Properties

Label 3864.41
Modulus $3864$
Conductor $483$
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(3864, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,11,11,12]))
 
pari: [g,chi] = znchar(Mod(41,3864))
 

Basic properties

Modulus: \(3864\)
Conductor: \(483\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{483}(41,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3864.cz

\(\chi_{3864}(41,\cdot)\) \(\chi_{3864}(209,\cdot)\) \(\chi_{3864}(377,\cdot)\) \(\chi_{3864}(545,\cdot)\) \(\chi_{3864}(1553,\cdot)\) \(\chi_{3864}(1889,\cdot)\) \(\chi_{3864}(2561,\cdot)\) \(\chi_{3864}(3065,\cdot)\) \(\chi_{3864}(3233,\cdot)\) \(\chi_{3864}(3569,\cdot)\)

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

\((967,1933,1289,2761,2857)\) → \((1,1,-1,-1,e\left(\frac{6}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 3864 }(41, a) \) \(1\)\(1\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{6}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3864 }(41,a) \;\) at \(\;a = \) e.g. 2