Properties

Label 800.571
Modulus $800$
Conductor $800$
Order $40$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, base_ring=CyclotomicField(40))
 
M = H._module
 
chi = DirichletCharacter(H, M([20,5,24]))
 
pari: [g,chi] = znchar(Mod(571,800))
 

Basic properties

Modulus: \(800\)
Conductor: \(800\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(40\)
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 800.cb

\(\chi_{800}(11,\cdot)\) \(\chi_{800}(91,\cdot)\) \(\chi_{800}(131,\cdot)\) \(\chi_{800}(171,\cdot)\) \(\chi_{800}(211,\cdot)\) \(\chi_{800}(291,\cdot)\) \(\chi_{800}(331,\cdot)\) \(\chi_{800}(371,\cdot)\) \(\chi_{800}(411,\cdot)\) \(\chi_{800}(491,\cdot)\) \(\chi_{800}(531,\cdot)\) \(\chi_{800}(571,\cdot)\) \(\chi_{800}(611,\cdot)\) \(\chi_{800}(691,\cdot)\) \(\chi_{800}(731,\cdot)\) \(\chi_{800}(771,\cdot)\)

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: \(\Q(\zeta_{40})\)
Fixed field: Number field defined by a degree 40 polynomial

Values on generators

\((351,101,577)\) → \((-1,e\left(\frac{1}{8}\right),e\left(\frac{3}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 800 }(571, a) \) \(-1\)\(1\)\(e\left(\frac{3}{40}\right)\)\(-i\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{29}{40}\right)\)\(e\left(\frac{11}{40}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{7}{40}\right)\)\(e\left(\frac{33}{40}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{9}{40}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 800 }(571,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 800 }(571,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 800 }(571,·),\chi_{ 800 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 800 }(571,·)) \;\) at \(\; a,b = \) e.g. 1,2