Properties

Modulus 82
Structure \(C_{40}\)
Order 40

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(82)
pari: g = idealstar(,82,2)

Character group

sage: G.order()
pari: g.no
Order = 40
sage: H.invariants()
pari: g.cyc
Structure = \(C_{40}\)
sage: H.gens()
pari: g.gen
Generators = $\chi_{82}(47,\cdot)$

First 32 of 40 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
\(\chi_{82}(1,\cdot)\) 82.a 1 No \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{82}(3,\cdot)\) 82.e 8 No \(-1\) \(1\) \(e\left(\frac{5}{8}\right)\) \(i\) \(e\left(\frac{5}{8}\right)\) \(i\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(i\)
\(\chi_{82}(5,\cdot)\) 82.g 20 No \(1\) \(1\) \(i\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{9}{20}\right)\) \(-1\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{7}{10}\right)\)
\(\chi_{82}(7,\cdot)\) 82.h 40 No \(-1\) \(1\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{1}{40}\right)\) \(i\) \(e\left(\frac{37}{40}\right)\) \(e\left(\frac{9}{40}\right)\) \(e\left(\frac{3}{40}\right)\) \(e\left(\frac{7}{40}\right)\) \(e\left(\frac{31}{40}\right)\) \(e\left(\frac{13}{20}\right)\)
\(\chi_{82}(9,\cdot)\) 82.c 4 No \(1\) \(1\) \(i\) \(-1\) \(i\) \(-1\) \(i\) \(i\) \(-i\) \(-i\) \(-i\) \(-1\)
\(\chi_{82}(11,\cdot)\) 82.h 40 No \(-1\) \(1\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{37}{40}\right)\) \(i\) \(e\left(\frac{9}{40}\right)\) \(e\left(\frac{13}{40}\right)\) \(e\left(\frac{31}{40}\right)\) \(e\left(\frac{19}{40}\right)\) \(e\left(\frac{27}{40}\right)\) \(e\left(\frac{1}{20}\right)\)
\(\chi_{82}(13,\cdot)\) 82.h 40 No \(-1\) \(1\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{9}{40}\right)\) \(i\) \(e\left(\frac{13}{40}\right)\) \(e\left(\frac{1}{40}\right)\) \(e\left(\frac{27}{40}\right)\) \(e\left(\frac{23}{40}\right)\) \(e\left(\frac{39}{40}\right)\) \(e\left(\frac{17}{20}\right)\)
\(\chi_{82}(15,\cdot)\) 82.h 40 No \(-1\) \(1\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{3}{40}\right)\) \(-i\) \(e\left(\frac{31}{40}\right)\) \(e\left(\frac{27}{40}\right)\) \(e\left(\frac{9}{40}\right)\) \(e\left(\frac{21}{40}\right)\) \(e\left(\frac{13}{40}\right)\) \(e\left(\frac{19}{20}\right)\)
\(\chi_{82}(17,\cdot)\) 82.h 40 No \(-1\) \(1\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{7}{40}\right)\) \(-i\) \(e\left(\frac{19}{40}\right)\) \(e\left(\frac{23}{40}\right)\) \(e\left(\frac{21}{40}\right)\) \(e\left(\frac{9}{40}\right)\) \(e\left(\frac{17}{40}\right)\) \(e\left(\frac{11}{20}\right)\)
\(\chi_{82}(19,\cdot)\) 82.h 40 No \(-1\) \(1\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{31}{40}\right)\) \(-i\) \(e\left(\frac{27}{40}\right)\) \(e\left(\frac{39}{40}\right)\) \(e\left(\frac{13}{40}\right)\) \(e\left(\frac{17}{40}\right)\) \(e\left(\frac{1}{40}\right)\) \(e\left(\frac{3}{20}\right)\)
\(\chi_{82}(21,\cdot)\) 82.g 20 No \(1\) \(1\) \(i\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{13}{20}\right)\) \(-1\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{9}{10}\right)\)
\(\chi_{82}(23,\cdot)\) 82.f 10 No \(1\) \(1\) \(-1\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{1}{10}\right)\) \(1\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{3}{5}\right)\)
\(\chi_{82}(25,\cdot)\) 82.f 10 No \(1\) \(1\) \(-1\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{9}{10}\right)\) \(1\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{2}{5}\right)\)
\(\chi_{82}(27,\cdot)\) 82.e 8 No \(-1\) \(1\) \(e\left(\frac{7}{8}\right)\) \(-i\) \(e\left(\frac{7}{8}\right)\) \(-i\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(-i\)
\(\chi_{82}(29,\cdot)\) 82.h 40 No \(-1\) \(1\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{33}{40}\right)\) \(i\) \(e\left(\frac{21}{40}\right)\) \(e\left(\frac{17}{40}\right)\) \(e\left(\frac{19}{40}\right)\) \(e\left(\frac{31}{40}\right)\) \(e\left(\frac{23}{40}\right)\) \(e\left(\frac{9}{20}\right)\)
