Properties

Label 2368.221
Modulus $2368$
Conductor $2368$
Order $16$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2368, base_ring=CyclotomicField(16))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,11,8]))
 
pari: [g,chi] = znchar(Mod(221,2368))
 

Basic properties

Modulus: \(2368\)
Conductor: \(2368\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
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 2368.cc

\(\chi_{2368}(221,\cdot)\) \(\chi_{2368}(517,\cdot)\) \(\chi_{2368}(813,\cdot)\) \(\chi_{2368}(1109,\cdot)\) \(\chi_{2368}(1405,\cdot)\) \(\chi_{2368}(1701,\cdot)\) \(\chi_{2368}(1997,\cdot)\) \(\chi_{2368}(2293,\cdot)\)

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: \(\Q(\zeta_{16})\)
Fixed field: 16.16.2123163551355495017117431991053058048.1

Values on generators

\((1407,1925,705)\) → \((1,e\left(\frac{11}{16}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 2368 }(221, a) \) \(1\)\(1\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{13}{16}\right)\)\(i\)\(-i\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{15}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2368 }(221,a) \;\) at \(\;a = \) e.g. 2