Properties

Label 461.400
Modulus $461$
Conductor $461$
Order $23$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(461, base_ring=CyclotomicField(46))
 
M = H._module
 
chi = DirichletCharacter(H, M([10]))
 
pari: [g,chi] = znchar(Mod(400,461))
 

Basic properties

Modulus: \(461\)
Conductor: \(461\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(23\)
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 461.g

\(\chi_{461}(14,\cdot)\) \(\chi_{461}(23,\cdot)\) \(\chi_{461}(30,\cdot)\) \(\chi_{461}(33,\cdot)\) \(\chi_{461}(68,\cdot)\) \(\chi_{461}(153,\cdot)\) \(\chi_{461}(167,\cdot)\) \(\chi_{461}(181,\cdot)\) \(\chi_{461}(196,\cdot)\) \(\chi_{461}(229,\cdot)\) \(\chi_{461}(262,\cdot)\) \(\chi_{461}(292,\cdot)\) \(\chi_{461}(298,\cdot)\) \(\chi_{461}(322,\cdot)\) \(\chi_{461}(348,\cdot)\) \(\chi_{461}(359,\cdot)\) \(\chi_{461}(400,\cdot)\) \(\chi_{461}(416,\cdot)\) \(\chi_{461}(420,\cdot)\) \(\chi_{461}(439,\cdot)\) \(\chi_{461}(440,\cdot)\) \(\chi_{461}(441,\cdot)\)

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: \(\Q(\zeta_{23})\)
Fixed field: Number field defined by a degree 23 polynomial

Values on generators

\(2\) → \(e\left(\frac{5}{23}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 461 }(400, a) \) \(1\)\(1\)\(e\left(\frac{5}{23}\right)\)\(e\left(\frac{3}{23}\right)\)\(e\left(\frac{10}{23}\right)\)\(e\left(\frac{10}{23}\right)\)\(e\left(\frac{8}{23}\right)\)\(e\left(\frac{7}{23}\right)\)\(e\left(\frac{15}{23}\right)\)\(e\left(\frac{6}{23}\right)\)\(e\left(\frac{15}{23}\right)\)\(e\left(\frac{8}{23}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 461 }(400,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 461 }(400,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 461 }(400,·),\chi_{ 461 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 461 }(400,·)) \;\) at \(\; a,b = \) e.g. 1,2