Properties

Modulus 61
Structure \(C_{60}\)
Order 60

Learn more about

Show commands for: SageMath / Pari/GP

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 \(-1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{12}\right)\) \(i\) \(1\) \(e\left(\frac{11}{12}\right)\) \(i\)