Properties

Label 4001.89
Modulus $4001$
Conductor $4001$
Order $2000$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4001, base_ring=CyclotomicField(2000))
 
M = H._module
 
chi = DirichletCharacter(H, M([1501]))
 
pari: [g,chi] = znchar(Mod(89,4001))
 

Basic properties

Modulus: \(4001\)
Conductor: \(4001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2000\)
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 4001.w

\(\chi_{4001}(9,\cdot)\) \(\chi_{4001}(18,\cdot)\) \(\chi_{4001}(29,\cdot)\) \(\chi_{4001}(31,\cdot)\) \(\chi_{4001}(33,\cdot)\) \(\chi_{4001}(36,\cdot)\) \(\chi_{4001}(45,\cdot)\) \(\chi_{4001}(47,\cdot)\) \(\chi_{4001}(61,\cdot)\) \(\chi_{4001}(62,\cdot)\) \(\chi_{4001}(66,\cdot)\) \(\chi_{4001}(72,\cdot)\) \(\chi_{4001}(83,\cdot)\) \(\chi_{4001}(89,\cdot)\) \(\chi_{4001}(90,\cdot)\) \(\chi_{4001}(94,\cdot)\) \(\chi_{4001}(97,\cdot)\) \(\chi_{4001}(111,\cdot)\) \(\chi_{4001}(116,\cdot)\) \(\chi_{4001}(117,\cdot)\) \(\chi_{4001}(118,\cdot)\) \(\chi_{4001}(122,\cdot)\) \(\chi_{4001}(124,\cdot)\) \(\chi_{4001}(126,\cdot)\) \(\chi_{4001}(137,\cdot)\) \(\chi_{4001}(138,\cdot)\) \(\chi_{4001}(145,\cdot)\) \(\chi_{4001}(149,\cdot)\) \(\chi_{4001}(155,\cdot)\) \(\chi_{4001}(165,\cdot)\) ...

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: $\Q(\zeta_{2000})$
Fixed field: Number field defined by a degree 2000 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{1501}{2000}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4001 }(89, a) \) \(1\)\(1\)\(e\left(\frac{23}{500}\right)\)\(e\left(\frac{1501}{2000}\right)\)\(e\left(\frac{23}{250}\right)\)\(e\left(\frac{59}{100}\right)\)\(e\left(\frac{1593}{2000}\right)\)\(e\left(\frac{57}{500}\right)\)\(e\left(\frac{69}{500}\right)\)\(e\left(\frac{501}{1000}\right)\)\(e\left(\frac{159}{250}\right)\)\(e\left(\frac{37}{80}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4001 }(89,a) \;\) at \(\;a = \) e.g. 2