Properties

Label 419.107
Modulus $419$
Conductor $419$
Order $19$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(419, base_ring=CyclotomicField(38))
 
M = H._module
 
chi = DirichletCharacter(H, M([18]))
 
pari: [g,chi] = znchar(Mod(107,419))
 

Basic properties

Modulus: \(419\)
Conductor: \(419\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(19\)
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 419.d

\(\chi_{419}(7,\cdot)\) \(\chi_{419}(47,\cdot)\) \(\chi_{419}(49,\cdot)\) \(\chi_{419}(60,\cdot)\) \(\chi_{419}(107,\cdot)\) \(\chi_{419}(114,\cdot)\) \(\chi_{419}(135,\cdot)\) \(\chi_{419}(136,\cdot)\) \(\chi_{419}(139,\cdot)\) \(\chi_{419}(199,\cdot)\) \(\chi_{419}(208,\cdot)\) \(\chi_{419}(215,\cdot)\) \(\chi_{419}(248,\cdot)\) \(\chi_{419}(306,\cdot)\) \(\chi_{419}(329,\cdot)\) \(\chi_{419}(330,\cdot)\) \(\chi_{419}(343,\cdot)\) \(\chi_{419}(379,\cdot)\)

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: \(\Q(\zeta_{19})\)
Fixed field: Number field defined by a degree 19 polynomial

Values on generators

\(2\) → \(e\left(\frac{9}{19}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 419 }(107, a) \) \(1\)\(1\)\(e\left(\frac{9}{19}\right)\)\(e\left(\frac{7}{19}\right)\)\(e\left(\frac{18}{19}\right)\)\(e\left(\frac{16}{19}\right)\)\(e\left(\frac{16}{19}\right)\)\(e\left(\frac{16}{19}\right)\)\(e\left(\frac{8}{19}\right)\)\(e\left(\frac{14}{19}\right)\)\(e\left(\frac{6}{19}\right)\)\(e\left(\frac{18}{19}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 419 }(107,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 419 }(107,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 419 }(107,·),\chi_{ 419 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 419 }(107,·)) \;\) at \(\; a,b = \) e.g. 1,2