Properties

Label 1792.149
Modulus $1792$
Conductor $1792$
Order $192$
Real no
Primitive yes
Minimal yes
Parity even

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([0,39,64]))
 
pari: [g,chi] = znchar(Mod(149,1792))
 

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()
 
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

\((1023,1541,1025)\) → \((1,e\left(\frac{13}{64}\right),e\left(\frac{1}{3}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(1\)\(1\)\(e\left(\frac{85}{192}\right)\)\(e\left(\frac{167}{192}\right)\)\(e\left(\frac{85}{96}\right)\)\(e\left(\frac{115}{192}\right)\)\(e\left(\frac{35}{64}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{1}{48}\right)\)\(e\left(\frac{65}{192}\right)\)\(e\left(\frac{49}{96}\right)\)\(e\left(\frac{71}{96}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1792 }(149,a) \;\) at \(\;a = \) e.g. 2