Properties

Modulus 86
Structure \(C_{42}\)
Order 42

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(86)
pari: g = idealstar(,86,2)

Character group

sage: G.order()
pari: g.no
Order = 42
sage: H.invariants()
pari: g.cyc
Structure = \(C_{42}\)
sage: H.gens()
pari: g.gen
Generators = $\chi_{86}(3,\cdot)$

First 32 of 42 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 3 5 7 9 11 13 15 17 19 21
\(\chi_{86}(1,\cdot)\) 86.a 1 No \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{86}(3,\cdot)\) 86.h 42 No \(-1\) \(1\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{6}{7}\right)\)
\(\chi_{86}(5,\cdot)\) 86.h 42 No \(-1\) \(1\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{3}{7}\right)\)
\(\chi_{86}(7,\cdot)\) 86.d 6 No \(-1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(1\)
\(\chi_{86}(9,\cdot)\) 86.g 21 No \(1\) \(1\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{5}{7}\right)\)
\(\chi_{86}(11,\cdot)\) 86.e 7 No \(1\) \(1\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(1\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{7}\right)\)
\(\chi_{86}(13,\cdot)\) 86.g 21 No \(1\) \(1\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{3}{7}\right)\)
\(\chi_{86}(15,\cdot)\) 86.g 21 No \(1\) \(1\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{2}{7}\right)\)
\(\chi_{86}(17,\cdot)\) 86.g 21 No \(1\) \(1\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{4}{7}\right)\)
\(\chi_{86}(19,\cdot)\) 86.h 42 No \(-1\) \(1\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{2}{7}\right)\)
\(\chi_{86}(21,\cdot)\) 86.e 7 No \(1\) \(1\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(1\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{6}{7}\right)\)
\(\chi_{86}(23,\cdot)\) 86.g 21 No \(1\) \(1\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{5}{7}\right)\)
\(\chi_{86}(25,\cdot)\) 86.g 21 No \(1\) \(1\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{6}{7}\right)\)
\(\chi_{86}(27,\cdot)\) 86.f 14 No \(-1\) \(1\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{11}{14}\right)\) \(-1\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{4}{7}\right)\)
\(\chi_{86}(29,\cdot)\) 86.h 42 No \(-1\) \(1\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{1}{7}\right)\)
\(\chi_{86}(31,\cdot)\) 86.g 21 No \(1\) \(1\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{1}{7}\right)\)
\(\chi_{86}(33,\cdot)\) 86.h 42 No \(-1\) \(1\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{4}{7}\right)\)
\(\chi_{86}(35,\cdot)\) 86.e 7 No \(1\) \(1\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(1\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{3}{7}\right)\)
\(\chi_{86}(37,\cdot)\) 86.d 6 No \(-1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(1\)
\(\chi_{86}(39,\cdot)\) 86.f 14 No \(-1\) \(1\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{9}{14}\right)\) \(-1\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{2}{7}\right)\)
\(\chi_{86}(41,\cdot)\) 86.e 7 No \(1\) \(1\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(1\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{1}{7}\right)\)
\(\chi_{86}(45,\cdot)\) 86.f 14 No \(-1\) \(1\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{1}{14}\right)\) \(-1\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{1}{7}\right)\)
\(\chi_{86}(47,\cdot)\) 86.e 7 No \(1\) \(1\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(1\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{2}{7}\right)\)
\(\chi_{86}(49,\cdot)\) 86.c 3 No \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\)
\(\chi_{86}(51,\cdot)\) 86.f 14 No \(-1\) \(1\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{3}{14}\right)\) \(-1\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{3}{7}\right)\)
\(\chi_{86}(53,\cdot)\) 86.g 21 No \(1\) \(1\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{4}{7}\right)\)
\(\chi_{86}(55,\cdot)\) 86.h 42 No \(-1\) \(1\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{1}{7}\right)\)
\(\chi_{86}(57,\cdot)\) 86.g 21 No \(1\) \(1\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{1}{7}\right)\)
\(\chi_{86}(59,\cdot)\) 86.e 7 No \(1\) \(1\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(1\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{4}{7}\right)\)
\(\chi_{86}(61,\cdot)\) 86.h 42 No \(-1\) \(1\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{6}{7}\right)\)
\(\chi_{86}(63,\cdot)\) 86.h 42 No \(-1\) \(1\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{5}{7}\right)\)
\(\chi_{86}(65,\cdot)\) 86.f 14 No \(-1\) \(1\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{13}{14}\right)\) \(-1\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{6}{7}\right)\)