Properties

Modulus $819$
Structure \(C_{12}\times C_{6}\times C_{6}\)
Order $432$

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Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(819)
 
pari: g = idealstar(,819,2)
 

Character group

sage: G.order()
 
pari: g.no
 
Order = 432
sage: H.invariants()
 
pari: g.cyc
 
Structure = \(C_{12}\times C_{6}\times C_{6}\)
sage: H.gens()
 
pari: g.gen
 
Generators = $\chi_{819}(92,\cdot)$, $\chi_{819}(703,\cdot)$, $\chi_{819}(379,\cdot)$

First 32 of 432 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 \(-1\) \(1\) \(2\) \(4\) \(5\) \(8\) \(10\) \(11\) \(16\) \(17\) \(19\) \(20\)
\(\chi_{819}(1,\cdot)\) 819.a 1 no \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{819}(2,\cdot)\) 819.gf 12 yes \(1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(i\) \(-i\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{12}\right)\)
\(\chi_{819}(4,\cdot)\) 819.cv 6 yes \(1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{819}(5,\cdot)\) 819.er 12 yes \(-1\) \(1\) \(i\) \(-1\) \(e\left(\frac{1}{12}\right)\) \(-i\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{12}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{7}{12}\right)\)
\(\chi_{819}(8,\cdot)\) 819.w 4 no \(1\) \(1\) \(-i\) \(-1\) \(-i\) \(i\) \(-1\) \(i\) \(1\) \(1\) \(i\) \(i\)
\(\chi_{819}(10,\cdot)\) 819.cx 6 no \(-1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(-1\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{819}(11,\cdot)\) 819.fe 12 yes \(1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{12}\right)\) \(i\) \(-1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(i\) \(e\left(\frac{7}{12}\right)\)
\(\chi_{819}(16,\cdot)\) 819.p 3 yes \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{819}(17,\cdot)\) 819.ei 6 no \(1\) \(1\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{819}(19,\cdot)\) 819.gh 12 no \(1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{12}\right)\) \(i\) \(1\) \(i\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(i\) \(e\left(\frac{1}{12}\right)\)
\(\chi_{819}(20,\cdot)\) 819.fq 12 yes \(-1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{7}{12}\right)\) \(i\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{12}\right)\) \(-i\)
\(\chi_{819}(22,\cdot)\) 819.t 3 no \(1\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{819}(23,\cdot)\) 819.bx 6 yes \(-1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(1\)
\(\chi_{819}(25,\cdot)\) 819.bk 6 yes \(1\) \(1\) \(-1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{819}(29,\cdot)\) 819.bi 6 no \(-1\) \(1\) \(-1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{819}(31,\cdot)\) 819.fk 12 yes \(1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(i\) \(i\) \(e\left(\frac{2}{3}\right)\) \(i\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{12}\right)\)
\(\chi_{819}(32,\cdot)\) 819.gf 12 yes \(1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(i\) \(-i\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{12}\right)\)
\(\chi_{819}(34,\cdot)\) 819.gl 12 yes \(1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{12}\right)\) \(-i\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(-i\) \(e\left(\frac{11}{12}\right)\)
\(\chi_{819}(37,\cdot)\) 819.eu 12 no \(-1\) \(1\) \(i\) \(-1\) \(e\left(\frac{11}{12}\right)\) \(-i\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{12}\right)\) \(1\) \(-1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{5}{12}\right)\)
\(\chi_{819}(38,\cdot)\) 819.eg 6 yes \(1\) \(1\) \(1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{819}(40,\cdot)\) 819.eh 6 no \(-1\) \(1\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{819}(41,\cdot)\) 819.fq 12 yes \(-1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{12}\right)\) \(-i\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{12}\right)\) \(i\)
\(\chi_{819}(43,\cdot)\) 819.dk 6 no \(1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(-1\)
\(\chi_{819}(44,\cdot)\) 819.fz 12 no \(1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{11}{12}\right)\) \(-i\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{12}\right)\) \(-i\)
\(\chi_{819}(46,\cdot)\) 819.eu 12 no \(-1\) \(1\) \(i\) \(-1\) \(e\left(\frac{7}{12}\right)\) \(-i\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{12}\right)\) \(1\) \(-1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{12}\right)\)
\(\chi_{819}(47,\cdot)\) 819.fp 12 yes \(-1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(i\) \(i\) \(e\left(\frac{1}{3}\right)\) \(i\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{12}\right)\)
\(\chi_{819}(50,\cdot)\) 819.fy 12 no \(1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{12}\right)\) \(i\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{11}{12}\right)\) \(i\)
\(\chi_{819}(53,\cdot)\) 819.dn 6 no \(-1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(-1\)
\(\chi_{819}(55,\cdot)\) 819.br 6 no \(-1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{819}(58,\cdot)\) 819.fg 12 yes \(-1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{12}\right)\) \(i\) \(-1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(-i\) \(e\left(\frac{11}{12}\right)\)
\(\chi_{819}(59,\cdot)\) 819.fj 12 yes \(-1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(i\) \(i\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{12}\right)\)
\(\chi_{819}(61,\cdot)\) 819.eb 6 yes \(-1\) \(1\) \(1\) \(1\) \(-1\) \(1\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(-1\)