Properties

Label 460.17
Modulus $460$
Conductor $115$
Order $44$
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(44))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,11,14]))
 
pari: [g,chi] = znchar(Mod(17,460))
 

Basic properties

Modulus: \(460\)
Conductor: \(115\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{115}(17,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 460.x

\(\chi_{460}(17,\cdot)\) \(\chi_{460}(33,\cdot)\) \(\chi_{460}(37,\cdot)\) \(\chi_{460}(53,\cdot)\) \(\chi_{460}(57,\cdot)\) \(\chi_{460}(97,\cdot)\) \(\chi_{460}(113,\cdot)\) \(\chi_{460}(153,\cdot)\) \(\chi_{460}(157,\cdot)\) \(\chi_{460}(217,\cdot)\) \(\chi_{460}(237,\cdot)\) \(\chi_{460}(273,\cdot)\) \(\chi_{460}(293,\cdot)\) \(\chi_{460}(297,\cdot)\) \(\chi_{460}(313,\cdot)\) \(\chi_{460}(333,\cdot)\) \(\chi_{460}(337,\cdot)\) \(\chi_{460}(373,\cdot)\) \(\chi_{460}(433,\cdot)\) \(\chi_{460}(457,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: \(\Q(\zeta_{115})^+\)

Values on generators

\((231,277,281)\) → \((1,i,e\left(\frac{7}{22}\right))\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\(1\)\(1\)\(e\left(\frac{37}{44}\right)\)\(e\left(\frac{13}{44}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{9}{44}\right)\)\(e\left(\frac{21}{44}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{23}{44}\right)\)\(e\left(\frac{5}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 460 }(17,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 460 }(17,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 460 }(17,·),\chi_{ 460 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 460 }(17,·)) \;\) at \(\; a,b = \) e.g. 1,2