Properties

Label 5520.1681
Modulus $5520$
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(5520, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,0,0,2]))
 
pari: [g,chi] = znchar(Mod(1681,5520))
 

Basic properties

Modulus: \(5520\)
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 \(\chi_{23}(2,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5520.dc

\(\chi_{5520}(721,\cdot)\) \(\chi_{5520}(961,\cdot)\) \(\chi_{5520}(1681,\cdot)\) \(\chi_{5520}(1921,\cdot)\) \(\chi_{5520}(2401,\cdot)\) \(\chi_{5520}(2881,\cdot)\) \(\chi_{5520}(3121,\cdot)\) \(\chi_{5520}(3361,\cdot)\) \(\chi_{5520}(3601,\cdot)\) \(\chi_{5520}(5041,\cdot)\)

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

\((4831,1381,1841,4417,1201)\) → \((1,1,1,1,e\left(\frac{1}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 5520 }(1681, a) \) \(1\)\(1\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{5}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5520 }(1681,a) \;\) at \(\;a = \) e.g. 2