Properties

Label 460.51
Modulus $460$
Conductor $92$
Order $22$
Real no
Primitive no
Minimal yes
Parity even

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([11,0,1]))
 
pari: [g,chi] = znchar(Mod(51,460))
 

Basic properties

Modulus: \(460\)
Conductor: \(92\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{92}(51,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 460.q

\(\chi_{460}(11,\cdot)\) \(\chi_{460}(51,\cdot)\) \(\chi_{460}(111,\cdot)\) \(\chi_{460}(171,\cdot)\) \(\chi_{460}(191,\cdot)\) \(\chi_{460}(251,\cdot)\) \(\chi_{460}(291,\cdot)\) \(\chi_{460}(411,\cdot)\) \(\chi_{460}(431,\cdot)\) \(\chi_{460}(451,\cdot)\)

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: \(\Q(\zeta_{92})^+\)

Values on generators

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

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\(1\)\(1\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{9}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 460 }(51,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 460 }(51,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 460 }(51,·),\chi_{ 460 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 460 }(51,·)) \;\) at \(\; a,b = \) e.g. 1,2