Properties

Label 5290.501
Modulus $5290$
Conductor $23$
Order $11$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5290, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,12]))
 
pari: [g,chi] = znchar(Mod(501,5290))
 

Basic properties

Modulus: \(5290\)
Conductor: \(23\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(11\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{23}(18,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5290.g

\(\chi_{5290}(501,\cdot)\) \(\chi_{5290}(1921,\cdot)\) \(\chi_{5290}(2371,\cdot)\) \(\chi_{5290}(2911,\cdot)\) \(\chi_{5290}(3111,\cdot)\) \(\chi_{5290}(3351,\cdot)\) \(\chi_{5290}(3661,\cdot)\) \(\chi_{5290}(3821,\cdot)\) \(\chi_{5290}(4631,\cdot)\) \(\chi_{5290}(4931,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{11})\)
Fixed field: \(\Q(\zeta_{23})^+\)

Values on generators

\((2117,2121)\) → \((1,e\left(\frac{6}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\( \chi_{ 5290 }(501, a) \) \(1\)\(1\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{9}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5290 }(501,a) \;\) at \(\;a = \) e.g. 2