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Properties

Label 1210.l
Modulus $1210$
Conductor $55$
Order $20$
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(1210, base_ring=CyclotomicField(20)) M = H._module chi = DirichletCharacter(H, M([15,2])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(233,1210)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(1210\)
Conductor: \(55\)
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: no, induced from 55.l
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: \(\Q(\zeta_{20})\)
Fixed field: \(\Q(\zeta_{55})^+\)

Characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(7\) \(9\) \(13\) \(17\) \(19\) \(21\) \(23\) \(27\) \(29\)
\(\chi_{1210}(233,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{4}{5}\right)\) \(-1\) \(i\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{1}{5}\right)\)
\(\chi_{1210}(403,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{3}{5}\right)\) \(-1\) \(i\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{2}{5}\right)\)
\(\chi_{1210}(457,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{1}{5}\right)\) \(-1\) \(-i\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{4}{5}\right)\)
\(\chi_{1210}(717,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{4}{5}\right)\) \(-1\) \(-i\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{1}{5}\right)\)
\(\chi_{1210}(723,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{2}{5}\right)\) \(-1\) \(i\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{3}{5}\right)\)
\(\chi_{1210}(887,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{3}{5}\right)\) \(-1\) \(-i\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{2}{5}\right)\)
\(\chi_{1210}(1183,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{1}{5}\right)\) \(-1\) \(i\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{4}{5}\right)\)
\(\chi_{1210}(1207,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{2}{5}\right)\) \(-1\) \(-i\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{3}{5}\right)\)