Properties

Modulus $114$
Structure \(C_{18}\times C_{2}\)
Order $36$

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Show commands for: Pari/GP / SageMath

sage: H = DirichletGroup(114)
 
pari: g = idealstar(,114,2)
 

Character group

sage: G.order()
 
pari: g.no
 
Order = 36
sage: H.invariants()
 
pari: g.cyc
 
Structure = \(C_{18}\times C_{2}\)
sage: H.gens()
 
pari: g.gen
 
Generators = $\chi_{114}(77,\cdot)$, $\chi_{114}(97,\cdot)$

First 32 of 36 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 \(-1\) \(1\) \(5\) \(7\) \(11\) \(13\) \(17\) \(23\) \(25\) \(29\) \(31\) \(35\)
\(\chi_{114}(1,\cdot)\) 114.a 1 no \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{114}(5,\cdot)\) 114.k 18 no \(-1\) \(1\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{18}\right)\)
\(\chi_{114}(7,\cdot)\) 114.e 3 no \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{114}(11,\cdot)\) 114.g 6 no \(-1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(1\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{114}(13,\cdot)\) 114.j 18 no \(-1\) \(1\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{9}\right)\)
\(\chi_{114}(17,\cdot)\) 114.k 18 no \(-1\) \(1\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{13}{18}\right)\)
\(\chi_{114}(23,\cdot)\) 114.k 18 no \(-1\) \(1\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{17}{18}\right)\)
\(\chi_{114}(25,\cdot)\) 114.i 9 no \(1\) \(1\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{9}\right)\)
\(\chi_{114}(29,\cdot)\) 114.l 18 no \(1\) \(1\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{18}\right)\)
\(\chi_{114}(31,\cdot)\) 114.f 6 no \(-1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{114}(35,\cdot)\) 114.k 18 no \(-1\) \(1\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{7}{18}\right)\)
\(\chi_{114}(37,\cdot)\) 114.d 2 no \(-1\) \(1\) \(1\) \(1\) \(1\) \(-1\) \(1\) \(1\) \(1\) \(-1\) \(-1\) \(1\)
\(\chi_{114}(41,\cdot)\) 114.l 18 no \(1\) \(1\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{7}{18}\right)\)
\(\chi_{114}(43,\cdot)\) 114.i 9 no \(1\) \(1\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{9}\right)\)
\(\chi_{114}(47,\cdot)\) 114.k 18 no \(-1\) \(1\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{18}\right)\)
\(\chi_{114}(49,\cdot)\) 114.e 3 no \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{114}(53,\cdot)\) 114.l 18 no \(1\) \(1\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{17}{18}\right)\)
\(\chi_{114}(55,\cdot)\) 114.i 9 no \(1\) \(1\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{9}\right)\)
\(\chi_{114}(59,\cdot)\) 114.l 18 no \(1\) \(1\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{13}{18}\right)\)
\(\chi_{114}(61,\cdot)\) 114.i 9 no \(1\) \(1\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{4}{9}\right)\)
\(\chi_{114}(65,\cdot)\) 114.h 6 no \(1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(1\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{114}(67,\cdot)\) 114.j 18 no \(-1\) \(1\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{7}{9}\right)\)
\(\chi_{114}(71,\cdot)\) 114.l 18 no \(1\) \(1\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{18}\right)\)
\(\chi_{114}(73,\cdot)\) 114.i 9 no \(1\) \(1\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{8}{9}\right)\)
\(\chi_{114}(77,\cdot)\) 114.c 2 no \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(-1\) \(1\) \(-1\) \(1\) \(-1\)
\(\chi_{114}(79,\cdot)\) 114.j 18 no \(-1\) \(1\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{8}{9}\right)\)
\(\chi_{114}(83,\cdot)\) 114.g 6 no \(-1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(1\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{114}(85,\cdot)\) 114.i 9 no \(1\) \(1\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{9}\right)\)
\(\chi_{114}(89,\cdot)\) 114.l 18 no \(1\) \(1\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{18}\right)\)
\(\chi_{114}(91,\cdot)\) 114.j 18 no \(-1\) \(1\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{4}{9}\right)\)
\(\chi_{114}(97,\cdot)\) 114.j 18 no \(-1\) \(1\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{9}\right)\)
\(\chi_{114}(101,\cdot)\) 114.k 18 no \(-1\) \(1\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{11}{18}\right)\)