Properties

Label 4026.569
Modulus $4026$
Conductor $2013$
Order $10$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4026, base_ring=CyclotomicField(10))
 
M = H._module
 
chi = DirichletCharacter(H, M([5,3,4]))
 
pari: [g,chi] = znchar(Mod(569,4026))
 

Basic properties

Modulus: \(4026\)
Conductor: \(2013\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(10\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2013}(569,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4026.bd

\(\chi_{4026}(569,\cdot)\) \(\chi_{4026}(827,\cdot)\) \(\chi_{4026}(1217,\cdot)\) \(\chi_{4026}(2327,\cdot)\)

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: \(\Q(\zeta_{5})\)
Fixed field: Number field defined by a degree 10 polynomial

Values on generators

\((1343,1465,3235)\) → \((-1,e\left(\frac{3}{10}\right),e\left(\frac{2}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 4026 }(569, a) \) \(1\)\(1\)\(-1\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{3}{10}\right)\)\(1\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{3}{10}\right)\)\(1\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{1}{5}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4026 }(569,a) \;\) at \(\;a = \) e.g. 2