Properties

Label 473.21
Modulus $473$
Conductor $473$
Order $14$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(473, base_ring=CyclotomicField(14))
 
M = H._module
 
chi = DirichletCharacter(H, M([7,12]))
 
pari: [g,chi] = znchar(Mod(21,473))
 

Basic properties

Modulus: \(473\)
Conductor: \(473\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(14\)
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 473.o

\(\chi_{473}(21,\cdot)\) \(\chi_{473}(54,\cdot)\) \(\chi_{473}(164,\cdot)\) \(\chi_{473}(219,\cdot)\) \(\chi_{473}(274,\cdot)\) \(\chi_{473}(428,\cdot)\)

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: \(\Q(\zeta_{7})\)
Fixed field: Number field defined by a degree 14 polynomial

Values on generators

\((431,89)\) → \((-1,e\left(\frac{6}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 473 }(21, a) \) \(-1\)\(1\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(-1\)\(-1\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{1}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 473 }(21,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 473 }(21,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 473 }(21,·),\chi_{ 473 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 473 }(21,·)) \;\) at \(\; a,b = \) e.g. 1,2