Properties

Label 3968.805
Modulus $3968$
Conductor $3968$
Order $32$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3968, base_ring=CyclotomicField(32))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,25,16]))
 
pari: [g,chi] = znchar(Mod(805,3968))
 

Basic properties

Modulus: \(3968\)
Conductor: \(3968\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(32\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3968.cf

\(\chi_{3968}(61,\cdot)\) \(\chi_{3968}(309,\cdot)\) \(\chi_{3968}(557,\cdot)\) \(\chi_{3968}(805,\cdot)\) \(\chi_{3968}(1053,\cdot)\) \(\chi_{3968}(1301,\cdot)\) \(\chi_{3968}(1549,\cdot)\) \(\chi_{3968}(1797,\cdot)\) \(\chi_{3968}(2045,\cdot)\) \(\chi_{3968}(2293,\cdot)\) \(\chi_{3968}(2541,\cdot)\) \(\chi_{3968}(2789,\cdot)\) \(\chi_{3968}(3037,\cdot)\) \(\chi_{3968}(3285,\cdot)\) \(\chi_{3968}(3533,\cdot)\) \(\chi_{3968}(3781,\cdot)\)

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: \(\Q(\zeta_{32})\)
Fixed field: 32.0.2283054469939826689646603434893544205211768622361793362696380065480141214463295488.1

Values on generators

\((2047,3845,2049)\) → \((1,e\left(\frac{25}{32}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 3968 }(805, a) \) \(-1\)\(1\)\(e\left(\frac{27}{32}\right)\)\(e\left(\frac{25}{32}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{29}{32}\right)\)\(e\left(\frac{7}{32}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{31}{32}\right)\)\(e\left(\frac{21}{32}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3968 }(805,a) \;\) at \(\;a = \) e.g. 2