Properties

Label 4600.51
Modulus $4600$
Conductor $184$
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(4600, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([11,11,0,1]))
 
pari: [g,chi] = znchar(Mod(51,4600))
 

Basic properties

Modulus: \(4600\)
Conductor: \(184\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{184}(51,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4600.bz

\(\chi_{4600}(51,\cdot)\) \(\chi_{4600}(251,\cdot)\) \(\chi_{4600}(451,\cdot)\) \(\chi_{4600}(651,\cdot)\) \(\chi_{4600}(1851,\cdot)\) \(\chi_{4600}(2251,\cdot)\) \(\chi_{4600}(3051,\cdot)\) \(\chi_{4600}(3651,\cdot)\) \(\chi_{4600}(3851,\cdot)\) \(\chi_{4600}(4251,\cdot)\)

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: 22.22.339058325839400057321133061640411938816.1

Values on generators

\((1151,2301,2577,1201)\) → \((-1,-1,1,e\left(\frac{1}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\( \chi_{ 4600 }(51, a) \) \(1\)\(1\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{7}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4600 }(51,a) \;\) at \(\;a = \) e.g. 2