Properties

Label 4000.837
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([0,25,138]))
 
pari: [g,chi] = znchar(Mod(837,4000))
 

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.de

\(\chi_{4000}(13,\cdot)\) \(\chi_{4000}(37,\cdot)\) \(\chi_{4000}(117,\cdot)\) \(\chi_{4000}(173,\cdot)\) \(\chi_{4000}(197,\cdot)\) \(\chi_{4000}(253,\cdot)\) \(\chi_{4000}(277,\cdot)\) \(\chi_{4000}(333,\cdot)\) \(\chi_{4000}(413,\cdot)\) \(\chi_{4000}(437,\cdot)\) \(\chi_{4000}(517,\cdot)\) \(\chi_{4000}(573,\cdot)\) \(\chi_{4000}(597,\cdot)\) \(\chi_{4000}(653,\cdot)\) \(\chi_{4000}(677,\cdot)\) \(\chi_{4000}(733,\cdot)\) \(\chi_{4000}(813,\cdot)\) \(\chi_{4000}(837,\cdot)\) \(\chi_{4000}(917,\cdot)\) \(\chi_{4000}(973,\cdot)\) \(\chi_{4000}(997,\cdot)\) \(\chi_{4000}(1053,\cdot)\) \(\chi_{4000}(1077,\cdot)\) \(\chi_{4000}(1133,\cdot)\) \(\chi_{4000}(1213,\cdot)\) \(\chi_{4000}(1237,\cdot)\) \(\chi_{4000}(1317,\cdot)\) \(\chi_{4000}(1373,\cdot)\) \(\chi_{4000}(1397,\cdot)\) \(\chi_{4000}(1453,\cdot)\) ...

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

\((2751,2501,1377)\) → \((1,e\left(\frac{1}{8}\right),e\left(\frac{69}{100}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 4000 }(837, a) \) \(-1\)\(1\)\(e\left(\frac{41}{200}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{41}{100}\right)\)\(e\left(\frac{13}{200}\right)\)\(e\left(\frac{157}{200}\right)\)\(e\left(\frac{87}{100}\right)\)\(e\left(\frac{59}{200}\right)\)\(e\left(\frac{21}{200}\right)\)\(e\left(\frac{7}{50}\right)\)\(e\left(\frac{123}{200}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4000 }(837,a) \;\) at \(\;a = \) e.g. 2