Properties

Label 3840.137
Modulus $3840$
Conductor $1920$
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([0,19,16,8]))
 
pari: [g,chi] = znchar(Mod(137,3840))
 

Basic properties

Modulus: \(3840\)
Conductor: \(1920\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(32\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1920}(317,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3840.dj

\(\chi_{3840}(137,\cdot)\) \(\chi_{3840}(473,\cdot)\) \(\chi_{3840}(617,\cdot)\) \(\chi_{3840}(953,\cdot)\) \(\chi_{3840}(1097,\cdot)\) \(\chi_{3840}(1433,\cdot)\) \(\chi_{3840}(1577,\cdot)\) \(\chi_{3840}(1913,\cdot)\) \(\chi_{3840}(2057,\cdot)\) \(\chi_{3840}(2393,\cdot)\) \(\chi_{3840}(2537,\cdot)\) \(\chi_{3840}(2873,\cdot)\) \(\chi_{3840}(3017,\cdot)\) \(\chi_{3840}(3353,\cdot)\) \(\chi_{3840}(3497,\cdot)\) \(\chi_{3840}(3833,\cdot)\)

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: 32.32.8052845212573000012543979797231296934933304854055472857088000000000000000000000000.2

Values on generators

\((511,2821,2561,1537)\) → \((1,e\left(\frac{19}{32}\right),-1,i)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 3840 }(137, a) \) \(1\)\(1\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{31}{32}\right)\)\(e\left(\frac{21}{32}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{5}{32}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{1}{32}\right)\)\(-i\)\(e\left(\frac{3}{32}\right)\)\(e\left(\frac{5}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3840 }(137,a) \;\) at \(\;a = \) e.g. 2