Properties

Modulus 81
Structure \(C_{54}\)
Order 54

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(81)
pari: g = idealstar(,81,2)

Character group

sage: G.order()
pari: g.no
Order = 54
sage: H.invariants()
pari: g.cyc
Structure = \(C_{54}\)
sage: H.gens()
pari: g.gen
Generators = $\chi_{81}(2,\cdot)$

First 32 of 54 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 4 5 7 8 10 11 13 14 16
\(\chi_{81}(1,\cdot)\) 81.a 1 No \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{81}(2,\cdot)\) 81.h 54 Yes \(-1\) \(1\) \(e\left(\frac{1}{54}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{23}{54}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{13}{54}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{17}{54}\right)\) \(e\left(\frac{2}{27}\right)\)
\(\chi_{81}(4,\cdot)\) 81.g 27 Yes \(1\) \(1\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{4}{27}\right)\)
\(\chi_{81}(5,\cdot)\) 81.h 54 Yes \(-1\) \(1\) \(e\left(\frac{23}{54}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{43}{54}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{29}{54}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{13}{54}\right)\) \(e\left(\frac{19}{27}\right)\)
\(\chi_{81}(7,\cdot)\) 81.g 27 Yes \(1\) \(1\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{5}{27}\right)\)
\(\chi_{81}(8,\cdot)\) 81.f 18 No \(-1\) \(1\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{2}{9}\right)\)
\(\chi_{81}(10,\cdot)\) 81.e 9 No \(1\) \(1\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{7}{9}\right)\)
\(\chi_{81}(11,\cdot)\) 81.h 54 Yes \(-1\) \(1\) \(e\left(\frac{13}{54}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{29}{54}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{7}{54}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{5}{54}\right)\) \(e\left(\frac{26}{27}\right)\)
\(\chi_{81}(13,\cdot)\) 81.g 27 Yes \(1\) \(1\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{16}{27}\right)\)
\(\chi_{81}(14,\cdot)\) 81.h 54 Yes \(-1\) \(1\) \(e\left(\frac{17}{54}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{13}{54}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{5}{54}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{19}{54}\right)\) \(e\left(\frac{7}{27}\right)\)
\(\chi_{81}(16,\cdot)\) 81.g 27 Yes \(1\) \(1\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{8}{27}\right)\)
\(\chi_{81}(17,\cdot)\) 81.f 18 No \(-1\) \(1\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{4}{9}\right)\)
\(\chi_{81}(19,\cdot)\) 81.e 9 No \(1\) \(1\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{5}{9}\right)\)
\(\chi_{81}(20,\cdot)\) 81.h 54 Yes \(-1\) \(1\) \(e\left(\frac{25}{54}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{35}{54}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{1}{54}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{47}{54}\right)\) \(e\left(\frac{23}{27}\right)\)
\(\chi_{81}(22,\cdot)\) 81.g 27 Yes \(1\) \(1\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{1}{27}\right)\)
\(\chi_{81}(23,\cdot)\) 81.h 54 Yes \(-1\) \(1\) \(e\left(\frac{11}{54}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{37}{54}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{35}{54}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{25}{54}\right)\) \(e\left(\frac{22}{27}\right)\)
\(\chi_{81}(25,\cdot)\) 81.g 27 Yes \(1\) \(1\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{11}{27}\right)\)
\(\chi_{81}(26,\cdot)\) 81.d 6 No \(-1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{81}(28,\cdot)\) 81.c 3 No \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{81}(29,\cdot)\) 81.h 54 Yes \(-1\) \(1\) \(e\left(\frac{37}{54}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{41}{54}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{49}{54}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{35}{54}\right)\) \(e\left(\frac{20}{27}\right)\)
\(\chi_{81}(31,\cdot)\) 81.g 27 Yes \(1\) \(1\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{13}{27}\right)\)
\(\chi_{81}(32,\cdot)\) 81.h 54 Yes \(-1\) \(1\) \(e\left(\frac{5}{54}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{7}{54}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{11}{54}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{31}{54}\right)\) \(e\left(\frac{10}{27}\right)\)
\(\chi_{81}(34,\cdot)\) 81.g 27 Yes \(1\) \(1\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{14}{27}\right)\)
\(\chi_{81}(35,\cdot)\) 81.f 18 No \(-1\) \(1\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{8}{9}\right)\)
\(\chi_{81}(37,\cdot)\) 81.e 9 No \(1\) \(1\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{1}{9}\right)\)
\(\chi_{81}(38,\cdot)\) 81.h 54 Yes \(-1\) \(1\) \(e\left(\frac{49}{54}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{47}{54}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{43}{54}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{23}{54}\right)\) \(e\left(\frac{17}{27}\right)\)
\(\chi_{81}(40,\cdot)\) 81.g 27 Yes \(1\) \(1\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{25}{27}\right)\)
\(\chi_{81}(41,\cdot)\) 81.h 54 Yes \(-1\) \(1\) \(e\left(\frac{53}{54}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{31}{54}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{41}{54}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{37}{54}\right)\) \(e\left(\frac{25}{27}\right)\)
\(\chi_{81}(43,\cdot)\) 81.g 27 Yes \(1\) \(1\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{17}{27}\right)\)
\(\chi_{81}(44,\cdot)\) 81.f 18 No \(-1\) \(1\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{1}{9}\right)\)
\(\chi_{81}(46,\cdot)\) 81.e 9 No \(1\) \(1\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{8}{9}\right)\)
\(\chi_{81}(47,\cdot)\) 81.h 54 Yes \(-1\) \(1\) \(e\left(\frac{7}{54}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{53}{54}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{37}{54}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{11}{54}\right)\) \(e\left(\frac{14}{27}\right)\)