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Properties

Label 51.i
Modulus $51$
Conductor $51$
Order $16$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(51, base_ring=CyclotomicField(16)) M = H._module chi = DirichletCharacter(H, M([8,5])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(5,51)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(51\)
Conductor: \(51\)
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: yes
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
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: \(\Q(\zeta_{51})^+\)

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(4\) \(5\) \(7\) \(8\) \(10\) \(11\) \(13\) \(14\) \(16\)
\(\chi_{51}(5,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{8}\right)\) \(-i\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{11}{16}\right)\) \(i\) \(e\left(\frac{5}{16}\right)\) \(-1\)
\(\chi_{51}(11,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{8}\right)\) \(i\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{9}{16}\right)\) \(-i\) \(e\left(\frac{7}{16}\right)\) \(-1\)
\(\chi_{51}(14,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{8}\right)\) \(-i\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{7}{16}\right)\) \(i\) \(e\left(\frac{9}{16}\right)\) \(-1\)
\(\chi_{51}(20,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{8}\right)\) \(-i\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{15}{16}\right)\) \(i\) \(e\left(\frac{1}{16}\right)\) \(-1\)
\(\chi_{51}(23,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{8}\right)\) \(i\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{1}{16}\right)\) \(-i\) \(e\left(\frac{15}{16}\right)\) \(-1\)
\(\chi_{51}(29,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{8}\right)\) \(-i\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{3}{16}\right)\) \(i\) \(e\left(\frac{13}{16}\right)\) \(-1\)
\(\chi_{51}(41,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{8}\right)\) \(i\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{5}{16}\right)\) \(-i\) \(e\left(\frac{11}{16}\right)\) \(-1\)
\(\chi_{51}(44,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{8}\right)\) \(i\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{13}{16}\right)\) \(-i\) \(e\left(\frac{3}{16}\right)\) \(-1\)