Properties

Label 1792.81
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([0,9,32]))
 
pari: [g,chi] = znchar(Mod(81,1792))
 

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 \(\chi_{448}(445,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1792.br

\(\chi_{1792}(81,\cdot)\) \(\chi_{1792}(177,\cdot)\) \(\chi_{1792}(305,\cdot)\) \(\chi_{1792}(401,\cdot)\) \(\chi_{1792}(529,\cdot)\) \(\chi_{1792}(625,\cdot)\) \(\chi_{1792}(753,\cdot)\) \(\chi_{1792}(849,\cdot)\) \(\chi_{1792}(977,\cdot)\) \(\chi_{1792}(1073,\cdot)\) \(\chi_{1792}(1201,\cdot)\) \(\chi_{1792}(1297,\cdot)\) \(\chi_{1792}(1425,\cdot)\) \(\chi_{1792}(1521,\cdot)\) \(\chi_{1792}(1649,\cdot)\) \(\chi_{1792}(1745,\cdot)\)

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

\((1023,1541,1025)\) → \((1,e\left(\frac{3}{16}\right),e\left(\frac{2}{3}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(1\)\(1\)\(e\left(\frac{11}{48}\right)\)\(e\left(\frac{25}{48}\right)\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{29}{48}\right)\)\(e\left(\frac{13}{16}\right)\)\(-i\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{31}{48}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{1}{24}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1792 }(81,a) \;\) at \(\;a = \) e.g. 2