# Properties

 Modulus 61 Structure $$C_{60}$$ Order 60

Show commands for: Pari/GP / SageMath

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

sage: H = DirichletGroup_conrey(61)

pari: g = idealstar(,61,2)

## Character group

 sage: G.order()  pari: g.no Order = 60 sage: H.invariants()  pari: g.cyc Structure = $$C_{60}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{61}(2,\cdot)$

## First 32 of 60 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 5 6 7 8 9 10 11
$$\chi_{61}(1,\cdot)$$ 61.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{61}(2,\cdot)$$ 61.l 60 yes $$-1$$ $$1$$ $$e\left(\frac{1}{60}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{7}{60}\right)$$ $$e\left(\frac{49}{60}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{23}{60}\right)$$ $$i$$
$$\chi_{61}(3,\cdot)$$ 61.g 10 yes $$1$$ $$1$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$-1$$
$$\chi_{61}(4,\cdot)$$ 61.k 30 yes $$1$$ $$1$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$-1$$
$$\chi_{61}(5,\cdot)$$ 61.k 30 yes $$1$$ $$1$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$-1$$
$$\chi_{61}(6,\cdot)$$ 61.l 60 yes $$-1$$ $$1$$ $$e\left(\frac{7}{60}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{49}{60}\right)$$ $$e\left(\frac{43}{60}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{41}{60}\right)$$ $$-i$$
$$\chi_{61}(7,\cdot)$$ 61.l 60 yes $$-1$$ $$1$$ $$e\left(\frac{49}{60}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{43}{60}\right)$$ $$e\left(\frac{1}{60}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{47}{60}\right)$$ $$i$$
$$\chi_{61}(8,\cdot)$$ 61.j 20 yes $$-1$$ $$1$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$-i$$
$$\chi_{61}(9,\cdot)$$ 61.e 5 yes $$1$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$1$$
$$\chi_{61}(10,\cdot)$$ 61.l 60 yes $$-1$$ $$1$$ $$e\left(\frac{23}{60}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{41}{60}\right)$$ $$e\left(\frac{47}{60}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{49}{60}\right)$$ $$-i$$
$$\chi_{61}(11,\cdot)$$ 61.d 4 yes $$-1$$ $$1$$ $$i$$ $$-1$$ $$-1$$ $$-1$$ $$-i$$ $$i$$ $$-i$$ $$1$$ $$-i$$ $$-i$$
$$\chi_{61}(12,\cdot)$$ 61.i 15 yes $$1$$ $$1$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$1$$
$$\chi_{61}(13,\cdot)$$ 61.c 3 yes $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$1$$
$$\chi_{61}(14,\cdot)$$ 61.f 6 yes $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$
$$\chi_{61}(15,\cdot)$$ 61.i 15 yes $$1$$ $$1$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$1$$
$$\chi_{61}(16,\cdot)$$ 61.i 15 yes $$1$$ $$1$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$1$$
$$\chi_{61}(17,\cdot)$$ 61.l 60 yes $$-1$$ $$1$$ $$e\left(\frac{47}{60}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{29}{60}\right)$$ $$e\left(\frac{23}{60}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{60}\right)$$ $$-i$$
$$\chi_{61}(18,\cdot)$$ 61.l 60 yes $$-1$$ $$1$$ $$e\left(\frac{13}{60}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{31}{60}\right)$$ $$e\left(\frac{37}{60}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{59}{60}\right)$$ $$i$$
$$\chi_{61}(19,\cdot)$$ 61.k 30 yes $$1$$ $$1$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$-1$$
$$\chi_{61}(20,\cdot)$$ 61.e 5 yes $$1$$ $$1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$1$$
$$\chi_{61}(21,\cdot)$$ 61.h 12 yes $$-1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$-i$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$-i$$
$$\chi_{61}(22,\cdot)$$ 61.i 15 yes $$1$$ $$1$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$1$$
$$\chi_{61}(23,\cdot)$$ 61.j 20 yes $$-1$$ $$1$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$i$$
$$\chi_{61}(24,\cdot)$$ 61.j 20 yes $$-1$$ $$1$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$i$$
$$\chi_{61}(25,\cdot)$$ 61.i 15 yes $$1$$ $$1$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$1$$
$$\chi_{61}(26,\cdot)$$ 61.l 60 yes $$-1$$ $$1$$ $$e\left(\frac{41}{60}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{47}{60}\right)$$ $$e\left(\frac{29}{60}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{43}{60}\right)$$ $$i$$
$$\chi_{61}(27,\cdot)$$ 61.g 10 yes $$1$$ $$1$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$-1$$
$$\chi_{61}(28,\cdot)$$ 61.j 20 yes $$-1$$ $$1$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$-i$$
$$\chi_{61}(29,\cdot)$$ 61.h 12 yes $$-1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$-i$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$-i$$
$$\chi_{61}(30,\cdot)$$ 61.l 60 yes $$-1$$ $$1$$ $$e\left(\frac{29}{60}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{23}{60}\right)$$ $$e\left(\frac{41}{60}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{7}{60}\right)$$ $$i$$
$$\chi_{61}(31,\cdot)$$ 61.l 60 yes $$-1$$ $$1$$ $$e\left(\frac{59}{60}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{53}{60}\right)$$ $$e\left(\frac{11}{60}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{37}{60}\right)$$ $$-i$$
$$\chi_{61}(32,\cdot)$$ 61.h 12 yes $$-1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$i$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$i$$