Properties

Label 4033.1561
Modulus $4033$
Conductor $4033$
Order $27$
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(54))
 
M = H._module
 
chi = DirichletCharacter(H, M([48,4]))
 
pari: [g,chi] = znchar(Mod(1561,4033))
 

Basic properties

Modulus: \(4033\)
Conductor: \(4033\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(27\)
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.dx

\(\chi_{4033}(182,\cdot)\) \(\chi_{4033}(514,\cdot)\) \(\chi_{4033}(789,\cdot)\) \(\chi_{4033}(860,\cdot)\) \(\chi_{4033}(996,\cdot)\) \(\chi_{4033}(1006,\cdot)\) \(\chi_{4033}(1329,\cdot)\) \(\chi_{4033}(1439,\cdot)\) \(\chi_{4033}(1529,\cdot)\) \(\chi_{4033}(1561,\cdot)\) \(\chi_{4033}(1607,\cdot)\) \(\chi_{4033}(1792,\cdot)\) \(\chi_{4033}(2051,\cdot)\) \(\chi_{4033}(2338,\cdot)\) \(\chi_{4033}(2734,\cdot)\) \(\chi_{4033}(3786,\cdot)\) \(\chi_{4033}(3820,\cdot)\) \(\chi_{4033}(3931,\cdot)\)

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: \(\Q(\zeta_{27})\)
Fixed field: Number field defined by a degree 27 polynomial

Values on generators

\((1963,2295)\) → \((e\left(\frac{8}{9}\right),e\left(\frac{2}{27}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4033 }(1561, a) \) \(1\)\(1\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{26}{27}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{2}{27}\right)\)\(e\left(\frac{2}{27}\right)\)\(e\left(\frac{11}{27}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{25}{27}\right)\)\(e\left(\frac{5}{27}\right)\)\(e\left(\frac{22}{27}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4033 }(1561,a) \;\) at \(\;a = \) e.g. 2