Properties

Label 4033.769
Modulus $4033$
Conductor $4033$
Order $108$
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(4033, base_ring=CyclotomicField(108))
 
M = H._module
 
chi = DirichletCharacter(H, M([63,1]))
 
pari: [g,chi] = znchar(Mod(769,4033))
 

Basic properties

Modulus: \(4033\)
Conductor: \(4033\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(108\)
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 4033.ii

\(\chi_{4033}(171,\cdot)\) \(\chi_{4033}(236,\cdot)\) \(\chi_{4033}(643,\cdot)\) \(\chi_{4033}(711,\cdot)\) \(\chi_{4033}(726,\cdot)\) \(\chi_{4033}(769,\cdot)\) \(\chi_{4033}(896,\cdot)\) \(\chi_{4033}(902,\cdot)\) \(\chi_{4033}(911,\cdot)\) \(\chi_{4033}(991,\cdot)\) \(\chi_{4033}(1213,\cdot)\) \(\chi_{4033}(1250,\cdot)\) \(\chi_{4033}(1361,\cdot)\) \(\chi_{4033}(1568,\cdot)\) \(\chi_{4033}(1679,\cdot)\) \(\chi_{4033}(1731,\cdot)\) \(\chi_{4033}(1784,\cdot)\) \(\chi_{4033}(2012,\cdot)\) \(\chi_{4033}(2021,\cdot)\) \(\chi_{4033}(2249,\cdot)\) \(\chi_{4033}(2302,\cdot)\) \(\chi_{4033}(2354,\cdot)\) \(\chi_{4033}(2465,\cdot)\) \(\chi_{4033}(2672,\cdot)\) \(\chi_{4033}(2783,\cdot)\) \(\chi_{4033}(2820,\cdot)\) \(\chi_{4033}(3042,\cdot)\) \(\chi_{4033}(3122,\cdot)\) \(\chi_{4033}(3131,\cdot)\) \(\chi_{4033}(3137,\cdot)\) ...

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: $\Q(\zeta_{108})$
Fixed field: Number field defined by a degree 108 polynomial (not computed)

Values on generators

\((1963,2295)\) → \((e\left(\frac{7}{12}\right),e\left(\frac{1}{108}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4033 }(769, a) \) \(1\)\(1\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{35}{54}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{13}{108}\right)\)\(e\left(\frac{41}{54}\right)\)\(e\left(\frac{1}{27}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{8}{27}\right)\)\(e\left(\frac{25}{108}\right)\)\(e\left(\frac{29}{108}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4033 }(769,a) \;\) at \(\;a = \) e.g. 2