Properties

Label 460.71
Modulus $460$
Conductor $92$
Order $22$
Real no
Primitive no
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([11,0,2]))
 
pari: [g,chi] = znchar(Mod(71,460))
 

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 \(\chi_{92}(71,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 460.t

\(\chi_{460}(31,\cdot)\) \(\chi_{460}(71,\cdot)\) \(\chi_{460}(131,\cdot)\) \(\chi_{460}(151,\cdot)\) \(\chi_{460}(211,\cdot)\) \(\chi_{460}(271,\cdot)\) \(\chi_{460}(311,\cdot)\) \(\chi_{460}(331,\cdot)\) \(\chi_{460}(351,\cdot)\) \(\chi_{460}(371,\cdot)\)

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: 22.0.7198079267989980836471065337135104.1

Values on generators

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

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\(-1\)\(1\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{7}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 460 }(71,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 460 }(71,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 460 }(71,·),\chi_{ 460 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 460 }(71,·)) \;\) at \(\; a,b = \) e.g. 1,2