Properties

Label 4000.3
Modulus $4000$
Conductor $4000$
Order $200$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4000, base_ring=CyclotomicField(200))
 
M = H._module
 
chi = DirichletCharacter(H, M([100,75,14]))
 
pari: [g,chi] = znchar(Mod(3,4000))
 

Basic properties

Modulus: \(4000\)
Conductor: \(4000\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(200\)
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 4000.cy

\(\chi_{4000}(3,\cdot)\) \(\chi_{4000}(27,\cdot)\) \(\chi_{4000}(83,\cdot)\) \(\chi_{4000}(163,\cdot)\) \(\chi_{4000}(187,\cdot)\) \(\chi_{4000}(267,\cdot)\) \(\chi_{4000}(323,\cdot)\) \(\chi_{4000}(347,\cdot)\) \(\chi_{4000}(403,\cdot)\) \(\chi_{4000}(427,\cdot)\) \(\chi_{4000}(483,\cdot)\) \(\chi_{4000}(563,\cdot)\) \(\chi_{4000}(587,\cdot)\) \(\chi_{4000}(667,\cdot)\) \(\chi_{4000}(723,\cdot)\) \(\chi_{4000}(747,\cdot)\) \(\chi_{4000}(803,\cdot)\) \(\chi_{4000}(827,\cdot)\) \(\chi_{4000}(883,\cdot)\) \(\chi_{4000}(963,\cdot)\) \(\chi_{4000}(987,\cdot)\) \(\chi_{4000}(1067,\cdot)\) \(\chi_{4000}(1123,\cdot)\) \(\chi_{4000}(1147,\cdot)\) \(\chi_{4000}(1203,\cdot)\) \(\chi_{4000}(1227,\cdot)\) \(\chi_{4000}(1283,\cdot)\) \(\chi_{4000}(1363,\cdot)\) \(\chi_{4000}(1387,\cdot)\) \(\chi_{4000}(1467,\cdot)\) ...

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: $\Q(\zeta_{200})$
Fixed field: Number field defined by a degree 200 polynomial (not computed)

Values on generators

\((2751,2501,1377)\) → \((-1,e\left(\frac{3}{8}\right),e\left(\frac{7}{100}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 4000 }(3, a) \) \(1\)\(1\)\(e\left(\frac{23}{200}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{23}{100}\right)\)\(e\left(\frac{139}{200}\right)\)\(e\left(\frac{71}{200}\right)\)\(e\left(\frac{61}{100}\right)\)\(e\left(\frac{77}{200}\right)\)\(e\left(\frac{63}{200}\right)\)\(e\left(\frac{23}{25}\right)\)\(e\left(\frac{69}{200}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4000 }(3,a) \;\) at \(\;a = \) e.g. 2