Properties

Label 4000.613
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([0,25,118]))
 
pari: [g,chi] = znchar(Mod(613,4000))
 

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.cz

\(\chi_{4000}(53,\cdot)\) \(\chi_{4000}(77,\cdot)\) \(\chi_{4000}(133,\cdot)\) \(\chi_{4000}(213,\cdot)\) \(\chi_{4000}(237,\cdot)\) \(\chi_{4000}(317,\cdot)\) \(\chi_{4000}(373,\cdot)\) \(\chi_{4000}(397,\cdot)\) \(\chi_{4000}(453,\cdot)\) \(\chi_{4000}(477,\cdot)\) \(\chi_{4000}(533,\cdot)\) \(\chi_{4000}(613,\cdot)\) \(\chi_{4000}(637,\cdot)\) \(\chi_{4000}(717,\cdot)\) \(\chi_{4000}(773,\cdot)\) \(\chi_{4000}(797,\cdot)\) \(\chi_{4000}(853,\cdot)\) \(\chi_{4000}(877,\cdot)\) \(\chi_{4000}(933,\cdot)\) \(\chi_{4000}(1013,\cdot)\) \(\chi_{4000}(1037,\cdot)\) \(\chi_{4000}(1117,\cdot)\) \(\chi_{4000}(1173,\cdot)\) \(\chi_{4000}(1197,\cdot)\) \(\chi_{4000}(1253,\cdot)\) \(\chi_{4000}(1277,\cdot)\) \(\chi_{4000}(1333,\cdot)\) \(\chi_{4000}(1413,\cdot)\) \(\chi_{4000}(1437,\cdot)\) \(\chi_{4000}(1517,\cdot)\) ...

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

\((2751,2501,1377)\) → \((1,e\left(\frac{1}{8}\right),e\left(\frac{59}{100}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 4000 }(613, a) \) \(-1\)\(1\)\(e\left(\frac{101}{200}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{1}{100}\right)\)\(e\left(\frac{93}{200}\right)\)\(e\left(\frac{177}{200}\right)\)\(e\left(\frac{57}{100}\right)\)\(e\left(\frac{99}{200}\right)\)\(e\left(\frac{181}{200}\right)\)\(e\left(\frac{1}{25}\right)\)\(e\left(\frac{103}{200}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4000 }(613,a) \;\) at \(\;a = \) e.g. 2