Properties

Label 367.2
Modulus $367$
Conductor $367$
Order $183$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(367, base_ring=CyclotomicField(366))
 
M = H._module
 
chi = DirichletCharacter(H, M([292]))
 
pari: [g,chi] = znchar(Mod(2,367))
 

Basic properties

Modulus: \(367\)
Conductor: \(367\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(183\)
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 367.g

\(\chi_{367}(2,\cdot)\) \(\chi_{367}(4,\cdot)\) \(\chi_{367}(13,\cdot)\) \(\chi_{367}(14,\cdot)\) \(\chi_{367}(16,\cdot)\) \(\chi_{367}(18,\cdot)\) \(\chi_{367}(23,\cdot)\) \(\chi_{367}(26,\cdot)\) \(\chi_{367}(28,\cdot)\) \(\chi_{367}(30,\cdot)\) \(\chi_{367}(31,\cdot)\) \(\chi_{367}(32,\cdot)\) \(\chi_{367}(33,\cdot)\) \(\chi_{367}(36,\cdot)\) \(\chi_{367}(37,\cdot)\) \(\chi_{367}(41,\cdot)\) \(\chi_{367}(50,\cdot)\) \(\chi_{367}(51,\cdot)\) \(\chi_{367}(53,\cdot)\) \(\chi_{367}(55,\cdot)\) \(\chi_{367}(57,\cdot)\) \(\chi_{367}(60,\cdot)\) \(\chi_{367}(61,\cdot)\) \(\chi_{367}(62,\cdot)\) \(\chi_{367}(66,\cdot)\) \(\chi_{367}(67,\cdot)\) \(\chi_{367}(73,\cdot)\) \(\chi_{367}(82,\cdot)\) \(\chi_{367}(85,\cdot)\) \(\chi_{367}(89,\cdot)\) ...

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: $\Q(\zeta_{183})$
Fixed field: Number field defined by a degree 183 polynomial (not computed)

Values on generators

\(6\) → \(e\left(\frac{146}{183}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 367 }(2, a) \) \(1\)\(1\)\(e\left(\frac{176}{183}\right)\)\(e\left(\frac{51}{61}\right)\)\(e\left(\frac{169}{183}\right)\)\(e\left(\frac{1}{61}\right)\)\(e\left(\frac{146}{183}\right)\)\(e\left(\frac{35}{61}\right)\)\(e\left(\frac{54}{61}\right)\)\(e\left(\frac{41}{61}\right)\)\(e\left(\frac{179}{183}\right)\)\(e\left(\frac{8}{183}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 367 }(2,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 367 }(2,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 367 }(2,·),\chi_{ 367 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 367 }(2,·)) \;\) at \(\; a,b = \) e.g. 1,2