Properties

Label 1792.97
Modulus $1792$
Conductor $224$
Order $8$
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(8))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,5,4]))
 
pari: [g,chi] = znchar(Mod(97,1792))
 

Basic properties

Modulus: \(1792\)
Conductor: \(224\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{224}(181,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1792.v

\(\chi_{1792}(97,\cdot)\) \(\chi_{1792}(545,\cdot)\) \(\chi_{1792}(993,\cdot)\) \(\chi_{1792}(1441,\cdot)\)

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: \(\Q(\zeta_{8})\)
Fixed field: 8.0.5156108238848.1

Values on generators

\((1023,1541,1025)\) → \((1,e\left(\frac{5}{8}\right),-1)\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(-1\)\(1\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(-i\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(-1\)\(1\)\(e\left(\frac{7}{8}\right)\)\(-i\)\(i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1792 }(97,a) \;\) at \(\;a = \) e.g. 2