Properties

Label 64.i
Modulus $64$
Conductor $64$
Order $16$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(64, base_ring=CyclotomicField(16))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,1]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(5,64))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(64\)
Conductor: \(64\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
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)
 

Related number fields

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

Characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(5\) \(7\) \(9\) \(11\) \(13\) \(15\) \(17\) \(19\) \(21\)
\(\chi_{64}(5,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{15}{16}\right)\) \(i\) \(-i\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{13}{16}\right)\)
\(\chi_{64}(13,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{1}{16}\right)\) \(-i\) \(i\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{3}{16}\right)\)
\(\chi_{64}(21,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{3}{16}\right)\) \(i\) \(-i\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{9}{16}\right)\)
\(\chi_{64}(29,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{5}{16}\right)\) \(-i\) \(i\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{15}{16}\right)\)
\(\chi_{64}(37,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{7}{16}\right)\) \(i\) \(-i\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{5}{16}\right)\)
\(\chi_{64}(45,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{9}{16}\right)\) \(-i\) \(i\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{11}{16}\right)\)
\(\chi_{64}(53,\cdot)\) \(1\) \(1\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{11}{16}\right)\) \(i\) \(-i\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{1}{16}\right)\)
\(\chi_{64}(61,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{13}{16}\right)\) \(-i\) \(i\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{7}{16}\right)\)