Properties

Label 475.83
Modulus $475$
Conductor $475$
Order $60$
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(475, base_ring=CyclotomicField(60))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([9,20]))
 
pari: [g,chi] = znchar(Mod(83,475))
 

Basic properties

Modulus: \(475\)
Conductor: \(475\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
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 475.bd

\(\chi_{475}(83,\cdot)\) \(\chi_{475}(87,\cdot)\) \(\chi_{475}(102,\cdot)\) \(\chi_{475}(163,\cdot)\) \(\chi_{475}(178,\cdot)\) \(\chi_{475}(197,\cdot)\) \(\chi_{475}(258,\cdot)\) \(\chi_{475}(273,\cdot)\) \(\chi_{475}(277,\cdot)\) \(\chi_{475}(292,\cdot)\) \(\chi_{475}(353,\cdot)\) \(\chi_{475}(372,\cdot)\) \(\chi_{475}(387,\cdot)\) \(\chi_{475}(448,\cdot)\) \(\chi_{475}(463,\cdot)\) \(\chi_{475}(467,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((77,401)\) → \((e\left(\frac{3}{20}\right),e\left(\frac{1}{3}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\(-1\)\(1\)\(e\left(\frac{29}{60}\right)\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{13}{15}\right)\)\(-i\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{31}{60}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 475 }(83,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 475 }(83,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 475 }(83,·),\chi_{ 475 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 475 }(83,·)) \;\) at \(\; a,b = \) e.g. 1,2