Properties

Label 2169.11
Modulus $2169$
Conductor $2169$
Order $48$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2169, base_ring=CyclotomicField(48))
 
M = H._module
 
chi = DirichletCharacter(H, M([8,5]))
 
pari: [g,chi] = znchar(Mod(11,2169))
 

Basic properties

Modulus: \(2169\)
Conductor: \(2169\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(48\)
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 2169.cu

\(\chi_{2169}(11,\cdot)\) \(\chi_{2169}(176,\cdot)\) \(\chi_{2169}(203,\cdot)\) \(\chi_{2169}(419,\cdot)\) \(\chi_{2169}(545,\cdot)\) \(\chi_{2169}(704,\cdot)\) \(\chi_{2169}(761,\cdot)\) \(\chi_{2169}(788,\cdot)\) \(\chi_{2169}(812,\cdot)\) \(\chi_{2169}(875,\cdot)\) \(\chi_{2169}(983,\cdot)\) \(\chi_{2169}(986,\cdot)\) \(\chi_{2169}(1676,\cdot)\) \(\chi_{2169}(1775,\cdot)\) \(\chi_{2169}(2081,\cdot)\) \(\chi_{2169}(2147,\cdot)\)

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: \(\Q(\zeta_{48})\)
Fixed field: Number field defined by a degree 48 polynomial

Values on generators

\((965,730)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{5}{48}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 2169 }(11, a) \) \(1\)\(1\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{37}{48}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{37}{48}\right)\)\(e\left(\frac{11}{48}\right)\)\(e\left(\frac{35}{48}\right)\)\(e\left(\frac{5}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2169 }(11,a) \;\) at \(\;a = \) e.g. 2