Properties

Label 2169.121
Modulus $2169$
Conductor $2169$
Order $24$
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(24))
 
M = H._module
 
chi = DirichletCharacter(H, M([8,5]))
 
pari: [g,chi] = znchar(Mod(121,2169))
 

Basic properties

Modulus: \(2169\)
Conductor: \(2169\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(24\)
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.bx

\(\chi_{2169}(121,\cdot)\) \(\chi_{2169}(484,\cdot)\) \(\chi_{2169}(610,\cdot)\) \(\chi_{2169}(1084,\cdot)\) \(\chi_{2169}(1237,\cdot)\) \(\chi_{2169}(2041,\cdot)\) \(\chi_{2169}(2137,\cdot)\) \(\chi_{2169}(2167,\cdot)\)

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: \(\Q(\zeta_{24})\)
Fixed field: Number field defined by a degree 24 polynomial

Values on generators

\((965,730)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{5}{24}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 2169 }(121, a) \) \(1\)\(1\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{13}{24}\right)\)\(-i\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{13}{24}\right)\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{2}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2169 }(121,a) \;\) at \(\;a = \) e.g. 2