Properties

Label 475.217
Modulus $475$
Conductor $475$
Order $60$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([39,10]))
 
pari: [g,chi] = znchar(Mod(217,475))
 

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

Galois orbit 475.be

\(\chi_{475}(8,\cdot)\) \(\chi_{475}(12,\cdot)\) \(\chi_{475}(27,\cdot)\) \(\chi_{475}(88,\cdot)\) \(\chi_{475}(103,\cdot)\) \(\chi_{475}(122,\cdot)\) \(\chi_{475}(183,\cdot)\) \(\chi_{475}(198,\cdot)\) \(\chi_{475}(202,\cdot)\) \(\chi_{475}(217,\cdot)\) \(\chi_{475}(278,\cdot)\) \(\chi_{475}(297,\cdot)\) \(\chi_{475}(312,\cdot)\) \(\chi_{475}(373,\cdot)\) \(\chi_{475}(388,\cdot)\) \(\chi_{475}(392,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ 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

\((77,401)\) → \((e\left(\frac{13}{20}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 475 }(217, a) \) \(1\)\(1\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{43}{60}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{8}{15}\right)\)\(i\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{11}{60}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 475 }(217,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 475 }(217,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 475 }(217,·),\chi_{ 475 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 475 }(217,·)) \;\) at \(\; a,b = \) e.g. 1,2