Properties

Label 193.22
Modulus $193$
Conductor $193$
Order $192$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(193, base_ring=CyclotomicField(192))
 
M = H._module
 
chi = DirichletCharacter(H, M([25]))
 
pari: [g,chi] = znchar(Mod(22,193))
 

Basic properties

Modulus: \(193\)
Conductor: \(193\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(192\)
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 193.n

\(\chi_{193}(5,\cdot)\) \(\chi_{193}(10,\cdot)\) \(\chi_{193}(15,\cdot)\) \(\chi_{193}(17,\cdot)\) \(\chi_{193}(19,\cdot)\) \(\chi_{193}(22,\cdot)\) \(\chi_{193}(26,\cdot)\) \(\chi_{193}(30,\cdot)\) \(\chi_{193}(34,\cdot)\) \(\chi_{193}(37,\cdot)\) \(\chi_{193}(38,\cdot)\) \(\chi_{193}(40,\cdot)\) \(\chi_{193}(41,\cdot)\) \(\chi_{193}(44,\cdot)\) \(\chi_{193}(45,\cdot)\) \(\chi_{193}(47,\cdot)\) \(\chi_{193}(51,\cdot)\) \(\chi_{193}(52,\cdot)\) \(\chi_{193}(53,\cdot)\) \(\chi_{193}(57,\cdot)\) \(\chi_{193}(58,\cdot)\) \(\chi_{193}(61,\cdot)\) \(\chi_{193}(66,\cdot)\) \(\chi_{193}(70,\cdot)\) \(\chi_{193}(73,\cdot)\) \(\chi_{193}(77,\cdot)\) \(\chi_{193}(78,\cdot)\) \(\chi_{193}(79,\cdot)\) \(\chi_{193}(80,\cdot)\) \(\chi_{193}(82,\cdot)\) ...

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: $\Q(\zeta_{192})$
Fixed field: Number field defined by a degree 192 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{25}{192}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 193 }(22, a) \) \(-1\)\(1\)\(e\left(\frac{41}{96}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{41}{48}\right)\)\(e\left(\frac{25}{192}\right)\)\(e\left(\frac{35}{96}\right)\)\(e\left(\frac{13}{24}\right)\)\(e\left(\frac{9}{32}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{107}{192}\right)\)\(e\left(\frac{53}{64}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 193 }(22,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 193 }(22,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 193 }(22,·),\chi_{ 193 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 193 }(22,·)) \;\) at \(\; a,b = \) e.g. 1,2