# Properties

 Modulus 50 Structure $$C_{20}$$ Order 20

Show commands for: SageMath / Pari/GP

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

sage: H = DirichletGroup_conrey(50)

pari: g = idealstar(,50,2)

## Character group

 sage: G.order()  pari: g.no Order = 20 sage: H.invariants()  pari: g.cyc Structure = $$C_{20}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{50}(27,\cdot)$

## 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 3 7 9 11 13 17 19 21 23 27
$$\chi_{50}(1,\cdot)$$ 50.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{50}(3,\cdot)$$ 50.f 20 no $$-1$$ $$1$$ $$e\left(\frac{9}{20}\right)$$ $$-i$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{7}{20}\right)$$
$$\chi_{50}(7,\cdot)$$ 50.c 4 no $$-1$$ $$1$$ $$-i$$ $$i$$ $$-1$$ $$1$$ $$-i$$ $$i$$ $$-1$$ $$1$$ $$-i$$ $$i$$
$$\chi_{50}(9,\cdot)$$ 50.e 10 no $$1$$ $$1$$ $$e\left(\frac{9}{10}\right)$$ $$-1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{50}(11,\cdot)$$ 50.d 5 no $$1$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{50}(13,\cdot)$$ 50.f 20 no $$-1$$ $$1$$ $$e\left(\frac{13}{20}\right)$$ $$-i$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{19}{20}\right)$$
$$\chi_{50}(17,\cdot)$$ 50.f 20 no $$-1$$ $$1$$ $$e\left(\frac{11}{20}\right)$$ $$i$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{13}{20}\right)$$
$$\chi_{50}(19,\cdot)$$ 50.e 10 no $$1$$ $$1$$ $$e\left(\frac{3}{10}\right)$$ $$-1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{50}(21,\cdot)$$ 50.d 5 no $$1$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$
$$\chi_{50}(23,\cdot)$$ 50.f 20 no $$-1$$ $$1$$ $$e\left(\frac{17}{20}\right)$$ $$-i$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{11}{20}\right)$$
$$\chi_{50}(27,\cdot)$$ 50.f 20 no $$-1$$ $$1$$ $$e\left(\frac{7}{20}\right)$$ $$i$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{1}{20}\right)$$
$$\chi_{50}(29,\cdot)$$ 50.e 10 no $$1$$ $$1$$ $$e\left(\frac{7}{10}\right)$$ $$-1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{50}(31,\cdot)$$ 50.d 5 no $$1$$ $$1$$ $$e\left(\frac{4}{5}\right)$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$
$$\chi_{50}(33,\cdot)$$ 50.f 20 no $$-1$$ $$1$$ $$e\left(\frac{1}{20}\right)$$ $$-i$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{3}{20}\right)$$
$$\chi_{50}(37,\cdot)$$ 50.f 20 no $$-1$$ $$1$$ $$e\left(\frac{3}{20}\right)$$ $$i$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{9}{20}\right)$$
$$\chi_{50}(39,\cdot)$$ 50.e 10 no $$1$$ $$1$$ $$e\left(\frac{1}{10}\right)$$ $$-1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{50}(41,\cdot)$$ 50.d 5 no $$1$$ $$1$$ $$e\left(\frac{2}{5}\right)$$ $$1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{50}(43,\cdot)$$ 50.c 4 no $$-1$$ $$1$$ $$i$$ $$-i$$ $$-1$$ $$1$$ $$i$$ $$-i$$ $$-1$$ $$1$$ $$i$$ $$-i$$
$$\chi_{50}(47,\cdot)$$ 50.f 20 no $$-1$$ $$1$$ $$e\left(\frac{19}{20}\right)$$ $$i$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{17}{20}\right)$$
$$\chi_{50}(49,\cdot)$$ 50.b 2 no $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$