Properties

Label 4000.659
Modulus $4000$
Conductor $4000$
Order $200$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4000, base_ring=CyclotomicField(200))
 
M = H._module
 
chi = DirichletCharacter(H, M([100,175,148]))
 
pari: [g,chi] = znchar(Mod(659,4000))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4000.dc

\(\chi_{4000}(19,\cdot)\) \(\chi_{4000}(59,\cdot)\) \(\chi_{4000}(139,\cdot)\) \(\chi_{4000}(179,\cdot)\) \(\chi_{4000}(219,\cdot)\) \(\chi_{4000}(259,\cdot)\) \(\chi_{4000}(339,\cdot)\) \(\chi_{4000}(379,\cdot)\) \(\chi_{4000}(419,\cdot)\) \(\chi_{4000}(459,\cdot)\) \(\chi_{4000}(539,\cdot)\) \(\chi_{4000}(579,\cdot)\) \(\chi_{4000}(619,\cdot)\) \(\chi_{4000}(659,\cdot)\) \(\chi_{4000}(739,\cdot)\) \(\chi_{4000}(779,\cdot)\) \(\chi_{4000}(819,\cdot)\) \(\chi_{4000}(859,\cdot)\) \(\chi_{4000}(939,\cdot)\) \(\chi_{4000}(979,\cdot)\) \(\chi_{4000}(1019,\cdot)\) \(\chi_{4000}(1059,\cdot)\) \(\chi_{4000}(1139,\cdot)\) \(\chi_{4000}(1179,\cdot)\) \(\chi_{4000}(1219,\cdot)\) \(\chi_{4000}(1259,\cdot)\) \(\chi_{4000}(1339,\cdot)\) \(\chi_{4000}(1379,\cdot)\) \(\chi_{4000}(1419,\cdot)\) \(\chi_{4000}(1459,\cdot)\) ...

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

\((2751,2501,1377)\) → \((-1,e\left(\frac{7}{8}\right),e\left(\frac{37}{50}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 4000 }(659, a) \) \(-1\)\(1\)\(e\left(\frac{61}{200}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{61}{100}\right)\)\(e\left(\frac{23}{200}\right)\)\(e\left(\frac{197}{200}\right)\)\(e\left(\frac{13}{25}\right)\)\(e\left(\frac{189}{200}\right)\)\(e\left(\frac{91}{200}\right)\)\(e\left(\frac{69}{100}\right)\)\(e\left(\frac{183}{200}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4000 }(659,a) \;\) at \(\;a = \) e.g. 2