Properties

Label 5472.439
Modulus $5472$
Conductor $2736$
Order $36$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5472, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([18,27,24,2]))
 
pari: [g,chi] = znchar(Mod(439,5472))
 

Basic properties

Modulus: \(5472\)
Conductor: \(2736\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2736}(1123,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5472.iv

\(\chi_{5472}(439,\cdot)\) \(\chi_{5472}(1447,\cdot)\) \(\chi_{5472}(1591,\cdot)\) \(\chi_{5472}(1687,\cdot)\) \(\chi_{5472}(2119,\cdot)\) \(\chi_{5472}(2407,\cdot)\) \(\chi_{5472}(3175,\cdot)\) \(\chi_{5472}(4183,\cdot)\) \(\chi_{5472}(4327,\cdot)\) \(\chi_{5472}(4423,\cdot)\) \(\chi_{5472}(4855,\cdot)\) \(\chi_{5472}(5143,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: Number field defined by a degree 36 polynomial

Values on generators

\((4447,2053,1217,3745)\) → \((-1,-i,e\left(\frac{2}{3}\right),e\left(\frac{1}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 5472 }(439, a) \) \(1\)\(1\)\(e\left(\frac{35}{36}\right)\)\(1\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{31}{36}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{31}{36}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{35}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5472 }(439,a) \;\) at \(\;a = \) e.g. 2