Properties

Label 4000.1309
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([0,75,68]))
 
pari: [g,chi] = znchar(Mod(1309,4000))
 

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.da

\(\chi_{4000}(29,\cdot)\) \(\chi_{4000}(69,\cdot)\) \(\chi_{4000}(109,\cdot)\) \(\chi_{4000}(189,\cdot)\) \(\chi_{4000}(229,\cdot)\) \(\chi_{4000}(269,\cdot)\) \(\chi_{4000}(309,\cdot)\) \(\chi_{4000}(389,\cdot)\) \(\chi_{4000}(429,\cdot)\) \(\chi_{4000}(469,\cdot)\) \(\chi_{4000}(509,\cdot)\) \(\chi_{4000}(589,\cdot)\) \(\chi_{4000}(629,\cdot)\) \(\chi_{4000}(669,\cdot)\) \(\chi_{4000}(709,\cdot)\) \(\chi_{4000}(789,\cdot)\) \(\chi_{4000}(829,\cdot)\) \(\chi_{4000}(869,\cdot)\) \(\chi_{4000}(909,\cdot)\) \(\chi_{4000}(989,\cdot)\) \(\chi_{4000}(1029,\cdot)\) \(\chi_{4000}(1069,\cdot)\) \(\chi_{4000}(1109,\cdot)\) \(\chi_{4000}(1189,\cdot)\) \(\chi_{4000}(1229,\cdot)\) \(\chi_{4000}(1269,\cdot)\) \(\chi_{4000}(1309,\cdot)\) \(\chi_{4000}(1389,\cdot)\) \(\chi_{4000}(1429,\cdot)\) \(\chi_{4000}(1469,\cdot)\) ...

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

\((2751,2501,1377)\) → \((1,e\left(\frac{3}{8}\right),e\left(\frac{17}{50}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 4000 }(1309, a) \) \(1\)\(1\)\(e\left(\frac{101}{200}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{1}{100}\right)\)\(e\left(\frac{143}{200}\right)\)\(e\left(\frac{177}{200}\right)\)\(e\left(\frac{8}{25}\right)\)\(e\left(\frac{149}{200}\right)\)\(e\left(\frac{31}{200}\right)\)\(e\left(\frac{79}{100}\right)\)\(e\left(\frac{103}{200}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4000 }(1309,a) \;\) at \(\;a = \) e.g. 2