Properties

Modulus 31
Structure \(C_{30}\)
Order 30

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(31)
pari: g = idealstar(,31,2)

Character group

sage: G.order()
pari: g.no
Order = 30
sage: H.invariants()
pari: g.cyc
Structure = \(C_{30}\)
sage: H.gens()
pari: g.gen
Generators = $\chi_{31}(3,\cdot)$

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