Properties

Label 4000.111
Modulus $4000$
Conductor $1000$
Order $50$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4000, base_ring=CyclotomicField(50))
 
M = H._module
 
chi = DirichletCharacter(H, M([25,25,18]))
 
pari: [g,chi] = znchar(Mod(111,4000))
 

Basic properties

Modulus: \(4000\)
Conductor: \(1000\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(50\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1000}(611,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4000.ci

\(\chi_{4000}(111,\cdot)\) \(\chi_{4000}(271,\cdot)\) \(\chi_{4000}(431,\cdot)\) \(\chi_{4000}(591,\cdot)\) \(\chi_{4000}(911,\cdot)\) \(\chi_{4000}(1071,\cdot)\) \(\chi_{4000}(1231,\cdot)\) \(\chi_{4000}(1391,\cdot)\) \(\chi_{4000}(1711,\cdot)\) \(\chi_{4000}(1871,\cdot)\) \(\chi_{4000}(2031,\cdot)\) \(\chi_{4000}(2191,\cdot)\) \(\chi_{4000}(2511,\cdot)\) \(\chi_{4000}(2671,\cdot)\) \(\chi_{4000}(2831,\cdot)\) \(\chi_{4000}(2991,\cdot)\) \(\chi_{4000}(3311,\cdot)\) \(\chi_{4000}(3471,\cdot)\) \(\chi_{4000}(3631,\cdot)\) \(\chi_{4000}(3791,\cdot)\)

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: \(\Q(\zeta_{25})\)
Fixed field: Number field defined by a degree 50 polynomial

Values on generators

\((2751,2501,1377)\) → \((-1,-1,e\left(\frac{9}{25}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 4000 }(111, a) \) \(-1\)\(1\)\(e\left(\frac{13}{25}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{1}{25}\right)\)\(e\left(\frac{9}{25}\right)\)\(e\left(\frac{27}{50}\right)\)\(e\left(\frac{7}{25}\right)\)\(e\left(\frac{12}{25}\right)\)\(e\left(\frac{31}{50}\right)\)\(e\left(\frac{33}{50}\right)\)\(e\left(\frac{14}{25}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4000 }(111,a) \;\) at \(\;a = \) e.g. 2