Properties

Label 944.273
Modulus $944$
Conductor $59$
Order $58$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(944, base_ring=CyclotomicField(58))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,55]))
 
pari: [g,chi] = znchar(Mod(273,944))
 

Basic properties

Modulus: \(944\)
Conductor: \(59\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(58\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{59}(37,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 944.n

\(\chi_{944}(33,\cdot)\) \(\chi_{944}(65,\cdot)\) \(\chi_{944}(97,\cdot)\) \(\chi_{944}(113,\cdot)\) \(\chi_{944}(129,\cdot)\) \(\chi_{944}(161,\cdot)\) \(\chi_{944}(209,\cdot)\) \(\chi_{944}(273,\cdot)\) \(\chi_{944}(305,\cdot)\) \(\chi_{944}(337,\cdot)\) \(\chi_{944}(385,\cdot)\) \(\chi_{944}(401,\cdot)\) \(\chi_{944}(465,\cdot)\) \(\chi_{944}(545,\cdot)\) \(\chi_{944}(561,\cdot)\) \(\chi_{944}(657,\cdot)\) \(\chi_{944}(673,\cdot)\) \(\chi_{944}(689,\cdot)\) \(\chi_{944}(705,\cdot)\) \(\chi_{944}(721,\cdot)\) \(\chi_{944}(769,\cdot)\) \(\chi_{944}(785,\cdot)\) \(\chi_{944}(801,\cdot)\) \(\chi_{944}(817,\cdot)\) \(\chi_{944}(849,\cdot)\) \(\chi_{944}(865,\cdot)\) \(\chi_{944}(881,\cdot)\) \(\chi_{944}(929,\cdot)\)

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: $\Q(\zeta_{29})$
Fixed field: Number field defined by a degree 58 polynomial

Values on generators

\((591,709,769)\) → \((1,1,e\left(\frac{55}{58}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 944 }(273, a) \) \(-1\)\(1\)\(e\left(\frac{12}{29}\right)\)\(e\left(\frac{20}{29}\right)\)\(e\left(\frac{2}{29}\right)\)\(e\left(\frac{24}{29}\right)\)\(e\left(\frac{41}{58}\right)\)\(e\left(\frac{39}{58}\right)\)\(e\left(\frac{3}{29}\right)\)\(e\left(\frac{27}{29}\right)\)\(e\left(\frac{1}{29}\right)\)\(e\left(\frac{14}{29}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 944 }(273,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 944 }(273,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 944 }(273,·),\chi_{ 944 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 944 }(273,·)) \;\) at \(\; a,b = \) e.g. 1,2