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Properties

Label 408.bi
Modulus $408$
Conductor $68$
Order $16$
Real no
Primitive no
Minimal no
Parity even

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(408, base_ring=CyclotomicField(16)) M = H._module chi = DirichletCharacter(H, M([8,0,0,11])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(7,408)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(408\)
Conductor: \(68\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(16\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from 68.i
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{16})\)
Fixed field: \(\Q(\zeta_{68})^+\)

Characters in Galois orbit

Character \(-1\) \(1\) \(5\) \(7\) \(11\) \(13\) \(19\) \(23\) \(25\) \(29\) \(31\) \(35\)
\(\chi_{408}(7,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{5}{16}\right)\) \(-i\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{11}{16}\right)\) \(-1\)
\(\chi_{408}(31,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{7}{16}\right)\) \(i\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{9}{16}\right)\) \(-1\)
\(\chi_{408}(79,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{9}{16}\right)\) \(-i\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{7}{16}\right)\) \(-1\)
\(\chi_{408}(175,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{11}{16}\right)\) \(i\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{5}{16}\right)\) \(-1\)
\(\chi_{408}(199,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{3}{16}\right)\) \(i\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{13}{16}\right)\) \(-1\)
\(\chi_{408}(295,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{1}{16}\right)\) \(-i\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{15}{16}\right)\) \(-1\)
\(\chi_{408}(343,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{15}{16}\right)\) \(i\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{1}{16}\right)\) \(-1\)
\(\chi_{408}(367,\cdot)\) \(1\) \(1\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{13}{16}\right)\) \(-i\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{3}{16}\right)\) \(-1\)