Show commands: Magma / Pari/GP / SageMath

Copy content comment:Define the Dirichlet group
 
Copy content sage:G = DirichletGroup(9)
 
Copy content gp:g = idealstar(,9,2)
 
Copy content magma:G := FullDirichletGroup(9);
 

Character group

Order = 6
Copy content comment:Order
 
Copy content sage:G.order()
 
Copy content gp:g.no
 
Copy content magma:Order(G);
 
Structure = \(C_{6}\)
Copy content comment:Group structure
 
Copy content sage:sorted(g.order() for g in G.gens())
 
Copy content gp:g.cyc
 
Copy content magma:PrimaryInvariants(AbelianGroup(G));
 
Generators = $\chi_{9}(2,\cdot)$
Copy content comment:Generators
 
Copy content sage:G.gens()
 
Copy content gp:g.gen
 
Copy content magma:Generators(G);
 

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 \(-1\) \(1\) \(2\) \(4\) \(5\) \(7\)
\(\chi_{9}(1,\cdot)\) 9.a 1 no \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{9}(2,\cdot)\) 9.d 6 yes \(-1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{9}(4,\cdot)\) 9.c 3 yes \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{9}(5,\cdot)\) 9.d 6 yes \(-1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{9}(7,\cdot)\) 9.c 3 yes \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{9}(8,\cdot)\) 9.b 2 no \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\)