Properties

Modulus 57
Structure \(C_{18}\times C_{2}\)
Order 36

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Show commands for: SageMath / Pari/GP

sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(57)
pari: g = idealstar(,57,2)

Character group

sage: G.order()
pari: g.no
Order = 36
sage: H.invariants()
pari: g.cyc
Structure = \(C_{18}\times C_{2}\)
sage: H.gens()
pari: g.gen
Generators = $\chi_{57}(40,\cdot)$, $\chi_{57}(20,\cdot)$

First 32 of 36 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
\(\chi_{57}(1,\cdot)\) 57.a 1 No \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{57}(2,\cdot)\) 57.j 18 Yes \(1\) \(1\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{2}{9}\right)\)
\(\chi_{57}(4,\cdot)\) 57.i 9 No \(1\) \(1\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{4}{9}\right)\)
\(\chi_{57}(5,\cdot)\) 57.l 18 Yes \(-1\) \(1\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{5}{9}\right)\)
\(\chi_{57}(7,\cdot)\) 57.e 3 No \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{57}(8,\cdot)\) 57.f 6 Yes \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{57}(10,\cdot)\) 57.k 18 No \(-1\) \(1\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{7}{9}\right)\)
\(\chi_{57}(11,\cdot)\) 57.h 6 Yes \(-1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{57}(13,\cdot)\) 57.k 18 No \(-1\) \(1\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{1}{9}\right)\)
\(\chi_{57}(14,\cdot)\) 57.j 18 Yes \(1\) \(1\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{5}{9}\right)\)
\(\chi_{57}(16,\cdot)\) 57.i 9 No \(1\) \(1\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{8}{9}\right)\)
\(\chi_{57}(17,\cdot)\) 57.l 18 Yes \(-1\) \(1\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{2}{9}\right)\)
\(\chi_{57}(20,\cdot)\) 57.b 2 No \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\)
\(\chi_{57}(22,\cdot)\) 57.k 18 No \(-1\) \(1\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{8}{9}\right)\)
\(\chi_{57}(23,\cdot)\) 57.l 18 Yes \(-1\) \(1\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{4}{9}\right)\)
\(\chi_{57}(25,\cdot)\) 57.i 9 No \(1\) \(1\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{1}{9}\right)\)
\(\chi_{57}(26,\cdot)\) 57.h 6 Yes \(-1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{57}(28,\cdot)\) 57.i 9 No \(1\) \(1\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{7}{9}\right)\)
\(\chi_{57}(29,\cdot)\) 57.j 18 Yes \(1\) \(1\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{7}{9}\right)\)
\(\chi_{57}(31,\cdot)\) 57.g 6 No \(-1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{57}(32,\cdot)\) 57.j 18 Yes \(1\) \(1\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{1}{9}\right)\)
\(\chi_{57}(34,\cdot)\) 57.k 18 No \(-1\) \(1\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{4}{9}\right)\)
\(\chi_{57}(35,\cdot)\) 57.l 18 Yes \(-1\) \(1\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{8}{9}\right)\)
\(\chi_{57}(37,\cdot)\) 57.c 2 No \(-1\) \(1\) \(-1\) \(1\) \(1\) \(1\) \(-1\) \(-1\) \(1\) \(-1\) \(-1\) \(1\)
\(\chi_{57}(40,\cdot)\) 57.k 18 No \(-1\) \(1\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{2}{9}\right)\)
\(\chi_{57}(41,\cdot)\) 57.j 18 Yes \(1\) \(1\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{8}{9}\right)\)
\(\chi_{57}(43,\cdot)\) 57.i 9 No \(1\) \(1\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{5}{9}\right)\)
\(\chi_{57}(44,\cdot)\) 57.l 18 Yes \(-1\) \(1\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{1}{9}\right)\)
\(\chi_{57}(46,\cdot)\) 57.g 6 No \(-1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{57}(47,\cdot)\) 57.l 18 Yes \(-1\) \(1\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{7}{9}\right)\)
\(\chi_{57}(49,\cdot)\) 57.e 3 No \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{57}(50,\cdot)\) 57.f 6 Yes \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\)