Properties

Label 803.588
Modulus $803$
Conductor $803$
Order $45$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(803, base_ring=CyclotomicField(90))
 
M = H._module
 
chi = DirichletCharacter(H, M([36,20]))
 
pari: [g,chi] = znchar(Mod(588,803))
 

Basic properties

Modulus: \(803\)
Conductor: \(803\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(45\)
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 803.bi

\(\chi_{803}(4,\cdot)\) \(\chi_{803}(16,\cdot)\) \(\chi_{803}(37,\cdot)\) \(\chi_{803}(75,\cdot)\) \(\chi_{803}(148,\cdot)\) \(\chi_{803}(201,\cdot)\) \(\chi_{803}(223,\cdot)\) \(\chi_{803}(235,\cdot)\) \(\chi_{803}(251,\cdot)\) \(\chi_{803}(256,\cdot)\) \(\chi_{803}(324,\cdot)\) \(\chi_{803}(367,\cdot)\) \(\chi_{803}(454,\cdot)\) \(\chi_{803}(493,\cdot)\) \(\chi_{803}(515,\cdot)\) \(\chi_{803}(543,\cdot)\) \(\chi_{803}(548,\cdot)\) \(\chi_{803}(566,\cdot)\) \(\chi_{803}(586,\cdot)\) \(\chi_{803}(588,\cdot)\) \(\chi_{803}(621,\cdot)\) \(\chi_{803}(746,\cdot)\) \(\chi_{803}(762,\cdot)\) \(\chi_{803}(785,\cdot)\)

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 45 polynomial

Values on generators

\((585,78)\) → \((e\left(\frac{2}{5}\right),e\left(\frac{2}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 803 }(588, a) \) \(1\)\(1\)\(e\left(\frac{8}{45}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{16}{45}\right)\)\(e\left(\frac{37}{45}\right)\)\(e\left(\frac{32}{45}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{1}{15}\right)\)\(1\)\(e\left(\frac{8}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 803 }(588,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 803 }(588,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 803 }(588,·),\chi_{ 803 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 803 }(588,·)) \;\) at \(\; a,b = \) e.g. 1,2