Properties

Label 2023.4
Modulus $2023$
Conductor $2023$
Order $204$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2023, base_ring=CyclotomicField(204))
 
M = H._module
 
chi = DirichletCharacter(H, M([136,81]))
 
pari: [g,chi] = znchar(Mod(4,2023))
 

Basic properties

Modulus: \(2023\)
Conductor: \(2023\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(204\)
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 2023.bg

\(\chi_{2023}(4,\cdot)\) \(\chi_{2023}(30,\cdot)\) \(\chi_{2023}(72,\cdot)\) \(\chi_{2023}(81,\cdot)\) \(\chi_{2023}(123,\cdot)\) \(\chi_{2023}(149,\cdot)\) \(\chi_{2023}(191,\cdot)\) \(\chi_{2023}(200,\cdot)\) \(\chi_{2023}(242,\cdot)\) \(\chi_{2023}(268,\cdot)\) \(\chi_{2023}(310,\cdot)\) \(\chi_{2023}(319,\cdot)\) \(\chi_{2023}(361,\cdot)\) \(\chi_{2023}(387,\cdot)\) \(\chi_{2023}(429,\cdot)\) \(\chi_{2023}(438,\cdot)\) \(\chi_{2023}(480,\cdot)\) \(\chi_{2023}(506,\cdot)\) \(\chi_{2023}(548,\cdot)\) \(\chi_{2023}(557,\cdot)\) \(\chi_{2023}(599,\cdot)\) \(\chi_{2023}(625,\cdot)\) \(\chi_{2023}(667,\cdot)\) \(\chi_{2023}(676,\cdot)\) \(\chi_{2023}(718,\cdot)\) \(\chi_{2023}(744,\cdot)\) \(\chi_{2023}(786,\cdot)\) \(\chi_{2023}(795,\cdot)\) \(\chi_{2023}(837,\cdot)\) \(\chi_{2023}(863,\cdot)\) ...

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: $\Q(\zeta_{204})$
Fixed field: Number field defined by a degree 204 polynomial (not computed)

Values on generators

\((290,1737)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{27}{68}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 2023 }(4, a) \) \(1\)\(1\)\(e\left(\frac{79}{102}\right)\)\(e\left(\frac{13}{204}\right)\)\(e\left(\frac{28}{51}\right)\)\(e\left(\frac{53}{204}\right)\)\(e\left(\frac{57}{68}\right)\)\(e\left(\frac{11}{34}\right)\)\(e\left(\frac{13}{102}\right)\)\(e\left(\frac{7}{204}\right)\)\(e\left(\frac{163}{204}\right)\)\(e\left(\frac{125}{204}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2023 }(4,a) \;\) at \(\;a = \) e.g. 2