Properties

Label 3264.41
Modulus $3264$
Conductor $1632$
Order $16$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3264, base_ring=CyclotomicField(16))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,14,8,11]))
 
pari: [g,chi] = znchar(Mod(41,3264))
 

Basic properties

Modulus: \(3264\)
Conductor: \(1632\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1632}(653,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3264.fc

\(\chi_{3264}(41,\cdot)\) \(\chi_{3264}(329,\cdot)\) \(\chi_{3264}(377,\cdot)\) \(\chi_{3264}(521,\cdot)\) \(\chi_{3264}(809,\cdot)\) \(\chi_{3264}(1049,\cdot)\) \(\chi_{3264}(1433,\cdot)\) \(\chi_{3264}(2105,\cdot)\)

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: 16.16.86609112753862292411929816663129369608192.2

Values on generators

\((511,2245,2177,2689)\) → \((1,e\left(\frac{7}{8}\right),-1,e\left(\frac{11}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 3264 }(41, a) \) \(1\)\(1\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{7}{8}\right)\)\(-i\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{1}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3264 }(41,a) \;\) at \(\;a = \) e.g. 2