Properties

Label 4027.13
Modulus $4027$
Conductor $4027$
Order $61$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4027, base_ring=CyclotomicField(122))
 
M = H._module
 
chi = DirichletCharacter(H, M([2]))
 
pari: [g,chi] = znchar(Mod(13,4027))
 

Basic properties

Modulus: \(4027\)
Conductor: \(4027\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(61\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4027.h

\(\chi_{4027}(13,\cdot)\) \(\chi_{4027}(67,\cdot)\) \(\chi_{4027}(153,\cdot)\) \(\chi_{4027}(169,\cdot)\) \(\chi_{4027}(210,\cdot)\) \(\chi_{4027}(211,\cdot)\) \(\chi_{4027}(224,\cdot)\) \(\chi_{4027}(326,\cdot)\) \(\chi_{4027}(327,\cdot)\) \(\chi_{4027}(372,\cdot)\) \(\chi_{4027}(462,\cdot)\) \(\chi_{4027}(538,\cdot)\) \(\chi_{4027}(756,\cdot)\) \(\chi_{4027}(762,\cdot)\) \(\chi_{4027}(809,\cdot)\) \(\chi_{4027}(834,\cdot)\) \(\chi_{4027}(871,\cdot)\) \(\chi_{4027}(1142,\cdot)\) \(\chi_{4027}(1466,\cdot)\) \(\chi_{4027}(1554,\cdot)\) \(\chi_{4027}(1565,\cdot)\) \(\chi_{4027}(1574,\cdot)\) \(\chi_{4027}(1607,\cdot)\) \(\chi_{4027}(1613,\cdot)\) \(\chi_{4027}(1695,\cdot)\) \(\chi_{4027}(1707,\cdot)\) \(\chi_{4027}(1774,\cdot)\) \(\chi_{4027}(1808,\cdot)\) \(\chi_{4027}(1852,\cdot)\) \(\chi_{4027}(1900,\cdot)\) ...

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: $\Q(\zeta_{61})$
Fixed field: Number field defined by a degree 61 polynomial

Values on generators

\(3\) → \(e\left(\frac{1}{61}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4027 }(13, a) \) \(1\)\(1\)\(e\left(\frac{9}{61}\right)\)\(e\left(\frac{1}{61}\right)\)\(e\left(\frac{18}{61}\right)\)\(e\left(\frac{47}{61}\right)\)\(e\left(\frac{10}{61}\right)\)\(e\left(\frac{52}{61}\right)\)\(e\left(\frac{27}{61}\right)\)\(e\left(\frac{2}{61}\right)\)\(e\left(\frac{56}{61}\right)\)\(e\left(\frac{32}{61}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4027 }(13,a) \;\) at \(\;a = \) e.g. 2