Properties

Modulus 53
Structure \(C_{52}\)
Order 52

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(53)
pari: g = idealstar(,53,2)

Character group

sage: G.order()
pari: g.no
Order = 52
sage: H.invariants()
pari: g.cyc
Structure = \(C_{52}\)
sage: H.gens()
pari: g.gen
Generators = $\chi_{53}(2,\cdot)$

First 32 of 52 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
\(\chi_{53}(1,\cdot)\) 53.a 1 No \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{53}(2,\cdot)\) 53.f 52 Yes \(-1\) \(1\) \(e\left(\frac{1}{52}\right)\) \(e\left(\frac{17}{52}\right)\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{47}{52}\right)\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{3}{52}\right)\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{3}{26}\right)\)
\(\chi_{53}(3,\cdot)\) 53.f 52 Yes \(-1\) \(1\) \(e\left(\frac{17}{52}\right)\) \(e\left(\frac{29}{52}\right)\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{19}{52}\right)\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{51}{52}\right)\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{25}{26}\right)\)
\(\chi_{53}(4,\cdot)\) 53.e 26 Yes \(1\) \(1\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{3}{13}\right)\)
\(\chi_{53}(5,\cdot)\) 53.f 52 Yes \(-1\) \(1\) \(e\left(\frac{47}{52}\right)\) \(e\left(\frac{19}{52}\right)\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{25}{52}\right)\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{37}{52}\right)\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{11}{26}\right)\)
\(\chi_{53}(6,\cdot)\) 53.e 26 Yes \(1\) \(1\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{1}{13}\right)\)
\(\chi_{53}(7,\cdot)\) 53.e 26 Yes \(1\) \(1\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{8}{13}\right)\)
\(\chi_{53}(8,\cdot)\) 53.f 52 Yes \(-1\) \(1\) \(e\left(\frac{3}{52}\right)\) \(e\left(\frac{51}{52}\right)\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{37}{52}\right)\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{9}{52}\right)\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{9}{26}\right)\)
\(\chi_{53}(9,\cdot)\) 53.e 26 Yes \(1\) \(1\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{12}{13}\right)\)
\(\chi_{53}(10,\cdot)\) 53.d 13 Yes \(1\) \(1\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{7}{13}\right)\)
\(\chi_{53}(11,\cdot)\) 53.e 26 Yes \(1\) \(1\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{9}{13}\right)\)
\(\chi_{53}(12,\cdot)\) 53.f 52 Yes \(-1\) \(1\) \(e\left(\frac{19}{52}\right)\) \(e\left(\frac{11}{52}\right)\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{9}{52}\right)\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{5}{52}\right)\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{5}{26}\right)\)
\(\chi_{53}(13,\cdot)\) 53.d 13 Yes \(1\) \(1\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{10}{13}\right)\)
\(\chi_{53}(14,\cdot)\) 53.f 52 Yes \(-1\) \(1\) \(e\left(\frac{15}{52}\right)\) \(e\left(\frac{47}{52}\right)\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{29}{52}\right)\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{45}{52}\right)\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{19}{26}\right)\)
\(\chi_{53}(15,\cdot)\) 53.d 13 Yes \(1\) \(1\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{5}{13}\right)\)
\(\chi_{53}(16,\cdot)\) 53.d 13 Yes \(1\) \(1\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{6}{13}\right)\)
\(\chi_{53}(17,\cdot)\) 53.e 26 Yes \(1\) \(1\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{2}{13}\right)\)
\(\chi_{53}(18,\cdot)\) 53.f 52 Yes \(-1\) \(1\) \(e\left(\frac{35}{52}\right)\) \(e\left(\frac{23}{52}\right)\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{33}{52}\right)\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{1}{52}\right)\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{1}{26}\right)\)
\(\chi_{53}(19,\cdot)\) 53.f 52 Yes \(-1\) \(1\) \(e\left(\frac{37}{52}\right)\) \(e\left(\frac{5}{52}\right)\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{23}{52}\right)\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{7}{52}\right)\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{7}{26}\right)\)
\(\chi_{53}(20,\cdot)\) 53.f 52 Yes \(-1\) \(1\) \(e\left(\frac{49}{52}\right)\) \(e\left(\frac{1}{52}\right)\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{15}{52}\right)\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{43}{52}\right)\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{17}{26}\right)\)
\(\chi_{53}(21,\cdot)\) 53.f 52 Yes \(-1\) \(1\) \(e\left(\frac{31}{52}\right)\) \(e\left(\frac{7}{52}\right)\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{1}{52}\right)\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{41}{52}\right)\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{15}{26}\right)\)
\(\chi_{53}(22,\cdot)\) 53.f 52 Yes \(-1\) \(1\) \(e\left(\frac{7}{52}\right)\) \(e\left(\frac{15}{52}\right)\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{17}{52}\right)\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{21}{52}\right)\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{21}{26}\right)\)
\(\chi_{53}(23,\cdot)\) 53.c 4 Yes \(-1\) \(1\) \(-i\) \(-i\) \(-1\) \(i\) \(-1\) \(-1\) \(i\) \(-1\) \(1\) \(-1\)
\(\chi_{53}(24,\cdot)\) 53.d 13 Yes \(1\) \(1\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{4}{13}\right)\)
\(\chi_{53}(25,\cdot)\) 53.e 26 Yes \(1\) \(1\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{11}{13}\right)\)
\(\chi_{53}(26,\cdot)\) 53.f 52 Yes \(-1\) \(1\) \(e\left(\frac{25}{52}\right)\) \(e\left(\frac{9}{52}\right)\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{31}{52}\right)\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{23}{52}\right)\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{23}{26}\right)\)
\(\chi_{53}(27,\cdot)\) 53.f 52 Yes \(-1\) \(1\) \(e\left(\frac{51}{52}\right)\) \(e\left(\frac{35}{52}\right)\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{5}{52}\right)\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{49}{52}\right)\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{23}{26}\right)\)
\(\chi_{53}(28,\cdot)\) 53.d 13 Yes \(1\) \(1\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{11}{13}\right)\)
\(\chi_{53}(29,\cdot)\) 53.e 26 Yes \(1\) \(1\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{4}{13}\right)\)
\(\chi_{53}(30,\cdot)\) 53.c 4 Yes \(-1\) \(1\) \(i\) \(i\) \(-1\) \(-i\) \(-1\) \(-1\) \(-i\) \(-1\) \(1\) \(-1\)
\(\chi_{53}(31,\cdot)\) 53.f 52 Yes \(-1\) \(1\) \(e\left(\frac{33}{52}\right)\) \(e\left(\frac{41}{52}\right)\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{43}{52}\right)\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{47}{52}\right)\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{21}{26}\right)\)
\(\chi_{53}(32,\cdot)\) 53.f 52 Yes \(-1\) \(1\) \(e\left(\frac{5}{52}\right)\) \(e\left(\frac{33}{52}\right)\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{27}{52}\right)\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{15}{52}\right)\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{15}{26}\right)\)