Properties

Label 1536.31
Modulus $1536$
Conductor $64$
Order $16$
Real no
Primitive no
Minimal no
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1536, base_ring=CyclotomicField(16))
 
M = H._module
 
chi = DirichletCharacter(H, M([8,1,0]))
 
pari: [g,chi] = znchar(Mod(31,1536))
 

Basic properties

Modulus: \(1536\)
Conductor: \(64\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{64}(59,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1536.t

\(\chi_{1536}(31,\cdot)\) \(\chi_{1536}(223,\cdot)\) \(\chi_{1536}(415,\cdot)\) \(\chi_{1536}(607,\cdot)\) \(\chi_{1536}(799,\cdot)\) \(\chi_{1536}(991,\cdot)\) \(\chi_{1536}(1183,\cdot)\) \(\chi_{1536}(1375,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{16})\)
Fixed field: 16.0.604462909807314587353088.1

Values on generators

\((511,517,1025)\) → \((-1,e\left(\frac{1}{16}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 1536 }(31, a) \) \(-1\)\(1\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{15}{16}\right)\)\(-i\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{11}{16}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1536 }(31,a) \;\) at \(\;a = \) e.g. 2