Properties

Label 1792.113
Modulus $1792$
Conductor $64$
Order $16$
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(16))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,1,0]))
 
pari: [g,chi] = znchar(Mod(113,1792))
 

Basic properties

Modulus: \(1792\)
Conductor: \(64\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{64}(5,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1792.bc

\(\chi_{1792}(113,\cdot)\) \(\chi_{1792}(337,\cdot)\) \(\chi_{1792}(561,\cdot)\) \(\chi_{1792}(785,\cdot)\) \(\chi_{1792}(1009,\cdot)\) \(\chi_{1792}(1233,\cdot)\) \(\chi_{1792}(1457,\cdot)\) \(\chi_{1792}(1681,\cdot)\)

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: \(\Q(\zeta_{16})\)
Fixed field: \(\Q(\zeta_{64})^+\)

Values on generators

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

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(1\)\(1\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{15}{16}\right)\)\(i\)\(-i\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{1}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1792 }(113,a) \;\) at \(\;a = \) e.g. 2