\(\chi_{82}(31,\cdot)\) 82.f 10 No \(1\) \(1\) \(-1\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{3}{10}\right)\) \(1\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{4}{5}\right)\)
\(\chi_{82}(33,\cdot)\) 82.g 20 No \(1\) \(1\) \(-i\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{11}{20}\right)\) \(-1\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{3}{10}\right)\)
\(\chi_{82}(35,\cdot)\) 82.h 40 No \(-1\) \(1\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{19}{40}\right)\) \(-i\) \(e\left(\frac{23}{40}\right)\) \(e\left(\frac{11}{40}\right)\) \(e\left(\frac{17}{40}\right)\) \(e\left(\frac{13}{40}\right)\) \(e\left(\frac{29}{40}\right)\) \(e\left(\frac{7}{20}\right)\)
\(\chi_{82}(37,\cdot)\) 82.d 5 No \(1\) \(1\) \(1\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{1}{5}\right)\) \(1\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{1}{5}\right)\)
\(\chi_{82}(39,\cdot)\) 82.g 20 No \(1\) \(1\) \(i\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{17}{20}\right)\) \(-1\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{1}{10}\right)\)
\(\chi_{82}(43,\cdot)\) 82.g 20 No \(1\) \(1\) \(-i\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{7}{20}\right)\) \(-1\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{1}{10}\right)\)
\(\chi_{82}(45,\cdot)\) 82.f 10 No \(1\) \(1\) \(-1\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{7}{10}\right)\) \(1\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{1}{5}\right)\)
\(\chi_{82}(47,\cdot)\) 82.h 40 No \(-1\) \(1\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{39}{40}\right)\) \(-i\) \(e\left(\frac{3}{40}\right)\) \(e\left(\frac{31}{40}\right)\) \(e\left(\frac{37}{40}\right)\) \(e\left(\frac{33}{40}\right)\) \(e\left(\frac{9}{40}\right)\) \(e\left(\frac{7}{20}\right)\)
\(\chi_{82}(49,\cdot)\) 82.g 20 No \(1\) \(1\) \(i\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{1}{20}\right)\) \(-1\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{3}{10}\right)\)
\(\chi_{82}(51,\cdot)\) 82.d 5 No \(1\) \(1\) \(1\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{4}{5}\right)\) \(1\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{4}{5}\right)\)
\(\chi_{82}(53,\cdot)\) 82.h 40 No \(-1\) \(1\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{13}{40}\right)\) \(i\) \(e\left(\frac{1}{40}\right)\) \(e\left(\frac{37}{40}\right)\) \(e\left(\frac{39}{40}\right)\) \(e\left(\frac{11}{40}\right)\) \(e\left(\frac{3}{40}\right)\) \(e\left(\frac{9}{20}\right)\)
\(\chi_{82}(55,\cdot)\) 82.e 8 No \(-1\) \(1\) \(e\left(\frac{3}{8}\right)\) \(-i\) \(e\left(\frac{3}{8}\right)\) \(-i\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(-i\)
\(\chi_{82}(57,\cdot)\) 82.d 5 No \(1\) \(1\) \(1\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(1\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{2}{5}\right)\)
\(\chi_{82}(59,\cdot)\) 82.d 5 No \(1\) \(1\) \(1\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{3}{5}\right)\) \(1\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{3}{5}\right)\)
\(\chi_{82}(61,\cdot)\) 82.g 20 No \(1\) \(1\) \(-i\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{3}{20}\right)\) \(-1\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{9}{10}\right)\)
\(\chi_{82}(63,\cdot)\) 82.h 40 No \(-1\) \(1\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{11}{40}\right)\) \(-i\) \(e\left(\frac{7}{40}\right)\) \(e\left(\frac{19}{40}\right)\) \(e\left(\frac{33}{40}\right)\) \(e\left(\frac{37}{40}\right)\) \(e\left(\frac{21}{40}\right)\) \(e\left(\frac{3}{20}\right)\)
\(\chi_{82}(65,\cdot)\) 82.h 40 No \(-1\) \(1\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{27}{40}\right)\) \(-i\) \(e\left(\frac{39}{40}\right)\) \(e\left(\frac{3}{40}\right)\) \(e\left(\frac{1}{40}\right)\) \(e\left(\frac{29}{40}\right)\) \(e\left(\frac{37}{40}\right)\) \(e\left(\frac{11}{20}\right)\)