Properties

Label 3072.641
Modulus $3072$
Conductor $96$
Order $8$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3072, base_ring=CyclotomicField(8))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,3,4]))
 
pari: [g,chi] = znchar(Mod(641,3072))
 

Basic properties

Modulus: \(3072\)
Conductor: \(96\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{96}(29,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3072.p

\(\chi_{3072}(641,\cdot)\) \(\chi_{3072}(1409,\cdot)\) \(\chi_{3072}(2177,\cdot)\) \(\chi_{3072}(2945,\cdot)\)

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: 8.0.173946175488.1

Values on generators

\((2047,2053,1025)\) → \((1,e\left(\frac{3}{8}\right),-1)\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{7}{8}\right)\)\(-i\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{5}{8}\right)\)\(1\)\(e\left(\frac{5}{8}\right)\)\(-i\)\(-i\)\(e\left(\frac{5}{8}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3072 }(641,a) \;\) at \(\;a = \) e.g. 2