Properties

Label 367.106
Modulus $367$
Conductor $367$
Order $61$
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(122))
 
M = H._module
 
chi = DirichletCharacter(H, M([44]))
 
pari: [g,chi] = znchar(Mod(106,367))
 

Basic properties

Modulus: \(367\)
Conductor: \(367\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(61\)
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.e

\(\chi_{367}(7,\cdot)\) \(\chi_{367}(8,\cdot)\) \(\chi_{367}(9,\cdot)\) \(\chi_{367}(15,\cdot)\) \(\chi_{367}(25,\cdot)\) \(\chi_{367}(46,\cdot)\) \(\chi_{367}(47,\cdot)\) \(\chi_{367}(49,\cdot)\) \(\chi_{367}(52,\cdot)\) \(\chi_{367}(56,\cdot)\) \(\chi_{367}(59,\cdot)\) \(\chi_{367}(63,\cdot)\) \(\chi_{367}(64,\cdot)\) \(\chi_{367}(72,\cdot)\) \(\chi_{367}(74,\cdot)\) \(\chi_{367}(81,\cdot)\) \(\chi_{367}(87,\cdot)\) \(\chi_{367}(101,\cdot)\) \(\chi_{367}(105,\cdot)\) \(\chi_{367}(106,\cdot)\) \(\chi_{367}(107,\cdot)\) \(\chi_{367}(114,\cdot)\) \(\chi_{367}(120,\cdot)\) \(\chi_{367}(122,\cdot)\) \(\chi_{367}(124,\cdot)\) \(\chi_{367}(132,\cdot)\) \(\chi_{367}(134,\cdot)\) \(\chi_{367}(135,\cdot)\) \(\chi_{367}(137,\cdot)\) \(\chi_{367}(145,\cdot)\) ...

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: $\Q(\zeta_{61})$
Fixed field: Number field defined by a degree 61 polynomial

Values on generators

\(6\) → \(e\left(\frac{22}{61}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 367 }(106, a) \) \(1\)\(1\)\(e\left(\frac{19}{61}\right)\)\(e\left(\frac{3}{61}\right)\)\(e\left(\frac{38}{61}\right)\)\(e\left(\frac{18}{61}\right)\)\(e\left(\frac{22}{61}\right)\)\(e\left(\frac{20}{61}\right)\)\(e\left(\frac{57}{61}\right)\)\(e\left(\frac{6}{61}\right)\)\(e\left(\frac{37}{61}\right)\)\(e\left(\frac{48}{61}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 367 }(106,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 367 }(106,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 367 }(106,·),\chi_{ 367 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 367 }(106,·)) \;\) at \(\; a,b = \) e.g. 1,2