Properties

Label 1792.739
Modulus $1792$
Conductor $1792$
Order $192$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1792, base_ring=CyclotomicField(192))
 
M = H._module
 
chi = DirichletCharacter(H, M([96,177,128]))
 
pari: [g,chi] = znchar(Mod(739,1792))
 

Basic properties

Modulus: \(1792\)
Conductor: \(1792\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(192\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1792.cd

\(\chi_{1792}(11,\cdot)\) \(\chi_{1792}(51,\cdot)\) \(\chi_{1792}(67,\cdot)\) \(\chi_{1792}(107,\cdot)\) \(\chi_{1792}(123,\cdot)\) \(\chi_{1792}(163,\cdot)\) \(\chi_{1792}(179,\cdot)\) \(\chi_{1792}(219,\cdot)\) \(\chi_{1792}(235,\cdot)\) \(\chi_{1792}(275,\cdot)\) \(\chi_{1792}(291,\cdot)\) \(\chi_{1792}(331,\cdot)\) \(\chi_{1792}(347,\cdot)\) \(\chi_{1792}(387,\cdot)\) \(\chi_{1792}(403,\cdot)\) \(\chi_{1792}(443,\cdot)\) \(\chi_{1792}(459,\cdot)\) \(\chi_{1792}(499,\cdot)\) \(\chi_{1792}(515,\cdot)\) \(\chi_{1792}(555,\cdot)\) \(\chi_{1792}(571,\cdot)\) \(\chi_{1792}(611,\cdot)\) \(\chi_{1792}(627,\cdot)\) \(\chi_{1792}(667,\cdot)\) \(\chi_{1792}(683,\cdot)\) \(\chi_{1792}(723,\cdot)\) \(\chi_{1792}(739,\cdot)\) \(\chi_{1792}(779,\cdot)\) \(\chi_{1792}(795,\cdot)\) \(\chi_{1792}(835,\cdot)\) ...

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: $\Q(\zeta_{192})$
Fixed field: Number field defined by a degree 192 polynomial (not computed)

Values on generators

\((1023,1541,1025)\) → \((-1,e\left(\frac{59}{64}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 1792 }(739, a) \) \(-1\)\(1\)\(e\left(\frac{83}{192}\right)\)\(e\left(\frac{49}{192}\right)\)\(e\left(\frac{83}{96}\right)\)\(e\left(\frac{101}{192}\right)\)\(e\left(\frac{21}{64}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{23}{48}\right)\)\(e\left(\frac{7}{192}\right)\)\(e\left(\frac{71}{96}\right)\)\(e\left(\frac{49}{96}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1792 }(739,a) \;\) at \(\;a = \) e.g. 2