Properties

Modulus $129$
Structure \(C_{42}\times C_{2}\)
Order $84$

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Show commands for: Pari/GP / SageMath

sage: H = DirichletGroup(129)
 
pari: g = idealstar(,129,2)
 

Character group

sage: G.order()
 
pari: g.no
 
Order = 84
sage: H.invariants()
 
pari: g.cyc
 
Structure = \(C_{42}\times C_{2}\)
sage: H.gens()
 
pari: g.gen
 
Generators = $\chi_{129}(44,\cdot)$, $\chi_{129}(46,\cdot)$

First 32 of 84 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.

Character Orbit Order Primitive \(-1\) \(1\) \(2\) \(4\) \(5\) \(7\) \(8\) \(10\) \(11\) \(13\) \(14\) \(16\)
\(\chi_{129}(1,\cdot)\) 129.a 1 no \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{129}(2,\cdot)\) 129.j 14 yes \(1\) \(1\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(-1\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{3}{7}\right)\)
\(\chi_{129}(4,\cdot)\) 129.i 7 no \(1\) \(1\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(1\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{6}{7}\right)\)
\(\chi_{129}(5,\cdot)\) 129.n 42 yes \(1\) \(1\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{2}{7}\right)\)
\(\chi_{129}(7,\cdot)\) 129.g 6 no \(-1\) \(1\) \(-1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\)
\(\chi_{129}(8,\cdot)\) 129.j 14 yes \(1\) \(1\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(-1\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{2}{7}\right)\)
\(\chi_{129}(10,\cdot)\) 129.m 21 no \(1\) \(1\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{5}{7}\right)\)
\(\chi_{129}(11,\cdot)\) 129.l 14 yes \(-1\) \(1\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{14}\right)\) \(1\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{1}{7}\right)\)
\(\chi_{129}(13,\cdot)\) 129.m 21 no \(1\) \(1\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{2}{7}\right)\)
\(\chi_{129}(14,\cdot)\) 129.o 42 yes \(-1\) \(1\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{3}{7}\right)\)
\(\chi_{129}(16,\cdot)\) 129.i 7 no \(1\) \(1\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(1\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{5}{7}\right)\)
\(\chi_{129}(17,\cdot)\) 129.o 42 yes \(-1\) \(1\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{5}{7}\right)\)
\(\chi_{129}(19,\cdot)\) 129.p 42 no \(-1\) \(1\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{6}{7}\right)\)
\(\chi_{129}(20,\cdot)\) 129.n 42 yes \(1\) \(1\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{1}{7}\right)\)
\(\chi_{129}(22,\cdot)\) 129.k 14 no \(-1\) \(1\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(-1\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{4}{7}\right)\)
\(\chi_{129}(23,\cdot)\) 129.o 42 yes \(-1\) \(1\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{1}{7}\right)\)
\(\chi_{129}(25,\cdot)\) 129.m 21 no \(1\) \(1\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{4}{7}\right)\)
\(\chi_{129}(26,\cdot)\) 129.n 42 yes \(1\) \(1\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{5}{7}\right)\)
\(\chi_{129}(28,\cdot)\) 129.p 42 no \(-1\) \(1\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{6}{7}\right)\)
\(\chi_{129}(29,\cdot)\) 129.n 42 yes \(1\) \(1\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{3}{7}\right)\)
\(\chi_{129}(31,\cdot)\) 129.m 21 no \(1\) \(1\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{3}{7}\right)\)
\(\chi_{129}(32,\cdot)\) 129.j 14 yes \(1\) \(1\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(-1\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{1}{7}\right)\)
\(\chi_{129}(34,\cdot)\) 129.p 42 no \(-1\) \(1\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{1}{7}\right)\)
\(\chi_{129}(35,\cdot)\) 129.l 14 yes \(-1\) \(1\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{3}{14}\right)\) \(1\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{2}{7}\right)\)
\(\chi_{129}(37,\cdot)\) 129.g 6 no \(-1\) \(1\) \(-1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\)
\(\chi_{129}(38,\cdot)\) 129.o 42 yes \(-1\) \(1\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{2}{7}\right)\)
\(\chi_{129}(40,\cdot)\) 129.m 21 no \(1\) \(1\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{4}{7}\right)\)
\(\chi_{129}(41,\cdot)\) 129.l 14 yes \(-1\) \(1\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(1\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{3}{7}\right)\)
\(\chi_{129}(44,\cdot)\) 129.c 2 no \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\)
\(\chi_{129}(46,\cdot)\) 129.p 42 no \(-1\) \(1\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{4}{7}\right)\)
\(\chi_{129}(47,\cdot)\) 129.l 14 yes \(-1\) \(1\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(1\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{6}{7}\right)\)
\(\chi_{129}(49,\cdot)\) 129.e 3 no \(1\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\)