Properties

Label 4003.1633
Modulus $4003$
Conductor $4003$
Order $29$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4003, base_ring=CyclotomicField(58))
 
M = H._module
 
chi = DirichletCharacter(H, M([54]))
 
pari: [g,chi] = znchar(Mod(1633,4003))
 

Basic properties

Modulus: \(4003\)
Conductor: \(4003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(29\)
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 4003.f

\(\chi_{4003}(102,\cdot)\) \(\chi_{4003}(168,\cdot)\) \(\chi_{4003}(203,\cdot)\) \(\chi_{4003}(413,\cdot)\) \(\chi_{4003}(691,\cdot)\) \(\chi_{4003}(1000,\cdot)\) \(\chi_{4003}(1124,\cdot)\) \(\chi_{4003}(1170,\cdot)\) \(\chi_{4003}(1179,\cdot)\) \(\chi_{4003}(1333,\cdot)\) \(\chi_{4003}(1633,\cdot)\) \(\chi_{4003}(1925,\cdot)\) \(\chi_{4003}(2080,\cdot)\) \(\chi_{4003}(2096,\cdot)\) \(\chi_{4003}(2118,\cdot)\) \(\chi_{4003}(2140,\cdot)\) \(\chi_{4003}(2398,\cdot)\) \(\chi_{4003}(2431,\cdot)\) \(\chi_{4003}(2443,\cdot)\) \(\chi_{4003}(2484,\cdot)\) \(\chi_{4003}(2564,\cdot)\) \(\chi_{4003}(2850,\cdot)\) \(\chi_{4003}(3160,\cdot)\) \(\chi_{4003}(3253,\cdot)\) \(\chi_{4003}(3560,\cdot)\) \(\chi_{4003}(3779,\cdot)\) \(\chi_{4003}(3867,\cdot)\) \(\chi_{4003}(3877,\cdot)\)

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: $\Q(\zeta_{29})$
Fixed field: Number field defined by a degree 29 polynomial

Values on generators

\(2\) → \(e\left(\frac{27}{29}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4003 }(1633, a) \) \(1\)\(1\)\(e\left(\frac{27}{29}\right)\)\(e\left(\frac{9}{29}\right)\)\(e\left(\frac{25}{29}\right)\)\(e\left(\frac{16}{29}\right)\)\(e\left(\frac{7}{29}\right)\)\(e\left(\frac{24}{29}\right)\)\(e\left(\frac{23}{29}\right)\)\(e\left(\frac{18}{29}\right)\)\(e\left(\frac{14}{29}\right)\)\(e\left(\frac{21}{29}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4003 }(1633,a) \;\) at \(\;a = \) e.g. 2