Properties

Label 1792.1069
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([0,21,160]))
 
pari: [g,chi] = znchar(Mod(1069,1792))
 

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.ca

\(\chi_{1792}(5,\cdot)\) \(\chi_{1792}(45,\cdot)\) \(\chi_{1792}(61,\cdot)\) \(\chi_{1792}(101,\cdot)\) \(\chi_{1792}(117,\cdot)\) \(\chi_{1792}(157,\cdot)\) \(\chi_{1792}(173,\cdot)\) \(\chi_{1792}(213,\cdot)\) \(\chi_{1792}(229,\cdot)\) \(\chi_{1792}(269,\cdot)\) \(\chi_{1792}(285,\cdot)\) \(\chi_{1792}(325,\cdot)\) \(\chi_{1792}(341,\cdot)\) \(\chi_{1792}(381,\cdot)\) \(\chi_{1792}(397,\cdot)\) \(\chi_{1792}(437,\cdot)\) \(\chi_{1792}(453,\cdot)\) \(\chi_{1792}(493,\cdot)\) \(\chi_{1792}(509,\cdot)\) \(\chi_{1792}(549,\cdot)\) \(\chi_{1792}(565,\cdot)\) \(\chi_{1792}(605,\cdot)\) \(\chi_{1792}(621,\cdot)\) \(\chi_{1792}(661,\cdot)\) \(\chi_{1792}(677,\cdot)\) \(\chi_{1792}(717,\cdot)\) \(\chi_{1792}(733,\cdot)\) \(\chi_{1792}(773,\cdot)\) \(\chi_{1792}(789,\cdot)\) \(\chi_{1792}(829,\cdot)\) ...

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

\((1023,1541,1025)\) → \((1,e\left(\frac{7}{64}\right),e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 1792 }(1069, a) \) \(-1\)\(1\)\(e\left(\frac{127}{192}\right)\)\(e\left(\frac{53}{192}\right)\)\(e\left(\frac{31}{96}\right)\)\(e\left(\frac{121}{192}\right)\)\(e\left(\frac{41}{64}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{43}{48}\right)\)\(e\left(\frac{131}{192}\right)\)\(e\left(\frac{19}{96}\right)\)\(e\left(\frac{53}{96}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1792 }(1069,a) \;\) at \(\;a = \) e.g. 2