Properties

Label 103.98
Modulus $103$
Conductor $103$
Order $51$
Real no
Primitive yes
Minimal yes
Parity even

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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([52]))
 
pari: [g,chi] = znchar(Mod(98,103))
 

Basic properties

Modulus: \(103\)
Conductor: \(103\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(51\)
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 103.g

\(\chi_{103}(2,\cdot)\) \(\chi_{103}(4,\cdot)\) \(\chi_{103}(7,\cdot)\) \(\chi_{103}(15,\cdot)\) \(\chi_{103}(16,\cdot)\) \(\chi_{103}(17,\cdot)\) \(\chi_{103}(18,\cdot)\) \(\chi_{103}(19,\cdot)\) \(\chi_{103}(25,\cdot)\) \(\chi_{103}(26,\cdot)\) \(\chi_{103}(28,\cdot)\) \(\chi_{103}(29,\cdot)\) \(\chi_{103}(32,\cdot)\) \(\chi_{103}(33,\cdot)\) \(\chi_{103}(36,\cdot)\) \(\chi_{103}(38,\cdot)\) \(\chi_{103}(41,\cdot)\) \(\chi_{103}(49,\cdot)\) \(\chi_{103}(50,\cdot)\) \(\chi_{103}(52,\cdot)\) \(\chi_{103}(55,\cdot)\) \(\chi_{103}(58,\cdot)\) \(\chi_{103}(59,\cdot)\) \(\chi_{103}(60,\cdot)\) \(\chi_{103}(63,\cdot)\) \(\chi_{103}(68,\cdot)\) \(\chi_{103}(82,\cdot)\) \(\chi_{103}(83,\cdot)\) \(\chi_{103}(91,\cdot)\) \(\chi_{103}(92,\cdot)\) ...

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 51 polynomial

Values on generators

\(5\) → \(e\left(\frac{26}{51}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 103 }(98, a) \) \(1\)\(1\)\(e\left(\frac{22}{51}\right)\)\(e\left(\frac{15}{17}\right)\)\(e\left(\frac{44}{51}\right)\)\(e\left(\frac{26}{51}\right)\)\(e\left(\frac{16}{51}\right)\)\(e\left(\frac{2}{51}\right)\)\(e\left(\frac{5}{17}\right)\)\(e\left(\frac{13}{17}\right)\)\(e\left(\frac{16}{17}\right)\)\(e\left(\frac{5}{51}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 103 }(98,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 103 }(98,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 103 }(98,·),\chi_{ 103 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 103 }(98,·)) \;\) at \(\; a,b = \) e.g. 1,2