Properties

Label 8033.4038
Modulus $8033$
Conductor $8033$
Order $21$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8033, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([18,28]))
 
pari: [g,chi] = znchar(Mod(4038,8033))
 

Basic properties

Modulus: \(8033\)
Conductor: \(8033\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(21\)
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 8033.y

\(\chi_{8033}(393,\cdot)\) \(\chi_{8033}(1822,\cdot)\) \(\chi_{8033}(2055,\cdot)\) \(\chi_{8033}(3761,\cdot)\) \(\chi_{8033}(4038,\cdot)\) \(\chi_{8033}(4315,\cdot)\) \(\chi_{8033}(4548,\cdot)\) \(\chi_{8033}(5700,\cdot)\) \(\chi_{8033}(6487,\cdot)\) \(\chi_{8033}(6764,\cdot)\) \(\chi_{8033}(7041,\cdot)\) \(\chi_{8033}(7362,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 21 polynomial

Values on generators

\((5541,1944)\) → \((e\left(\frac{3}{7}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8033 }(4038, a) \) \(1\)\(1\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{8}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8033 }(4038,a) \;\) at \(\;a = \) e.g. 2