Properties

Modulus 69
Structure \(C_{22}\times C_{2}\)
Order 44

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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(69)
pari: g = idealstar(,69,2)

Character group

sage: G.order()
pari: g.no
Order = 44
sage: H.invariants()
pari: g.cyc
Structure = \(C_{22}\times C_{2}\)
sage: H.gens()
pari: g.gen
Generators = $\chi_{69}(28,\cdot)$, $\chi_{69}(47,\cdot)$

First 32 of 44 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
\(\chi_{69}(1,\cdot)\) 69.a 1 No \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{69}(2,\cdot)\) 69.h 22 Yes \(-1\) \(1\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{8}{11}\right)\)
\(\chi_{69}(4,\cdot)\) 69.e 11 No \(1\) \(1\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{5}{11}\right)\)
\(\chi_{69}(5,\cdot)\) 69.g 22 Yes \(1\) \(1\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{4}{11}\right)\)
\(\chi_{69}(7,\cdot)\) 69.f 22 No \(-1\) \(1\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{10}{11}\right)\)
\(\chi_{69}(8,\cdot)\) 69.h 22 Yes \(-1\) \(1\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{2}{11}\right)\)
\(\chi_{69}(10,\cdot)\) 69.f 22 No \(-1\) \(1\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{1}{11}\right)\)
\(\chi_{69}(11,\cdot)\) 69.g 22 Yes \(1\) \(1\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{3}{11}\right)\)
\(\chi_{69}(13,\cdot)\) 69.e 11 No \(1\) \(1\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{1}{11}\right)\)
\(\chi_{69}(14,\cdot)\) 69.g 22 Yes \(1\) \(1\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{7}{11}\right)\)
\(\chi_{69}(16,\cdot)\) 69.e 11 No \(1\) \(1\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{10}{11}\right)\)
\(\chi_{69}(17,\cdot)\) 69.g 22 Yes \(1\) \(1\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{6}{11}\right)\)
\(\chi_{69}(19,\cdot)\) 69.f 22 No \(-1\) \(1\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{5}{11}\right)\)
\(\chi_{69}(20,\cdot)\) 69.g 22 Yes \(1\) \(1\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{9}{11}\right)\)
\(\chi_{69}(22,\cdot)\) 69.d 2 No \(-1\) \(1\) \(1\) \(1\) \(-1\) \(-1\) \(1\) \(-1\) \(-1\) \(1\) \(-1\) \(1\)
\(\chi_{69}(25,\cdot)\) 69.e 11 No \(1\) \(1\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{8}{11}\right)\)
\(\chi_{69}(26,\cdot)\) 69.h 22 Yes \(-1\) \(1\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{9}{11}\right)\)
\(\chi_{69}(28,\cdot)\) 69.f 22 No \(-1\) \(1\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{4}{11}\right)\)
\(\chi_{69}(29,\cdot)\) 69.h 22 Yes \(-1\) \(1\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{6}{11}\right)\)
\(\chi_{69}(31,\cdot)\) 69.e 11 No \(1\) \(1\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{2}{11}\right)\)
\(\chi_{69}(32,\cdot)\) 69.h 22 Yes \(-1\) \(1\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{7}{11}\right)\)
\(\chi_{69}(34,\cdot)\) 69.f 22 No \(-1\) \(1\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{3}{11}\right)\)
\(\chi_{69}(35,\cdot)\) 69.h 22 Yes \(-1\) \(1\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{3}{11}\right)\)
\(\chi_{69}(37,\cdot)\) 69.f 22 No \(-1\) \(1\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{7}{11}\right)\)
\(\chi_{69}(38,\cdot)\) 69.g 22 Yes \(1\) \(1\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{2}{11}\right)\)
\(\chi_{69}(40,\cdot)\) 69.f 22 No \(-1\) \(1\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{6}{11}\right)\)
\(\chi_{69}(41,\cdot)\) 69.h 22 Yes \(-1\) \(1\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{4}{11}\right)\)
\(\chi_{69}(43,\cdot)\) 69.f 22 No \(-1\) \(1\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{9}{11}\right)\)
\(\chi_{69}(44,\cdot)\) 69.g 22 Yes \(1\) \(1\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{8}{11}\right)\)
\(\chi_{69}(47,\cdot)\) 69.b 2 No \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\)
\(\chi_{69}(49,\cdot)\) 69.e 11 No \(1\) \(1\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{9}{11}\right)\)
\(\chi_{69}(50,\cdot)\) 69.h 22 Yes \(-1\) \(1\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{5}{11}\right)\)