Properties

Label 4000.1919
Modulus $4000$
Conductor $500$
Order $50$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4000, base_ring=CyclotomicField(50))
 
M = H._module
 
chi = DirichletCharacter(H, M([25,0,39]))
 
pari: [g,chi] = znchar(Mod(1919,4000))
 

Basic properties

Modulus: \(4000\)
Conductor: \(500\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(50\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{500}(419,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4000.cf

\(\chi_{4000}(159,\cdot)\) \(\chi_{4000}(319,\cdot)\) \(\chi_{4000}(479,\cdot)\) \(\chi_{4000}(639,\cdot)\) \(\chi_{4000}(959,\cdot)\) \(\chi_{4000}(1119,\cdot)\) \(\chi_{4000}(1279,\cdot)\) \(\chi_{4000}(1439,\cdot)\) \(\chi_{4000}(1759,\cdot)\) \(\chi_{4000}(1919,\cdot)\) \(\chi_{4000}(2079,\cdot)\) \(\chi_{4000}(2239,\cdot)\) \(\chi_{4000}(2559,\cdot)\) \(\chi_{4000}(2719,\cdot)\) \(\chi_{4000}(2879,\cdot)\) \(\chi_{4000}(3039,\cdot)\) \(\chi_{4000}(3359,\cdot)\) \(\chi_{4000}(3519,\cdot)\) \(\chi_{4000}(3679,\cdot)\) \(\chi_{4000}(3839,\cdot)\)

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

\((2751,2501,1377)\) → \((-1,1,e\left(\frac{39}{50}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 4000 }(1919, a) \) \(-1\)\(1\)\(e\left(\frac{24}{25}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{23}{25}\right)\)\(e\left(\frac{39}{50}\right)\)\(e\left(\frac{21}{50}\right)\)\(e\left(\frac{47}{50}\right)\)\(e\left(\frac{27}{50}\right)\)\(e\left(\frac{19}{25}\right)\)\(e\left(\frac{17}{25}\right)\)\(e\left(\frac{22}{25}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4000 }(1919,a) \;\) at \(\;a = \) e.g. 2