Properties

Label 1792.3
Modulus $1792$
Conductor $1792$
Order $192$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: 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([96,105,32]))
 
pari: [g,chi] = znchar(Mod(3,1792))
 

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.cc

\(\chi_{1792}(3,\cdot)\) \(\chi_{1792}(19,\cdot)\) \(\chi_{1792}(59,\cdot)\) \(\chi_{1792}(75,\cdot)\) \(\chi_{1792}(115,\cdot)\) \(\chi_{1792}(131,\cdot)\) \(\chi_{1792}(171,\cdot)\) \(\chi_{1792}(187,\cdot)\) \(\chi_{1792}(227,\cdot)\) \(\chi_{1792}(243,\cdot)\) \(\chi_{1792}(283,\cdot)\) \(\chi_{1792}(299,\cdot)\) \(\chi_{1792}(339,\cdot)\) \(\chi_{1792}(355,\cdot)\) \(\chi_{1792}(395,\cdot)\) \(\chi_{1792}(411,\cdot)\) \(\chi_{1792}(451,\cdot)\) \(\chi_{1792}(467,\cdot)\) \(\chi_{1792}(507,\cdot)\) \(\chi_{1792}(523,\cdot)\) \(\chi_{1792}(563,\cdot)\) \(\chi_{1792}(579,\cdot)\) \(\chi_{1792}(619,\cdot)\) \(\chi_{1792}(635,\cdot)\) \(\chi_{1792}(675,\cdot)\) \(\chi_{1792}(691,\cdot)\) \(\chi_{1792}(731,\cdot)\) \(\chi_{1792}(747,\cdot)\) \(\chi_{1792}(787,\cdot)\) \(\chi_{1792}(803,\cdot)\) ...

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

\((1023,1541,1025)\) → \((-1,e\left(\frac{35}{64}\right),e\left(\frac{1}{6}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(1\)\(1\)\(e\left(\frac{155}{192}\right)\)\(e\left(\frac{73}{192}\right)\)\(e\left(\frac{59}{96}\right)\)\(e\left(\frac{125}{192}\right)\)\(e\left(\frac{13}{64}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{23}{48}\right)\)\(e\left(\frac{175}{192}\right)\)\(e\left(\frac{47}{96}\right)\)\(e\left(\frac{73}{96}\right)\)
value at e.g. 2