Properties

Label 83.58
Modulus $83$
Conductor $83$
Order $82$
Real no
Primitive yes
Minimal yes
Parity odd

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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([13]))
 
pari: [g,chi] = znchar(Mod(58,83))
 

Basic properties

Modulus: \(83\)
Conductor: \(83\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(82\)
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 83.d

\(\chi_{83}(2,\cdot)\) \(\chi_{83}(5,\cdot)\) \(\chi_{83}(6,\cdot)\) \(\chi_{83}(8,\cdot)\) \(\chi_{83}(13,\cdot)\) \(\chi_{83}(14,\cdot)\) \(\chi_{83}(15,\cdot)\) \(\chi_{83}(18,\cdot)\) \(\chi_{83}(19,\cdot)\) \(\chi_{83}(20,\cdot)\) \(\chi_{83}(22,\cdot)\) \(\chi_{83}(24,\cdot)\) \(\chi_{83}(32,\cdot)\) \(\chi_{83}(34,\cdot)\) \(\chi_{83}(35,\cdot)\) \(\chi_{83}(39,\cdot)\) \(\chi_{83}(42,\cdot)\) \(\chi_{83}(43,\cdot)\) \(\chi_{83}(45,\cdot)\) \(\chi_{83}(46,\cdot)\) \(\chi_{83}(47,\cdot)\) \(\chi_{83}(50,\cdot)\) \(\chi_{83}(52,\cdot)\) \(\chi_{83}(53,\cdot)\) \(\chi_{83}(54,\cdot)\) \(\chi_{83}(55,\cdot)\) \(\chi_{83}(56,\cdot)\) \(\chi_{83}(57,\cdot)\) \(\chi_{83}(58,\cdot)\) \(\chi_{83}(60,\cdot)\) ...

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 82 polynomial

Values on generators

\(2\) → \(e\left(\frac{13}{82}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 83 }(58, a) \) \(-1\)\(1\)\(e\left(\frac{13}{82}\right)\)\(e\left(\frac{17}{41}\right)\)\(e\left(\frac{13}{41}\right)\)\(e\left(\frac{23}{82}\right)\)\(e\left(\frac{47}{82}\right)\)\(e\left(\frac{11}{41}\right)\)\(e\left(\frac{39}{82}\right)\)\(e\left(\frac{34}{41}\right)\)\(e\left(\frac{18}{41}\right)\)\(e\left(\frac{33}{41}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 83 }(58,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 83 }(58,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 83 }(58,·),\chi_{ 83 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 83 }(58,·)) \;\) at \(\; a,b = \) e.g. 1,2