Properties

Label 1792.923
Modulus $1792$
Conductor $1792$
Order $64$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1792, base_ring=CyclotomicField(64))
 
M = H._module
 
chi = DirichletCharacter(H, M([32,9,32]))
 
pari: [g,chi] = znchar(Mod(923,1792))
 

Basic properties

Modulus: \(1792\)
Conductor: \(1792\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(64\)
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 1792.bv

\(\chi_{1792}(27,\cdot)\) \(\chi_{1792}(83,\cdot)\) \(\chi_{1792}(139,\cdot)\) \(\chi_{1792}(195,\cdot)\) \(\chi_{1792}(251,\cdot)\) \(\chi_{1792}(307,\cdot)\) \(\chi_{1792}(363,\cdot)\) \(\chi_{1792}(419,\cdot)\) \(\chi_{1792}(475,\cdot)\) \(\chi_{1792}(531,\cdot)\) \(\chi_{1792}(587,\cdot)\) \(\chi_{1792}(643,\cdot)\) \(\chi_{1792}(699,\cdot)\) \(\chi_{1792}(755,\cdot)\) \(\chi_{1792}(811,\cdot)\) \(\chi_{1792}(867,\cdot)\) \(\chi_{1792}(923,\cdot)\) \(\chi_{1792}(979,\cdot)\) \(\chi_{1792}(1035,\cdot)\) \(\chi_{1792}(1091,\cdot)\) \(\chi_{1792}(1147,\cdot)\) \(\chi_{1792}(1203,\cdot)\) \(\chi_{1792}(1259,\cdot)\) \(\chi_{1792}(1315,\cdot)\) \(\chi_{1792}(1371,\cdot)\) \(\chi_{1792}(1427,\cdot)\) \(\chi_{1792}(1483,\cdot)\) \(\chi_{1792}(1539,\cdot)\) \(\chi_{1792}(1595,\cdot)\) \(\chi_{1792}(1651,\cdot)\) ...

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: $\Q(\zeta_{64})$
Fixed field: Number field defined by a degree 64 polynomial

Values on generators

\((1023,1541,1025)\) → \((-1,e\left(\frac{9}{64}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 1792 }(923, a) \) \(1\)\(1\)\(e\left(\frac{59}{64}\right)\)\(e\left(\frac{41}{64}\right)\)\(e\left(\frac{27}{32}\right)\)\(e\left(\frac{29}{64}\right)\)\(e\left(\frac{7}{64}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{15}{64}\right)\)\(e\left(\frac{15}{32}\right)\)\(e\left(\frac{9}{32}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1792 }(923,a) \;\) at \(\;a = \) e.g. 2