Properties

Label 235.127
Modulus $235$
Conductor $235$
Order $92$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(235, base_ring=CyclotomicField(92))
 
M = H._module
 
chi = DirichletCharacter(H, M([23,54]))
 
pari: [g,chi] = znchar(Mod(127,235))
 

Basic properties

Modulus: \(235\)
Conductor: \(235\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(92\)
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 235.l

\(\chi_{235}(13,\cdot)\) \(\chi_{235}(22,\cdot)\) \(\chi_{235}(23,\cdot)\) \(\chi_{235}(33,\cdot)\) \(\chi_{235}(38,\cdot)\) \(\chi_{235}(43,\cdot)\) \(\chi_{235}(52,\cdot)\) \(\chi_{235}(57,\cdot)\) \(\chi_{235}(58,\cdot)\) \(\chi_{235}(62,\cdot)\) \(\chi_{235}(67,\cdot)\) \(\chi_{235}(73,\cdot)\) \(\chi_{235}(77,\cdot)\) \(\chi_{235}(78,\cdot)\) \(\chi_{235}(82,\cdot)\) \(\chi_{235}(87,\cdot)\) \(\chi_{235}(88,\cdot)\) \(\chi_{235}(92,\cdot)\) \(\chi_{235}(107,\cdot)\) \(\chi_{235}(113,\cdot)\) \(\chi_{235}(117,\cdot)\) \(\chi_{235}(123,\cdot)\) \(\chi_{235}(127,\cdot)\) \(\chi_{235}(132,\cdot)\) \(\chi_{235}(133,\cdot)\) \(\chi_{235}(137,\cdot)\) \(\chi_{235}(138,\cdot)\) \(\chi_{235}(152,\cdot)\) \(\chi_{235}(163,\cdot)\) \(\chi_{235}(167,\cdot)\) ...

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: $\Q(\zeta_{92})$
Fixed field: Number field defined by a degree 92 polynomial

Values on generators

\((142,146)\) → \((i,e\left(\frac{27}{46}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 235 }(127, a) \) \(1\)\(1\)\(e\left(\frac{75}{92}\right)\)\(e\left(\frac{45}{92}\right)\)\(e\left(\frac{29}{46}\right)\)\(e\left(\frac{7}{23}\right)\)\(e\left(\frac{3}{92}\right)\)\(e\left(\frac{41}{92}\right)\)\(e\left(\frac{45}{46}\right)\)\(e\left(\frac{5}{46}\right)\)\(e\left(\frac{11}{92}\right)\)\(e\left(\frac{19}{92}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 235 }(127,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 235 }(127,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 235 }(127,·),\chi_{ 235 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 235 }(127,·)) \;\) at \(\; a,b = \) e.g. 1,2