Properties

Label 955.136
Modulus $955$
Conductor $191$
Order $19$
Real no
Primitive no
Minimal yes
Parity even

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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([0,8]))
 
pari: [g,chi] = znchar(Mod(136,955))
 

Basic properties

Modulus: \(955\)
Conductor: \(191\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(19\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{191}(136,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 955.k

\(\chi_{955}(6,\cdot)\) \(\chi_{955}(36,\cdot)\) \(\chi_{955}(121,\cdot)\) \(\chi_{955}(136,\cdot)\) \(\chi_{955}(196,\cdot)\) \(\chi_{955}(216,\cdot)\) \(\chi_{955}(221,\cdot)\) \(\chi_{955}(316,\cdot)\) \(\chi_{955}(341,\cdot)\) \(\chi_{955}(351,\cdot)\) \(\chi_{955}(371,\cdot)\) \(\chi_{955}(451,\cdot)\) \(\chi_{955}(536,\cdot)\) \(\chi_{955}(726,\cdot)\) \(\chi_{955}(796,\cdot)\) \(\chi_{955}(816,\cdot)\) \(\chi_{955}(871,\cdot)\) \(\chi_{955}(941,\cdot)\)

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: Number field defined by a degree 19 polynomial

Values on generators

\((192,401)\) → \((1,e\left(\frac{4}{19}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 955 }(136, a) \) \(1\)\(1\)\(e\left(\frac{5}{19}\right)\)\(e\left(\frac{8}{19}\right)\)\(e\left(\frac{10}{19}\right)\)\(e\left(\frac{13}{19}\right)\)\(1\)\(e\left(\frac{15}{19}\right)\)\(e\left(\frac{16}{19}\right)\)\(e\left(\frac{17}{19}\right)\)\(e\left(\frac{18}{19}\right)\)\(e\left(\frac{11}{19}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 955 }(136,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 955 }(136,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 955 }(136,·),\chi_{ 955 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 955 }(136,·)) \;\) at \(\; a,b = \) e.g. 1,2