Properties

Label 1792.1117
Modulus $1792$
Conductor $1792$
Order $192$
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(192))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,129,128]))
 
pari: [g,chi] = znchar(Mod(1117,1792))
 

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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1792.cb

\(\chi_{1792}(37,\cdot)\) \(\chi_{1792}(53,\cdot)\) \(\chi_{1792}(93,\cdot)\) \(\chi_{1792}(109,\cdot)\) \(\chi_{1792}(149,\cdot)\) \(\chi_{1792}(165,\cdot)\) \(\chi_{1792}(205,\cdot)\) \(\chi_{1792}(221,\cdot)\) \(\chi_{1792}(261,\cdot)\) \(\chi_{1792}(277,\cdot)\) \(\chi_{1792}(317,\cdot)\) \(\chi_{1792}(333,\cdot)\) \(\chi_{1792}(373,\cdot)\) \(\chi_{1792}(389,\cdot)\) \(\chi_{1792}(429,\cdot)\) \(\chi_{1792}(445,\cdot)\) \(\chi_{1792}(485,\cdot)\) \(\chi_{1792}(501,\cdot)\) \(\chi_{1792}(541,\cdot)\) \(\chi_{1792}(557,\cdot)\) \(\chi_{1792}(597,\cdot)\) \(\chi_{1792}(613,\cdot)\) \(\chi_{1792}(653,\cdot)\) \(\chi_{1792}(669,\cdot)\) \(\chi_{1792}(709,\cdot)\) \(\chi_{1792}(725,\cdot)\) \(\chi_{1792}(765,\cdot)\) \(\chi_{1792}(781,\cdot)\) \(\chi_{1792}(821,\cdot)\) \(\chi_{1792}(837,\cdot)\) ...

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

\((1023,1541,1025)\) → \((1,e\left(\frac{43}{64}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 1792 }(1117, a) \) \(1\)\(1\)\(e\left(\frac{35}{192}\right)\)\(e\left(\frac{1}{192}\right)\)\(e\left(\frac{35}{96}\right)\)\(e\left(\frac{149}{192}\right)\)\(e\left(\frac{37}{64}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{23}{48}\right)\)\(e\left(\frac{151}{192}\right)\)\(e\left(\frac{71}{96}\right)\)\(e\left(\frac{1}{96}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1792 }(1117,a) \;\) at \(\;a = \) e.g. 2