Properties

Label 1792.47
Modulus $1792$
Conductor $448$
Order $48$
Real no
Primitive no
Minimal no
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(48))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([24,33,40]))
 
pari: [g,chi] = znchar(Mod(47,1792))
 

Basic properties

Modulus: \(1792\)
Conductor: \(448\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(48\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{448}(355,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1792.bq

\(\chi_{1792}(47,\cdot)\) \(\chi_{1792}(143,\cdot)\) \(\chi_{1792}(271,\cdot)\) \(\chi_{1792}(367,\cdot)\) \(\chi_{1792}(495,\cdot)\) \(\chi_{1792}(591,\cdot)\) \(\chi_{1792}(719,\cdot)\) \(\chi_{1792}(815,\cdot)\) \(\chi_{1792}(943,\cdot)\) \(\chi_{1792}(1039,\cdot)\) \(\chi_{1792}(1167,\cdot)\) \(\chi_{1792}(1263,\cdot)\) \(\chi_{1792}(1391,\cdot)\) \(\chi_{1792}(1487,\cdot)\) \(\chi_{1792}(1615,\cdot)\) \(\chi_{1792}(1711,\cdot)\)

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: \(\Q(\zeta_{48})\)
Fixed field: Number field defined by a degree 48 polynomial

Values on generators

\((1023,1541,1025)\) → \((-1,e\left(\frac{11}{16}\right),e\left(\frac{5}{6}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(1\)\(1\)\(e\left(\frac{19}{48}\right)\)\(e\left(\frac{41}{48}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{13}{48}\right)\)\(e\left(\frac{13}{16}\right)\)\(i\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{23}{48}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{17}{24}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1792 }(47,a) \;\) at \(\;a = \) e.g. 2