Properties

Label 4003.25
Modulus $4003$
Conductor $4003$
Order $2001$
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(4002))
 
M = H._module
 
chi = DirichletCharacter(H, M([2942]))
 
pari: [g,chi] = znchar(Mod(25,4003))
 

Basic properties

Modulus: \(4003\)
Conductor: \(4003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2001\)
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.o

\(\chi_{4003}(4,\cdot)\) \(\chi_{4003}(7,\cdot)\) \(\chi_{4003}(9,\cdot)\) \(\chi_{4003}(16,\cdot)\) \(\chi_{4003}(19,\cdot)\) \(\chi_{4003}(24,\cdot)\) \(\chi_{4003}(25,\cdot)\) \(\chi_{4003}(29,\cdot)\) \(\chi_{4003}(37,\cdot)\) \(\chi_{4003}(40,\cdot)\) \(\chi_{4003}(42,\cdot)\) \(\chi_{4003}(44,\cdot)\) \(\chi_{4003}(46,\cdot)\) \(\chi_{4003}(49,\cdot)\) \(\chi_{4003}(52,\cdot)\) \(\chi_{4003}(53,\cdot)\) \(\chi_{4003}(54,\cdot)\) \(\chi_{4003}(59,\cdot)\) \(\chi_{4003}(60,\cdot)\) \(\chi_{4003}(62,\cdot)\) \(\chi_{4003}(63,\cdot)\) \(\chi_{4003}(67,\cdot)\) \(\chi_{4003}(68,\cdot)\) \(\chi_{4003}(77,\cdot)\) \(\chi_{4003}(79,\cdot)\) \(\chi_{4003}(81,\cdot)\) \(\chi_{4003}(82,\cdot)\) \(\chi_{4003}(83,\cdot)\) \(\chi_{4003}(89,\cdot)\) \(\chi_{4003}(91,\cdot)\) ...

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: $\Q(\zeta_{2001})$
Fixed field: Number field defined by a degree 2001 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{1471}{2001}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4003 }(25, a) \) \(1\)\(1\)\(e\left(\frac{1471}{2001}\right)\)\(e\left(\frac{1631}{2001}\right)\)\(e\left(\frac{941}{2001}\right)\)\(e\left(\frac{760}{2001}\right)\)\(e\left(\frac{367}{667}\right)\)\(e\left(\frac{1546}{2001}\right)\)\(e\left(\frac{137}{667}\right)\)\(e\left(\frac{1261}{2001}\right)\)\(e\left(\frac{10}{87}\right)\)\(e\left(\frac{289}{667}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4003 }(25,a) \;\) at \(\;a = \) e.g. 2