Properties

Label 293.91
Modulus $293$
Conductor $293$
Order $73$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(293, base_ring=CyclotomicField(146))
 
M = H._module
 
chi = DirichletCharacter(H, M([82]))
 
pari: [g,chi] = znchar(Mod(91,293))
 

Basic properties

Modulus: \(293\)
Conductor: \(293\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(73\)
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 293.d

\(\chi_{293}(16,\cdot)\) \(\chi_{293}(17,\cdot)\) \(\chi_{293}(22,\cdot)\) \(\chi_{293}(24,\cdot)\) \(\chi_{293}(26,\cdot)\) \(\chi_{293}(33,\cdot)\) \(\chi_{293}(36,\cdot)\) \(\chi_{293}(38,\cdot)\) \(\chi_{293}(39,\cdot)\) \(\chi_{293}(40,\cdot)\) \(\chi_{293}(46,\cdot)\) \(\chi_{293}(53,\cdot)\) \(\chi_{293}(54,\cdot)\) \(\chi_{293}(55,\cdot)\) \(\chi_{293}(56,\cdot)\) \(\chi_{293}(57,\cdot)\) \(\chi_{293}(59,\cdot)\) \(\chi_{293}(60,\cdot)\) \(\chi_{293}(65,\cdot)\) \(\chi_{293}(69,\cdot)\) \(\chi_{293}(73,\cdot)\) \(\chi_{293}(77,\cdot)\) \(\chi_{293}(81,\cdot)\) \(\chi_{293}(82,\cdot)\) \(\chi_{293}(84,\cdot)\) \(\chi_{293}(90,\cdot)\) \(\chi_{293}(91,\cdot)\) \(\chi_{293}(94,\cdot)\) \(\chi_{293}(95,\cdot)\) \(\chi_{293}(100,\cdot)\) ...

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: $\Q(\zeta_{73})$
Fixed field: Number field defined by a degree 73 polynomial

Values on generators

\(2\) → \(e\left(\frac{41}{73}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 293 }(91, a) \) \(1\)\(1\)\(e\left(\frac{41}{73}\right)\)\(e\left(\frac{13}{73}\right)\)\(e\left(\frac{9}{73}\right)\)\(e\left(\frac{12}{73}\right)\)\(e\left(\frac{54}{73}\right)\)\(e\left(\frac{46}{73}\right)\)\(e\left(\frac{50}{73}\right)\)\(e\left(\frac{26}{73}\right)\)\(e\left(\frac{53}{73}\right)\)\(e\left(\frac{43}{73}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 293 }(91,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 293 }(91,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 293 }(91,·),\chi_{ 293 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 293 }(91,·)) \;\) at \(\; a,b = \) e.g. 1,2