Properties

Label 4003.1358
Modulus $4003$
Conductor $4003$
Order $23$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4003, base_ring=CyclotomicField(46))
 
M = H._module
 
chi = DirichletCharacter(H, M([8]))
 
pari: [g,chi] = znchar(Mod(1358,4003))
 

Basic properties

Modulus: \(4003\)
Conductor: \(4003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(23\)
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.e

\(\chi_{4003}(551,\cdot)\) \(\chi_{4003}(703,\cdot)\) \(\chi_{4003}(835,\cdot)\) \(\chi_{4003}(848,\cdot)\) \(\chi_{4003}(1081,\cdot)\) \(\chi_{4003}(1173,\cdot)\) \(\chi_{4003}(1358,\cdot)\) \(\chi_{4003}(1840,\cdot)\) \(\chi_{4003}(1960,\cdot)\) \(\chi_{4003}(2567,\cdot)\) \(\chi_{4003}(2723,\cdot)\) \(\chi_{4003}(2784,\cdot)\) \(\chi_{4003}(2900,\cdot)\) \(\chi_{4003}(3065,\cdot)\) \(\chi_{4003}(3153,\cdot)\) \(\chi_{4003}(3187,\cdot)\) \(\chi_{4003}(3251,\cdot)\) \(\chi_{4003}(3376,\cdot)\) \(\chi_{4003}(3552,\cdot)\) \(\chi_{4003}(3688,\cdot)\) \(\chi_{4003}(3700,\cdot)\) \(\chi_{4003}(3743,\cdot)\)

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: \(\Q(\zeta_{23})\)
Fixed field: Number field defined by a degree 23 polynomial

Values on generators

\(2\) → \(e\left(\frac{4}{23}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4003 }(1358, a) \) \(1\)\(1\)\(e\left(\frac{4}{23}\right)\)\(e\left(\frac{8}{23}\right)\)\(e\left(\frac{8}{23}\right)\)\(e\left(\frac{19}{23}\right)\)\(e\left(\frac{12}{23}\right)\)\(e\left(\frac{3}{23}\right)\)\(e\left(\frac{12}{23}\right)\)\(e\left(\frac{16}{23}\right)\)\(1\)\(e\left(\frac{5}{23}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4003 }(1358,a) \;\) at \(\;a = \) e.g. 2