Properties

Modulus 49
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(49)
pari: g = idealstar(,49,2)

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 = $\chi_{49}(3,\cdot)$

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 2 3 4 5 6 8 9 10 11 12
\(\chi_{49}(1,\cdot)\) 49.a 1 No \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{49}(2,\cdot)\) 49.g 21 Yes \(1\) \(1\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{17}{21}\right)\)
\(\chi_{49}(3,\cdot)\) 49.h 42 Yes \(-1\) \(1\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{11}{42}\right)\)
\(\chi_{49}(4,\cdot)\) 49.g 21 Yes \(1\) \(1\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{13}{21}\right)\)
\(\chi_{49}(5,\cdot)\) 49.h 42 Yes \(-1\) \(1\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{25}{42}\right)\)
\(\chi_{49}(6,\cdot)\) 49.f 14 Yes \(-1\) \(1\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{1}{14}\right)\)
\(\chi_{49}(8,\cdot)\) 49.e 7 Yes \(1\) \(1\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{3}{7}\right)\)
\(\chi_{49}(9,\cdot)\) 49.g 21 Yes \(1\) \(1\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{11}{21}\right)\)
\(\chi_{49}(10,\cdot)\) 49.h 42 Yes \(-1\) \(1\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{17}{42}\right)\)
\(\chi_{49}(11,\cdot)\) 49.g 21 Yes \(1\) \(1\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{10}{21}\right)\)
\(\chi_{49}(12,\cdot)\) 49.h 42 Yes \(-1\) \(1\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{37}{42}\right)\)
\(\chi_{49}(13,\cdot)\) 49.f 14 Yes \(-1\) \(1\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{9}{14}\right)\)
\(\chi_{49}(15,\cdot)\) 49.e 7 Yes \(1\) \(1\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{6}{7}\right)\)
\(\chi_{49}(16,\cdot)\) 49.g 21 Yes \(1\) \(1\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{5}{21}\right)\)
\(\chi_{49}(17,\cdot)\) 49.h 42 Yes \(-1\) \(1\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{23}{42}\right)\)
\(\chi_{49}(18,\cdot)\) 49.c 3 No \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{49}(19,\cdot)\) 49.d 6 No \(-1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{49}(20,\cdot)\) 49.f 14 Yes \(-1\) \(1\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{3}{14}\right)\)
\(\chi_{49}(22,\cdot)\) 49.e 7 Yes \(1\) \(1\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{2}{7}\right)\)
\(\chi_{49}(23,\cdot)\) 49.g 21 Yes \(1\) \(1\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{20}{21}\right)\)
\(\chi_{49}(24,\cdot)\) 49.h 42 Yes \(-1\) \(1\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{29}{42}\right)\)
\(\chi_{49}(25,\cdot)\) 49.g 21 Yes \(1\) \(1\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{4}{21}\right)\)
\(\chi_{49}(26,\cdot)\) 49.h 42 Yes \(-1\) \(1\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{19}{42}\right)\)
\(\chi_{49}(27,\cdot)\) 49.f 14 Yes \(-1\) \(1\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{11}{14}\right)\)
\(\chi_{49}(29,\cdot)\) 49.e 7 Yes \(1\) \(1\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{5}{7}\right)\)
\(\chi_{49}(30,\cdot)\) 49.c 3 No \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{49}(31,\cdot)\) 49.d 6 No \(-1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{49}(32,\cdot)\) 49.g 21 Yes \(1\) \(1\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{1}{21}\right)\)
\(\chi_{49}(33,\cdot)\) 49.h 42 Yes \(-1\) \(1\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{31}{42}\right)\)
\(\chi_{49}(34,\cdot)\) 49.f 14 Yes \(-1\) \(1\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{14}\right)\)
\(\chi_{49}(36,\cdot)\) 49.e 7 Yes \(1\) \(1\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{1}{7}\right)\)
\(\chi_{49}(37,\cdot)\) 49.g 21 Yes \(1\) \(1\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{8}{21}\right)\)