Properties

Label 367.11
Modulus $367$
Conductor $367$
Order $366$
Real no
Primitive yes
Minimal yes
Parity odd

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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([163]))
 
pari: [g,chi] = znchar(Mod(11,367))
 

Basic properties

Modulus: \(367\)
Conductor: \(367\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(366\)
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 367.h

\(\chi_{367}(6,\cdot)\) \(\chi_{367}(10,\cdot)\) \(\chi_{367}(11,\cdot)\) \(\chi_{367}(12,\cdot)\) \(\chi_{367}(17,\cdot)\) \(\chi_{367}(19,\cdot)\) \(\chi_{367}(20,\cdot)\) \(\chi_{367}(22,\cdot)\) \(\chi_{367}(34,\cdot)\) \(\chi_{367}(39,\cdot)\) \(\chi_{367}(42,\cdot)\) \(\chi_{367}(43,\cdot)\) \(\chi_{367}(48,\cdot)\) \(\chi_{367}(54,\cdot)\) \(\chi_{367}(58,\cdot)\) \(\chi_{367}(65,\cdot)\) \(\chi_{367}(69,\cdot)\) \(\chi_{367}(70,\cdot)\) \(\chi_{367}(71,\cdot)\) \(\chi_{367}(76,\cdot)\) \(\chi_{367}(77,\cdot)\) \(\chi_{367}(78,\cdot)\) \(\chi_{367}(79,\cdot)\) \(\chi_{367}(80,\cdot)\) \(\chi_{367}(88,\cdot)\) \(\chi_{367}(90,\cdot)\) \(\chi_{367}(93,\cdot)\) \(\chi_{367}(96,\cdot)\) \(\chi_{367}(97,\cdot)\) \(\chi_{367}(99,\cdot)\) ...

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 366 polynomial (not computed)

Values on generators

\(6\) → \(e\left(\frac{163}{366}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 367 }(11, a) \) \(-1\)\(1\)\(e\left(\frac{8}{183}\right)\)\(e\left(\frac{49}{122}\right)\)\(e\left(\frac{16}{183}\right)\)\(e\left(\frac{111}{122}\right)\)\(e\left(\frac{163}{366}\right)\)\(e\left(\frac{21}{61}\right)\)\(e\left(\frac{8}{61}\right)\)\(e\left(\frac{49}{61}\right)\)\(e\left(\frac{349}{366}\right)\)\(e\left(\frac{217}{366}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 367 }(11,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 367 }(11,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 367 }(11,·),\chi_{ 367 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 367 }(11,·)) \;\) at \(\; a,b = \) e.g. 1,2