Properties

Label 6027.1030
Modulus $6027$
Conductor $41$
Order $20$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6027, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,11]))
 
pari: [g,chi] = znchar(Mod(1030,6027))
 

Basic properties

Modulus: \(6027\)
Conductor: \(41\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{41}(5,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6027.bw

\(\chi_{6027}(295,\cdot)\) \(\chi_{6027}(736,\cdot)\) \(\chi_{6027}(1030,\cdot)\) \(\chi_{6027}(1471,\cdot)\) \(\chi_{6027}(1765,\cdot)\) \(\chi_{6027}(2206,\cdot)\) \(\chi_{6027}(3382,\cdot)\) \(\chi_{6027}(5146,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: Number field defined by a degree 20 polynomial

Values on generators

\((4019,493,2794)\) → \((1,1,e\left(\frac{11}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 6027 }(1030, a) \) \(1\)\(1\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{19}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6027 }(1030,a) \;\) at \(\;a = \) e.g. 2