Properties

Label 191.6
Modulus $191$
Conductor $191$
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(191, base_ring=CyclotomicField(38))
 
M = H._module
 
chi = DirichletCharacter(H, M([32]))
 
pari: [g,chi] = znchar(Mod(6,191))
 

Basic properties

Modulus: \(191\)
Conductor: \(191\)
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 191.e

\(\chi_{191}(5,\cdot)\) \(\chi_{191}(6,\cdot)\) \(\chi_{191}(25,\cdot)\) \(\chi_{191}(30,\cdot)\) \(\chi_{191}(32,\cdot)\) \(\chi_{191}(36,\cdot)\) \(\chi_{191}(52,\cdot)\) \(\chi_{191}(69,\cdot)\) \(\chi_{191}(107,\cdot)\) \(\chi_{191}(121,\cdot)\) \(\chi_{191}(125,\cdot)\) \(\chi_{191}(136,\cdot)\) \(\chi_{191}(150,\cdot)\) \(\chi_{191}(153,\cdot)\) \(\chi_{191}(154,\cdot)\) \(\chi_{191}(160,\cdot)\) \(\chi_{191}(177,\cdot)\) \(\chi_{191}(180,\cdot)\)

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

\(19\) → \(e\left(\frac{16}{19}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 191 }(6, a) \) \(1\)\(1\)\(e\left(\frac{1}{19}\right)\)\(e\left(\frac{13}{19}\right)\)\(e\left(\frac{2}{19}\right)\)\(e\left(\frac{2}{19}\right)\)\(e\left(\frac{14}{19}\right)\)\(1\)\(e\left(\frac{3}{19}\right)\)\(e\left(\frac{7}{19}\right)\)\(e\left(\frac{3}{19}\right)\)\(e\left(\frac{11}{19}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 191 }(6,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 191 }(6,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 191 }(6,·),\chi_{ 191 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 191 }(6,·)) \;\) at \(\; a,b = \) e.g. 1,2