Properties

Label 193.60
Modulus $193$
Conductor $193$
Order $64$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(193, base_ring=CyclotomicField(64))
 
M = H._module
 
chi = DirichletCharacter(H, M([51]))
 
pari: [g,chi] = znchar(Mod(60,193))
 

Basic properties

Modulus: \(193\)
Conductor: \(193\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(64\)
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.l

\(\chi_{193}(11,\cdot)\) \(\chi_{193}(13,\cdot)\) \(\chi_{193}(20,\cdot)\) \(\chi_{193}(29,\cdot)\) \(\chi_{193}(33,\cdot)\) \(\chi_{193}(35,\cdot)\) \(\chi_{193}(39,\cdot)\) \(\chi_{193}(60,\cdot)\) \(\chi_{193}(68,\cdot)\) \(\chi_{193}(71,\cdot)\) \(\chi_{193}(74,\cdot)\) \(\chi_{193}(76,\cdot)\) \(\chi_{193}(87,\cdot)\) \(\chi_{193}(88,\cdot)\) \(\chi_{193}(89,\cdot)\) \(\chi_{193}(94,\cdot)\) \(\chi_{193}(99,\cdot)\) \(\chi_{193}(104,\cdot)\) \(\chi_{193}(105,\cdot)\) \(\chi_{193}(106,\cdot)\) \(\chi_{193}(117,\cdot)\) \(\chi_{193}(119,\cdot)\) \(\chi_{193}(122,\cdot)\) \(\chi_{193}(125,\cdot)\) \(\chi_{193}(133,\cdot)\) \(\chi_{193}(154,\cdot)\) \(\chi_{193}(158,\cdot)\) \(\chi_{193}(160,\cdot)\) \(\chi_{193}(164,\cdot)\) \(\chi_{193}(173,\cdot)\) ...

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: $\Q(\zeta_{64})$
Fixed field: Number field defined by a degree 64 polynomial

Values on generators

\(5\) → \(e\left(\frac{51}{64}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 193 }(60, a) \) \(-1\)\(1\)\(e\left(\frac{3}{32}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{51}{64}\right)\)\(e\left(\frac{1}{32}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{9}{32}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{57}{64}\right)\)\(e\left(\frac{53}{64}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 193 }(60,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 193 }(60,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 193 }(60,·),\chi_{ 193 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 193 }(60,·)) \;\) at \(\; a,b = \) e.g. 1,2