Properties

Modulus $420$
Structure \(C_{12}\times C_{2}\times C_{2}\times C_{2}\)
Order $96$

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Show commands for: Pari/GP / SageMath

sage: H = DirichletGroup(420)
 
pari: g = idealstar(,420,2)
 

Character group

sage: G.order()
 
pari: g.no
 
Order = 96
sage: H.invariants()
 
pari: g.cyc
 
Structure = \(C_{12}\times C_{2}\times C_{2}\times C_{2}\)
sage: H.gens()
 
pari: g.gen
 
Generators = $\chi_{420}(211,\cdot)$, $\chi_{420}(281,\cdot)$, $\chi_{420}(337,\cdot)$, $\chi_{420}(241,\cdot)$

First 32 of 96 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 \(-1\) \(1\) \(11\) \(13\) \(17\) \(19\) \(23\) \(29\) \(31\) \(37\) \(41\) \(43\)
\(\chi_{420}(1,\cdot)\) 420.a 1 no \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{420}(11,\cdot)\) 420.bf 6 no \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(-1\)
\(\chi_{420}(13,\cdot)\) 420.x 4 no \(1\) \(1\) \(1\) \(-i\) \(i\) \(1\) \(i\) \(-1\) \(-1\) \(-i\) \(-1\) \(i\)
\(\chi_{420}(17,\cdot)\) 420.bt 12 no \(-1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(i\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{7}{12}\right)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{7}{12}\right)\) \(1\) \(-i\)
\(\chi_{420}(19,\cdot)\) 420.bk 6 no \(1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(1\)
\(\chi_{420}(23,\cdot)\) 420.bp 12 yes \(-1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(i\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{11}{12}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{12}\right)\) \(-1\) \(-i\)
\(\chi_{420}(29,\cdot)\) 420.e 2 no \(-1\) \(1\) \(-1\) \(-1\) \(1\) \(1\) \(1\) \(-1\) \(1\) \(-1\) \(-1\) \(-1\)
\(\chi_{420}(31,\cdot)\) 420.bi 6 no \(1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(-1\)
\(\chi_{420}(37,\cdot)\) 420.bq 12 no \(-1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(-i\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{12}\right)\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{11}{12}\right)\) \(1\) \(-i\)
\(\chi_{420}(41,\cdot)\) 420.d 2 no \(1\) \(1\) \(-1\) \(-1\) \(1\) \(-1\) \(-1\) \(-1\) \(-1\) \(1\) \(1\) \(1\)
\(\chi_{420}(43,\cdot)\) 420.t 4 no \(1\) \(1\) \(-1\) \(i\) \(-i\) \(1\) \(-i\) \(-1\) \(-1\) \(-i\) \(1\) \(-i\)
\(\chi_{420}(47,\cdot)\) 420.br 12 yes \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(i\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{12}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{11}{12}\right)\) \(1\) \(i\)
\(\chi_{420}(53,\cdot)\) 420.bv 12 no \(1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(i\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{12}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{12}\right)\) \(-1\) \(i\)
\(\chi_{420}(59,\cdot)\) 420.be 6 yes \(-1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\)
\(\chi_{420}(61,\cdot)\) 420.bc 6 no \(-1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(1\)
\(\chi_{420}(67,\cdot)\) 420.bs 12 no \(1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(-i\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{7}{12}\right)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{7}{12}\right)\) \(1\) \(i\)
\(\chi_{420}(71,\cdot)\) 420.n 2 no \(1\) \(1\) \(1\) \(1\) \(-1\) \(-1\) \(1\) \(-1\) \(-1\) \(1\) \(-1\) \(-1\)
\(\chi_{420}(73,\cdot)\) 420.bo 12 no \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(-i\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{7}{12}\right)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{12}\right)\) \(-1\) \(i\)
\(\chi_{420}(79,\cdot)\) 420.bj 6 no \(-1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(1\)
\(\chi_{420}(83,\cdot)\) 420.w 4 yes \(1\) \(1\) \(1\) \(-i\) \(-i\) \(-1\) \(i\) \(1\) \(1\) \(-i\) \(1\) \(-i\)
\(\chi_{420}(89,\cdot)\) 420.bn 6 no \(1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(-1\)
\(\chi_{420}(97,\cdot)\) 420.x 4 no \(1\) \(1\) \(1\) \(i\) \(-i\) \(1\) \(-i\) \(-1\) \(-1\) \(i\) \(-1\) \(-i\)
\(\chi_{420}(101,\cdot)\) 420.bh 6 no \(1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)
\(\chi_{420}(103,\cdot)\) 420.bu 12 no \(-1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(-i\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{12}\right)\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{12}\right)\) \(-1\) \(-i\)
\(\chi_{420}(107,\cdot)\) 420.bp 12 yes \(-1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(-i\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{12}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{11}{12}\right)\) \(-1\) \(i\)
\(\chi_{420}(109,\cdot)\) 420.bb 6 no \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(-1\)
\(\chi_{420}(113,\cdot)\) 420.s 4 no \(1\) \(1\) \(-1\) \(i\) \(i\) \(-1\) \(-i\) \(1\) \(1\) \(-i\) \(-1\) \(i\)
\(\chi_{420}(121,\cdot)\) 420.q 3 no \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)
\(\chi_{420}(127,\cdot)\) 420.t 4 no \(1\) \(1\) \(-1\) \(-i\) \(i\) \(1\) \(i\) \(-1\) \(-1\) \(i\) \(1\) \(i\)
\(\chi_{420}(131,\cdot)\) 420.z 6 no \(-1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(-1\)
\(\chi_{420}(137,\cdot)\) 420.bv 12 no \(1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(-i\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{7}{12}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{12}\right)\) \(-1\) \(-i\)
\(\chi_{420}(139,\cdot)\) 420.i 2 no \(1\) \(1\) \(-1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(-1\) \(-1\) \(1\)