Properties

Label 4830.659
Modulus $4830$
Conductor $345$
Order $22$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4830, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([11,11,0,17]))
 
pari: [g,chi] = znchar(Mod(659,4830))
 

Basic properties

Modulus: \(4830\)
Conductor: \(345\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{345}(314,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4830.cd

\(\chi_{4830}(659,\cdot)\) \(\chi_{4830}(1079,\cdot)\) \(\chi_{4830}(1709,\cdot)\) \(\chi_{4830}(1919,\cdot)\) \(\chi_{4830}(2549,\cdot)\) \(\chi_{4830}(3179,\cdot)\) \(\chi_{4830}(3599,\cdot)\) \(\chi_{4830}(3809,\cdot)\) \(\chi_{4830}(4019,\cdot)\) \(\chi_{4830}(4229,\cdot)\)

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

\((3221,967,2761,1891)\) → \((-1,-1,1,e\left(\frac{17}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)\(47\)
\( \chi_{ 4830 }(659, a) \) \(1\)\(1\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{4}{11}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4830 }(659,a) \;\) at \(\;a = \) e.g. 2