Properties

Modulus $111$
Structure \(C_{36}\times C_{2}\)
Order $72$

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Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(111)
 
pari: g = idealstar(,111,2)
 

Character group

sage: G.order()
 
pari: g.no
 
Order = 72
sage: H.invariants()
 
pari: g.cyc
 
Structure = \(C_{36}\times C_{2}\)
sage: H.gens()
 
pari: g.gen
 
Generators = $\chi_{111}(38,\cdot)$, $\chi_{111}(76,\cdot)$

First 32 of 72 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 \(-1\) \(1\) \(2\) \(4\) \(5\) \(7\) \(8\) \(10\) \(11\) \(13\) \(14\) \(16\)
\(\chi_{111}(1,\cdot)\) 111.a 1 no \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{111}(2,\cdot)\) 111.q 36 yes \(1\) \(1\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{9}\right)\)
\(\chi_{111}(4,\cdot)\) 111.o 18 no \(1\) \(1\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{9}\right)\)
\(\chi_{111}(5,\cdot)\) 111.q 36 yes \(1\) \(1\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{5}{9}\right)\)
\(\chi_{111}(7,\cdot)\) 111.k 9 no \(1\) \(1\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{9}\right)\)
\(\chi_{111}(8,\cdot)\) 111.m 12 yes \(1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(-i\) \(1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(i\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{111}(10,\cdot)\) 111.e 3 no \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{111}(11,\cdot)\) 111.h 6 yes \(-1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{111}(13,\cdot)\) 111.r 36 no \(-1\) \(1\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{2}{9}\right)\)
\(\chi_{111}(14,\cdot)\) 111.m 12 yes \(1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(i\) \(1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(-i\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{111}(16,\cdot)\) 111.k 9 no \(1\) \(1\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{4}{9}\right)\)
\(\chi_{111}(17,\cdot)\) 111.q 36 yes \(1\) \(1\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{7}{9}\right)\)
\(\chi_{111}(19,\cdot)\) 111.r 36 no \(-1\) \(1\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{8}{9}\right)\)
\(\chi_{111}(20,\cdot)\) 111.q 36 yes \(1\) \(1\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{7}{9}\right)\)
\(\chi_{111}(22,\cdot)\) 111.r 36 no \(-1\) \(1\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{4}{9}\right)\)
\(\chi_{111}(23,\cdot)\) 111.m 12 yes \(1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(-i\) \(1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(i\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{111}(25,\cdot)\) 111.o 18 no \(1\) \(1\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{9}\right)\)
\(\chi_{111}(26,\cdot)\) 111.i 6 yes \(-1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(1\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{111}(28,\cdot)\) 111.o 18 no \(1\) \(1\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{7}{9}\right)\)
\(\chi_{111}(29,\cdot)\) 111.m 12 yes \(1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(i\) \(1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(-i\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{111}(31,\cdot)\) 111.f 4 no \(-1\) \(1\) \(i\) \(-1\) \(-i\) \(1\) \(-i\) \(1\) \(-1\) \(-i\) \(i\) \(1\)
\(\chi_{111}(32,\cdot)\) 111.q 36 yes \(1\) \(1\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{5}{9}\right)\)
\(\chi_{111}(34,\cdot)\) 111.k 9 no \(1\) \(1\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{8}{9}\right)\)
\(\chi_{111}(35,\cdot)\) 111.q 36 yes \(1\) \(1\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{9}\right)\)
\(\chi_{111}(38,\cdot)\) 111.b 2 no \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\)
\(\chi_{111}(40,\cdot)\) 111.o 18 no \(1\) \(1\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{8}{9}\right)\)
\(\chi_{111}(41,\cdot)\) 111.n 18 yes \(-1\) \(1\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{9}\right)\)
\(\chi_{111}(43,\cdot)\) 111.f 4 no \(-1\) \(1\) \(-i\) \(-1\) \(i\) \(1\) \(i\) \(1\) \(-1\) \(i\) \(-i\) \(1\)
\(\chi_{111}(44,\cdot)\) 111.p 18 yes \(-1\) \(1\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{9}\right)\)
\(\chi_{111}(46,\cdot)\) 111.k 9 no \(1\) \(1\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{9}\right)\)
\(\chi_{111}(47,\cdot)\) 111.i 6 yes \(-1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(1\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{111}(49,\cdot)\) 111.k 9 no \(1\) \(1\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{9}\right)\)