Properties

Label 3840.103
Modulus $3840$
Conductor $640$
Order $32$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3840, base_ring=CyclotomicField(32))
 
M = H._module
 
chi = DirichletCharacter(H, M([16,17,0,24]))
 
pari: [g,chi] = znchar(Mod(103,3840))
 

Basic properties

Modulus: \(3840\)
Conductor: \(640\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(32\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{640}(443,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3840.di

\(\chi_{3840}(103,\cdot)\) \(\chi_{3840}(247,\cdot)\) \(\chi_{3840}(583,\cdot)\) \(\chi_{3840}(727,\cdot)\) \(\chi_{3840}(1063,\cdot)\) \(\chi_{3840}(1207,\cdot)\) \(\chi_{3840}(1543,\cdot)\) \(\chi_{3840}(1687,\cdot)\) \(\chi_{3840}(2023,\cdot)\) \(\chi_{3840}(2167,\cdot)\) \(\chi_{3840}(2503,\cdot)\) \(\chi_{3840}(2647,\cdot)\) \(\chi_{3840}(2983,\cdot)\) \(\chi_{3840}(3127,\cdot)\) \(\chi_{3840}(3463,\cdot)\) \(\chi_{3840}(3607,\cdot)\)

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: 32.32.187072209578355573530071658587684226515959365500928000000000000000000000000.1

Values on generators

\((511,2821,2561,1537)\) → \((-1,e\left(\frac{17}{32}\right),1,-i)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 3840 }(103, a) \) \(1\)\(1\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{21}{32}\right)\)\(e\left(\frac{7}{32}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{7}{32}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{27}{32}\right)\)\(-i\)\(e\left(\frac{1}{32}\right)\)\(e\left(\frac{15}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3840 }(103,a) \;\) at \(\;a = \) e.g. 2