Properties

Label 1792.657
Modulus $1792$
Conductor $448$
Order $16$
Real no
Primitive no
Minimal no
Parity odd

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([0,15,8]))
 
pari: [g,chi] = znchar(Mod(657,1792))
 

Basic properties

Modulus: \(1792\)
Conductor: \(448\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{448}(13,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1792.bf

\(\chi_{1792}(209,\cdot)\) \(\chi_{1792}(433,\cdot)\) \(\chi_{1792}(657,\cdot)\) \(\chi_{1792}(881,\cdot)\) \(\chi_{1792}(1105,\cdot)\) \(\chi_{1792}(1329,\cdot)\) \(\chi_{1792}(1553,\cdot)\) \(\chi_{1792}(1777,\cdot)\)

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: 16.0.3484608386920116940487669055488.4

Values on generators

\((1023,1541,1025)\) → \((1,e\left(\frac{15}{16}\right),-1)\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(-1\)\(1\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{9}{16}\right)\)\(-i\)\(-i\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{7}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1792 }(657,a) \;\) at \(\;a = \) e.g. 2