Properties

Label 103.5
Modulus $103$
Conductor $103$
Order $102$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(103, base_ring=CyclotomicField(102))
 
M = H._module
 
chi = DirichletCharacter(H, M([1]))
 
pari: [g,chi] = znchar(Mod(5,103))
 

Basic properties

Modulus: \(103\)
Conductor: \(103\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(102\)
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 103.h

\(\chi_{103}(5,\cdot)\) \(\chi_{103}(6,\cdot)\) \(\chi_{103}(11,\cdot)\) \(\chi_{103}(12,\cdot)\) \(\chi_{103}(20,\cdot)\) \(\chi_{103}(21,\cdot)\) \(\chi_{103}(35,\cdot)\) \(\chi_{103}(40,\cdot)\) \(\chi_{103}(43,\cdot)\) \(\chi_{103}(44,\cdot)\) \(\chi_{103}(45,\cdot)\) \(\chi_{103}(48,\cdot)\) \(\chi_{103}(51,\cdot)\) \(\chi_{103}(53,\cdot)\) \(\chi_{103}(54,\cdot)\) \(\chi_{103}(62,\cdot)\) \(\chi_{103}(65,\cdot)\) \(\chi_{103}(67,\cdot)\) \(\chi_{103}(70,\cdot)\) \(\chi_{103}(71,\cdot)\) \(\chi_{103}(74,\cdot)\) \(\chi_{103}(75,\cdot)\) \(\chi_{103}(77,\cdot)\) \(\chi_{103}(78,\cdot)\) \(\chi_{103}(84,\cdot)\) \(\chi_{103}(85,\cdot)\) \(\chi_{103}(86,\cdot)\) \(\chi_{103}(87,\cdot)\) \(\chi_{103}(88,\cdot)\) \(\chi_{103}(96,\cdot)\) ...

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: $\Q(\zeta_{51})$
Fixed field: Number field defined by a degree 102 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{1}{102}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 103 }(5, a) \) \(-1\)\(1\)\(e\left(\frac{22}{51}\right)\)\(e\left(\frac{13}{34}\right)\)\(e\left(\frac{44}{51}\right)\)\(e\left(\frac{1}{102}\right)\)\(e\left(\frac{83}{102}\right)\)\(e\left(\frac{2}{51}\right)\)\(e\left(\frac{5}{17}\right)\)\(e\left(\frac{13}{17}\right)\)\(e\left(\frac{15}{34}\right)\)\(e\left(\frac{61}{102}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 103 }(5,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 103 }(5,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 103 }(5,·),\chi_{ 103 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 103 }(5,·)) \;\) at \(\; a,b = \) e.g. 1,2