16 yes $$1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$-i$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$1$$
$$\chi_{85}(29,\cdot)$$ 85.p 16 yes $$-1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$-i$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$-i$$
$$\chi_{85}(31,\cdot)$$ 85.q 16 no $$-1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$-i$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$i$$
$$\chi_{85}(32,\cdot)$$ 85.k 8 yes $$-1$$ $$1$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$-1$$ $$i$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$i$$
$$\chi_{85}(33,\cdot)$$ 85.g 4 yes $$-1$$ $$1$$ $$-i$$ $$-i$$ $$-1$$ $$-1$$ $$i$$ $$i$$ $$-1$$ $$-1$$ $$i$$ $$i$$
$$\chi_{85}(36,\cdot)$$ 85.l 8 no $$1$$ $$1$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$-1$$
$$\chi_{85}(37,\cdot)$$ 85.r 16 yes $$1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$i$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$1$$
$$\chi_{85}(38,\cdot)$$ 85.i 4 yes $$-1$$ $$1$$ $$i$$ $$1$$ $$-1$$ $$i$$ $$1$$ $$-i$$ $$1$$ $$i$$ $$-1$$ $$i$$
$$\chi_{85}(39,\cdot)$$ 85.p 16 yes $$-1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$-i$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$-i$$
$$\chi_{85}(41,\cdot)$$ 85.q 16 no $$-1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$i$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$-i$$
$$\chi_{85}(42,\cdot)$$ 85.n 8 yes $$-1$$ $$1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$1$$ $$-i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$i$$