Properties

Label 193.67
Modulus $193$
Conductor $193$
Order $32$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(193, base_ring=CyclotomicField(32))
 
M = H._module
 
chi = DirichletCharacter(H, M([3]))
 
pari: [g,chi] = znchar(Mod(67,193))
 

Basic properties

Modulus: \(193\)
Conductor: \(193\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(32\)
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 193.j

\(\chi_{193}(8,\cdot)\) \(\chi_{193}(14,\cdot)\) \(\chi_{193}(23,\cdot)\) \(\chi_{193}(24,\cdot)\) \(\chi_{193}(42,\cdot)\) \(\chi_{193}(67,\cdot)\) \(\chi_{193}(69,\cdot)\) \(\chi_{193}(72,\cdot)\) \(\chi_{193}(121,\cdot)\) \(\chi_{193}(124,\cdot)\) \(\chi_{193}(126,\cdot)\) \(\chi_{193}(151,\cdot)\) \(\chi_{193}(169,\cdot)\) \(\chi_{193}(170,\cdot)\) \(\chi_{193}(179,\cdot)\) \(\chi_{193}(185,\cdot)\)

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: \(\Q(\zeta_{32})\)
Fixed field: Number field defined by a degree 32 polynomial

Values on generators

\(5\) → \(e\left(\frac{3}{32}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 193 }(67, a) \) \(1\)\(1\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{3}{32}\right)\)\(e\left(\frac{1}{16}\right)\)\(-i\)\(e\left(\frac{9}{16}\right)\)\(-i\)\(e\left(\frac{9}{32}\right)\)\(e\left(\frac{5}{32}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 193 }(67,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 193 }(67,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 193 }(67,·),\chi_{ 193 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 193 }(67,·)) \;\) at \(\; a,b = \) e.g. 1,2