Properties

Label 2176.237
Modulus $2176$
Conductor $2176$
Order $32$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2176, base_ring=CyclotomicField(32))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,23,16]))
 
pari: [g,chi] = znchar(Mod(237,2176))
 

Basic properties

Modulus: \(2176\)
Conductor: \(2176\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(32\)
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 2176.ej

\(\chi_{2176}(101,\cdot)\) \(\chi_{2176}(237,\cdot)\) \(\chi_{2176}(373,\cdot)\) \(\chi_{2176}(509,\cdot)\) \(\chi_{2176}(645,\cdot)\) \(\chi_{2176}(781,\cdot)\) \(\chi_{2176}(917,\cdot)\) \(\chi_{2176}(1053,\cdot)\) \(\chi_{2176}(1189,\cdot)\) \(\chi_{2176}(1325,\cdot)\) \(\chi_{2176}(1461,\cdot)\) \(\chi_{2176}(1597,\cdot)\) \(\chi_{2176}(1733,\cdot)\) \(\chi_{2176}(1869,\cdot)\) \(\chi_{2176}(2005,\cdot)\) \(\chi_{2176}(2141,\cdot)\)

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: \(\Q(\zeta_{32})\)
Fixed field: 32.32.152725625984366375633872344931707463035520335286488090842213114237373265215488.1

Values on generators

\((511,1157,513)\) → \((1,e\left(\frac{23}{32}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(19\)\(21\)\(23\)
\( \chi_{ 2176 }(237, a) \) \(1\)\(1\)\(e\left(\frac{21}{32}\right)\)\(e\left(\frac{7}{32}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{19}{32}\right)\)\(e\left(\frac{25}{32}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{17}{32}\right)\)\(e\left(\frac{11}{32}\right)\)\(e\left(\frac{9}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2176 }(237,a) \;\) at \(\;a = \) e.g. 2