Properties

Label 473.14
Modulus $473$
Conductor $473$
Order $105$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(473, base_ring=CyclotomicField(210))
 
M = H._module
 
chi = DirichletCharacter(H, M([168,100]))
 
pari: [g,chi] = znchar(Mod(14,473))
 

Basic properties

Modulus: \(473\)
Conductor: \(473\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(105\)
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 473.bc

\(\chi_{473}(9,\cdot)\) \(\chi_{473}(14,\cdot)\) \(\chi_{473}(15,\cdot)\) \(\chi_{473}(25,\cdot)\) \(\chi_{473}(31,\cdot)\) \(\chi_{473}(38,\cdot)\) \(\chi_{473}(53,\cdot)\) \(\chi_{473}(58,\cdot)\) \(\chi_{473}(60,\cdot)\) \(\chi_{473}(81,\cdot)\) \(\chi_{473}(103,\cdot)\) \(\chi_{473}(124,\cdot)\) \(\chi_{473}(126,\cdot)\) \(\chi_{473}(146,\cdot)\) \(\chi_{473}(152,\cdot)\) \(\chi_{473}(169,\cdot)\) \(\chi_{473}(181,\cdot)\) \(\chi_{473}(185,\cdot)\) \(\chi_{473}(196,\cdot)\) \(\chi_{473}(203,\cdot)\) \(\chi_{473}(212,\cdot)\) \(\chi_{473}(224,\cdot)\) \(\chi_{473}(225,\cdot)\) \(\chi_{473}(229,\cdot)\) \(\chi_{473}(240,\cdot)\) \(\chi_{473}(246,\cdot)\) \(\chi_{473}(267,\cdot)\) \(\chi_{473}(268,\cdot)\) \(\chi_{473}(273,\cdot)\) \(\chi_{473}(289,\cdot)\) ...

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: $\Q(\zeta_{105})$
Fixed field: Number field defined by a degree 105 polynomial (not computed)

Values on generators

\((431,89)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{10}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 473 }(14, a) \) \(1\)\(1\)\(e\left(\frac{23}{35}\right)\)\(e\left(\frac{92}{105}\right)\)\(e\left(\frac{11}{35}\right)\)\(e\left(\frac{11}{105}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{34}{35}\right)\)\(e\left(\frac{79}{105}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{4}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 473 }(14,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 473 }(14,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 473 }(14,·),\chi_{ 473 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 473 }(14,·)) \;\) at \(\; a,b = \) e.g. 1,2