Properties

Label 1920.533
Modulus $1920$
Conductor $1920$
Order $32$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1920, base_ring=CyclotomicField(32))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,13,16,24]))
 
pari: [g,chi] = znchar(Mod(533,1920))
 

Basic properties

Modulus: \(1920\)
Conductor: \(1920\)
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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1920.dj

\(\chi_{1920}(53,\cdot)\) \(\chi_{1920}(77,\cdot)\) \(\chi_{1920}(293,\cdot)\) \(\chi_{1920}(317,\cdot)\) \(\chi_{1920}(533,\cdot)\) \(\chi_{1920}(557,\cdot)\) \(\chi_{1920}(773,\cdot)\) \(\chi_{1920}(797,\cdot)\) \(\chi_{1920}(1013,\cdot)\) \(\chi_{1920}(1037,\cdot)\) \(\chi_{1920}(1253,\cdot)\) \(\chi_{1920}(1277,\cdot)\) \(\chi_{1920}(1493,\cdot)\) \(\chi_{1920}(1517,\cdot)\) \(\chi_{1920}(1733,\cdot)\) \(\chi_{1920}(1757,\cdot)\)

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

\((511,901,641,1537)\) → \((1,e\left(\frac{13}{32}\right),-1,-i)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 1920 }(533, a) \) \(1\)\(1\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{1}{32}\right)\)\(e\left(\frac{11}{32}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{27}{32}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{31}{32}\right)\)\(i\)\(e\left(\frac{29}{32}\right)\)\(e\left(\frac{11}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1920 }(533,a) \;\) at \(\;a = \) e.g. 2