Properties

Label 4000.771
Modulus $4000$
Conductor $4000$
Order $200$
Real no
Primitive yes
Minimal yes
Parity odd

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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([100,75,184]))
 
pari: [g,chi] = znchar(Mod(771,4000))
 

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.db

\(\chi_{4000}(11,\cdot)\) \(\chi_{4000}(91,\cdot)\) \(\chi_{4000}(131,\cdot)\) \(\chi_{4000}(171,\cdot)\) \(\chi_{4000}(211,\cdot)\) \(\chi_{4000}(291,\cdot)\) \(\chi_{4000}(331,\cdot)\) \(\chi_{4000}(371,\cdot)\) \(\chi_{4000}(411,\cdot)\) \(\chi_{4000}(491,\cdot)\) \(\chi_{4000}(531,\cdot)\) \(\chi_{4000}(571,\cdot)\) \(\chi_{4000}(611,\cdot)\) \(\chi_{4000}(691,\cdot)\) \(\chi_{4000}(731,\cdot)\) \(\chi_{4000}(771,\cdot)\) \(\chi_{4000}(811,\cdot)\) \(\chi_{4000}(891,\cdot)\) \(\chi_{4000}(931,\cdot)\) \(\chi_{4000}(971,\cdot)\) \(\chi_{4000}(1011,\cdot)\) \(\chi_{4000}(1091,\cdot)\) \(\chi_{4000}(1131,\cdot)\) \(\chi_{4000}(1171,\cdot)\) \(\chi_{4000}(1211,\cdot)\) \(\chi_{4000}(1291,\cdot)\) \(\chi_{4000}(1331,\cdot)\) \(\chi_{4000}(1371,\cdot)\) \(\chi_{4000}(1411,\cdot)\) \(\chi_{4000}(1491,\cdot)\) ...

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

\((2751,2501,1377)\) → \((-1,e\left(\frac{3}{8}\right),e\left(\frac{23}{25}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 4000 }(771, a) \) \(-1\)\(1\)\(e\left(\frac{13}{200}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{13}{100}\right)\)\(e\left(\frac{59}{200}\right)\)\(e\left(\frac{101}{200}\right)\)\(e\left(\frac{33}{50}\right)\)\(e\left(\frac{137}{200}\right)\)\(e\left(\frac{103}{200}\right)\)\(e\left(\frac{27}{100}\right)\)\(e\left(\frac{39}{200}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4000 }(771,a) \;\) at \(\;a = \) e.g. 2