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Properties

Label 656.bl
Modulus $656$
Conductor $656$
Order $20$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(656, base_ring=CyclotomicField(20)) M = H._module chi = DirichletCharacter(H, M([0,5,9])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(197,656)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(656\)
Conductor: \(656\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(20\)
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: \(\Q(\zeta_{20})\)
Fixed field: 20.20.158322645890088916737377685620382389712556392448.2

Characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(5\) \(7\) \(9\) \(11\) \(13\) \(15\) \(17\) \(19\) \(21\)
\(\chi_{656}(197,\cdot)\) \(1\) \(1\) \(-1\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{1}{20}\right)\) \(1\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{11}{20}\right)\)
\(\chi_{656}(285,\cdot)\) \(1\) \(1\) \(-1\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{7}{20}\right)\) \(1\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{17}{20}\right)\)
\(\chi_{656}(333,\cdot)\) \(1\) \(1\) \(-1\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{19}{20}\right)\) \(1\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{9}{20}\right)\)
\(\chi_{656}(349,\cdot)\) \(1\) \(1\) \(-1\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{3}{20}\right)\) \(1\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{13}{20}\right)\)
\(\chi_{656}(389,\cdot)\) \(1\) \(1\) \(-1\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{13}{20}\right)\) \(1\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{3}{20}\right)\)
\(\chi_{656}(405,\cdot)\) \(1\) \(1\) \(-1\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{9}{20}\right)\) \(1\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{19}{20}\right)\)
\(\chi_{656}(453,\cdot)\) \(1\) \(1\) \(-1\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{17}{20}\right)\) \(1\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{7}{20}\right)\)
\(\chi_{656}(541,\cdot)\) \(1\) \(1\) \(-1\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{11}{20}\right)\) \(1\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{1}{20}\right)\)