Show commands: Magma / Pari/GP / SageMath

Copy content comment:Define the Dirichlet group
 
Copy content sage:G = DirichletGroup(30)
 
Copy content gp:g = idealstar(,30,2)
 
Copy content magma:G := FullDirichletGroup(30);
 

Character group

Order = 8
Copy content comment:Order
 
Copy content sage:G.order()
 
Copy content gp:g.no
 
Copy content magma:Order(G);
 
Structure = \(C_{2}\times C_{4}\)
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 = $\chi_{30}(11,\cdot)$, $\chi_{30}(7,\cdot)$
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 \(-1\) \(1\) \(7\) \(11\) \(13\) \(17\) \(19\) \(23\)
\(\chi_{30}(1,\cdot)\) 30.a 1 no \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{30}(7,\cdot)\) 30.f 4 no \(-1\) \(1\) \(i\) \(1\) \(-i\) \(i\) \(-1\) \(-i\)
\(\chi_{30}(11,\cdot)\) 30.d 2 no \(-1\) \(1\) \(1\) \(-1\) \(1\) \(-1\) \(1\) \(-1\)
\(\chi_{30}(13,\cdot)\) 30.f 4 no \(-1\) \(1\) \(-i\) \(1\) \(i\) \(-i\) \(-1\) \(i\)
\(\chi_{30}(17,\cdot)\) 30.e 4 no \(1\) \(1\) \(i\) \(-1\) \(-i\) \(-i\) \(-1\) \(i\)
\(\chi_{30}(19,\cdot)\) 30.c 2 no \(1\) \(1\) \(-1\) \(1\) \(-1\) \(-1\) \(1\) \(-1\)
\(\chi_{30}(23,\cdot)\) 30.e 4 no \(1\) \(1\) \(-i\) \(-1\) \(i\) \(i\) \(-1\) \(-i\)
\(\chi_{30}(29,\cdot)\) 30.b 2 no \(-1\) \(1\) \(-1\) \(-1\) \(-1\) \(1\) \(1\) \(1\)