Properties

Modulus 41
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(41)
pari: g = idealstar(,41,2)

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 = $\chi_{41}(6,\cdot)$

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 2 3 4 5 6 7 8 9 10 11
\(\chi_{41}(1,\cdot)\) 41.a 1 No \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{41}(2,\cdot)\) 41.g 20 Yes \(1\) \(1\) \(e\left(\frac{9}{10}\right)\) \(-i\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(-1\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{19}{20}\right)\)
\(\chi_{41}(3,\cdot)\) 41.e 8 Yes \(-1\) \(1\) \(-i\) \(e\left(\frac{5}{8}\right)\) \(-1\) \(i\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(i\) \(i\) \(1\) \(e\left(\frac{1}{8}\right)\)
\(\chi_{41}(4,\cdot)\) 41.f 10 Yes \(1\) \(1\) \(e\left(\frac{4}{5}\right)\) \(-1\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{2}{5}\right)\) \(1\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{9}{10}\right)\)
\(\chi_{41}(5,\cdot)\) 41.g 20 Yes \(1\) \(1\) \(e\left(\frac{3}{10}\right)\) \(i\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(-1\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{13}{20}\right)\)
\(\chi_{41}(6,\cdot)\) 41.h 40 Yes \(-1\) \(1\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{1}{40}\right)\) \(e\left(\frac{39}{40}\right)\) \(e\left(\frac{19}{20}\right)\) \(-i\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{3}{40}\right)\)
\(\chi_{41}(7,\cdot)\) 41.h 40 Yes \(-1\) \(1\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{39}{40}\right)\) \(e\left(\frac{1}{40}\right)\) \(e\left(\frac{1}{20}\right)\) \(i\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{37}{40}\right)\)
\(\chi_{41}(8,\cdot)\) 41.g 20 Yes \(1\) \(1\) \(e\left(\frac{7}{10}\right)\) \(i\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(-1\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{17}{20}\right)\)
\(\chi_{41}(9,\cdot)\) 41.c 4 Yes \(1\) \(1\) \(-1\) \(i\) \(1\) \(-1\) \(-i\) \(i\) \(-1\) \(-1\) \(1\) \(i\)
\(\chi_{41}(10,\cdot)\) 41.d 5 Yes \(1\) \(1\) \(e\left(\frac{1}{5}\right)\) \(1\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{3}{5}\right)\) \(1\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{3}{5}\right)\)
\(\chi_{41}(11,\cdot)\) 41.h 40 Yes \(-1\) \(1\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{3}{40}\right)\) \(e\left(\frac{37}{40}\right)\) \(e\left(\frac{17}{20}\right)\) \(i\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{9}{40}\right)\)
\(\chi_{41}(12,\cdot)\) 41.h 40 Yes \(-1\) \(1\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{27}{40}\right)\) \(e\left(\frac{13}{40}\right)\) \(e\left(\frac{13}{20}\right)\) \(i\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{1}{40}\right)\)
\(\chi_{41}(13,\cdot)\) 41.h 40 Yes \(-1\) \(1\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{31}{40}\right)\) \(e\left(\frac{9}{40}\right)\) \(e\left(\frac{9}{20}\right)\) \(i\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{13}{40}\right)\)
\(\chi_{41}(14,\cdot)\) 41.e 8 Yes \(-1\) \(1\) \(i\) \(e\left(\frac{3}{8}\right)\) \(-1\) \(-i\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(-i\) \(-i\) \(1\) \(e\left(\frac{7}{8}\right)\)
\(\chi_{41}(15,\cdot)\) 41.h 40 Yes \(-1\) \(1\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{37}{40}\right)\) \(e\left(\frac{3}{40}\right)\) \(e\left(\frac{3}{20}\right)\) \(-i\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{31}{40}\right)\)
\(\chi_{41}(16,\cdot)\) 41.d 5 Yes \(1\) \(1\) \(e\left(\frac{3}{5}\right)\) \(1\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{4}{5}\right)\) \(1\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{4}{5}\right)\)
