Properties

Label 5520.41
Modulus $5520$
Conductor $552$
Order $22$
Real no
Primitive no
Minimal no
Parity odd

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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([0,11,11,0,12]))
 
pari: [g,chi] = znchar(Mod(41,5520))
 

Basic properties

Modulus: \(5520\)
Conductor: \(552\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{552}(317,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5520.di

\(\chi_{5520}(41,\cdot)\) \(\chi_{5520}(761,\cdot)\) \(\chi_{5520}(1001,\cdot)\) \(\chi_{5520}(1481,\cdot)\) \(\chi_{5520}(1961,\cdot)\) \(\chi_{5520}(2201,\cdot)\) \(\chi_{5520}(2441,\cdot)\) \(\chi_{5520}(2681,\cdot)\) \(\chi_{5520}(4121,\cdot)\) \(\chi_{5520}(5321,\cdot)\)

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: Number field defined by a degree 22 polynomial

Values on generators

\((4831,1381,1841,4417,1201)\) → \((1,-1,-1,1,e\left(\frac{6}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 5520 }(41, a) \) \(-1\)\(1\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{5}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5520 }(41,a) \;\) at \(\;a = \) e.g. 2