Properties

Label 1408.1321
Modulus $1408$
Conductor $64$
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(1408, base_ring=CyclotomicField(16))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,15,0]))
 
pari: [g,chi] = znchar(Mod(1321,1408))
 

Basic properties

Modulus: \(1408\)
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 \(\chi_{64}(13,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1408.z

\(\chi_{1408}(89,\cdot)\) \(\chi_{1408}(265,\cdot)\) \(\chi_{1408}(441,\cdot)\) \(\chi_{1408}(617,\cdot)\) \(\chi_{1408}(793,\cdot)\) \(\chi_{1408}(969,\cdot)\) \(\chi_{1408}(1145,\cdot)\) \(\chi_{1408}(1321,\cdot)\)

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: \(\Q(\zeta_{64})^+\)

Values on generators

\((639,133,1025)\) → \((1,e\left(\frac{15}{16}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(13\)\(15\)\(17\)\(19\)\(21\)\(23\)
\( \chi_{ 1408 }(1321, a) \) \(1\)\(1\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{1}{16}\right)\)\(-i\)\(i\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{1}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1408 }(1321,a) \;\) at \(\;a = \) e.g. 2