Properties

Modulus $156$
Structure \(C_{12}\times C_{2}\times C_{2}\)
Order $48$

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Show commands for: Pari/GP / SageMath

sage: H = DirichletGroup(156)
 
pari: g = idealstar(,156,2)
 

Character group

sage: G.order()
 
pari: g.no
 
Order = 48
sage: H.invariants()
 
pari: g.cyc
 
Structure = \(C_{12}\times C_{2}\times C_{2}\)
sage: H.gens()
 
pari: g.gen
 
Generators = $\chi_{156}(79,\cdot)$, $\chi_{156}(53,\cdot)$, $\chi_{156}(145,\cdot)$

First 32 of 48 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 \(-1\) \(1\) \(5\) \(7\) \(11\) \(17\) \(19\) \(23\) \(25\) \(29\) \(31\) \(35\)
\(\chi_{156}(1,\cdot)\) 156.a 1 no \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{156}(5,\cdot)\) 156.m 4 no \(1\) \(1\) \(i\) \(i\) \(-i\) \(1\) \(-i\) \(1\) \(-1\) \(-1\) \(-i\) \(-1\)
\(\chi_{156}(7,\cdot)\) 156.w 12 no \(1\) \(1\) \(i\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(-i\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{156}(11,\cdot)\) 156.v 12 yes \(-1\) \(1\) \(-i\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(-i\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{156}(17,\cdot)\) 156.s 6 no \(-1\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{156}(19,\cdot)\) 156.w 12 no \(1\) \(1\) \(-i\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(i\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{156}(23,\cdot)\) 156.r 6 yes \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{156}(25,\cdot)\) 156.b 2 no \(1\) \(1\) \(-1\) \(-1\) \(-1\) \(1\) \(-1\) \(1\) \(1\) \(1\) \(-1\) \(1\)
\(\chi_{156}(29,\cdot)\) 156.o 6 no \(-1\) \(1\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(1\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{156}(31,\cdot)\) 156.k 4 no \(1\) \(1\) \(-i\) \(-i\) \(-i\) \(-1\) \(i\) \(1\) \(-1\) \(1\) \(i\) \(-1\)
\(\chi_{156}(35,\cdot)\) 156.p 6 yes \(1\) \(1\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{156}(37,\cdot)\) 156.x 12 no \(-1\) \(1\) \(i\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(i\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{156}(41,\cdot)\) 156.u 12 no \(1\) \(1\) \(i\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(-i\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{156}(43,\cdot)\) 156.n 6 no \(-1\) \(1\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{156}(47,\cdot)\) 156.l 4 yes \(-1\) \(1\) \(-i\) \(i\) \(-i\) \(1\) \(-i\) \(-1\) \(-1\) \(-1\) \(-i\) \(1\)
\(\chi_{156}(49,\cdot)\) 156.q 6 no \(1\) \(1\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{156}(53,\cdot)\) 156.d 2 no \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\) \(-1\)
\(\chi_{156}(55,\cdot)\) 156.t 6 no \(-1\) \(1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{156}(59,\cdot)\) 156.v 12 yes \(-1\) \(1\) \(-i\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(-i\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{156}(61,\cdot)\) 156.i 3 no \(1\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{156}(67,\cdot)\) 156.w 12 no \(1\) \(1\) \(-i\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(i\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{156}(71,\cdot)\) 156.v 12 yes \(-1\) \(1\) \(i\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(i\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{156}(73,\cdot)\) 156.j 4 no \(-1\) \(1\) \(i\) \(-i\) \(-i\) \(-1\) \(i\) \(-1\) \(-1\) \(1\) \(i\) \(1\)
\(\chi_{156}(77,\cdot)\) 156.g 2 no \(-1\) \(1\) \(1\) \(-1\) \(1\) \(-1\) \(-1\) \(-1\) \(1\) \(-1\) \(-1\) \(-1\)
\(\chi_{156}(79,\cdot)\) 156.f 2 no \(-1\) \(1\) \(1\) \(-1\) \(-1\) \(1\) \(-1\) \(-1\) \(1\) \(1\) \(-1\) \(-1\)
\(\chi_{156}(83,\cdot)\) 156.l 4 yes \(-1\) \(1\) \(i\) \(-i\) \(i\) \(1\) \(i\) \(-1\) \(-1\) \(-1\) \(i\) \(1\)
\(\chi_{156}(85,\cdot)\) 156.x 12 no \(-1\) \(1\) \(i\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(i\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{156}(89,\cdot)\) 156.u 12 no \(1\) \(1\) \(-i\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(i\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{156}(95,\cdot)\) 156.r 6 yes \(1\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{156}(97,\cdot)\) 156.x 12 no \(-1\) \(1\) \(-i\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(-i\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{156}(101,\cdot)\) 156.s 6 no \(-1\) \(1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{156}(103,\cdot)\) 156.e 2 no \(-1\) \(1\) \(-1\) \(1\) \(1\) \(1\) \(1\) \(-1\) \(1\) \(1\) \(1\) \(-1\)