Properties

Label 460.187
Modulus $460$
Conductor $460$
Order $44$
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(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([22,11,32]))
 
pari: [g,chi] = znchar(Mod(187,460))
 

Basic properties

Modulus: \(460\)
Conductor: \(460\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
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.w

\(\chi_{460}(3,\cdot)\) \(\chi_{460}(27,\cdot)\) \(\chi_{460}(87,\cdot)\) \(\chi_{460}(123,\cdot)\) \(\chi_{460}(127,\cdot)\) \(\chi_{460}(147,\cdot)\) \(\chi_{460}(163,\cdot)\) \(\chi_{460}(167,\cdot)\) \(\chi_{460}(187,\cdot)\) \(\chi_{460}(223,\cdot)\) \(\chi_{460}(243,\cdot)\) \(\chi_{460}(303,\cdot)\) \(\chi_{460}(307,\cdot)\) \(\chi_{460}(347,\cdot)\) \(\chi_{460}(363,\cdot)\) \(\chi_{460}(403,\cdot)\) \(\chi_{460}(407,\cdot)\) \(\chi_{460}(423,\cdot)\) \(\chi_{460}(427,\cdot)\) \(\chi_{460}(443,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: Number field defined by a degree 44 polynomial

Values on generators

\((231,277,281)\) → \((-1,i,e\left(\frac{8}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\( \chi_{ 460 }(187, a) \) \(1\)\(1\)\(e\left(\frac{39}{44}\right)\)\(e\left(\frac{25}{44}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{41}{44}\right)\)\(e\left(\frac{15}{44}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{29}{44}\right)\)\(e\left(\frac{13}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 460 }(187,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 460 }(187,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 460 }(187,·),\chi_{ 460 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 460 }(187,·)) \;\) at \(\; a,b = \) e.g. 1,2