Properties

Label 83.10
Modulus $83$
Conductor $83$
Order $41$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(83, base_ring=CyclotomicField(82))
 
M = H._module
 
chi = DirichletCharacter(H, M([28]))
 
pari: [g,chi] = znchar(Mod(10,83))
 

Basic properties

Modulus: \(83\)
Conductor: \(83\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(41\)
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 83.c

\(\chi_{83}(3,\cdot)\) \(\chi_{83}(4,\cdot)\) \(\chi_{83}(7,\cdot)\) \(\chi_{83}(9,\cdot)\) \(\chi_{83}(10,\cdot)\) \(\chi_{83}(11,\cdot)\) \(\chi_{83}(12,\cdot)\) \(\chi_{83}(16,\cdot)\) \(\chi_{83}(17,\cdot)\) \(\chi_{83}(21,\cdot)\) \(\chi_{83}(23,\cdot)\) \(\chi_{83}(25,\cdot)\) \(\chi_{83}(26,\cdot)\) \(\chi_{83}(27,\cdot)\) \(\chi_{83}(28,\cdot)\) \(\chi_{83}(29,\cdot)\) \(\chi_{83}(30,\cdot)\) \(\chi_{83}(31,\cdot)\) \(\chi_{83}(33,\cdot)\) \(\chi_{83}(36,\cdot)\) \(\chi_{83}(37,\cdot)\) \(\chi_{83}(38,\cdot)\) \(\chi_{83}(40,\cdot)\) \(\chi_{83}(41,\cdot)\) \(\chi_{83}(44,\cdot)\) \(\chi_{83}(48,\cdot)\) \(\chi_{83}(49,\cdot)\) \(\chi_{83}(51,\cdot)\) \(\chi_{83}(59,\cdot)\) \(\chi_{83}(61,\cdot)\) ...

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: $\Q(\zeta_{41})$
Fixed field: Number field defined by a degree 41 polynomial

Values on generators

\(2\) → \(e\left(\frac{14}{41}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 83 }(10, a) \) \(1\)\(1\)\(e\left(\frac{14}{41}\right)\)\(e\left(\frac{24}{41}\right)\)\(e\left(\frac{28}{41}\right)\)\(e\left(\frac{9}{41}\right)\)\(e\left(\frac{38}{41}\right)\)\(e\left(\frac{30}{41}\right)\)\(e\left(\frac{1}{41}\right)\)\(e\left(\frac{7}{41}\right)\)\(e\left(\frac{23}{41}\right)\)\(e\left(\frac{8}{41}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 83 }(10,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 83 }(10,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 83 }(10,·),\chi_{ 83 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 83 }(10,·)) \;\) at \(\; a,b = \) e.g. 1,2