Properties

Label 460.259
Modulus $460$
Conductor $460$
Order $22$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(460, base_ring=CyclotomicField(22))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([11,11,18]))
 
pari: [g,chi] = znchar(Mod(259,460))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 460.n

\(\chi_{460}(39,\cdot)\) \(\chi_{460}(59,\cdot)\) \(\chi_{460}(119,\cdot)\) \(\chi_{460}(179,\cdot)\) \(\chi_{460}(219,\cdot)\) \(\chi_{460}(239,\cdot)\) \(\chi_{460}(259,\cdot)\) \(\chi_{460}(279,\cdot)\) \(\chi_{460}(399,\cdot)\) \(\chi_{460}(439,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{11})\)
Fixed field: 22.0.351468714257323283030813737164800000000000.1

Values on generators

\((231,277,281)\) → \((-1,-1,e\left(\frac{9}{11}\right))\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\(-1\)\(1\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{8}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 460 }(259,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 460 }(259,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 460 }(259,·),\chi_{ 460 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 460 }(259,·)) \;\) at \(\; a,b = \) e.g. 1,2