# Properties

 Modulus 82 Structure $$C_{40}$$ 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(82)

pari: g = idealstar(,82,2)

## Character group

 sage: G.order()  pari: g.no Order = 40 sage: H.invariants()  pari: g.cyc Structure = $$C_{40}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{82}(47,\cdot)$

## 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 3 5 7 9 11 13 15 17 19 21
$$\chi_{82}(1,\cdot)$$ 82.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{82}(3,\cdot)$$ 82.e 8 no $$-1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$i$$
$$\chi_{82}(5,\cdot)$$ 82.g 20 no $$1$$ $$1$$ $$i$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$-1$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$
$$\chi_{82}(7,\cdot)$$ 82.h 40 no $$-1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{1}{40}\right)$$ $$i$$ $$e\left(\frac{37}{40}\right)$$ $$e\left(\frac{9}{40}\right)$$ $$e\left(\frac{3}{40}\right)$$ $$e\left(\frac{7}{40}\right)$$ $$e\left(\frac{31}{40}\right)$$ $$e\left(\frac{13}{20}\right)$$
$$\chi_{82}(9,\cdot)$$ 82.c 4 no $$1$$ $$1$$ $$i$$ $$-1$$ $$i$$ $$-1$$ $$i$$ $$i$$ $$-i$$ $$-i$$ $$-i$$ $$-1$$
$$\chi_{82}(11,\cdot)$$ 82.h 40 no $$-1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{37}{40}\right)$$ $$i$$ $$e\left(\frac{9}{40}\right)$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{31}{40}\right)$$ $$e\left(\frac{19}{40}\right)$$ $$e\left(\frac{27}{40}\right)$$ $$e\left(\frac{1}{20}\right)$$
$$\chi_{82}(13,\cdot)$$ 82.h 40 no $$-1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{9}{40}\right)$$ $$i$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{27}{40}\right)$$ $$e\left(\frac{23}{40}\right)$$ $$e\left(\frac{39}{40}\right)$$ $$e\left(\frac{17}{20}\right)$$
$$\chi_{82}(15,\cdot)$$ 82.h 40 no $$-1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{3}{40}\right)$$ $$-i$$ $$e\left(\frac{31}{40}\right)$$ $$e\left(\frac{27}{40}\right)$$ $$e\left(\frac{9}{40}\right)$$ $$e\left(\frac{21}{40}\right)$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{19}{20}\right)$$
$$\chi_{82}(17,\cdot)$$ 82.h 40 no $$-1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{7}{40}\right)$$ $$-i$$ $$e\left(\frac{19}{40}\right)$$ $$e\left(\frac{23}{40}\right)$$ $$e\left(\frac{21}{40}\right)$$ $$e\left(\frac{9}{40}\right)$$ $$e\left(\frac{17}{40}\right)$$ $$e\left(\frac{11}{20}\right)$$
$$\chi_{82}(19,\cdot)$$ 82.h 40 no $$-1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{31}{40}\right)$$ $$-i$$ $$e\left(\frac{27}{40}\right)$$ $$e\left(\frac{39}{40}\right)$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{17}{40}\right)$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{3}{20}\right)$$
$$\chi_{82}(21,\cdot)$$ 82.g 20 no $$1$$ $$1$$ $$i$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$-1$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{82}(23,\cdot)$$ 82.f 10 no $$1$$ $$1$$ $$-1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$1$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$
$$\chi_{82}(25,\cdot)$$ 82.f 10 no $$1$$ $$1$$ $$-1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$1$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$
$$\chi_{82}(27,\cdot)$$ 82.e 8 no $$-1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$-i$$
$$\chi_{82}(29,\cdot)$$ 82.h 40 no $$-1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{33}{40}\right)$$ $$i$$ $$e\left(\frac{21}{40}\right)$$ $$e\left(\frac{17}{40}\right)$$ $$e\left(\frac{19}{40}\right)$$ $$e\left(\frac{31}{40}\right)$$ $$e\left(\frac{23}{40}\right)$$ $$e\left(\frac{9}{20}\right)$$
$$\chi_{82}(31,\cdot)$$ 82.f 10 no $$1$$ $$1$$ $$-1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$1$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{82}(33,\cdot)$$ 82.g 20 no $$1$$ $$1$$ $$-i$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$-1$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{82}(35,\cdot)$$ 82.h 40 no $$-1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{19}{40}\right)$$ $$-i$$ $$e\left(\frac{23}{40}\right)$$ $$e\left(\frac{11}{40}\right)$$ $$e\left(\frac{17}{40}\right)$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{29}{40}\right)$$ $$e\left(\frac{7}{20}\right)$$
$$\chi_{82}(37,\cdot)$$ 82.d 5 no $$1$$ $$1$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{82}(39,\cdot)$$ 82.g 20 no $$1$$ $$1$$ $$i$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$-1$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{82}(43,\cdot)$$ 82.g 20 no $$1$$ $$1$$ $$-i$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$-1$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$
$$\chi_{82}(45,\cdot)$$ 82.f 10 no $$1$$ $$1$$ $$-1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$1$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$
$$\chi_{82}(47,\cdot)$$ 82.h 40 no $$-1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{39}{40}\right)$$ $$-i$$ $$e\left(\frac{3}{40}\right)$$ $$e\left(\frac{31}{40}\right)$$ $$e\left(\frac{37}{40}\right)$$ $$e\left(\frac{33}{40}\right)$$ $$e\left(\frac{9}{40}\right)$$ $$e\left(\frac{7}{20}\right)$$
$$\chi_{82}(49,\cdot)$$ 82.g 20 no $$1$$ $$1$$ $$i$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$-1$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{82}(51,\cdot)$$ 82.d 5 no $$1$$ $$1$$ $$1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$
$$\chi_{82}(53,\cdot)$$ 82.h 40 no $$-1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{13}{40}\right)$$ $$i$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{37}{40}\right)$$ $$e\left(\frac{39}{40}\right)$$ $$e\left(\frac{11}{40}\right)$$ $$e\left(\frac{3}{40}\right)$$ $$e\left(\frac{9}{20}\right)$$
$$\chi_{82}(55,\cdot)$$ 82.e 8 no $$-1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$-i$$
$$\chi_{82}(57,\cdot)$$ 82.d 5 no $$1$$ $$1$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$
$$\chi_{82}(59,\cdot)$$ 82.d 5 no $$1$$ $$1$$ $$1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$
$$\chi_{82}(61,\cdot)$$ 82.g 20 no $$1$$ $$1$$ $$-i$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$-1$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$
$$\chi_{82}(63,\cdot)$$ 82.h 40 no $$-1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{11}{40}\right)$$ $$-i$$ $$e\left(\frac{7}{40}\right)$$ $$e\left(\frac{19}{40}\right)$$ $$e\left(\frac{33}{40}\right)$$ $$e\left(\frac{37}{40}\right)$$ $$e\left(\frac{21}{40}\right)$$ $$e\left(\frac{3}{20}\right)$$
$$\chi_{82}(65,\cdot)$$ 82.h 40 no $$-1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{27}{40}\right)$$ $$-i$$ $$e\left(\frac{39}{40}\right)$$ $$e\left(\frac{3}{40}\right)$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{29}{40}\right)$$ $$e\left(\frac{37}{40}\right)$$ $$e\left(\frac{11}{20}\right)$$