Properties

Label 955.739
Modulus $955$
Conductor $955$
Order $38$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(955, base_ring=CyclotomicField(38))
 
M = H._module
 
chi = DirichletCharacter(H, M([19,1]))
 
pari: [g,chi] = znchar(Mod(739,955))
 

Basic properties

Modulus: \(955\)
Conductor: \(955\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(38\)
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 955.p

\(\chi_{955}(14,\cdot)\) \(\chi_{955}(84,\cdot)\) \(\chi_{955}(139,\cdot)\) \(\chi_{955}(159,\cdot)\) \(\chi_{955}(229,\cdot)\) \(\chi_{955}(419,\cdot)\) \(\chi_{955}(504,\cdot)\) \(\chi_{955}(584,\cdot)\) \(\chi_{955}(604,\cdot)\) \(\chi_{955}(614,\cdot)\) \(\chi_{955}(639,\cdot)\) \(\chi_{955}(734,\cdot)\) \(\chi_{955}(739,\cdot)\) \(\chi_{955}(759,\cdot)\) \(\chi_{955}(819,\cdot)\) \(\chi_{955}(834,\cdot)\) \(\chi_{955}(919,\cdot)\) \(\chi_{955}(949,\cdot)\)

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: \(\Q(\zeta_{19})\)
Fixed field: 38.0.47716070387122491878768794713669776106334434794159463718664766104756555702804321854228973388671875.1

Values on generators

\((192,401)\) → \((-1,e\left(\frac{1}{38}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 955 }(739, a) \) \(-1\)\(1\)\(e\left(\frac{25}{38}\right)\)\(e\left(\frac{21}{38}\right)\)\(e\left(\frac{6}{19}\right)\)\(e\left(\frac{4}{19}\right)\)\(1\)\(e\left(\frac{37}{38}\right)\)\(e\left(\frac{2}{19}\right)\)\(e\left(\frac{9}{38}\right)\)\(e\left(\frac{33}{38}\right)\)\(e\left(\frac{17}{38}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 955 }(739,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 955 }(739,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 955 }(739,·),\chi_{ 955 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 955 }(739,·)) \;\) at \(\; a,b = \) e.g. 1,2