Properties

Label 191.87
Modulus $191$
Conductor $191$
Order $190$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(191, base_ring=CyclotomicField(190))
 
M = H._module
 
chi = DirichletCharacter(H, M([149]))
 
pari: [g,chi] = znchar(Mod(87,191))
 

Basic properties

Modulus: \(191\)
Conductor: \(191\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(190\)
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 191.h

\(\chi_{191}(19,\cdot)\) \(\chi_{191}(21,\cdot)\) \(\chi_{191}(22,\cdot)\) \(\chi_{191}(28,\cdot)\) \(\chi_{191}(29,\cdot)\) \(\chi_{191}(33,\cdot)\) \(\chi_{191}(35,\cdot)\) \(\chi_{191}(42,\cdot)\) \(\chi_{191}(44,\cdot)\) \(\chi_{191}(47,\cdot)\) \(\chi_{191}(53,\cdot)\) \(\chi_{191}(56,\cdot)\) \(\chi_{191}(57,\cdot)\) \(\chi_{191}(58,\cdot)\) \(\chi_{191}(61,\cdot)\) \(\chi_{191}(62,\cdot)\) \(\chi_{191}(63,\cdot)\) \(\chi_{191}(71,\cdot)\) \(\chi_{191}(73,\cdot)\) \(\chi_{191}(74,\cdot)\) \(\chi_{191}(76,\cdot)\) \(\chi_{191}(83,\cdot)\) \(\chi_{191}(87,\cdot)\) \(\chi_{191}(88,\cdot)\) \(\chi_{191}(89,\cdot)\) \(\chi_{191}(91,\cdot)\) \(\chi_{191}(93,\cdot)\) \(\chi_{191}(94,\cdot)\) \(\chi_{191}(95,\cdot)\) \(\chi_{191}(99,\cdot)\) ...

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 190 polynomial (not computed)

Values on generators

\(19\) → \(e\left(\frac{149}{190}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 191 }(87, a) \) \(-1\)\(1\)\(e\left(\frac{48}{95}\right)\)\(e\left(\frac{92}{95}\right)\)\(e\left(\frac{1}{95}\right)\)\(e\left(\frac{4}{19}\right)\)\(e\left(\frac{9}{19}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{49}{95}\right)\)\(e\left(\frac{89}{95}\right)\)\(e\left(\frac{68}{95}\right)\)\(e\left(\frac{25}{38}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 191 }(87,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 191 }(87,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 191 }(87,·),\chi_{ 191 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 191 }(87,·)) \;\) at \(\; a,b = \) e.g. 1,2