\(\chi_{41}(17,\cdot)\) 41.h 40 Yes \(-1\) \(1\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{33}{40}\right)\) \(e\left(\frac{7}{40}\right)\) \(e\left(\frac{7}{20}\right)\) \(-i\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{19}{40}\right)\)
\(\chi_{41}(18,\cdot)\) 41.d 5 Yes \(1\) \(1\) \(e\left(\frac{2}{5}\right)\) \(1\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{1}{5}\right)\) \(1\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{1}{5}\right)\)
\(\chi_{41}(19,\cdot)\) 41.h 40 Yes \(-1\) \(1\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{9}{40}\right)\) \(e\left(\frac{31}{40}\right)\) \(e\left(\frac{11}{20}\right)\) \(-i\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{27}{40}\right)\)
\(\chi_{41}(20,\cdot)\) 41.g 20 Yes \(1\) \(1\) \(e\left(\frac{1}{10}\right)\) \(-i\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(-1\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{11}{20}\right)\)
\(\chi_{41}(21,\cdot)\) 41.g 20 Yes \(1\) \(1\) \(e\left(\frac{1}{10}\right)\) \(i\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(-1\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{1}{20}\right)\)
\(\chi_{41}(22,\cdot)\) 41.h 40 Yes \(-1\) \(1\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{29}{40}\right)\) \(e\left(\frac{11}{40}\right)\) \(e\left(\frac{11}{20}\right)\) \(-i\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{7}{40}\right)\)
\(\chi_{41}(23,\cdot)\) 41.f 10 Yes \(1\) \(1\) \(e\left(\frac{2}{5}\right)\) \(-1\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{1}{5}\right)\) \(1\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{7}{10}\right)\)
\(\chi_{41}(24,\cdot)\) 41.h 40 Yes \(-1\) \(1\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{13}{40}\right)\) \(e\left(\frac{27}{40}\right)\) \(e\left(\frac{7}{20}\right)\) \(-i\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{39}{40}\right)\)
\(\chi_{41}(25,\cdot)\) 41.f 10 Yes \(1\) \(1\) \(e\left(\frac{3}{5}\right)\) \(-1\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{4}{5}\right)\) \(1\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{3}{10}\right)\)
\(\chi_{41}(26,\cdot)\) 41.h 40 Yes \(-1\) \(1\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{17}{40}\right)\) \(e\left(\frac{23}{40}\right)\) \(e\left(\frac{3}{20}\right)\) \(-i\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{11}{40}\right)\)
\(\chi_{41}(27,\cdot)\) 41.e 8 Yes \(-1\) \(1\) \(i\) \(e\left(\frac{7}{8}\right)\) \(-1\) \(-i\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(-i\) \(-i\) \(1\) \(e\left(\frac{3}{8}\right)\)
\(\chi_{41}(28,\cdot)\) 41.h 40 Yes \(-1\) \(1\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{11}{40}\right)\) \(e\left(\frac{29}{40}\right)\) \(e\left(\frac{9}{20}\right)\) \(i\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{33}{40}\right)\)
\(\chi_{41}(29,\cdot)\) 41.h 40 Yes \(-1\) \(1\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{7}{40}\right)\) \(e\left(\frac{33}{40}\right)\) \(e\left(\frac{13}{20}\right)\) \(i\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{21}{40}\right)\)
\(\chi_{41}(30,\cdot)\) 41.h 40 Yes \(-1\) \(1\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{23}{40}\right)\) \(e\left(\frac{17}{40}\right)\) \(e\left(\frac{17}{20}\right)\) \(i\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{29}{40}\right)\)
\(\chi_{41}(31,\cdot)\) 41.f 10 Yes \(1\) \(1\) \(e\left(\frac{1}{5}\right)\) \(-1\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{3}{5}\right)\) \(1\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{1}{10}\right)\)
\(\chi_{41}(32,\cdot)\) 41.c 4 Yes \(1\) \(1\) \(-1\) \(-i\) \(1\) \(-1\) \(i\) \(-i\) \(-1\) \(-1\) \(1\) \(-i\)