Properties

Modulus $392$
Structure \(C_{42}\times C_{2}\times C_{2}\)
Order $168$

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Show commands for: Pari/GP / SageMath

sage: H = DirichletGroup(392)
 
pari: g = idealstar(,392,2)
 

Character group

sage: G.order()
 
pari: g.no
 
Order = 168
sage: H.invariants()
 
pari: g.cyc
 
Structure = \(C_{42}\times C_{2}\times C_{2}\)
sage: H.gens()
 
pari: g.gen
 
Generators = $\chi_{392}(295,\cdot)$, $\chi_{392}(197,\cdot)$, $\chi_{392}(297,\cdot)$

First 32 of 168 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 \(-1\) \(1\) \(3\) \(5\) \(9\) \(11\) \(13\) \(15\) \(17\) \(19\) \(23\) \(25\)
\(\chi_{392}(1,\cdot)\) 392.a 1 no \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{392}(3,\cdot)\) 392.bc 42 yes \(1\) \(1\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{8}{21}\right)\)
\(\chi_{392}(5,\cdot)\) 392.bf 42 yes \(-1\) \(1\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{1}{21}\right)\)
\(\chi_{392}(9,\cdot)\) 392.y 21 no \(1\) \(1\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{16}{21}\right)\)
\(\chi_{392}(11,\cdot)\) 392.be 42 yes \(-1\) \(1\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{5}{21}\right)\)
\(\chi_{392}(13,\cdot)\) 392.r 14 yes \(-1\) \(1\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(1\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{4}{7}\right)\)
\(\chi_{392}(15,\cdot)\) 392.v 14 no \(-1\) \(1\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(-1\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{3}{7}\right)\)
\(\chi_{392}(17,\cdot)\) 392.ba 42 no \(-1\) \(1\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{11}{21}\right)\)
\(\chi_{392}(19,\cdot)\) 392.m 6 no \(1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{392}(23,\cdot)\) 392.bb 42 no \(-1\) \(1\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{10}{21}\right)\)
\(\chi_{392}(25,\cdot)\) 392.y 21 no \(1\) \(1\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{2}{21}\right)\)
\(\chi_{392}(27,\cdot)\) 392.u 14 yes \(1\) \(1\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{11}{14}\right)\) \(-1\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{1}{7}\right)\)
\(\chi_{392}(29,\cdot)\) 392.x 14 yes \(1\) \(1\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(-1\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{6}{7}\right)\)
\(\chi_{392}(31,\cdot)\) 392.l 6 no \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{392}(33,\cdot)\) 392.ba 42 no \(-1\) \(1\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{13}{21}\right)\)
\(\chi_{392}(37,\cdot)\) 392.z 42 yes \(1\) \(1\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{4}{21}\right)\)
\(\chi_{392}(39,\cdot)\) 392.bb 42 no \(-1\) \(1\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{20}{21}\right)\)
\(\chi_{392}(41,\cdot)\) 392.w 14 no \(-1\) \(1\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(-1\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{7}\right)\)
\(\chi_{392}(43,\cdot)\) 392.s 14 yes \(-1\) \(1\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(1\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{2}{7}\right)\)
\(\chi_{392}(45,\cdot)\) 392.bf 42 yes \(-1\) \(1\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{17}{21}\right)\)
\(\chi_{392}(47,\cdot)\) 392.bd 42 no \(1\) \(1\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{19}{21}\right)\)
\(\chi_{392}(51,\cdot)\) 392.be 42 yes \(-1\) \(1\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{19}{21}\right)\)
\(\chi_{392}(53,\cdot)\) 392.z 42 yes \(1\) \(1\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{17}{21}\right)\)
\(\chi_{392}(55,\cdot)\) 392.t 14 no \(1\) \(1\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{1}{14}\right)\) \(1\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{2}{7}\right)\)
\(\chi_{392}(57,\cdot)\) 392.q 7 no \(1\) \(1\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(1\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{7}\right)\)
\(\chi_{392}(59,\cdot)\) 392.bc 42 yes \(1\) \(1\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{20}{21}\right)\)
\(\chi_{392}(61,\cdot)\) 392.bf 42 yes \(-1\) \(1\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{4}{21}\right)\)
\(\chi_{392}(65,\cdot)\) 392.y 21 no \(1\) \(1\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{13}{21}\right)\)
\(\chi_{392}(67,\cdot)\) 392.k 6 no \(-1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{392}(69,\cdot)\) 392.r 14 yes \(-1\) \(1\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{3}{14}\right)\) \(1\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{6}{7}\right)\)
\(\chi_{392}(71,\cdot)\) 392.v 14 no \(-1\) \(1\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(-1\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{1}{7}\right)\)
\(\chi_{392}(73,\cdot)\) 392.ba 42 no \(-1\) \(1\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{2}{21}\right)\)