Properties

Label 4033.2075
Modulus $4033$
Conductor $4033$
Order $18$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4033, base_ring=CyclotomicField(18))
 
M = H._module
 
chi = DirichletCharacter(H, M([13,1]))
 
pari: [g,chi] = znchar(Mod(2075,4033))
 

Basic properties

Modulus: \(4033\)
Conductor: \(4033\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
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.cr

\(\chi_{4033}(289,\cdot)\) \(\chi_{4033}(1172,\cdot)\) \(\chi_{4033}(1242,\cdot)\) \(\chi_{4033}(1619,\cdot)\) \(\chi_{4033}(1669,\cdot)\) \(\chi_{4033}(2075,\cdot)\)

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: \(\Q(\zeta_{9})\)
Fixed field: Number field defined by a degree 18 polynomial

Values on generators

\((1963,2295)\) → \((e\left(\frac{13}{18}\right),e\left(\frac{1}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4033 }(2075, a) \) \(1\)\(1\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{5}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4033 }(2075,a) \;\) at \(\;a = \) e.g. 2