Properties

Label 1792.1283
Modulus $1792$
Conductor $1792$
Order $192$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1792, base_ring=CyclotomicField(192))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([96,105,64]))
 
pari: [g,chi] = znchar(Mod(1283,1792))
 

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()
 
pari: order = charorder(g,chi)
 
pari: [ 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

\((1023,1541,1025)\) → \((-1,e\left(\frac{35}{64}\right),e\left(\frac{1}{3}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(-1\)\(1\)\(e\left(\frac{187}{192}\right)\)\(e\left(\frac{41}{192}\right)\)\(e\left(\frac{91}{96}\right)\)\(e\left(\frac{61}{192}\right)\)\(e\left(\frac{45}{64}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{31}{48}\right)\)\(e\left(\frac{143}{192}\right)\)\(e\left(\frac{79}{96}\right)\)\(e\left(\frac{41}{96}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1792 }(1283,a) \;\) at \(\;a = \) e.g. 2