Properties

Modulus $784$
Structure \(C_{84}\times C_{2}\times C_{2}\)
Order $336$

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Show commands for: Pari/GP / SageMath

sage: H = DirichletGroup(784)
 
pari: g = idealstar(,784,2)
 

Character group

sage: G.order()
 
pari: g.no
 
Order = 336
sage: H.invariants()
 
pari: g.cyc
 
Structure = \(C_{84}\times C_{2}\times C_{2}\)
sage: H.gens()
 
pari: g.gen
 
Generators = $\chi_{784}(687,\cdot)$, $\chi_{784}(197,\cdot)$, $\chi_{784}(689,\cdot)$

First 32 of 336 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 \(-1\) \(1\) \(3\) \(5\) \(9\) \(11\) \(13\) \(15\) \(17\) \(19\) \(23\) \(25\)
\(\chi_{784}(1,\cdot)\) 784.a 1 no \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{784}(3,\cdot)\) 784.bu 84 yes \(1\) \(1\) \(e\left(\frac{65}{84}\right)\) \(e\left(\frac{37}{84}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{17}{84}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{37}{42}\right)\)
\(\chi_{784}(5,\cdot)\) 784.bs 84 yes \(-1\) \(1\) \(e\left(\frac{37}{84}\right)\) \(e\left(\frac{23}{84}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{73}{84}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{23}{42}\right)\)
\(\chi_{784}(9,\cdot)\) 784.bl 42 no \(1\) \(1\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{16}{21}\right)\)
\(\chi_{784}(11,\cdot)\) 784.bv 84 yes \(-1\) \(1\) \(e\left(\frac{17}{84}\right)\) \(e\left(\frac{73}{84}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{71}{84}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{31}{42}\right)\)
\(\chi_{784}(13,\cdot)\) 784.bi 28 yes \(-1\) \(1\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(-i\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{1}{14}\right)\)
\(\chi_{784}(15,\cdot)\) 784.bd 14 no \(-1\) \(1\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(-1\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{3}{7}\right)\)
\(\chi_{784}(17,\cdot)\) 784.bm 42 no \(-1\) \(1\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{11}{21}\right)\)
\(\chi_{784}(19,\cdot)\) 784.w 12 no \(1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{7}{12}\right)\) \(-i\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{784}(23,\cdot)\) 784.bq 42 no \(-1\) \(1\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{10}{21}\right)\)
\(\chi_{784}(25,\cdot)\) 784.bl 42 no \(1\) \(1\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{2}{21}\right)\)
\(\chi_{784}(27,\cdot)\) 784.bk 28 yes \(1\) \(1\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{11}{14}\right)\) \(-i\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{9}{14}\right)\)
\(\chi_{784}(29,\cdot)\) 784.bh 28 yes \(1\) \(1\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(i\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{5}{14}\right)\)
\(\chi_{784}(31,\cdot)\) 784.p 6 no \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{784}(33,\cdot)\) 784.bm 42 no \(-1\) \(1\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{13}{21}\right)\)
\(\chi_{784}(37,\cdot)\) 784.bt 84 yes \(1\) \(1\) \(e\left(\frac{43}{84}\right)\) \(e\left(\frac{29}{84}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{61}{84}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{29}{42}\right)\)
\(\chi_{784}(39,\cdot)\) 784.bq 42 no \(-1\) \(1\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{20}{21}\right)\)
\(\chi_{784}(41,\cdot)\) 784.z 14 no \(-1\) \(1\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(1\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{7}\right)\)
\(\chi_{784}(43,\cdot)\) 784.bj 28 yes \(-1\) \(1\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(i\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{11}{14}\right)\)
\(\chi_{784}(45,\cdot)\) 784.bs 84 yes \(-1\) \(1\) \(e\left(\frac{83}{84}\right)\) \(e\left(\frac{13}{84}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{23}{84}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{13}{42}\right)\)
\(\chi_{784}(47,\cdot)\) 784.bp 42 no \(1\) \(1\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{19}{21}\right)\)
\(\chi_{784}(51,\cdot)\) 784.bv 84 yes \(-1\) \(1\) \(e\left(\frac{31}{84}\right)\) \(e\left(\frac{59}{84}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{1}{84}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{17}{42}\right)\)
\(\chi_{784}(53,\cdot)\) 784.bt 84 yes \(1\) \(1\) \(e\left(\frac{83}{84}\right)\) \(e\left(\frac{13}{84}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{65}{84}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{13}{42}\right)\)
\(\chi_{784}(55,\cdot)\) 784.bc 14 no \(1\) \(1\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{1}{14}\right)\) \(-1\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{2}{7}\right)\)
\(\chi_{784}(57,\cdot)\) 784.bf 14 no \(1\) \(1\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(-1\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{7}\right)\)
\(\chi_{784}(59,\cdot)\) 784.bu 84 yes \(1\) \(1\) \(e\left(\frac{47}{84}\right)\) \(e\left(\frac{19}{84}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{11}{84}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{19}{42}\right)\)
\(\chi_{784}(61,\cdot)\) 784.bs 84 yes \(-1\) \(1\) \(e\left(\frac{43}{84}\right)\) \(e\left(\frac{29}{84}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{19}{84}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{29}{42}\right)\)
\(\chi_{784}(65,\cdot)\) 784.bg 21 no \(1\) \(1\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{13}{21}\right)\)
\(\chi_{784}(67,\cdot)\) 784.v 12 no \(-1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{11}{12}\right)\) \(i\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{784}(69,\cdot)\) 784.bi 28 yes \(-1\) \(1\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{3}{14}\right)\) \(i\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{5}{14}\right)\)
\(\chi_{784}(71,\cdot)\) 784.ba 14 no \(-1\) \(1\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(1\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{1}{7}\right)\)
\(\chi_{784}(73,\cdot)\) 784.br 42 no \(-1\) \(1\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{2}{21}\right)\)