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Properties

Modulus $19$
Structure \(C_{18}\)
Order $18$

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Show commands: Pari/GP / SageMath

Copy content sage:H = DirichletGroup(19)
 
Copy content pari:g = idealstar(,19,2)
 

Character group

Copy content sage:G.order()
 
Copy content pari:g.no
 
Order = 18
Copy content sage:H.invariants()
 
Copy content pari:g.cyc
 
Structure = \(C_{18}\)
Copy content sage:H.gens()
 
Copy content pari:g.gen
 
Generators = $\chi_{19}(2,\cdot)$

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 \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(11\)
\(\chi_{19}(1,\cdot)\) 19.a 1 no \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{19}(2,\cdot)\) 19.f 18 yes \(-1\) \(1\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{19}(3,\cdot)\) 19.f 18 yes \(-1\) \(1\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{19}(4,\cdot)\) 19.e 9 yes \(1\) \(1\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{19}(5,\cdot)\) 19.e 9 yes \(1\) \(1\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{19}(6,\cdot)\) 19.e 9 yes \(1\) \(1\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{19}(7,\cdot)\) 19.c 3 yes \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\)
\(\chi_{19}(8,\cdot)\) 19.d 6 yes \(-1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(1\)
\(\chi_{19}(9,\cdot)\) 19.e 9 yes \(1\) \(1\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{19}(10,\cdot)\) 19.f 18 yes \(-1\) \(1\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{19}(11,\cdot)\) 19.c 3 yes \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\)
\(\chi_{19}(12,\cdot)\) 19.d 6 yes \(-1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(1\)
\(\chi_{19}(13,\cdot)\) 19.f 18 yes \(-1\) \(1\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{19}(14,\cdot)\) 19.f 18 yes \(-1\) \(1\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{19}(15,\cdot)\) 19.f 18 yes \(-1\) \(1\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{19}(16,\cdot)\) 19.e 9 yes \(1\) \(1\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{19}(17,\cdot)\) 19.e 9 yes \(1\) \(1\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{19}(18,\cdot)\) 19.b 2 yes \(-1\) \(1\) \(-1\) \(-1\) \(1\) \(1\) \(1\) \(1\) \(-1\) \(1\) \(-1\) \(1\)