Properties

Modulus $185$
Structure \(C_{36}\times C_{4}\)
Order $144$

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Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(185)
 
pari: g = idealstar(,185,2)
 

Character group

sage: G.order()
 
pari: g.no
 
Order = 144
sage: H.invariants()
 
pari: g.cyc
 
Structure = \(C_{36}\times C_{4}\)
sage: H.gens()
 
pari: g.gen
 
Generators = $\chi_{185}(112,\cdot)$, $\chi_{185}(76,\cdot)$

First 32 of 144 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 \(-1\) \(1\) \(2\) \(3\) \(4\) \(6\) \(7\) \(8\) \(9\) \(11\) \(12\) \(13\)
\(\chi_{185}(1,\cdot)\) 185.a 1 no \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{185}(2,\cdot)\) 185.bc 36 yes \(1\) \(1\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{5}{9}\right)\) \(-i\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{1}{18}\right)\)
\(\chi_{185}(3,\cdot)\) 185.y 36 yes \(-1\) \(1\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{17}{18}\right)\) \(-1\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{7}{36}\right)\)
\(\chi_{185}(4,\cdot)\) 185.v 18 yes \(1\) \(1\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{1}{9}\right)\) \(-1\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{1}{9}\right)\)
\(\chi_{185}(6,\cdot)\) 185.g 4 no \(-1\) \(1\) \(-i\) \(-1\) \(-1\) \(i\) \(1\) \(i\) \(1\) \(-1\) \(1\) \(i\)
\(\chi_{185}(7,\cdot)\) 185.bd 36 yes \(-1\) \(1\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{5}{18}\right)\) \(1\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{19}{36}\right)\)
\(\chi_{185}(8,\cdot)\) 185.u 12 yes \(1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(i\) \(e\left(\frac{5}{12}\right)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{185}(9,\cdot)\) 185.x 18 yes \(1\) \(1\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{8}{9}\right)\) \(1\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{7}{18}\right)\)
\(\chi_{185}(11,\cdot)\) 185.m 6 no \(1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{185}(12,\cdot)\) 185.bd 36 yes \(-1\) \(1\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{1}{18}\right)\) \(1\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{11}{36}\right)\)
\(\chi_{185}(13,\cdot)\) 185.bc 36 yes \(1\) \(1\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{1}{9}\right)\) \(i\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{11}{18}\right)\)
\(\chi_{185}(14,\cdot)\) 185.q 12 yes \(-1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(-i\) \(e\left(\frac{5}{6}\right)\) \(i\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{7}{12}\right)\)
\(\chi_{185}(16,\cdot)\) 185.o 9 no \(1\) \(1\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(1\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{2}{9}\right)\)
\(\chi_{185}(17,\cdot)\) 185.z 36 yes \(1\) \(1\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{8}{9}\right)\) \(i\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{8}{9}\right)\)
\(\chi_{185}(18,\cdot)\) 185.z 36 yes \(1\) \(1\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{4}{9}\right)\) \(-i\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{4}{9}\right)\)
\(\chi_{185}(19,\cdot)\) 185.ba 36 yes \(-1\) \(1\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{17}{18}\right)\) \(i\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{7}{36}\right)\)
\(\chi_{185}(21,\cdot)\) 185.w 18 no \(1\) \(1\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(-1\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{13}{18}\right)\)
\(\chi_{185}(22,\cdot)\) 185.z 36 yes \(1\) \(1\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{2}{9}\right)\) \(i\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{2}{9}\right)\)
\(\chi_{185}(23,\cdot)\) 185.u 12 yes \(1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(i\) \(e\left(\frac{1}{12}\right)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{185}(24,\cdot)\) 185.ba 36 yes \(-1\) \(1\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{11}{18}\right)\) \(-i\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{13}{36}\right)\)
\(\chi_{185}(26,\cdot)\) 185.e 3 no \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{185}(27,\cdot)\) 185.r 12 yes \(-1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(e\left(\frac{7}{12}\right)\) \(i\) \(e\left(\frac{1}{6}\right)\) \(1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{7}{12}\right)\)
\(\chi_{185}(28,\cdot)\) 185.y 36 yes \(-1\) \(1\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{7}{18}\right)\) \(-1\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{23}{36}\right)\)
\(\chi_{185}(29,\cdot)\) 185.q 12 yes \(-1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(-i\) \(e\left(\frac{1}{6}\right)\) \(i\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{11}{12}\right)\)
\(\chi_{185}(31,\cdot)\) 185.g 4 no \(-1\) \(1\) \(i\) \(-1\) \(-1\) \(-i\) \(1\) \(-i\) \(1\) \(-1\) \(1\) \(-i\)
\(\chi_{185}(32,\cdot)\) 185.bc 36 yes \(1\) \(1\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{7}{9}\right)\) \(-i\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{5}{18}\right)\)
\(\chi_{185}(33,\cdot)\) 185.bd 36 yes \(-1\) \(1\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{11}{18}\right)\) \(1\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{13}{36}\right)\)
\(\chi_{185}(34,\cdot)\) 185.x 18 yes \(1\) \(1\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{4}{9}\right)\) \(1\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{17}{18}\right)\)
\(\chi_{185}(36,\cdot)\) 185.c 2 no \(1\) \(1\) \(-1\) \(1\) \(1\) \(-1\) \(1\) \(-1\) \(1\) \(1\) \(1\) \(-1\)
\(\chi_{185}(38,\cdot)\) 185.i 4 no \(-1\) \(1\) \(-i\) \(i\) \(-1\) \(1\) \(-i\) \(i\) \(-1\) \(1\) \(-i\) \(i\)
\(\chi_{185}(39,\cdot)\) 185.ba 36 yes \(-1\) \(1\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{1}{18}\right)\) \(-i\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{29}{36}\right)\)
\(\chi_{185}(41,\cdot)\) 185.w 18 no \(1\) \(1\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(-1\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{11}{18}\right)\)