Properties

Label 235.122
Modulus $235$
Conductor $235$
Order $92$
Real no
Primitive yes
Minimal yes
Parity odd

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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([23,44]))
 
pari: [g,chi] = znchar(Mod(122,235))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 235.k

\(\chi_{235}(2,\cdot)\) \(\chi_{235}(3,\cdot)\) \(\chi_{235}(7,\cdot)\) \(\chi_{235}(8,\cdot)\) \(\chi_{235}(12,\cdot)\) \(\chi_{235}(17,\cdot)\) \(\chi_{235}(18,\cdot)\) \(\chi_{235}(27,\cdot)\) \(\chi_{235}(28,\cdot)\) \(\chi_{235}(32,\cdot)\) \(\chi_{235}(37,\cdot)\) \(\chi_{235}(42,\cdot)\) \(\chi_{235}(53,\cdot)\) \(\chi_{235}(63,\cdot)\) \(\chi_{235}(68,\cdot)\) \(\chi_{235}(72,\cdot)\) \(\chi_{235}(83,\cdot)\) \(\chi_{235}(97,\cdot)\) \(\chi_{235}(98,\cdot)\) \(\chi_{235}(102,\cdot)\) \(\chi_{235}(103,\cdot)\) \(\chi_{235}(108,\cdot)\) \(\chi_{235}(112,\cdot)\) \(\chi_{235}(118,\cdot)\) \(\chi_{235}(122,\cdot)\) \(\chi_{235}(128,\cdot)\) \(\chi_{235}(143,\cdot)\) \(\chi_{235}(147,\cdot)\) \(\chi_{235}(148,\cdot)\) \(\chi_{235}(153,\cdot)\) ...

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

\((142,146)\) → \((i,e\left(\frac{11}{23}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 235 }(122, a) \) \(-1\)\(1\)\(e\left(\frac{79}{92}\right)\)\(e\left(\frac{29}{92}\right)\)\(e\left(\frac{33}{46}\right)\)\(e\left(\frac{4}{23}\right)\)\(e\left(\frac{51}{92}\right)\)\(e\left(\frac{53}{92}\right)\)\(e\left(\frac{29}{46}\right)\)\(e\left(\frac{8}{23}\right)\)\(e\left(\frac{3}{92}\right)\)\(e\left(\frac{1}{92}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 235 }(122,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 235 }(122,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 235 }(122,·),\chi_{ 235 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 235 }(122,·)) \;\) at \(\; a,b = \) e.g. 1,2