Properties

Label 4000.2051
Modulus $4000$
Conductor $800$
Order $40$
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(40))
 
M = H._module
 
chi = DirichletCharacter(H, M([20,15,32]))
 
pari: [g,chi] = znchar(Mod(2051,4000))
 

Basic properties

Modulus: \(4000\)
Conductor: \(800\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(40\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{800}(611,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4000.cc

\(\chi_{4000}(51,\cdot)\) \(\chi_{4000}(451,\cdot)\) \(\chi_{4000}(651,\cdot)\) \(\chi_{4000}(851,\cdot)\) \(\chi_{4000}(1051,\cdot)\) \(\chi_{4000}(1451,\cdot)\) \(\chi_{4000}(1651,\cdot)\) \(\chi_{4000}(1851,\cdot)\) \(\chi_{4000}(2051,\cdot)\) \(\chi_{4000}(2451,\cdot)\) \(\chi_{4000}(2651,\cdot)\) \(\chi_{4000}(2851,\cdot)\) \(\chi_{4000}(3051,\cdot)\) \(\chi_{4000}(3451,\cdot)\) \(\chi_{4000}(3651,\cdot)\) \(\chi_{4000}(3851,\cdot)\)

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: \(\Q(\zeta_{40})\)
Fixed field: Number field defined by a degree 40 polynomial

Values on generators

\((2751,2501,1377)\) → \((-1,e\left(\frac{3}{8}\right),e\left(\frac{4}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 4000 }(2051, a) \) \(-1\)\(1\)\(e\left(\frac{9}{40}\right)\)\(i\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{7}{40}\right)\)\(e\left(\frac{33}{40}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{21}{40}\right)\)\(e\left(\frac{19}{40}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{27}{40}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4000 }(2051,a) \;\) at \(\;a = \) e.g. 2