Properties

Label 4033.919
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([27,17]))
 
pari: [g,chi] = znchar(Mod(919,4033))
 

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.ij

\(\chi_{4033}(179,\cdot)\) \(\chi_{4033}(364,\cdot)\) \(\chi_{4033}(660,\cdot)\) \(\chi_{4033}(820,\cdot)\) \(\chi_{4033}(882,\cdot)\) \(\chi_{4033}(919,\cdot)\) \(\chi_{4033}(931,\cdot)\) \(\chi_{4033}(1005,\cdot)\) \(\chi_{4033}(1104,\cdot)\) \(\chi_{4033}(1141,\cdot)\) \(\chi_{4033}(1252,\cdot)\) \(\chi_{4033}(1264,\cdot)\) \(\chi_{4033}(1326,\cdot)\) \(\chi_{4033}(1375,\cdot)\) \(\chi_{4033}(1622,\cdot)\) \(\chi_{4033}(1733,\cdot)\) \(\chi_{4033}(1893,\cdot)\) \(\chi_{4033}(1992,\cdot)\) \(\chi_{4033}(2041,\cdot)\) \(\chi_{4033}(2140,\cdot)\) \(\chi_{4033}(2300,\cdot)\) \(\chi_{4033}(2411,\cdot)\) \(\chi_{4033}(2658,\cdot)\) \(\chi_{4033}(2707,\cdot)\) \(\chi_{4033}(2769,\cdot)\) \(\chi_{4033}(2781,\cdot)\) \(\chi_{4033}(2892,\cdot)\) \(\chi_{4033}(2929,\cdot)\) \(\chi_{4033}(3028,\cdot)\) \(\chi_{4033}(3102,\cdot)\) ...

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

\((1963,2295)\) → \((i,e\left(\frac{17}{108}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4033 }(919, a) \) \(1\)\(1\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{37}{54}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{77}{108}\right)\)\(e\left(\frac{49}{54}\right)\)\(e\left(\frac{8}{27}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{10}{27}\right)\)\(e\left(\frac{101}{108}\right)\)\(e\left(\frac{61}{108}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4033 }(919,a) \;\) at \(\;a = \) e.g. 2