Properties

Label 191.17
Modulus $191$
Conductor $191$
Order $95$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(191, base_ring=CyclotomicField(190))
 
M = H._module
 
chi = DirichletCharacter(H, M([98]))
 
pari: [g,chi] = znchar(Mod(17,191))
 

Basic properties

Modulus: \(191\)
Conductor: \(191\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(95\)
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.g

\(\chi_{191}(2,\cdot)\) \(\chi_{191}(3,\cdot)\) \(\chi_{191}(4,\cdot)\) \(\chi_{191}(8,\cdot)\) \(\chi_{191}(9,\cdot)\) \(\chi_{191}(10,\cdot)\) \(\chi_{191}(12,\cdot)\) \(\chi_{191}(13,\cdot)\) \(\chi_{191}(15,\cdot)\) \(\chi_{191}(16,\cdot)\) \(\chi_{191}(17,\cdot)\) \(\chi_{191}(18,\cdot)\) \(\chi_{191}(20,\cdot)\) \(\chi_{191}(23,\cdot)\) \(\chi_{191}(24,\cdot)\) \(\chi_{191}(26,\cdot)\) \(\chi_{191}(27,\cdot)\) \(\chi_{191}(34,\cdot)\) \(\chi_{191}(40,\cdot)\) \(\chi_{191}(43,\cdot)\) \(\chi_{191}(45,\cdot)\) \(\chi_{191}(46,\cdot)\) \(\chi_{191}(48,\cdot)\) \(\chi_{191}(50,\cdot)\) \(\chi_{191}(51,\cdot)\) \(\chi_{191}(54,\cdot)\) \(\chi_{191}(59,\cdot)\) \(\chi_{191}(60,\cdot)\) \(\chi_{191}(64,\cdot)\) \(\chi_{191}(65,\cdot)\) ...

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: $\Q(\zeta_{95})$
Fixed field: Number field defined by a degree 95 polynomial

Values on generators

\(19\) → \(e\left(\frac{49}{95}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 191 }(17, a) \) \(1\)\(1\)\(e\left(\frac{66}{95}\right)\)\(e\left(\frac{79}{95}\right)\)\(e\left(\frac{37}{95}\right)\)\(e\left(\frac{15}{19}\right)\)\(e\left(\frac{10}{19}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{8}{95}\right)\)\(e\left(\frac{63}{95}\right)\)\(e\left(\frac{46}{95}\right)\)\(e\left(\frac{16}{19}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 191 }(17,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 191 }(17,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 191 }(17,·),\chi_{ 191 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 191 }(17,·)) \;\) at \(\; a,b = \) e.g. 1,2