Properties

Label 1792.1147
Modulus $1792$
Conductor $1792$
Order $64$
Real no
Primitive yes
Minimal yes
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(64))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([32,33,32]))
 
pari: [g,chi] = znchar(Mod(1147,1792))
 

Basic properties

Modulus: \(1792\)
Conductor: \(1792\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(64\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1792.bv

\(\chi_{1792}(27,\cdot)\) \(\chi_{1792}(83,\cdot)\) \(\chi_{1792}(139,\cdot)\) \(\chi_{1792}(195,\cdot)\) \(\chi_{1792}(251,\cdot)\) \(\chi_{1792}(307,\cdot)\) \(\chi_{1792}(363,\cdot)\) \(\chi_{1792}(419,\cdot)\) \(\chi_{1792}(475,\cdot)\) \(\chi_{1792}(531,\cdot)\) \(\chi_{1792}(587,\cdot)\) \(\chi_{1792}(643,\cdot)\) \(\chi_{1792}(699,\cdot)\) \(\chi_{1792}(755,\cdot)\) \(\chi_{1792}(811,\cdot)\) \(\chi_{1792}(867,\cdot)\) \(\chi_{1792}(923,\cdot)\) \(\chi_{1792}(979,\cdot)\) \(\chi_{1792}(1035,\cdot)\) \(\chi_{1792}(1091,\cdot)\) \(\chi_{1792}(1147,\cdot)\) \(\chi_{1792}(1203,\cdot)\) \(\chi_{1792}(1259,\cdot)\) \(\chi_{1792}(1315,\cdot)\) \(\chi_{1792}(1371,\cdot)\) \(\chi_{1792}(1427,\cdot)\) \(\chi_{1792}(1483,\cdot)\) \(\chi_{1792}(1539,\cdot)\) \(\chi_{1792}(1595,\cdot)\) \(\chi_{1792}(1651,\cdot)\) ...

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: $\Q(\zeta_{64})$
Fixed field: Number field defined by a degree 64 polynomial

Values on generators

\((1023,1541,1025)\) → \((-1,e\left(\frac{33}{64}\right),-1)\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(1\)\(1\)\(e\left(\frac{3}{64}\right)\)\(e\left(\frac{1}{64}\right)\)\(e\left(\frac{3}{32}\right)\)\(e\left(\frac{21}{64}\right)\)\(e\left(\frac{47}{64}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{55}{64}\right)\)\(e\left(\frac{23}{32}\right)\)\(e\left(\frac{1}{32}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1792 }(1147,a) \;\) at \(\;a = \) e.g. 2