Properties

Modulus $32$
Structure \(C_{2}\times C_{8}\)
Order $16$

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Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(32)
 
pari: g = idealstar(,32,2)
 

Character group

sage: G.order()
 
pari: g.no
 
Order = 16
sage: H.invariants()
 
pari: g.cyc
 
Structure = \(C_{2}\times C_{8}\)
sage: H.gens()
 
pari: g.gen
 
Generators = $\chi_{32}(31,\cdot)$, $\chi_{32}(5,\cdot)$

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 \(-1\) \(1\) \(3\) \(5\) \(7\) \(9\) \(11\) \(13\) \(15\) \(17\) \(19\) \(21\)
\(\chi_{32}(1,\cdot)\) 32.a 1 no \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{32}(3,\cdot)\) 32.h 8 yes \(-1\) \(1\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(i\) \(i\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(1\) \(-1\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{8}\right)\)
\(\chi_{32}(5,\cdot)\) 32.g 8 yes \(1\) \(1\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(i\) \(-i\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(-1\) \(-1\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{5}{8}\right)\)
\(\chi_{32}(7,\cdot)\) 32.f 4 no \(-1\) \(1\) \(i\) \(i\) \(1\) \(-1\) \(-i\) \(-i\) \(-1\) \(1\) \(i\) \(i\)
\(\chi_{32}(9,\cdot)\) 32.e 4 no \(1\) \(1\) \(i\) \(-i\) \(-1\) \(-1\) \(-i\) \(i\) \(1\) \(1\) \(i\) \(-i\)
\(\chi_{32}(11,\cdot)\) 32.h 8 yes \(-1\) \(1\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(-i\) \(-i\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(1\) \(-1\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{1}{8}\right)\)
\(\chi_{32}(13,\cdot)\) 32.g 8 yes \(1\) \(1\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(-i\) \(i\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(-1\) \(-1\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{3}{8}\right)\)
\(\chi_{32}(15,\cdot)\) 32.d 2 no \(-1\) \(1\) \(1\) \(-1\) \(-1\) \(1\) \(1\) \(-1\) \(-1\) \(1\) \(1\) \(-1\)
\(\chi_{32}(17,\cdot)\) 32.b 2 no \(1\) \(1\) \(-1\) \(-1\) \(1\) \(1\) \(-1\) \(-1\) \(1\) \(1\) \(-1\) \(-1\)
\(\chi_{32}(19,\cdot)\) 32.h 8 yes \(-1\) \(1\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(i\) \(i\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(1\) \(-1\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{8}\right)\)
\(\chi_{32}(21,\cdot)\) 32.g 8 yes \(1\) \(1\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(i\) \(-i\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(-1\) \(-1\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{1}{8}\right)\)
\(\chi_{32}(23,\cdot)\) 32.f 4 no \(-1\) \(1\) \(-i\) \(-i\) \(1\) \(-1\) \(i\) \(i\) \(-1\) \(1\) \(-i\) \(-i\)
\(\chi_{32}(25,\cdot)\) 32.e 4 no \(1\) \(1\) \(-i\) \(i\) \(-1\) \(-1\) \(i\) \(-i\) \(1\) \(1\) \(-i\) \(i\)
\(\chi_{32}(27,\cdot)\) 32.h 8 yes \(-1\) \(1\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(-i\) \(-i\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(1\) \(-1\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{8}\right)\)
\(\chi_{32}(29,\cdot)\) 32.g 8 yes \(1\) \(1\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(-i\) \(i\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(-1\) \(-1\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{7}{8}\right)\)
\(\chi_{32}(31,\cdot)\) 32.c 2 no \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\)