Properties

Label 837.293
Modulus $837$
Conductor $837$
Order $90$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(837, base_ring=CyclotomicField(90))
 
M = H._module
 
chi = DirichletCharacter(H, M([55,66]))
 
pari: [g,chi] = znchar(Mod(293,837))
 

Basic properties

Modulus: \(837\)
Conductor: \(837\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(90\)
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 837.ci

\(\chi_{837}(14,\cdot)\) \(\chi_{837}(20,\cdot)\) \(\chi_{837}(50,\cdot)\) \(\chi_{837}(59,\cdot)\) \(\chi_{837}(131,\cdot)\) \(\chi_{837}(164,\cdot)\) \(\chi_{837}(173,\cdot)\) \(\chi_{837}(227,\cdot)\) \(\chi_{837}(293,\cdot)\) \(\chi_{837}(299,\cdot)\) \(\chi_{837}(329,\cdot)\) \(\chi_{837}(338,\cdot)\) \(\chi_{837}(410,\cdot)\) \(\chi_{837}(443,\cdot)\) \(\chi_{837}(452,\cdot)\) \(\chi_{837}(506,\cdot)\) \(\chi_{837}(572,\cdot)\) \(\chi_{837}(578,\cdot)\) \(\chi_{837}(608,\cdot)\) \(\chi_{837}(617,\cdot)\) \(\chi_{837}(689,\cdot)\) \(\chi_{837}(722,\cdot)\) \(\chi_{837}(731,\cdot)\) \(\chi_{837}(785,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 90 polynomial

Values on generators

\((218,406)\) → \((e\left(\frac{11}{18}\right),e\left(\frac{11}{15}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 837 }(293, a) \) \(-1\)\(1\)\(e\left(\frac{19}{90}\right)\)\(e\left(\frac{19}{45}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{14}{45}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{73}{90}\right)\)\(e\left(\frac{43}{45}\right)\)\(e\left(\frac{47}{90}\right)\)\(e\left(\frac{38}{45}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 837 }(293,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 837 }(293,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 837 }(293,·),\chi_{ 837 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 837 }(293,·)) \;\) at \(\; a,b = \) e.g. 1,2