Properties

Label 460.419
Modulus $460$
Conductor $460$
Order $22$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(460, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([11,11,1]))
 
pari: [g,chi] = znchar(Mod(419,460))
 

Basic properties

Modulus: \(460\)
Conductor: \(460\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 460.o

\(\chi_{460}(19,\cdot)\) \(\chi_{460}(79,\cdot)\) \(\chi_{460}(99,\cdot)\) \(\chi_{460}(159,\cdot)\) \(\chi_{460}(199,\cdot)\) \(\chi_{460}(319,\cdot)\) \(\chi_{460}(339,\cdot)\) \(\chi_{460}(359,\cdot)\) \(\chi_{460}(379,\cdot)\) \(\chi_{460}(419,\cdot)\)

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: 22.22.8083780427918435509708715954790400000000000.1

Values on generators

\((231,277,281)\) → \((-1,-1,e\left(\frac{1}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\( \chi_{ 460 }(419, a) \) \(1\)\(1\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{9}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 460 }(419,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 460 }(419,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 460 }(419,·),\chi_{ 460 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 460 }(419,·)) \;\) at \(\; a,b = \) e.g. 1,2